University of Southern California Dissertations and Theses

Heart Rate Response To Stress: A Mathematical Model

(USC Thesis Other)

Heart Rate Response To Stress: A Mathematical Model

PDF

Download a page range

Download transcript

Copy asset link

Request this asset

Transcript (if available)

Content 72-3775 FREELAND, Thomas Edward, 1940- HEART RATE RESPONSE TO STRESS: A MATHEMATICAL MODEL. University of Southern California, Ph.D., 1971 Education, physical University Microfilms, A X ERO X C om pany, Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED HEART RATE RESPONSE TO STRESS: A MATHEMATICAL MODEL by Thomas Edward F re e la n d A D i s s e r t a t i o n P re s e n te d to th e FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In P a r t i a l F u l f i l l m e n t of th e R equirem ents f o r th e Degree DOCTOR OF PHILOSOPHY ( P h y s ic a l E d ucatio n) June 1971 UNIVERSITY O F SO U T H E R N CALIFORNIA TH E GRADUATE SCH O O L UNIVERSITY PARK LOS ANGELES. CA LIFO RN IA 0 0 0 0 7 This dissertation, written by Thomas Edward F r e e la n d under the direction of Dissertation Com mittee, and approved by all its members, has been presented to and accepted by The Gradu ate School, in partial fulfillment of require ments of the degree of D O C T O R OF P H I L O S O P H Y O Dean Date J m e . . ! 9 Z ! DISSERTATION COMMITTEE / --- Chairman V PLEASE NOTE: Some Pages have i n d i s t i n c t p r i n t . Filmed as received. UNIVERSITY MICROFILMS ACKNOWLEDGMENTS The w r i t e r w ish e s t o acknow ledge th e a s s i s t a n c e of Mr. J a c k M ankiewciz, Dr. Thomas C u lle n , and h i s Guidance Com m ittee, w ith o u t whose a s s i s t a n c e and su p p o rt th e s tu d y c o u ld not have been c o n c lu d e d . I n a d d i t i o n , t h e w r i t e r B • Purn,mi - — would l i k e t o acknow ledge t h e encouragem ent and s u p p o rt of h i s w ife , J o a n , and Dr. K enneth L e r s te n , i i TABLE OF CONTENTS Page ACKNOWLEDGMENTS........................................................................................ l i LIST OF T A B L E S ........................................................................................ v LIST OF FIGURES........................................................................................ Vi Chapter I . INTRODUCTION .......................................................................... 1 Purpose of th e Study Statem ent of th e Problem Im portance of th e Study Procedure D e f in i t io n s of Terms Used O rg a n iz a tio n of th e Remainder o f th e Study I I . REVIEW OF RELATED LITERATURE................................... 10 P a rt One—Why Model? P a r t Two— Process of C o n stru c tin g a Model P a rt T hree—A Review o f Heart R ate Response Models P a rt F ou r—Review of H eart Rate L i t e r a t u r e I I I . DERIVATION OF THE MODEL.................................................. 44 Model I IV. PROCEDURES............................................................................... 59 S u b je c ts V. RESULTS AND D ISCU SSIO N .................................................. 7^ V e r i f i c a t i o n of Model I T r i a l Problem V e r i f i c a t i o n of Model I I VI. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS . . 125 Summary C onclusions Recommendations i l l Page BIBLIOGRAPHY .............................................................................................. 130 APPEN D IC ES................................................................................................... 138 APPENDIX I . Trend A n a l y s i s —Time Growth . . . 139 I I . D e riv a tio n of th e M o d e l ....... l 4 l I I I . A c t i v i t y C l a s s i f i c a t i o n Index . . 146 IV. P h y s ic a l Working C a p ac ity E stim a te d and P r e d ic te d . . .. 148 V. A H e art R ate f o r T hree L evels of S t r e s s ................................... 179 i v LIST OF TABLES T ab le . Page 1. P o s t u r a l Changes A f f e c t i n g H e a rt H ate ................... 36 2. P r e d i c t i v e Value o f T e s ts Based on H e a rt R a te f o r P r e d i c t i n g Maximum Oxygen I n t a k e ......................................................... 42 3. Summary o f S e le c te d C h a r a c t e r i s t i c s f o r a l l S u b j e c t s ................................................... 60 4. D ata Summary f o r S u b je c t 2 8 ............................................... 85 5 - 1 - A H e a rt R a te f o r S u b je c ts 1-30 8 8 - 5 -3 0 . P r e d i c t e d (A) and O bserved ( B ) .................. 117 6 . P r e d i c t e d (A) and E s tim a te d (B) PWC-^q . .. . 119 7 . P a ra m e te r E s tim a te s f o r a l l Work Loads .. . . 121 v LIST OF FIGURES F ig u re Page 1. H eart R ate Response Curves .............................................. 2 2. N orm alized H e art Rate Response D ata from S c h lip p and Brodan and K u h n ........................................ 26 3. Block Diagram of "P ic k e rin g " Model .......................... 32 4. Diagram matic P r e s e n t a t io n o f R e s u lts of P ic k e r in g M o d e l ......................................... ........................ 33 5. Diagrammatic P r e s e n t a t io n o f H om eostatic S y s t e m ................................................................ 48 6. H eart R ate Response Curves f o r V ario u s L ev els of S t r e s s ................................................................. 49 7. H eart R ate as a F u n c tio n of Work L o a d ..................... 51 8. Course o f an E x p o n e n tia l C u r v e .................................... 53 9. E x p o n e n tia l Curve A d justed f o r P la te a u . . . . 5 4 - 10. A pproxim ation o r H eart Rate Response v i a Model I ............................................................................... 56 11. H eart R ate Response to S t r e s s ......................................... 57 1 2 .1 - P h y s ic a l Working C a p a c ity E stim a te d 149- 12.30 and P r e d i c t e d ....................................................................... 178 1 3 .1 - A H e art R ate f o r Three L evels of S t r e s s . . . . 180- 13.30 209 v i CHAPTER I INTRODUCTION I t I s p e rh a p s d i f f i c u l t to b e l i e v e t h a t d e s p i t e many y e a r s o f i n t e n s i v e i n v e s t i g a t i o n we do n o t y e t have a s a t i s f a c t o r y e x p la n a tio n f o r th e most im p o rta n t p h y s i o l o g i c a l re s p o n se of a l l , i . e . , t h e re s p o n s e to e x e r c i s e . F red S. G rodins ( 9 ) Man i s c a p a b le o f p e r c e iv in g many ty p e s of s t r e s s . A u n i v e r s a l r e a c t i o n to s t r e s s , r e g a r d l e s s o f I t s m o d a lity , i s a change i n h e a r t r a t e fre q u e n c y . The change i n h e a r t r a t e fre q u e n c y may be p o s i t i v e o r n e g a t iv e , th e r a t e of change may be g r a d u a l o r p r e c i p i t o u s , and th e a l t e r a t i o n i n r a t e may be t r a n s i e n t o r p r o t r a c t e d . In a l l p r o b a b i l i t y , t h e o b s e r v a tio n t h a t t h e h e a r t r a t e fre q u e n c y v a r i e s was f i r s t n o ted p r i o r t o re c o rd e d h i s t o r y . With th e e x p a n sio n of m an's knowledge r e l a t i n g t o h i s body s t r u c t u r e and f u n c t i o n , numerous e x p e rim e n ts were and a re s t i l l b e in g condu cted which e x p lo r e th e r e l a t i o n s h ip which e x i s t s betw een h e a r t r a t e and th e s t r e s s of work o r p h y s i c a l a c t i v i t y . I t has long been a common o b s e r v a t i o n t h a t when an i n d i v i d u a l p erfo rm s work o r i s p h y s i c a l l y a c t i v e , t h e fre q u e n c y of h i s h e a r t b e a t i n c r e a s e s . The r a t e and e x te n t of i n c r e a s e a p p e a rs t o be d epend ent upon 1 th e i n t e n s i t y and d u ra tio n of th e a c t i v i t y o r work p e rio d and on th e p h y s ic a l and p h y s io lo g ic a l c h a r a c t e r i s t i c s of th e i n d i v i d u a l ( 5 6, 6 3 ). Because th e a c tio n of th e h e a r t i s p u l s a t i l e , an obvious l i m i t a t i o n of th e e x te n t to which th e h e a r t r a t e may i n c r e a s e i s th e tim e r e q u ir e d f o r a c a rd ia c c y c le . The l i t e r a t u r e r e l a t e d to h e a r t r a t e and p h y s ic a l a c t i v i t y and work i n d ic a t e s t h a t a h ig h degree of s i m i l i a r - i t y e x i s t s among h e a r t r a t e resp onse c u rv e s. This o b se rv a t i o n i s most a s s o c i a t e d w ith s p e c i f i c work lo a d s ; however, th e b a s ic shape o f th e resp o n se curve i n d i c a t e s t h a t a h ig h degree o f g e n e r a l i t y may e x is t f o r work lo ad s of v a rio u s i n t e n s i t i e s . 6^0** s boP 150 140 p 130 u 120 L 110 S 100 E 90 80 P ig . 1. c ? M inutes ( 3 8 ) The g e n e ra l p a t t e r n or conform ity o f th e c u rv e s p o rtra y e d i n th e above f i g u r e a ll u d e s to th e s u p p o s itio n t h a t t h e h e a r t r a t e resp o n se curve may be r e p r e s e n te d a n d /o r d e sc rib e d by a m athem atical model. 3 Purpose o f th e Study The p rim a ry p u rp o se of t h i s stu d y was to c o n c e p t u a l i z e a fu n d am e n ta l model which may be used t o b oth d e s c r i b e a n d /o r p r e d i c t an i n d i v i d u a l 's h e a r t r a t e re s p o n se c u rv e . A se co n d a ry p u rp o se o f th e stu d y was to e v a lu a te t h e u s e f u l n e s s of t h e model i n p r e d i c t i n g o r e s t im a t i n g an i n d i v i d u a l ' s p h y s i c a l w orking c a p a c i t y from a s h o r t , low l e v e l s t r e s s t e s t . S tatem en t o f th e Problem I s i t p o s s i b l e t o d e f i n e t h e h e a r t r a t e r e s p o n s e cu rve by way o f a b a s ic model and, i f so, does t h e e m p iric a l d a t a f i t t h e model? I s th e model g e n e r a l i z a b l e t o a l l mem b e r s of a s p e c i f i c segment o f th e p o p u la tio n ? And f i n a l l y , i s t h e r e a s i g n i f i c a n t r e l a t i o n s h i p between th e h e a r t r a t e re s p o n s e cu rve and an i n d i v i d u a l 's p h y s i c a l w orking c a p a c i t y ? The f o llo w in g h y p o th e s e s were t e s t e d : ( l ) e m p ir ic a l d a t a w i l l v e r i f y th e pro p o sed model f o r a segment o f th e p o p u l a t i o n , and ( 2 ) s p e c i f i c segm ents of t h e h e a r t r a t e re s p o n s e cu rve w i l l be i n d i c a t o r s o f an i n d i v i d u a l ' s p h y s i c a l w orking c a p a c i t y . Im po rtance o f th e Study The t h e o r e t i c a l and p r a c t i c a l u s e f u l n e s s o f a h e a r t r a t e re s p o n se model a r e i n f i n i t e . S e v e ra l s u g g e ste d u se s o f th e model a r e : ( l ) i t may be u sed to p r e d i c t a s u b j e c t 's 4 resp o n se t o a measured work lo a d . The p r a c t i c a l a p p l i c a t i o n of th e model would en ab le one to c o n s tr u c t a low l e v e l s t r e s s t e s t f o r a s s e s s in g th e c a p a b i l i t i e s of th e c a r d i o r e s p i r a t o r y system . P h y s i o lo g i c a l ly , a minimal s t r e s s t e s t a p p ears to be j u s t i f i e d on th e fo llo w in g b a se s : th e admin i s t r a t i o n of a t e s t b a t t e r y i s o fte n unw ise, s in c e th e p e r formance of th e t e s t s u s u a l l y demands a maximal o r n e a r maximal e f f o r t from an i n d iv i d u a l who may be c o m p le te ly un t r a i n e d or m e d ic a lly u n s u ite d f o r such a t e s t i n g regim en; (2) i t i s r e l a t i v e l y s a fe to assume t h a t a m a jo rity of c l i n i c i a n s and a p p lie d p h y s i o l o g is t s a re i n t e r e s t e d in m easuring a change i n th e c a p a c ity of th e c a r d i o r e s p i r a t o r y system , t h a t i s , an i n c r e a s e i n the c a p a b i l i t y o f th e s y s tem t o exchange and t r a n s p o r t oxygen. I f t h i s assum ption i s v a l i d , i t i s t o t a l l y unw arran ted t o conclude t h a t a p r e t e s t based on a maximal e f f o r t i s a t r u e and r e l i a b l e i n d i c a t o r of th e c a p a b i l i t y of th e c a r d i o r e s p i r a t o r y system when th e l i m i t i n g f a c t o r of th e t e s t may be m uscular weak n e s s ; ( 3 ) t h e r e i s always th e p o s s i b i l i t y t h a t red u n d an cies w ith in th e human organism may f u n c tio n as o v e rlo a d g o v e r no rs and p re v e n t a t r u l y maximal e f f o r t . Numerous i n v e s t i g a t o r s have im p lie d t h a t a maximal e f f o r t i s p s y c h o lo g ic a lly im p o s sib le in a l a b o r a to r y t e s t ing p ro c e d u re . Were i t p o s s ib l e , however, th e p r a c t i c a l d is a d v a n ta g e s of a maximal perform ance t e s t in c lu d e ( l ) ex c e s s iv e tim e re q u ire m e n t, (2) th e n e c e s s i t y of c o s t l y 5 equipm ent and s u p p l i e s , ( 3 ) th e use o f h i g h ly t r a i n e d a s s i s t a n t s , and (4) th e d i f f i c u l t y of m a in ta in in g a r e l i a b l e t e s t i n g p r o t o c o l . The l i m i t a t i o n s of maximal t e s t i n g have been summarized by d e V ries and K la fs : The measurement o f human p h y s ic a l w orking c a p a c i t y a s commonly p e rfo rm ed th ro u g h m easuring maximal oxygen i n t a k e w h ile th e s u b je c t i s s t r e s s e d on a b ic y c le e r - gom eter o r t r e a d m i l l i s an e x tre m e ly r i g o r o u s and d e manding one f o r th e s u b j e c t . F u rth e rm o re , a w e ll equipped l a b o r a t o r y w ith s k i l l e d t e c h n i c i a n s i s a p r e r e q u i s i t e f o r any such t e s t i n g program w hich r e q u i r e s a t l e a s t s e v e r a l h o u rs of s u b je c t tim e p lu s s e v e r a l h o u rs o f t e c h n i c i a n tim e p e r s u b j e c t . (48) Each of t h e fo re g o in g p a ra g ra p h s i n d i c a t e s th e v a lu e , i f i t i s p o s s i b l e to do so , of a p p ly in g t h e h e a r t r a t e re s p o n se model as a p r e d i c t o r of h e a r t r a t e re s p o n se to s p e c i f i e d work lo a d s . A d d itio n a l u se s o f th e model cou ld be made i n s t u d i e s which a tte m p t t o r e l a t e p h y s i o l o g i c a l p a ra m e te rs to e m p i r i c a l l y o b se rv e d e f f e c t s on th e e x e r c i s e re s p o n s e , such as e f f e c t s due t o age, f i t n e s s , f a t i g u e , p h a rm a c o lo g ic a l a g e n ts and so on. F i n a l l y , i t i s p ro b a b le t h a t a s u b j e c t ' s c o r r e s pondence to t h e model may be u se d as a means of c l a s s i f i c a t i o n o f h i s c a p a c i t y t o perfo rm work. P ro cedu re The s tu d y was c o n d u cted as f o llo w s : 1 . A v a ila b le l i t e r a t u r e was s tu d ie d w hich was r e l a t e d t o : ( l ) t h e u se of b i o l o g i c a l and p h y s io l o g i c a l m odels, (2) t h e development of m odels, and (3) t h e u se of h e a r t r a t e as an i n d ic a n t o f s t r e s s . T his l i t e r a t u r e was review ed to d e t e r mine: a) an a c c e p ta b le r a t i o n a l e from which to p ro ce ed , b) c r i t e r i a t o be u sed in e x p e rim e n ta t i o n , and c) th e s e l e c t i o n o f an a p p r o p r i a te r e s e a r c h d e s ig n and a p p r o p r i a te s t a t i s t i c a l and a n a l y t i c a l te c h n iq u e s . 2. A ll of th e n e c e s s a ry equipm ent was assem bled and p r e lim in a r y t e s t s were made f o r f a m i l i a r i z a t i o n and r e l i a b i l i t y p u rp o se s. 3. The s u b j e c t s were c o n ta c te d and o r i e n t e d t o th e t e s t i n g p ro c e d u re . 4. The d a ta were c o l l e c t e d . 5. An a n a l y s i s of th e model was com pleted. Ths use of th e model as a p r e d i c t o r was e v a lu a te d . 6. The r e s u l t s of th e p a p er were d is c u s s e d . 7. C o nclusions r e g a r d in g t h e im portance of the f i n d in g s were drawn, and recom mendations were made f o r f u r t h e r stu d y . D e f i n i t i o n s of Terms Used A nalog. —A system which i s s i m i l a r i n f u n c t i o n to a biosystem . As u sed w ith in th e c o n fin e s of t h i s s tu d y , i t i s synonomous w ith th e te rm "m odel." 7 C om posite m odel. —A model o r s i m u l a t i o n w hich i s composed of one o r more p a r t s o f a p a r t i c u l a r ty p e o f model. C o n tr o l t h e o r y . —As u se d i n t h i s p a p e r , r e f e r s to t h e a u to m a tic c o n t r o l o f t h e b io s y s te m ; u se d h e r e i n a s a m in im iz a tio n t e c h n iq u e . D e t e r m i n i s t i c m odel. —A model f o r w hich t h e r e i s a u n iq u e s e t o f r e s u l t s f o r each s e t o f d a t a . E s tim a te d PWC^q . —As u se d i n t h i s s tu d y , r e p r e s e n t s p h y s i c a l w orking c a p a c i t y c a l c u l a t e d by way o f th e S j o s t r a n d te c h n iq u e . E xp ected v a lu e m o d el.- - A c l a s s o f m odels i n which a n t i c i p a t e d o r e x p e c te d v a lu e s a r e a s s ig n e d t o chance p a r a m e te r s . F eedback l o o p . —A su b sy stem o f a c o n t r o l sy ste m w hich r e g u l a t e s t h e s y s t e m 's o u t p u t. M odel, — A s y n t h e t i c r e p r e s e n t a t i o n o f a p a r t of th e b io s y s te m . N o n - l i n e a r l e a s t s q u a r e s . —A method of s o lv in g r e g r e s s i o n e q u a t io n s . P r e d i c t e d PW C-L7Q .— As used- t h i s s tu d y , i t d e f i n e s p h y s i c a l w orking c a p a c i t y which i s p r e d i c t e d on th e b a s i s of t h e d e s c r i b e d m a th e m a tic a l m odel. 8 Problem s e t s . —A s e r i e s of un iq u e e q u a tio n s f o r c a l c u l a t i n g p a ra m eter e s tim a te s f o r any g iv e n s u b j e c t . S to c h a s tic m odel. —A model which i s based on p ro b a b i l i t y . Systems t h e o r y . —The d e s c r i p t i o n o f a system o r sub system as a netw ork o f i n t e r a c t i n g components which a r e s u b je c t t o d is tu r b a n c e s and r e a c t i o n to th e s e d is tu r b a n c e s w i t h in s p e c i f i e d l i m i t a t i o n s . O rg a n iz a tio n o f th e Remainder of t h e Study A review of s e l e c t e d l i t e r a t u r e r e l a t e d to t h i s s tu d y i s p r e s e n te d i n C hap ter I I . C hapter I I was o rg a n iz e d I n to f o u r p r i n c i p a l p a r t s : P a rt One i s a review of th e l i t e r a t u r e p e r t a i n i n g t o th e u se of b i o l o g i c a l and m ath em atical m odels; P a r t Two p r e s e n t s a d i s c u s s i o n o f t h e p r i n c i p l e s of d e v elo p in g a model; P a r t Three c o n s i s t s of a review of s e l e c t e d models p u b lis h e d i n th e l i t e r a t u r e ; and, f i n a l l y , P a r t F o u r—review of l i t e r a t u r e —p r e s e n ts an overview of s e l e c t e d l i t e r a t u r e I n which th e r e l a t i o n s h i p of h e a r t r a t e and work lo ad i s d is c u s s e d . The d e r i v a t i o n of th e model, along w ith a d is c u s s io n o f th e a ssu m ptions, l i m i t a t i o n s , and r a t i o n a l e , a r e d i s c u sse d i n C hapter I I I . I n C hapter IV, th e e x p e rim e n ta l methods and p r o c e d u re s u sed i n c o n d u ctin g t h i s i n v e s t i g a t i o n a re examined. 9 C h a p te r V re v ie w s t h e m ethods u se d t o re d u c e and p r e p a r e th e d a t a f o r a n a l y s i s . The r e s u l t s and a n a l y s i s a r e a l s o d i s c u s s e d t h e r e i n . I n C h a p te r V I, a summary and th e c o n c lu s io n s drawn from th e f i n d i n g s , and recom m endations f o r f u r t h e r s tu d y a r e p r e s e n t e d . CHAPTER I I REVIEW OF RELATED LITERATURE The l i t e r a t u r e review ed i n t h i s c h a p te r was s e le c te d f o r t h e pu rp ose o f p ro v id in g summaries w ith in t h e a re a s s p e c i f i c a l l y r e l a t e d t o th e p r e s e n t stu d y . These a re a s were: ( l ) a summary of s e l e c t e d l i t e r a t u r e r e l a t i n g to t h e u se and p u rp o ses of models i n sim u la tin g p h y s i o l o g ic a l s y s tems s p e c i f i c a l l y , and a b r i e f review of th e use o f c o n tr o l th e o r y ; (2) a rev iew o f th e g e n e r a l p r i n c i p l e s o r p ro c e s s e s r e l a t e d t o t h e development o f m athem atical a n d /o r b i o l o g i c a l m odels; ( 3 ) a review o f p re v io u s e x p erim e n ts, an o u t come o f which was a model o f th e h e a r t r a t e re sp o n se to e x e r c i s e ; and (4) a b r i e f review o f h e a r t r a t e l i t e r a t u r e , t h e pu rp ose of which was to review e x i s t i n g in f o r m a tio n p e r t i n e n t to th e development and v a l i d a t i o n o f a m athem ati c a l model. I t sh ould be emphasized t h a t w h ile a g r e a t d e a l o f in fo rm a tio n r e l a t e d t o th e c a r d io v a s c u la r system s c o n t r o l and r e g u l a t i o n h as been p u b lis h e d , a p a u c i ty of in f o r m a tio n p e r t a i n i n g to h e a r t r a t e models e x i s t s . The re a so n f o r t h i s " in fo rm a tio n gap" i s not known. 10 11 P a r t One— Why M odel? One can t r a c e th e u s e o f m odels back t o e a r l y c i v i l i z a t i o n . A nalog m odels h a v e p r o b a b ly b e en u s e d s i n c e r e c o rd e d tim e b e g an . U n d o u b ted ly , e a r l y m odels of t h i s ty p e w ere r e l a t e d t o p r i m i t i v e m an 's need f o r fo o d , s h e l t e r and p r o t e c t i o n . M a th e m a tic a l m odels c a n be t r a c e d t o t h e e a r l y b e g in n in g s o f m a th e m a tic s ; a n example o f one o f t h e e a r l i e s t f o r m a l m odels p ro p o s e d was t h e P y th a g o re a n Theorem, w hich d a t e s from t h e s i x t h c e n t u r y B.C. i n G reece ( l 8 ). The i d e a o f a p p ly in g m a th e m a tic a l m ethods t o b i o l o g i c a l system s i s g e n e r a l l y c r e d i t e d t o W iener (19^9)* b u t a s so o f t e n h a p p e n s i n t h e h i s t o r y of s c i e n c e , i t i s c l e a r t h a t th e same i d e a was t a k i n g shape i n v a r i o u s o t h e r p a r t s o f th e w o rld ( 1 3 ) . At t h e p r e s e n t tim e , t h e u s e o f m o d elin g and s i m u l a t i o n t e c h n i q u e s a r e r a p i d l y e x p a n d in g due t o i n c r e a s e d te c h n o lo g y and t h e a v a i l a b i l i t y o f e l e c t r o n i c com p u t e r s . The p r o c e s s o f m o deling a n d / o r s i m u l a t i o n may g e n e r a l l y be d e f i n e d a s form in g a r e p r e s e n t a t i o n , r e a l o r a r t i f i c i a l , o f a p r o c e s s o r o p e r a t i o n - f o r w hich a s o l u t i o n may be o b t a i n e d ( 1 3 , 1 8 , 1 9, 2 2 , 3 ^ > ^ 7 ). T hese t e c h n i q u e s may be a p p l i e d t o v i r t u a l l y e v e r y d i s c i p l i n e i n w hich phenomena can be m easu red o r q u a n t i f i e d . The s p e c i f i c u se o f th e m odelin g te c h n iq u e depends on t h e f o c u s o f t h e i n v e s t i g a t i o n and on t h e i n v e s t i g a t o r ' s l e v e l o f o p e r a t i o n . K a n to r and h i s c o -w o rk e rs h a v e u t i l i z e d models as p a t t e r n s o r paradigm s f o r th e a c q u i s i t i o n of d a ta and f o r th e c o n t r o l of d a ta a n a l y s i s (5*0. Kalman s t a t e d , "A model i s a summary of e x p e rim e n ta l d a ta ; r e p e a tin g an experim ent on th e model should y i e l d e x a c tly th e same d a ta as was assumed i n c o n s t r u c t in g th e model" (1 3 ). I n most ca se s modeling i s re g a rd e d as an e x te n s io n o f th e i n v e s t i g a t o r 's t h in k in g , i t s p rim a ry p urpose being to develop o r c l a r i f y th e I n d i v i d u a l 's u n d e rs ta n d in g o f th e system o r p ro c e ss modeled ( 1 9 ,2 3 ,4 6 ,4 7 ) . Models may be used t o in c r e a s e our u n d e rs ta n d in g of th e a c t u a l system . A b i l i t y t o p r e d i c t th e b e h a v io r o f th e a c t u a l system i s a t e s t of our d egree of u n d e r s ta n d in g . Models may be used to e s tim a te i n a c c e s s i b l e q u a n t i t i e s (p a ra m e te rs or v a r i a b l e s ) on th e b a s i s o f measured q u a n t i t i e s . An im p o rta n t use of models i s t h e e v a l u a t i o n o f th e in f lu e n c e of a p a r t i c u l a r param e t e r on t h e im p o rta n t v a r i a b l e s of th e system . (46) The v a lu e of s im u la tio n i s i n th e in c r e a s e d u n d e r s ta n d in g which i t makes p o s s i b l e of p h y s i o l o g ic a l f u n c t i o n , which le a d s t o th e p r e d i c t i o n of s t r u c t u r a l p r o p e r t i e s , new e x p e rim e n ta l f i n d i n g s , and i n c l a r i f y i n g th e dynamic beh av i o r of th e sy stem — t h a t i s , p ro v id in g u n d e rs ta n d a b le re a so n s f o r what p r e v i o u s ly a p p eared to be u n p r e d ic ta b le e v e n ts (2 3 ,4 2 ,5 0 ). The e x te n t to which a model o r s im u la tio n c l a r i f i e s , i l l u m in a t e s o r r e d i r e c t s t h e i n v e s t i g a t i v e p ro c e ss i s d e pendent on th e v u l n e r a b i l i t y o f th e model. The v u l n e r a b i l i t y o f a model i s based on th e v a l i d i t y of th e assu m p tio n s r e q u ir e d to c o n s t r u c t th e model and on th e im p lie d h a z a rd s 13 o f th e m odel. I t sh o u ld be em phasized t h a t t h e h i g h e r t h e v u l n e r a b i l i t y o f a model t h e h i g h e r I s t h e p o t e n t i a l g a in , and th e l e s s l i k e l i h o o d f o r s u c c e s s ; on t h e o t h e r hand, th e lo w e r t h e v u l n e r a b i l i t y o f t h e model t h e h i g h e r t h e p r o b a b i l i t y o f o b t a i n i n g modest g a in s ( 4 6 ,4 7 ) . The more complex th e system b e in g s c r u t i n i z e d th e more c o m p e llin g i s th e re a s o n f o r u s in g a n d /o r d e v e lo p in g a t h e o r e t i c a l m odel. T h is i s p a r t i c u l a r l y t r u e i n th e s tu d y o f p h y s io lo g y f o r th e f o llo w in g r e a s o n : I f one t r i e s t o change o n ly one f a c t o r a t a tim e , a number o f o t h e r f a c t o r s may change i n a p o o r ly known m anner, so t h a t I t i s . . . w e l l - n i g h im p o s s ib le to ru n a s t r i c t l y c o n t r o l l e d p a r a m e tr ic e x p e rim e n t. (47) However, th e b u lk o f t h e model must n o t be so l a r g e (e x c e s s o f p a r a m e te r s , v a r i a b l e s , and e q u a t io n s ) a s t o n e g a te i t s u s e . C a ro l Newton s t a t e d , . . . when i t i s d i r e c t e d t o b i o l o g i c a l mechanisms I n I n t a c t l i v i n g sy ste m s, t h e number of components i n t h e ensem ble and t h e i r p o s s i b l e m o d a l i t i e s of I n t e r a c t i o n a r e so g r e a t t h a t i t i s u n l i k e l y t h a t enough e x p e r i m e n ta l p a ra m e te r s can p e n e t r a t e t h e u n c e r t a i n t i e s o f measurem ent e r r o r , b i o l o g i c a l v a r i a t i o n , e t c . t o v a l i d a t e t h e m odel. Two avenues f o r a p p ro a c h in g t h i s problem a r e o b v io u s, d e c r e a s e t h e number o f model p a r a m e te r s o r i n c r e a s e th e number o f d i s c e r n i b l e e x p e r i m e n ta l p a r a m e te r s . (49) The t h e o r y o f how a p a r t i c u l a r sy stem f u n c t i o n s i s shaped I n t o a model o r im age. The model t h e n r e p r e s e n t s a p a r t i c u l a r segment o f r e a l i t y . I n i t s m a th e m a tic a l f o r m u la tio n t h e t h e o r y o r model d e a l s w ith s e l e c t e d p r o p e r t i e s of t h e r e a l w o rld , th u s s i m p l i f y i n g t h e r e a l w o rld (2 3 ). I t would 14 app ear, th e n , t h a t i n o rd e r to model a biosystem , i t must be s im p li f i e d ( 9 ,2 2 ,3 4 ,6 2 ) . The num erical s o l u t io n of th e problem s e t s c o n tain ed w ith in th e model may be solved v ia s e v e r a l methods. F o r t u n a t e l y , i t i s now p o s s ib le to u se h ig h speed computers i n s e t t i n g up c u r r e n t co n cep ts as models, t e s t i n g t h e i r v a l i d i t y a g a in s t th e b e s t d a ta a v a i l a b l e , and u sin g th e r e s u l t s to a s s i s t i n e x ten d in g o r m odifying th e model. The use of a computer g r e a t l y f a c i l i t a t e s th e p ro g re s s in ex tendin g the b o u n d a rie s of man's knowledge of f u n c tio n (65). As p r e v io u s ly s t a t e d , th e use of a model to de s c r ib e or stu d y a sub-system of man i s most a p p ro p r ia te . A n a t u r a l e x te n s io n of t h i s pro ced u re i s th e use of c o n tr o l th e o ry in d e riv in g th e c o n cep ts of th e model. This combi n a tio n i s p a r t i c u l a r l y s u i t a b l e f o r stu d y in g th e m u lt i f a c e te d mechanisms which c o n tr o l and r e g u l a t e man's "in n er" e nvironm ent. P h y s io lo g ic a l r e g u l a t i o n has b oth th e s h o r t e s t tim e s c a le and u s u a l ly i t s own c h a r a c t e r i s t i c m achinery. Such c o n tr o l in c lu d e s th e m aintenance of th e p h y sio l o g i c a l ste a d y s t a t e . But h o m eostasis I t s e l f i s t r u l y dynamic i n n a tu re d e s p i te i t s p ro p e r d i s t i n c t i o n from th e r e f e r e n c e — i n p u t —fo llo w in g se rv o -ty p e c o n tr o l . T his i s so because th e p h y s io lo g ic a l stead y s t a t e i s n e i t h e r j u s t a p a s s iv e r e s i s t a n c e to change nor a mere com pliance t o p a t t e r n imposed m ainly from th e o u ts id e . R ather i t r e s u l t s p r im a r ily from com pensating a d j u s t ments a c t i v e l y programmed w ith in th e organism i n r e sponse to th e t o t a l r e l e v a n t in fo rm a tio n i t has a v a i l a b le . Thus in s p i t e of an e x q u is it e s e n s i t i v i t y to changes of many k in d s, organism s m ain tain t h e i r s t a b le s t a t e w ith rem arkable th oroug hness and p r e c is io n . ( 1 9) G-rodin has d e fin e d a system "as a c o l l e c t i o n of 15 components a rra n g e d and c o n n ec te d i n a d e f i n i t e way" ( 9 ). He c o n tin u e d t o s t a t e , "A d i s t i n g u i s h i n g f e a t u r e o f such a system i s t h a t i t h a s an in p u t and an o u tp u t" ( 9 ). W ater man h as d e f in e d system s a n a l y s i s as t h e a p p l i c a t i o n of o rg a n iz e d a n a l y t i c a l m odeling t e c h n iq u e s a p p r o p r i a t e t o e x p la in in g m u l t i v a r i a b l e system s many o f whose f u n c t i o n a l components may be i n i t i a l l y q u i t e i m p e r f e c t l y m easured o r even l a r g e l y u n i d e n t i f i e d . ( 1 9 ) T here a r e two p rim a ry ty p e s o f c o n t r o l sy stem s: an i n t r i n s i c c o n t r o l system o r what i s commonly c a l l e d an open loop system , and th e e x t r i n s i c c o n t r o l sy stem o r c lo s e d lo o p . The d i s t i n c t i o n betw een t h e s e two system s i s shown g r a p h i c a l l y : INTRINSIC CONTROL SYSTEM System O utput EXTRINSIC CONTROL SYSTEM Output System In p u t R e g u la to r As th e diagram i l l u s t r a t e s , t h e p rim a ry d i f f e r e n c e between th e two system s i s t h e p o in t where r e g u l a t i o n o c c u r s . The i n t r i n s i c c o n t r o l system r e q u i r e s o r demands t h a t th e s y s tem i t s e l f o v e r r i d e , a d j u s t o r accommodate t o th e i n p u t - 16 o u tp u t mechanisms I n t e r n a l l y . I n th e e x t r i n s i c c o n t r o l s y s tem t h i s a d ju s tm e n t i s made by a r e g u l a t o r o u t s i d e o f t h e system ( 9 j 13*18 j 19 j 2 3 ,2 5 ) . B ecause o f t h e Im p o rta n ce o f t h i s c o n c e p t I n d e v e lo p in g a m odel, an I l l u s t r a t i o n i s a p p r o p r i a t e . F o r exam ple, l e t us suppose t h a t t h i s l i n e |-------------------j i s a p i e c e o f e l e c t r i c a l w i r e . T h is w ire i s a system , t h e f u n c t i o n o f w hich I s t o conduct c u r r e n t . The amount o f c u r r e n t w hich t h i s system i s c a p a b le of c o n d u c tin g depends on: t h e le n g t h of t h e w ir e , th e d ia m e te r of t h e w ir e , t h e p h y s i c a l c o m p o s itio n o f th e w ir e , th e te m p e r a tu r e of i t s e n v iro n m e n t, and i t s s p a t i a l r e l a t i o n s h i p t o o t h e r s i m i l a r s y s te m s . Assuming none o f th e fo r e g o in g p a ra m e te r s a r e a l t e r e d , t h e amount o f c u r r e n t w hich t h i s sy stem w i l l p a s s i s d ep en d e n t on t h e I n p u t - o u t p u t d e v ic e s . I f in p u t and o u tp u t a r e n o t b a la n c e d , th e sy stem w i l l accommodate i t s f u n c t i o n ( t o some s p e c i f i c p o i n t ) by a d j u s t i n g i t s h e a t o u t p u t. I f t h e l i m i t s of t h e system a r e exceed ed , system f a i l u r e r e s u l t s . I f we a t t a c h a r e g u l a t o r t o th e system and I f i t i s programmed t o p a s s a g iv e n amount of c u r r e n t , t h e n t h e i n p u t t o t h e system i s a d j u s t e d a c c o rd in g t o t h e c a p a b i l i t i e s i n h e r e n t i n t h e sy stem r e s u l t i n g i n a c o n s t a n t o u t p u t . The c o m b in a tio n of c o n t r o l t h e o r y o r system s t h i n k in g and th e u s e o f a t h e o r e t i c a l model sh o u ld e x te n d t h e b o u n d a rie s o f e x p e r im e n ta l i n v e s t i g a t i o n i n p h y s i o l o g i c a l r e s e a r c h . T hese combined p ro c e d u r e s w i l l e n a b le one t o 17 examine t h e i n t e r a c t i o n o f v a rio u s sub -system s a t any p a r t i c u l a r tim e r e f e r e n c e ; th e y a ls o perm it t h e i n v e s t i g a t o r a wide d egree of l a t i t u d e i n m a n ip u latin g and c o n t r o l l i n g th e p a ra m e te rs and v a r i a b l e s b eing i n v e s t i g a t e d . P a r t Two—P ro c ess of C o n s tru c tin g a Model The c o n s t r u c t i o n o f a model of any b i o l o g i c a l s y s tem must always be based on a l a r g e number of s im p lif y in g a ssu m p tio n s. The ju d ic io u s s e l e c t i o n of th e s e assu m p tio n s r e p r e s e n t s a c r i t i c a l s te p in t h e development of a model. These assum ption s a re e s s e n t i a l because o f p r e s e n t la c k of knowledge and becau se of th e la c k o f ad eq uate m athem atical te c h n iq u e s f o r s o lv in g more r e a l i s t i c e q u a tio n s (1 1 ,2 5 ) . A t t in g e r has summarized th o se f a c t o r s which should be con s id e r e d i n s t a t i n g o r s e l e c t i n g a ssu m p tio n s: 1. The model must be p h y s i c a l l y r e a l i z a b l e and must be e s t a b l i s h e d in term s o f p a ra m e te rs which a re s i g n i f i c a n t and m easurable i n th e l i v i n g system , 2. The model has to in c lu d e a l l th e a v a i l a b l e p e r t i n e n t in f o rm a tio n of th e system under stu d y . Assum ptions which c o n f l i c t w ith e x i s t i n g d a ta must be e v a lu a te d and j u s t i f i e d w ith p a r t i c u l a r c a r e . 3. The s im p le r a model, th e e a s i e r i t i s to e v a lu a te i t s b e h a v io r and th e c o n tr i b u ti o n s of i t s i n d i v id u a l components t o d i f f e r e n c e s betw een t h e p e r form ance o f th e b io sy ste m and i t s a n a lo g . 4 . I t i s p r e f e r a b l e , a lth o u g h n o t always p o s s i b l e , t o c o n s tr u c t th e model i n a form which p e rm its a l t e r a t i o n s o f assu m p tio n s and system p a ra m e te rs w ith o u t e x c e s s iv e e f f o r t . The model m ust, of c o u rs e , be sim p le r t o m an ip u late t h a n th e e x p e r i m en tal p r e p a r a t i o n . 18 5. A good model s h o u ld s e r v e a s a g u id e t o t h e e x p e r i m e n ta l I n v e s t i g a t o r , s u g g e s t c e r t a i n e x p e rim e n ts and r u l e o u t t h e n e c e s s i t y o f o t h e r s . I t s h o u ld a l s o form a b a s i s f o r e x t r a p o l a t i o n , b oth beyond t h e ra n g e o f o b s e rv e d d a t a and to g e n e r a l p r o p e r t i e s o f b i o l o g i c a l s y s te m s . (34) R e g a r d le s s o f t h e t y p e o f model d e v e lo p e d , t h e r e a r e f o u r b a s i c c r i t e r i a w hich must be met i f t h e m odeling t e c h n iq u e i s t o be e f f i c i e n t and e f f e c t i v e : t h e model must h a v e v a l i d i t y , g e n e r a l i t y , p r e d i c t a b i l i t y , and c o m p u t a b i l i t y ( 6 2 ). A model i s v a l i d d e s c r i p t i o n o f a r e a l s i t u a t i o n i f a l l known c o n se q u e n c e s o f t h e model a r e c o m p a tib le w ith a l l v e r i f i e d know ledge o f t h e r e a l s i t u a t i o n ( 1 3 , 1 8, 1 9 ). ". . . t h e v a l i d i t y o f a m odel d e pend s on o u r know ledge o f r e a l i t y and on t h e c l e v e r n e s s we d i s p l a y i n d e r i v i n g c o n se q u e n c e s o f th e model . . . " ( 6 2 ). Prom a p r a c t i c a l s t a n d p o i n t , i t i s u s u a l l y more e c o n o m ic a l t o d e v e lo p and a n a ly z e one model t h a n t o d e v e lo p and m a n ip u la te s e v e r a l m odels f o r t h e s tu d y of r e l a t e d s i t u a t i o n s . " G e n e r a l i t y i s d e s i r a b l e on t h e g ro u n d s o f b ro a d s c i e n t i f i c p r i n c i p l e a s w e l l a s on a e s t h e t i c g ro u n d s" ( 6 2 ) . Thus, i n f o r m u l a t i n g t h e model i t s a p p l i c a b i l i t y t o r e l a t e d p ro b lem s s h o u ld be c o n s i d e r e d . Q u ite o f t e n , g e n e r a l i t y o f a m a th e m a tic a l m odel o c c u r s a s a r e s u l t o f d i s c o v e r in g d i f f e r e n t s i t u a t i o n s w hich have t h e same m a th e m a ti c a l d e s c r i p t i o n ( 4 7 ) . A t h i r d i m p o r t a n t q u a l i t y i n th e r e l a t i o n o f t h e 19 model to r e a l i t y I s th e p r e d i c t i v e a b i l i t y of th e model. "To be u s e f u l , a model must p ro v id e more th a n a co n v en ien t hook on which to hang a th e o ry and I t s s u p p o rtin g d a ta ; I t must p ro v id e a f r e s h , h o p e f u lly f e r t i l e view of r e a l i t y or p ro v id e a means of e x p lo rin g r e a l i t y " (6 2 ). The l a t t e r u s u a l l y c o n s i s t s of p r e d i c t i n g h i t h e r t o unseen o r unmeasured phenomena o r d is c o v e rin g c o n c e a le d r e l a t i o n s . T his p r e d i c t i o n may le a d i n t o a s e r i e s o f v e r i f y i n g experim en ts or p e rh ap s th e e x t r a p o l a t i o n of th e knowledge i n t o a re a s p r e s e n t l y i n a c c e s s i b l e to experim ent (4 2 ,6 2 ). I f a model i s t o be a n a ly z e d r e a s o n a b ly q u ic k ly , e a s i l y , and a c c u r a t e l y , i t must have a h ig h deg ree of com p u t a b i l i t y ; t h a t i s , i t must conform t o m ath em a tica l t r a d i t i o n s . One must be a b le t o e x t r a c t from th e model a s u f f i c i e n t amount and ty p e of n u m e ric a l in f o r m a tio n i n o r d e r to c h a r a c t e r i z e d im ensions or a s p e c t s o f th e r e a l s i t u a t i o n (4 9 ,6 2 ). A ll o f th e afo re m en tio n e d c h a r a c t e r i s t i c s o r p ro p e r t i e s a r e I n t e r r e l a t e d ; each e x e r t s a s tro n g i n f lu e n c e on t h e o t h e r s . T his i n t e g r a t i v e r e l a t i o n s h i p r e f l e c t s a c h a r a c t e r i s t i c of t h e m ath em atical m odeling p r o c e s s , t h a t of th e c o n ti n u a l c y c l i c a l p ro c e s s o f a b s t r a c t i o n , p r e d i c t i o n , v e r i f i c a t i o n and re f in e m e n t, t h i s p ro c e s s p o s s ib l y b ein g r e i t e r a t e d u n t i l th e ap p ro x im atio n s of th e model converge. With each s p i r a l th e model sh o u ld become more and more a t r u e r e f l e c t i o n of r e a l i t y ( 1 3 ,1 9 ,2 2 ,2 5 ) . 20 A p r o p e r l y c o n s t r u c t e d model i s an e x tre m e ly f l e x i b l e and u s e f u l d e v ic e . Model i n p u t s can be s y s t e m a t i c a l l y and e a s i l y changed t o t e s t any d e s i r e d c o m b in a tio n of sy stem p a ra m e te r s and i n p u t v a r i a b l e s . ( 1 8 ) F o r some ty p e s o f m odeling problem s i t i s o f t e n an a d v a n ta g e t o a p p ro a c h t h e u l t i m a t e s o l u t i o n i t e r a t i v e l y by c o n s t r u c t i n g more th a n one g e n e r a t i o n o f th e m odel. The I n i t i a l model (a lo n g w ith t h e r e s u l t s o f i t s a p p l i c a t i o n ) i n f l u e n c e s o r p r o v i d e s g u i d e l i n e s f o r th e c o n tin u e d d e v e lo p ment o f t h i s p a r t i c u l a r model o r f o r th e d e r i v a t i o n o f su b se q u e n t m odels. T h is i t e r a t i v e a p p ro a c h t o t h e s o l u t i o n i s c o n tin u e d u n t i l a model i s d e v e lo p e d which i s c o n s id e r e d to be t h e b e s t a p p ro x im a tio n of r e a l i t y a t t h e s p e c i f i c p o i n t i n tim e ( 2 5 ). Even th o u g h s i m u l a t i o n s d i f f e r and a l t e r n a t i v e a p p ro a c h e s i n problem s o l u t i o n a r e p o s s i b l e , t h e r e a r e common p ro c e d u r e s and t a s k s i n problem s o l u t i o n by com puter s i m u l a t i o n . . . . We have g ro u p ed th e t a s k s i n t o t h r e e m ajor p h a s e s : (a ) c o n c e p t u a l i z a t i o n , (b) im p le m e n ta tio n , and (c) r e s u l t s . ( l 8 ) The m ajo r p u rp o se o f p h a se 1 i s t o d e f i n e and a n a ly z e t h e pro b lem . T h is p r o c e s s i s i n i t i a t e d w ith a s e a r c h o f r e l a t e d l i t e r a t u r e and an e v a l u a t i o n o f t h e q u a l i t y o f i n f o r m a ti o n and d a t a w hich i s a v a i l a b l e and p e r t i n e n t t o t h e p ro b lem . The n e x t s t e p i s t o d e v e lo p h y p o th e s e s , r e c o g n iz e a s s u m p tio n s , and e s t a b l i s h s u i t a b l e r a t i o n a l e f o r th e m odel. The p a r a m e te r s and v a r i a b l e s and t h e m ethods w hich w i l l be u s e d t o q u a n t i f y and c o l l e c t t h e d a t a a r e th e n d e te rm in e d and d e f i n e d . P hase 1 p r o c e s s e s a r e com p l e t e d w ith a d e s c r i p t i o n o f t h e f i n a l i z e d m odel. P r i o r to 21 proceeding w ith th e next phase, th e v a l i d i t y of th e concepts on which th e models a re based and a re th e n e v a lu a te d a n d /o r v e r i f i e d ( 1 8, 2 2). The purpose of phase 2 i s to t r a n s l a t e th e concep t u a l model in to symbolic and num erical term s. Next, th e m ath em atical e q u a tio n s a re s t a t e d and examined f o r com p u t a b i l i t y ( 18, 2 2). In th e t h i r d and f i n a l phase of model c o n s tr u c tio n one e s tim a te s a n a l y t i c a l methods, s t a t i s t i c a l tre a tm e n t of th e d a ta , and p r o j e c t s th e expected outcomes of th e m odel's s o l u t i o n s e t s . I n a d d it i o n , one should examine f u t u r e ap p l i c a t i o n s and th e p o s s i b i l i t i e s of p o t e n t i a l growth of th e model ( l 8 ,2 2 ) . A review and th e a p p l i c a t i o n of th e s e s te p s a re d is c u s s e d i n g r e a t e r d e t a i l i n Chapter I I I —D e riv a tio n of th e Model. P a rt Three A Review of H eart Rate Response Models As p r e v io u s ly s t a t e d , a thorough review of th e l i t e r a t u r e r e s u l t e d In th e fo llo w in g c o n c lu sio n : numerous c a r d io v a s c u la r models have been proposed; however, few i n v e s t i g a t o r s have used th e modeling te c h n iq u e t o i n v e s t i g a t e th e h e a r t r a t e resp o n se to e x e rc is e . Of t h e s i x h e a r t r a t e models r e p o r t e d i n th e l i t e r a tu r e , f o u r may be c l a s s i f i e d as d e te r m i n i s t ic models; one as a s t o c h a s t i c model, and one as an ex pected v a lu e model. 22 A d e t e r m i n i s t i c model i s an a n a l y t i c a l r e p r e s e n t a t i o n o f a system o r o p e r a t i o n f o r w hich t h e r e a r e u n iq u e o u t p u ts f o r a g iv e n s e t o f i n p u t s . A s t o c h a s t i c model i s "based on p r o b a b i l i t y . The f u n c t i o n a l r e l a t i o n s h i p s s t a t e d i n th e model a r e d e p e n d e n t on ch an ce p a ra m e te r s . The r e s u l t s from a g iv e n s e t o f i n p u t s ’ c an o n ly be p r e d i c t e d on a p r o b a b i l i t y b a s i s . The t h i r d c l a s s o f m odels i s one i n which t h e e x p e c te d v a lu e s a r e a s s i g n e d t o chance p a r a m e te r s . A model may be composed o f numerous f e a t u r e s drawn from each c l a s s o f model d e s c r i b e d above. The f o l lo w in g d i s c u s s i o n o f t h e m odels r e p o r t e d i n t h e l i t e r a t u r e i s o r g a n iz e d on a c h r o n o l o g i c a l b a s i s ; t h e o r d e r i n w hich t h e models w ere r e p o r t e d i n t h e l i t e r a t u r e . The e a r l i e s t r e p o r t e d a tte m p t to f o r m u la te a h e a r t r a t e model was p ro p o se d by R o b ert S c h lip p . Only a re v ie w o f t h e e x e r c i s e p o r t i o n o f t h i s model i s g iv e n h e r e . c > I t was S c h l i p p rs c o n t e n t i o n t h a t t h e amount o f i n c r e a s e i n p u l s e r a t e due t o e x e r c i s e c o u ld be a c c u r a t e l y d e s c r i b e d by a l o g a r i th m i c f o rm u la . He s u g g e s te d t h a t t h e p u l s e r a t e c u rv e i s a two component sy ste m ; t h e f i r s t component o f t h e system d e s c r i b e s t h e r a p i d i n c r e a s e i n th e p u l s e r a t e which o c c u rs d u rin g t h e f i r s t 10 t o 15 seconds o f e x e r c is e ^ t h e second component a c c o u n ts f o r t h e main p a r t o f t h e p u l s e r a t e i n c r e a s e d u r in g e x e r c i s e and i t h a s a much slo w e r v e l o c i t y c o n s t a n t and i s o f l o n g e r d u r a t i o n . H is model f o l l o w s : 23 yt = A i(l - <rk lt) + M l - e'*2*) (67) y = I n t e r c e p t v a lu e = I n c r e a s e i n h e a r t r a t e Aq = A^ + Ag = 80 = th e p o in ts where th e c u rv e d e f l e c t s k^ = V e lo c ity c o n s ta n t kg = V e lo c ity c o n s ta n t t = Time e = Base of a n a t u r a l log As a r e s u l t of h i s i n v e s t i g a t i o n , th e fo llo w in g v a lu e s were o b ta in e d f o r th e c o n s ta n ts , k^ = 0,385^ kg = 0.0 233 . The p o in t a t which th e v e l o c i t y c o n s ta n t was d e fin e d by param e t e r s A^ and Ag i n t h i s case were 19 and 61 r e s p e c t i v e l y (67). S c h l ip p 's model combines f e a t u r e s of b o th d e te rm in i s t i c and exp ected v a lu e m odels. I n d e v elo p in g and e v a l u a t ing t h i s model, S c h lip p made s e v e r a l assum ptions which ap p e a r t o be u n te n a b le . These assum ptions were: ( l ) t h a t an ex p ec te d v a lu e model was an a p p r o p r i a te c h o ic e o f models; (2) t h a t t h e work lo ad (bench ste p p in g ) was a c o n s ta n t; t h a t i s , th e bench ste p p in g e x e r c i s e was o f th e same approx im ate s t r e s s f o r a l l s u b j e c t s ; and ( 3 ) t h a t th e curve was b e s t d e s c r ib e d by two components. T his t h i r d assum ption i s c h a lle n g e d due to th e fo llo w in g sta te m e n t and th e r e s u l t s d is p la y e d In F ig u re 1. S c h lip p s t a t e d , "The r a t e i s s t i l l going up a t th e end o f two m in u te s, and th e i n d i c a t i o n s ■ a r e t h a t i t would r e q u i r e about 4 m inutes f o r a ste a d y 24 s t a t e t o be re a c h e d " ( 6 8 ). S c h l i p p 1s c u rv e i n d i c a t e s t h a t t h e a c c e l e r a t i o n o f h e a r t r a t e had r e a c h e d an a sy m p to te b e f o r e f o u r m in u te s . I n a d d itio n * th e c u rv e may c o n t a i n a s many com ponents a s one w ish e s to f a c t o r ; t h e c r i t i c a l f e a t u r e b e in g t h e d i f f e r e n c e betw een o r among t h e d e r i v e d co n s t a n t s . The c h o ic e o f an e x p e c te d v a lu e model does n o t p e rm it t h e i n v e s t i g a t o r t o p a r t i a l o u t any v a r i a n c e w hich may be due t o i n d i v i d u a l d i f f e r e n c e s . I n r e g a r d t o th e second assum ption* s i n c e t h e r a t e i n c r e a s e was n o t n o rm a l i z e d as f u n c t i o n of work load* th e g ro up r e s p o n s e c u rv e must c o n t a i n a l a r g e amount o f i n d i v i d u a l v a r i a n c e ; t h e r e fore* t h e a s s u m p tio n o f e q u al work lo a d I s n o t t e n a b l e . Also* t h e a s s u m p tio n Aq = 80 a p p e a rs t o b i a s t h e model a g a i n s t f i t I n d i v i d u a l s . A model v e ry s i m i l a r t o S c h l i p p 's was p ro p o se d by Brodan and Kuhn i n 1 9 6 8 . T h e ir model s t a t e d : = 0^(1 - e~k l t ) + 0 , ( 1 - e - k l l t ) + C (38) p = I n c r e a s e I n h e a r t r a t e a t s p e c i f i c tim e a l + P 2 = Th e p o i n t s where t h e c u rv e d e f l e c t s k]_ = R a te c o n s t a n t k n = R ate c o n s ta n t t = Time e = Base o f a n a t u r a l lo g a r ith m C = I n t e g r a t i o n c o n s ta n t 25 Reduced to common term s th e s e models a re as fo llo w s: yt = A ( 1 - + B ( 1 - e _ i 2 t ) t P = A(l - e k l t ) + B(l - e“kat) + C 7*- t P = H eart r a t e in c r e a s e a t t A + B - The p o in ts where th e curve d e f l e c t s k^ + k 2 = R ate c o n s ta n ts t = Time e = Base of a n a tu r a l lo g arith m C = I n t e g r a t i o n c o n s ta n t As one can see, th e p rim ary d i s t i n c t i o n I s th e a d d itio n of an i n t e g r a t i o n c o n s ta n t. Since t h i s c o n sta n t i s a c t u a l l y equal t o zero , t h e r e a re no d i f f e r e n c e s between th e s e two models. N a tu ra lly , th e models sh are the same d is a d v a n ta g e s which were p r e v io u s ly d is c u s s e d . F ig ure 2 p r e s e n t s th e d a ta from th e s e preceding r e p o r t s . I t should be p o in te d out t h a t th e d a ta have been tran sfo rm e d i n to s i m i l a r dim ensions and t h a t no a d ju stm en ts were made due t o d i f f e r e n t work lo a d s . I n 1968 Suggs r e p o r te d th e d e r iv a tio n and e v a lu a t i o n of an e x p o n e n tia l h e a r t r a t e response eq u atio n . Suggs' model combined some of th e f e a tu r e s of a d e te r m in is t i c model w ith f e a t u r e s of an ex pected value model. The h e a r t r a t e p r e d i c t i o n e q u a tio n which r e s u l te d from t h i s i n v e s t i g a t i o n was: H E A R T R A T E 80 60 40 20 6 m inutes 0 X = 1 .5 ( o /k g j 11=10 ( 3 8 ) ■ = 2.0 w/kgj N=10 (38) # = Bench. stepping, 36 S.P.M./12"; N=20 (6 6) F ig . 2 . —N orm alized h e a r t r a t e re s p o n s e d a ta from S c h lip p (6 7 ) and Brodan and Kuhn ( 3 8 ) ro < ? \ 27 H = He - Ae - T ^ t (69) H = Heart r a t e He = E q u ilib riu m h e a r t r a t e , th e p o in t a t which a b alan ce between energy in p u t and energy o u t put i s a ch iev ed A = Change i n h e a r t r a t e e = Base o f a n a t u r a l lo g arith m k]_ = Hate o r v e l o c i t y c o n sta n t t = Time The s o l u t io n of t h i s e q u a tio n i s not r e a d i l y d e d u c ib le from th e o ry but may be d e riv e d from ex p erim en tal d a ta . The k]_ v alu es f o r each group of d a ta were c a l c u l a t e d v i a r e g r e s sio n a n a l y s i s . The a u th o r dem onstrated th e r e s u l t s of th e model on one p a r t i c u l a r group of d a ta ; t h e f i t was q u ite good and th e re p o rte d c o r r e l a t i o n c o e f f i c i e n t was O.9 8 5 which i n d ic a te d t h a t k^ was s i g n i f i c a n t a t th e 0,01 l e v e l . For th e rem aining groups of d a ta th e c o r r e l a t i o n c o e f f i c ie n ts ranged from O.8 3 to 0.99» However, th e a u th o r d id r e p o r t a b a sic flaw of th e model: For th e com posite re sp o n se to e x e rc is e a t l i g h t and medium lo ad s and th e re c o v e ry respon se of RM a t medium and heavy lo a d s , th e c o e f f i c i e n t (k]_) in c r e a s e d as th e e x e r c is e l e v e l in c r e a s e d . For th e rem ainder of th e re s p o n se s, t h e c o e f f i c i e n t d ecreased as th e e x e rc is e l e v e l i n c r e a s e d . That i s , f o r th e rem ainder of th e d a ta , th e tim e c o n s ta n t, l / k ^ , in c re a s e d w ith th e ex e r c i s e l e v e l so t h a t e q u ilib riu m i s approached more slow ly. Ho e x p la n a tio n i s o ff e r e d re g a rd in g t h i s i n c o n s is te n c y i n th e d a ta . ( 6 9 ) Two p o in ts should be made i n d e fe n se of th e a u th o r; f i r s t , th e d a ta under d is c u s s io n were based on only one s u b je c t, and second and most im p o rta n t was t h a t t h e v a l i d i t y and hence t h e i n t e g r i t y of th e model was compromised by t h e f o llo w in g s e t o f a ssu m p tio n s: 1. C = S + R C = R ate o f oxygen consum ption S = R ate o f oxygen su p p ly R = Oxygen r e s e r v e s 2. R = - a 0 e- k t a Q = Change i n su p p ly e = Base of a n a t u r a l lo g a r ith m k = C o nstan t t = Time One may re c o g n iz e th e r i g h t h a l f o f t h e e q u a tio n as a p r o d u c t o f H e n ry ’s c l a s s i c a l s tu d y e n t i t l e d "A erobic Oxygen Consumption and A l a c t ic Debt i n M u scu lar Work." The a u th o r i n t e r p r e t e d H e n ry 's work a s f o llo w s : The a c c u m u la tio n o f s u b s t r a t e i s e q u iv a le n t t o th e d e p l e t i o n o f th e r e s e r v e s ; t h a t i s , t h e amount o f su b s t r a t e , x, p lu s th e amount o f m a t e r i a l l e f t i n t h e r e s e r v e s , Q . (b o th i n oxygen u n i t s ) , must be a c o n s ta n t e q u a l to th e i n i t i a l amount o f m a t e r i a l i n t h e r e s e r v e s . ( 6 9 ) I n h i s a r t i c l e Henry s t a t e d , " . . . th e t h e o r y i n i t s p r e s e n t form a p p l i e s o n ly to e x e r c i s e t h a t i s s u f f i c i e n t l y m oderate t h a t th e slow component d e b t i s n o t q u a n t i t a t i v e l y im p o r ta n t" (5 2 ). He c o n tin u e d : . . . th e d e r i v a t i o n o f t h e fo rm u la p o s t u l a t e d su b s t r a t e p r o d u c tio n i n p r o p o r t i o n t o r a t e o f work, p r e d i c t i n g a s t e a d y - s t a t e oxygen i n t a k e a Q d e te rm in e d 29 o n ly by th e r a t e of work, even though th e amount o f s u b s t r a t e a 0/ k can v a ry as a r e s u l t of i n d i v i d u a l d i f f e r e n c e s i n k. I t may be m entioned t h a t t h i s f i n d in g m ight have been ob scured i f th e r a t e of work had been m easured as e x te r n a l work perform ed , s in c e energy would be expended and s u b s t r a t e formed w hether th e muscle c o n t r a c t i o n s tu rn e d th e b ic y c le w heel, opposed i t s t u r n i n g due t o im p e rfe c t c o o r d in a tio n , o r were used t o m a in ta in p o s t u r e . . . . At h ig h e r work lo ad s i n d i v id u a l d i f f e r e n c e s i n m e ta b o lic e f f i c i e n c y may occur because th e r e g io n o f l im i t e d oxygen i n t a k e i s p ro b ab ly n o t t h e same f o r everyone. I f i t v a r i e s , th e p r o p o r t i o n o f a n a e ro b ic to a e ro b ic o x id a tio n w i l l v a ry , r e s u l t i n g in d i f f e r e n c e s i n e f f i c i e n c y . ( 5 2 ) As Henry h as p o in te d o u t, th e assum ption t h a t R = c t Q e ~ ^ i s n o t t e n a b le f o r a l l o f th e v a r io u s c o n d itio n s under which th e p r e s e n t model was t e s t e d . I n summary, th e model c o n s tr u c te d by Suggs was s u p e r i o r t o th e p re v io u s models d is c u s s e d ; however, th e g e n e r a l i t y o f t h e model was poor due to th e a u t h o r 's e r roneous assu m p tio n s i n re g a rd to energy r e s e r v e s . I n 1968 Cardus and Z e i g le r a ls o proposed a model of h e a r t r a t e re s p o n s e ; t h e i r model s t a t e d : a = C onstant b = C onstant c = C onstant w = Work load t = Time (in) y I n c r e a s e i n h e a r t r a t e 30 T h e ir fo rm u la i s a p p li c a b l e i f , and o n ly i f , t Q i s g r e a t e r t h a n t and t i s g r e a t e r th a n z e r o . The p a r t i c u l a r a d v a n ta g e o f t h i s model was t h e u se of a fe e d b a c k loop which i n c o r p o r a t e s t h e change p e r u n i t tim e of h e a r t r a t e re s p o n se and t h e t o t a l tim e of th e work p e r i o d i n th e c a l c u l a t i o n o f y*3. T h is p a r t i c u l a r f e a t u r e e n a b le s th e model t o more a c c u r a t e l y d e s c r i b e th e change i n h e a r t r a t e t o work lo a d s of v a r i o u s i n t e n s i t i e s . The weak n e ss o f t h i s p a r t i c u l a r model i s : The model does n o t p r e d i c t th e bending o f t h e " s te a d y s t a t e " c u rv e r e l a t i n g h e a r t b e a t fre q u e n c y and work lo a d a t work lo a d s a p p ro a c h in g maximum h e a r t r a t e s . T h is i s due, of c o u rs e , t o th e a ssu m p tio n F(w) = W . (4l) T h e r e fo r e , t h i s model b re a k s down i n t h e same manner as d id th e p r e v io u s model. I n a d d itio n , t h e p h y s i o l o g i c a l s i g n i f i cance o f th e c o n s t a n t s has n o t , a s y e t , been a s c e r t a i n e d . I n a d d i t i o n t o th e d e t e r m i n i s t i c model w hich was d e s c r ib e d i n t h e p re c e d in g p a r a g r a p h s , t h e a u th o r s a l s o p ro p o se d a s t o c h a s t i c model o f h e a r t r a t e r e s p o n s e .b ased on th e P o is s o n p r o b a b i l i t y p r o c e s s . U n f o r tu n a te ly , i n t h e i r f i n a l a n a l y s i s o f t h e s t o c h a s t i c model t h e a u th o r s u t i l i z e d a v a lu e o b ta in e d by way of t h e d e t e r m i n i s t i c model f o r th e 1 / v a r ia n c e of y . While th e s t o c h a s t i c model of t h e P o is so n p r o c e s s i s a t t r a c t i v e from a p r o b a b i l i s t i c v ie w p o in t, i t does have c e r t a i n l i m i t a t i o n s . I n p a r t i c u l a r , s in c e t h e v a r ia n c e I s g iv e n by a^, t h i s would i n d i c a t e t h a t , In t h e e x e r c i s e p h a se , th e v a r ia n c e would become l a r g e r a s t i n c r e a s e s . T here a re I n d i c a t i o n s t h a t t h i s i s n o t gen e r a l l y t h e c a s e . (4 l) 31 The f i n a l model re v ie w e d h e r e i s th e v e ry s o p h i s t i c a t e d model d e v e lo p e d by P i c k e r i n g and h i s c o -w o rk e rs . I t i s i l l u s t r a t e d i n t h e f o llo w in g b lo c k d iagram (s e e P ig . 3)* O b v io u sly , t h e i r model i s b a se d on numerous assu m p tio n s and a p p ro x im a tio n s . Not so o b v io u s , a r e th e e x tre m e ly good r e s u l t s o f t h e s e a p p ro x im a tio n s . F ig u r e 4 d e p i c t s a p o r t i o n o f t h e d a t a from one s u b j e c t . U n dou b ted ly , t h e m ajor d i s c r e p a n c i e s o f th e model a r e due t o : "The e q u a tio n s u se d i n t h e model w ere f o r t h e most p a r t l i n e a r and a l l c o e f f i c i e n t s w ere assumed t o be c o n s t a n t f o r a l l work lo a d s " (6 4 ). The l e v e l o f s o p h i s t i c a t i o n o f t h i s l a s t model i s f a r b e yond t h a t w hich was r e q u i r e d f o r t h e p r e s e n t s tu d y . T his f i n a l model was p r e s e n t e d m e re ly t o a c q u a in t th e r e a d e r w ith t h e u n l i m i t e d p o s s i b i l i t i e s o f t h i s te c h n iq u e . P a r t Four Review of H e a rt R a te L i t e r a t u r e No o t h e r p h y s i o l o g i c a l p a ra m e te r h a s been m easured o r s t u d i e d more o f t e n t h a n h a s t h e r e s p o n s e of h e a r t r a t e t o e x e r c i s e o r work lo a d s o f v a r i o u s i n t e n s i t i e s . I t has been w e l l e s t a b l i s h e d t h a t h e a r t r a t e i s l i n e a r l y r e l a t e d , o r p r o p o r t i o n a l , t o any g iv e n work lo a d ( 1 ,2 ,3 * 4 ,5 * 1 0 ,1 1 , 1 2 ,1 4 ,1 5 ,2 0 ) . I n a d d i t i o n , numerous i n v e s t i g a t o r s have d e m o n s tr a te d t h e e x i s t e n c e o f a h i g h p o s i t i v e c o r r e l a t i o n betw een h e a r t r a t e and oxygen c o n su m p tio n ( 4 5 ,5 7 ,6 8 ) . F ig . 3 . —Block diagram of " P ic k e rin g " model (64) 0. 1 2 3 4 5 6 7 8 = Work load 9. = Multiplier = Pressure Reference function 10. = Multiplier Constant Integrator Carotid Sinus Baroreceptor function Central Nervous System Heart Rate Frequency Constant Stroke Volume function 11. = Arterial Peripheral Resistance function 12. = Oxygen Need 1 3. - Integrator 14. = Effective 02 debt function 15. = Arterial/Venous 02 difference function 16. = Multiplier 1 7. = Minute Volume 02 required 100 A c tu a l P r e d ic te d t n o t r e p e a te d 300 KPM 600 KPM 900 KPM 1050 KPM F ig . 4.—Diagram m atic p r e s e n t a t i o n o f r e s u l t s of P ic k e r in g model (64) 00 34 . . . because of i t s c lo s e r e l a t i o n t o c a rd ia c o u tp u t and oxygen consum ption, h e a r t r a t e can be u t i l i z e d to e v a lu a te th e s t r e s s imposed by m u scu lar a c t i v i t y upon th e h e a r t and th e c i r c u l a t i o n w ith a minimum amount of i n t e r f e r e n c e w ith th e s u b j e c t 's freedom of m otion and perform ance a b i l i t y . As a s i n g l e f a c t o r , i t q u ite a c c u r a t e l y d e p ic ts th e c a r d i o v a s c u l a r a d ju stm en t o f th e i n d i v i d u a l t o m uscular a c t i v i t y . (12) I n a d d i t i o n , B erggren and C h r is te n s e n s t a t e d : The I n c r e a s e d Og consum ption d u rin g work i s c l o s e l y r e l a t e d to a sim u lta n eo u s i n c r e a s e in c i r c u l a t i o n r a t e which d e m o n stra te s i t s e l f th ro u g h p u ls e a c c e l e r a t i o n . Even d u rin g se v ere work t r a i n e d s u b j e c t s w i l l show a p r a c t i c a l l i n e a r r e l a t i o n betw een work i n t e n s i t y and p u ls e r a t e , and a ls o betw een m e ta b o lic r a t e and p u ls e , as e f f i c i e n c y i s p r a c t i c a l l y c o n s ta n t over a wide ra n g e . C onsequently p u ls e c o u n ts d u rin g work ought to g iv e r a t h e r dependable in f o rm a tio n s [ s i c ] about th e m e ta b o lic r a t e . (37) As was a lr e a d y s t a t e d , th e a e r o b ic c a p a c i ty can be d e te rm in e d . U n f o rtu n a te ly , th e method i s r a t h e r tim e consuming, and r e q u i r e s c o m p lic a te d la b o r a to r y methods. As maximal work f o r a t l e a s t 5 m inutes i s n e c e s s a ry f o r d i r e c t d e te rm in a tio n , c o l l a b o r a t i o n on th e p a r t o f th e s u b j e c t i s u r g e n t l y needed. There i s r i s k of o v e r s t r a i n , e s p e c i a l l y w ith u n t r a i n e d o r o ld e r s u b j e c t s j w ith p a t i e n t s t h i s method i s c e r t a i n l y i n a d v i s a b le . To what e x te n t, th e n , i s i t p o s s i b l e t o draw any c o n c lu s io n s about r e s p i r a t o r y - c i r c u l a t o r y f i t n e s s from i n v e s t i g a t i o n s a t r e s t , d u rin g submaximal work, o r r e covery? (3 2 ) F a c to r s A f f e c tin g R estin g H e art R ate R e stin g h e a r t r a t e i s in f lu e n c e d by many f a c t o r s i n c l u d i n g : age, sex, p o s tu r e , food i n t a k e , smoking, body te m p e r a tu re , th e com bination of e n v iro n m e n ta l te m p e ra tu re and h u m id ity , d i u r n a l v a r i a t i o n , and p s y c h i a l f a c t o r s (5, 1 2 ,1 4 ,2 0 ,3 3 ,7 0 ) . I t i s h i g h ly p o s s i b l e t h a t f a c t o r s , i n a d d i t i o n t o th o s e p r e v io u s ly m entioned, may in f lu e n c e r e s t i n g h e a r t r a t e . I t h a s b een w e l l docum ented t h a t th e " r e s t i n g " h e a r t r a t e d e c l i n e s w ith ag e. T h is d e c l i n e o c c u rs up t o e a r l y a d o le s c e n c e a f t e r w hich t h e v a r i a t i o n i s no lo n g e r o b s e rv e d ( l 4 ) . A lso , numerous a u t h o r i t i e s have r e p o r t e d t h a t t h e r e i s a d e c l i n e w ith age i n t h e maximal a t t a i n a b l e h e a r t r a t e . R ecent s t u d i e s c o n d u c te d by A s tra n d have r e p o r t e d t h a t t h e maximum h e a r t r a t e d e c l i n e s on th e a v e ra g e o f 1 .3 p e r c e n t p e r y e a r ( 3 2 ). T ab le 1 p r e s e n t s a summary o f t h e d a t a a s r e p o r t e d by Morehouse o f an e a r l y s tu d y c o n d u c te d by S c h n e id e r and T r u e s d e l l w hich e x p lo r e d th e r e l a t i o n s h i p betw een p o s t u r e and r e s t i n g h e a r t r a t e . Most o b s e r v a t i o n s hav e shown t h a t p u l s e r a t e I s d e f i n i t e l y a f f e c t e d by body p o s i t i o n . The e x a c t meaning o r s i g n i f i c a n c e o f t h i s change h a s n o t been w e l l d e f i n e d . I t i s g e n e r a l l y th o u g h t t h a t th e change i n p u l s e r a t e i s a co m p en sato ry f a c t o r r e l a t e d t o t h e h y d r o s t a t i c chang es Im posed on t h e v a s c u l a r system j t h e r e a p p e a r s t o be no s i g n i f i c a n c e I n t h e m agnitude o f th e p u l s e r a t e change f o r h e a l t h y s u b j e c t s ( 5 3 ) . O b v io u sly , a g r o s s d i f f e r e n c e i n p u l s e r a t e a s a r e s u l t o f a change i n p o s t u r e may i n d i c a t e t h e e x i s t e n c e o f a p a t h o l o g i c a l c o n d i t i o n ( 2 0 ). An i n c r e a s e i n te m p e r a tu re r e s u l t s i n an i n c r e a s e i n r e s t i n g p u l s e r a t e . T h is r e l a t i o n s h i p h o ld s t r u e w h e th e r t h e i n c r e a s e i n body t e m p e r a tu r e i s due t o a f e b r i l e s t a t e o r an I n c r e a s e I n t h e e n v iro n m e n ta l t e m p e r a t u r e . Of c o u r s e , an I n c r e a s e i n body t e m p e r a tu re h a s a much g r e a t e r 36 a f f e c t on r e s t i n g h e a r t r a t e th a n does an i n c r e a s e i n t h e e n v iro n m en ta l te m p e ra tu re . The e f f e c t s of e n v iro n m en tal te m p e ra tu re a re m o dified by h u m id ity and a i r movement. The r e s u l t s o f s e v e r a l s t u d i e s have i n d ic a t e d t h a t th e h e a r t r a t e re sp o n se to e x e r c is e i s a more v i a b l e measurement th a n oxygen consum ption when work c a p a c ity i n a warm environm ent i s t o be measured ( 2 ,2 7 ,2 8 ,3 9 ,4 0 ,6 0 ) . TABLE 1 POSTURAL CHANGES AEEECTING HEART RATE (20) Lying S tanding D Average s u b je c t 74 92 18 A th le te 66 83 17 I t has long been a common o b s e rv a tio n th a t th e h e a r t r a t e has a d i u r n a l v a r i a t i o n . I n most i n d i v i d u a l s th e r e s t i n g p u ls e i s a t i t s low est j u s t b e fo re awakening. T his ebb p o in t i s u s u a l l y fo llo w ed by an i n c r e a s e , a p la te a u and a n o th e r i n c r e a s e which peaks i n th e l a t e a fte r n o o n ( l 4 ) . P s y c h ia l f a c t o r s , such as em otional s t r e s s , have a v e ry stro n g e f f e c t on r e s t i n g h e a r t r a t e (5 5 ). deV ries s t a t e d , "Em otional s t r e s s b r in g s about a c a r d io v a s c u la r r e sponse t h a t i s q u i te s i m i l a r t o th e resp o n se to e x e r c is e " ( 5 ). deV ries a ls o s t a t e d t h a t th e e f f e c t s of em otional ex cite m en t a r e most e a s i l y ob serv ed a t r e s t . 37 I n t a k i n g a r e s t i n g p u l s e r a t e b e f o r e an e x e r c i s e i n v o lv in g an elem en t o f c o n t e s t , one sh o u ld remember t h a t i n s t e a d o f a " r e s t i n g " r a t e t h e r e m ight be a " s t a r t " . . . p u l s e a c c e l e r a t e d by t h e e x c ite m e n t o f a n t i c i p a t i o n . . . . One sh o u ld n o t f o r g e t t h a t a p u l s e r a t e o b ta in e d d u r in g a p e r i o d o f a p p a re n t r e s t may not n e c e s s a r i l y be a r e s t i n g p u l s e . ( 5 ) A n te l and Cummlng ( 2 9 )j i n t h e i r r e c e n t s tu d y , c o n c lu d e d t h a t t h e e f f e c t s of em otion on th e h e a r t r a t e a r e n o t b lo c k e d d u rin g e x e r c i s e ; r a t h e r , th e e f f e c t s o f em otion and e x e r c i s e a r e a d d i t i v e , t h e r e b y g r e a t l y I n c r e a s i n g th e h e a r t r a t e and r e s u l t i n g i n t h e p r e d i c t i o n o f an e rro n e o u s and low v a lu e o f t h e s u b j e c t ' s work c a p a c i t y (5 ). One o f t h e g r e a t e s t d i f f i c u l t i e s i n a s s e s s i n g th e p h y s i o l o g i c a l s i g n i f i c a n c e o f th e r e s t i n g h e a r t r a t e h a s been th e d i f f i c u l t y i n c o n t r o l l i n g o t h e r v a r i a b l e s w hich most i n f l u e n c e th e r e s t i n g p u l s e r a t e . F u rth e rm o re , t h e r e h a s been a la c k o f c o n se n s u s among a p p li e d p h y s i o l o g i s t s i n d e f i n i n g t h e te rm " r e s t i n g h e a r t r a t e . " From th e r e s u l t s o f an e a r l y stu d y Brohua c o n c lu d e d , "m easurem ents t a k e n a t r e s t hav e l i t t l e o r no r e l a t i o n t o t h e p e rfo rm an c e c a p a c i t y of t h e i n d i v i d u a l " ( 2 , 3 5 ) . He f u r t h e r s t a t e d t h a t no s a t i s f a c t o r y r e l a t i o n was found b e tw een b a s a l o r s i t t i n g p u l s e r a t e and th e c a p a c i t y t o p e r form h a rd w ork. F u rth e rm o re , d i f f e r e n c e s betw een s i t t i n g h e a r t r a t e and l y in g h e a r t r a t e , a s w e ll a s t h o s e betw een t h e s ta n d in g h e a r t r a t e and l y in g h e a r t r a t e , show no r e l a t i o n t o an i n d i v i d u a l ' s c a p a c i t y t o p erfo rm work ( 2 ) . 38 R eco v ery H e a rt R a te as an I n d e x o f S t r e s s M endelsohn (1901) was one o f t h e e a r l i e s t i n v e s t i g a t o r s t o r e c o r d t h e p u l s e change f o l l o w i n g e x e r c i s e ; he u s e d t h i s m easure a s an i n d i c a t i o n o f th e amount o f s t r e s s t h e body can t o l e r a t e ( 5 9 ) . P ro b a b ly t h e b e s t known s tu d y u s in g t h e r e c o v e r y h e a r t r a t e a s a c r i t e r i o n m easure f o r p r e d i c t i n g work c a p a c i t y was p u b l i s h e d by M a x f ie ld and B rouha. They s t a t e d , "A r e l a t i o n betw een th e p h y s i o l o g i c a l s t r a i n in d u c e d by t h e s t r e s s e s o f work a n d / o r h e a t and t h e r e c o v e r y from t h a t s t r a i n h a s b een a c c e p t e d f o r many y e a r s " ( 6 0 ) . I n an e a r l i e r p u b l i c a t i o n B rouha s t a t e d . As p r e v i o u s l y i n d i c a t e d , i t i s s e e n from t h e s e m e a s u re m ents t h a t t h e h e a v i e r t h e work lo a d t h e h i g h e r t h e h e a r t r a t e d u r i n g r e c o v e r y and t h e more s lo w ly i t r e t u r n s t o t h e r e s t i n g l e v e l . I n a d d i t i o n , t h e b e t t e r t h e p h y s i c a l c a p a c i t y o f a n i n d i v i d u a l , t h e s m a ll e r t h e i n c r e a s e i n h i s h e a r t r a t e f o r a s t a n d a r d work lo a d and th e more r a p i d i t s r e t u r n to i t s r e s t i n g v a l u e . Thus, p h y s i o l o g i c a l s t r e s s and p h y s i c a l a p t i t u d e f o r a s p e c i f i c jo b c a n be d e te rm in e d from t h e r e c o v e ry p u l s e r a t e s . ( 2 ) I n c o n t r a s t , A s tr a n d s t a t e d : E x p e rie n c e shows t h a t t h e i n v e s t i g a t i o n o f d i f f e r e n t f u n c t i o n s d u r in g r e c o v e r y from m u s c u la r work d oes n o t g i v e r e l i a b l e i n f o r m a t i o n a b o u t t h e r e a c t i o n o f t h e s e sy ste m s t o t h e w ork. F o r one and t h e same p e r s o n th e c o r r e l a t i o n b e tw e e n , e . g . , t h e p u l s e r a t e d u r in g a c o n s t a n t work and t h e p u l s e r a t e a f t e r a c e r t a i n tim e a f t e r t h e end o f t h e w ork i s r e l a t i v e l y h ig h , b u t t h e d e c l i n e from a c e r t a i n l e v e l o f p u l s e r a t e f o llo w s w i t h v a ry in g sp e e d i n one i n d i v i d u a l com pared w i t h a n o t h e r , even i f th e p h y s i c a l c o n d i t i o n i s r o u g h ly t h e same, . . . I t seems a lm o s t c e r t a i n t h a t i f t h e aim i s t o examine t h e p h y s i c a l w ork c a p a c i t y o f an i n d i v i d u a l , 39 t h e ex am in a tio n sh o u ld be made d u rin g m u scular work. (32) A s tra n d p ro v id e s a n o th e r m ajor c r i t i c i s m o f t h i s th e o r y : As a m easure o f th e s t r e s s on c i r c u l a t i o n some t e s t s u se th e number of p u l s e b e a ts beyond r e s t l e v e l d u rin g work and re c o v e ry from t h i s l o a d — A rbeitspulssum m e, APS. The low er t h i s sum, t h e l i g h t e r th e lo a d i s con s i d e r e d t o be on th e c i r c u l a t i o n . I t must be re g a rd e d as t h e o r e t i c a l l y i n c o r r e c t t o e v a l u a t e th e e f f i c i e n c y o f th e c i r c u l a t o r y system by th e n e t p u ls e , i . e . , th e p u ls e r a t e d u rin g work minus th e p u ls e r a t e a t r e s t , as a b a s i s , s in c e 10 p u ls e b e a t s a t r e s t , w ith a s tr o k e volume o f p e rh ap s 60 m i l l i l i t e r s , a r e not e q u iv a le n t t o 10 work b e a ts a t a s t r o k e volume o f p e rh a p s 150 m i l l i l i t e r s . For s i m i l a r re a s o n s th e r e c o v e r y p u ls e r a t e sh o u ld h a r d l y be "mixed up" w ith th e work p u ls e r a t e when form ing APS. (32) An exp erim en t co nd ucted by D enolin e t a l . s u p p o rts t h e p o s i t i o n ta k e n by A stra n d . D eno lin and h i s co -w o rk ers s t u d i e d th e h e a r t r a t e re c o v e r y c u rv e a f t e r t h r e e s t a n d a r d iz e d b o u ts of e x e r c i s e . They c o r r e l a t e d th e work c a p a c i t y r e s u l t s , as i n d i c a t e d by th e r e c o v e r y c u rv e s , w ith t h e r e s u l t s o f a PWCjjjQ t e s t . They c o n clu d ed t h a t , "The c o r r e l a t i o n betw een th e r e s u l t s th u s o b ta in e d and th o s e g iv e n by t h e t e s t b a sed on t h e h e a r t r a t e d u rin g work (PWCqyo) i s low" (1 5 ). The c o r r e l a t i o n between t h e s e two t e s t s was - 0 .4 0 . I n s t u d i e s on th e same i n d i v i d u a l a c o r r e l a t i o n b e tween r e c o v e r y p u ls e and work p u l s e o f O.9 6 has been r e p o r te d ; however, when th e same m easurement was made on a group of s u b j e c t s th e c o r r e l a t i o n was re d u c e d to 0 .7 7 ( 1 5 ). A s tra n d h as s t a t e d t h a t some c o r r e l a t i o n betw een t h e r e s u l t s o b ta in e d from any t e s t , even a q u e s t i o n n a i r e , and t h e 4o a e r o b i c c a p a c i t y s h o u ld be assumed ( 3 2 ) . A s t r a n d 's p o in t I s w e ll made., a c o r r e l a t i o n does n o t im ply a c au se and e f f e c t r e l a t i o n s h i p ; I t I s e n t i r e l y p o s s i b l e t h a t an a d d i t i o n a l f a c t o r I s r e s p o n s i b l e f o r t h e ob s e r v e d c o r r e l a t i o n s . He s t a t e d f u r t h e r t h a t " f o r an e v a l u a t i o n o f an i n d i v i d u a l from i n d i r e c t m ethods th e s t a n d a r d d e v i a t i o n from t h e r e g r e s s i o n l i n e i n d i c a t e s t h e a c c u ra c y o f th e p r e d i c t i o n " ( l ) . A d d i t i o n a l s u p p o r t f o r A s t r a n d ' s v ie w p o in t comes from t h e d a t a o f a r e c e n t s tu d y made by McArdle e t a l . who c o n c lu d e d t h a t " th e r a t e of r e c o v e r y i s not s i m i l a r f o r a l l i n d i v i d u a l s . T h is v a r i a b i l i t y may i n tr o d u c e s e r i o u s e r r o r s when t h e p o s t - e x e r c i s e h e a r t r a t e i s u se d t o i n f e r th e h e a r t r a t e d u rin g e x e r c i s e " ( 6 l ) . Submaximal Work Volumes of s t u d i e s have been p u b l is h e d w hich r e p o r t r e l a t i o n s h i p betw een h e a r t r a t e d u r in g submaxlmal work and an i n d i v i d u a l ' s maximal a e r o b i c c a p a c i t y ( 3 1 ) . M a ste r and Oppenheim er r e p o r t e d t h e u s e o f a f u n c t i o n a l s t r e s s t e s t i n 1 9 2 9; t h e y s t a te d * "by c a r e f u l u se of t h i s t e s t t h e p h y s i c i a n may d i s c o v e r t h e e a r l y s t a g e s o f c i r c u l a t o r y i n s u f f i c i e n c y * o r may t r a c e t h e p r o g r e s s o f an o r g a n ic c o n d i t i o n o f t h e h e a r t . . . " ( 5 9 ). A s tu d y p u b l is h e d by d e V rie s and K la f s ( 1 9 6 5 ) e v a l u a te d f i v e o f t h e commonly u se d subm axlm al work t e s t s . 4 l They c o n clu d ed t h a t p h y s i c a l w orking c a p a c i ty c o u ld he p r e d i c t e d from submaximal t e s t s (4 8 ). T able 2 i s a p r e s e n t a t i o n o f t h e i r f i n d i n g s . I t should be n o te d t h a t t h e i r c o r r e l a t i o n c o e f f i c i e n t s were based on th e r e l a t i o n s h i p b e tween a submaximal t e s t and maximal oxygen I n t a k e , Davis r e p o r t e d a 7 p e rc e n t e r r o r of p r e d i c t i o n when u s in g fre q u e n c y o f h e a r t r a t e m easures t o p r e d i c t maximal oxygen consum ption (4 4 ). The r e s u l t s o f h i s I n v e s t i g a t i o n c o r r o b r a t e t h e f i n d i n g s of d eV ries and K la fs (4 8 ). I n a r e c e n t a r t i c l e Alderman s t a t e d , "The q u e s t io n a s t o w h e th er a common f a c t o r was o p e r a t in g w ith r e s p e c t t o h e a r t r a t e re s p o n s e t o two d i f f e r e n t work lo a d s w i t h i n th e same t a s k was answ ered i n th e a f f i r m a t i v e " (2 6 ). T h is le d to th e fo llo w in g c o n c lu s io n : "On t h e b a s i s o f t h i s h ig h com m onality of i n d i v i d u a l d if f e r e n c e s , i t would a p p e a r t h a t h e a r t r a t e t e s t s s i m i l a r to th e one u se d I n t h i s s tu d y a re p o w e rfu l enough to d i f f e r e n t i a t e betw een I n d i v i d u a l s w ith r e s p e c t t o work c a p a c i ty " ( 2 6 ). H e lls tro m and Holmgren c o n clu d ed t h a t " th e r e p e a t a b i l i t y of (work lo a d a t h e a r t r a t e 1 7 0 ) and t h e h e a r t r a t e a t 900 KPM/min. was 4 .9 and 4 .0 r e s p e c t i v e l y , e x p re sse d as t h e s ta n d a r d e r r o r o f th e s i n g l e d e te r m in a tio n " ( 5 1 ). M a ritz e t a l . co n clu d ed from t h e i r stu d y o f maximum oxygen i n t a k e and maximum h e a r t r a t e d u rin g s tre n u o u s work t h a t a t low r a t e s o f work a s t r a i g h t l i n e f i t s b o th oxygen i n t a k e and h e a r t r a t e p l o t s a g a i n s t work r a t e , b u t a t h ig h 42 TABLE 2 PREDICTIVE VALUE OP TESTS BASED ON HEART RATE FOR PREDICTING MAXIMUM OXIGEN INTAKE Sjostrand Harvard Step Test Progressive Pulse Ratio Astrand- Ryhming Nomogram Correlation * CD • 766 •711 .736 Standard Error + 4-. 7^ ml/kg ± 6.35 ± 6.93 +3*59 L/min. Standard Error (percent) + 9 .^ + 12.5 ± 13.7 ± 9.3 43 work r a t e s t h e c u rv es te n d tow ard an asym pto te, w ith th e oxygen i n ta k e curve re a c h in g i t s p l a t e a u more slo w ly th an th e h e a r t r a t e curve ( 5 8, 6 6 ). CHAPTER I I I DERIVATION OF THE MODEL Model I The model w hich was d e v elo p ed I n t h i s s tu d y was b a se d on t h e p r i n c i p l e o f c o n t r o l t h e o r y . Due t o t h e i n h e r e n t c o m p le x ity o f t h e c a r d i o v a s c u l a r system and to th e many i n t e r a c t i n g r e g u l a t o r s o f t h e system i n a d d i t i o n to t h e l a c k o f s p e c i f i c knowledge r e l a t e d t o c o n t r o l f u n c t i o n s , i t was d e c id e d t h a t o n ly a s i m p l i f i e d model o f t h e h e a r t r a t e r e s p o n s e t o s t r e s s would be s y n t h e s i z e d . The model d e v elo p ed a s a r e s u l t o f t h i s i n v e s t i g a t i o n was a c o m b in a tio n o f t h e e x p e c te d v a lu e and d e t e r m i n i s t i c m odeling p r o c e d u r e s . I n a d d i t i o n , t h e model was d e v e lo p e d w ith a c l o s e d loop ( e x t r i n s i c c o n t r o l ) , o r fe e d b a c k lo o p f e a t u r e . I n t h i s p a r t i c u l a r c a s e t h e fe e d b a c k loop I s a n e g a t i v e fe e d b a c k lo o p j t h a t I s , a change i n one param e t e r i n f l u e n c e s t h e change I n a n o th e r p a r a m e te r w hich i n t u r n r e g u l a t e s o r l i m i t s th e r a t e o f change a n d /o r t h e amount o f change o f th e f i r s t p a ra m e te r. I t i s t h e o p in io n o f numerous a u th o r s t h a t a n e g a t iv e fe e d b a c k system g o v e rn s a l l o f m an 's i n t e r n a l c o n t r o l mechanisms (9 ^ 2 2 ,2 4 ). P r i o r t o p ro c e e d in g w ith a d i s c u s s i o n o f th e model, 44 45 a b r i e f review of models proposed In th e l i t e r a t u r e along w ith t h e i r b a s ic flaw s i s p re s e n te d . F irst., th e model which was proposed by S chlipp and l a t e r r e s t a t e d by Brodan and Kuhn: t h i s model proposed t h a t t h e in c r e a s e i n e x e rc is e h e a r t r a t e i s a f u n c tio n o f two r a t e c o n s ta n ts and a th e o r e t i c a l maximum h e a r t r a t e which can be p r o p o r t io n a ll y con ceded to each of th e two r a t e c o n s ta n ts . T his model i g n o res two w e ll known f a c t s : ( l ) th e wide range of in d iv id u a l v a r i a t i o n which occurs i n r e s t i n g o r p r e - e x e r c i s e h e a r t r a t e , and (2) th e model assumed t h a t th e upper l i m i t of th e e x e rc is e h e a r t r a t e i s a f ix e d q u a n ti t y which i s a d d itiv e to r e s t i n g h e a r t r a t e , th e re b y e q u a llin g any g iven i n d i v i d u a l ' s maximum h e a r t r a t e . For example, Schlipp assumed t h a t A0, th e p o in ts o f g r e a t e s t curve d e f l e c t i o n , was equal t o p lu s 8 0; i f t h i s be t r u e , th en th e maximal h e a r t r a t e would be e q u al to i n i t i a l o r r e s t i n g h e a r t r a t e p lu s 8 0 . Assuming t h a t a mean r e s t i n g h e a r t r a t e of 70 b e a ts p er m inute i s th e mean v alue f o r t h i s segment of th e p o p u latio n , th e n th e summed maximal h e a r t r a t e would be o n ly 150 b e ats p e r m inute. This v alu e of maximal h e a r t r a t e i s not i n agreement w ith th e values re p o r te d i n th e l i t e r a t u r e o r th o se commonly observed. The model proposed by Suggs i s a f a i r l y good a p p ro x im a tio n o f r e a l i t y . However, i n a d d it i o n to th e com ments i n C hapter I I , th e p r e s e n t w r i t e r would l i k e to add an a d d i t i o n a l comment: t h i s model does not a d ju s t th e 46 a c c e l e r a t i o n r a t e , o r th e r a t e c o n s t a n t , i n r e l a t i o n s h i p to v a r i o u s i n t e n s i t i e s o f s t r e s s . The fo rm u la r e p o r t e d by Cardus and Z e i g l e r was t h e r e s u l t of a v e ry s o p h i s t i c a t e d n o n - l i n e a r r e g r e s s i o n a n a l y s i s ; how ever, th e c o n s t a n t s w hich were u s e d i n t h i s fo rm u la were s t a t i s t i c a l c o n s t a n t s . The p h y s i o l o g i c a l s i g n i f i c a n c e o f t h e s e c o n s t a n t s was not a s c e r t a i n e d . I f a model i s t o be o f any p r a c t i c a l v a lu e , t h e p a ra m e te r s , v a r i a b l e s and c o n s t a n t s sh o u ld be r e a s o n a b ly c o n s i s t e n t and sh o u ld p o s s e s s a h ig h d e g re e o f i n t u i t i v e p h y s i o l o g i c a l m eaning. The p rim a ry p u rp o se o f t h i s b r i e f re v ie w was t o h i g h l i g h t s e v e r a l im p o r ta n t f a c t o r s which must be co n s i d e r e d i n d e r i v i n g a t h e o r e t i c a l model o f h e a r t r a t e r e spo nse t o s t r e s s . T hese f a c t o r s a r e : ( l ) th e model sh o u ld p r o v id e f o r a wide ra n g e of i n d i v i d u a l v a r i a t i o n i n a t l e a s t two p a r a m e t e r s - - i n i t i a l h e a r t r a t e and maximum h e a r t r a t e ; (2) th e m odel sh o u ld p r o v id e f o r v a r i a t i o n s i n p a ra m e te r s r e l a t i n g t o th e i n t e n s i t y o f s t r e s s , th e tim e p e r i o d f o r w hich th e s t r e s s i s a p p l i e d , and t h e t o t a l amount o f work p e rfo rm ed ; and ( 3 ) t h e d e s c r i p t i o n and l a b e l i n g o f c o n s ta n ts w i t h i n t h e fo rm u la s h o u ld be c o n s i s t e n t and p o s s e s s i n n a t e p h y s i o l o g i c a l v a lu e . I t was th e o b j e c t i v e o f t h e p r e s e n t s t u d y t o d e a l w i t h h e a r t fre q u e n c y a s a dynamic sy stem , a sy stem con s t r a i n e d by p h y s i c a l law s and p h y s i o l o g i c a l r e g u l a t i n g s y s tem s. The h e a r t r a t e re s p o n s e t o e x e r c i s e s o f v a r i o u s 47 i n t e n s i t i e s of s t r e s s were d e riv e d t h e o r e t i c a l l y , t h e n com p ared t o e x p e rim e n ta l o b s e r v a tio n s i n o r d e r to e v a lu a te and check th e a cc u ra c y o f th e p ro posed model. As p r e v io u s ly s t a t e d , th e proposed model i s b ased on m acroscopic p r i n c i p l e s and i s u n a f f e c te d by many o f th e i n t r i c a c i e s of th e p h y s i o l o g ic a l c o n t r o l system . I t sho uld be n o ted t h a t p h y s i o l o g ic a l s p e c i f i c i t i e s a r e r e f l e c t e d i n th e g e n e ric p r i n c i p l e s of c o n t r o l and r e g u l a t i o n on which th e model i s based. The model d e riv e d i n t h i s i n v e s t i g a t i o n i s b ased on th e fundam ental p r i n c i p l e o f h o m e o s ta s is . For t h e purp o se o f t h i s stu d y i t was assumed t h a t th e h e a r t r a t e c o n t r o l system i s c o n t i n u a l l y s t r i v i n g f o r b a la n c e between energy e x p e n d itu re , as m easured by p h y s i c a l work being perfo rm ed, and energy in p u t as r e f l e c t e d by changes i n h e a r t r a t e . S ta te d more sim ply, o r Ei n = E £. The r e a d e r may re c o g n iz e t h i s e q u a tio n as a g e n e r a l i z e d sta te m e n t of th e f i r s t and second laws o f therm odynam ics. I t I s a x io m atic t h a t when E^n = th e system i s i n a s t a t e o f dynamic e q u ilib riu m o r h o m e o s ta s is j i t i s as e q u a lly s e l f e v id e n t t h a t when E^n / Eo u t J ^ 6 B y s ^ ern l s ou't b a i l e e and w i l l a d ju s t i t s perform ance p a ra m e te rs In o r d e r to r e e s t a b l i s h h o m eo sta sis a t a new l e v e l . F ig u re 5 d e p ic ts a b lock diagram of th e system . 48 work F i g . 5 .— D iagram m atic p r e s e n t a t i o n o f h o m e o s ta tic system The r a t i o n a l e f o r u t i l i z i n g h e a r t r a t e a s an i n d i c a t o r of t h e u n i t s of oxygen consumed w hich i n t u r n i s a m easurem ent of e n e rg y i n p u t w ere amply d i s c u s s e d i n C h a p te r I I . To b r i e f l y r e i t e r a t e , h e a r t r a t e i s l i n e a r l y r e l a t e d t o oxygen c o n su m p tio n . I n su b s e q u e n t s e l e c t i o n s o f t h i s s tu d y h e a r t r a t e was u se d a s a c r i t e r i o n m easure; t h e co n t i n u e d u s e o f h e a r t r a t e d a t a i s j u s t i f i a b l e on two p o i n t s ; f i r s t , c o n s i s t e n c y w i t h i n t h e s tu d y and se co n d , c a l o r i m e t r y a p p a r a t u s was n o t a v a i l a b l e . A gain, l e t u s i n t e r r u p t t h e developm ent o f our model t o exam ine b a s i c d a t a w hich d e s c r i b e s r e l a t i o n s h i p s w hich sh o u ld be i n c o r p o r a t e d i n th e d e v e lo p e d m odel. F i g u r e 6 d e p i c t s a s e t o f t y p i c a l h e a r t r a t e r e s p o n s e c u rv e s H E A R T R A T E 1200 KPM 900 KPM 200 ( P r o je c te d ) 150 600 KPM 300 KPM : 100 10 4o 20 50 Time P ig , 6 . —H eart r a t e re s p o n s e c u rv e s f o r v a r io u s l e v e l s of s t r e s s v o ! 50 f o r s e v e r a l l e v e l s o f s t r e s s f o r se le c te d , tim e i n t e r v a l s up t o one m in u te . A c r i t i c a l e x a m in a tio n o f t h e s e c u rv e s r e v e a l s f o u r d i s t i n g u i s h i n g c h a r a c t e r i s t i c s which a p p e a r t o be common t o a l l o t h e r c u rv e s o f t h i s t y p e : ( l ) t h e su b j e c t ' s h e a r t fre q u e n c y h a s a b a s a l r a t e j t h a t i s , no s u b j e c t can h av e a h e a r t r a t e o f z e r o ; (2) th e h e a r t r a t e r e s p o n s e c u rv e s f o r each s t r e s s l e v e l f o llo w s a s i m i l a r p a t t e r n — t h e r e i s a grow th c u rv e fo llo w e d by an a sy m p to te ; (3) t h e r e i s a c r i t i c a l s t r e s s f a c t o r w hich i f exceeded a l t e r s t h e p a t t e r n o f t h e c u rv e ; and (4) t h e r e i s an u p p e r l i m i t o f ih e a rt r a t e w hich c a n n o t be ex ce ed e d . The p r e c i s e d e t e r m i n a n t o f t h i s maximum r a t e h a s n o t been e s t a b l i s h e d ; u n d o u b te d ly , i t i s c l o s e l y a l l i e d w ith e i t h e r t h e n e u r a l co n t r o l mechanism o r t h e u l t r a s t r u c t u r e o f t h e h e a r t ' s m uscle t i s s u e . R e g a rd le s s o f c a u s e , i t would be i n t e r e s t i n g t o d e te r m in e t h e mechanism w hich r e s u l t s i n a d e c r e a s e i n maximum h e a r t r a t e w ith age. F ig u r e 7 p o r t r a y s a t y p i c a l c u rv e of one s u b j e c t ' s h e a r t r a t e a s a f u n c t i o n o f work load. I f one exam ines t h i s c u rv e c r i t i c a l l y , s e v e r a l s i g n i f i c a n t c h a r a c t e r i s t i c s a r e o b s e rv e d : ( l ) th e h e a r t r a t e r e s p o n s e , o f t h i s p a r t i c u l a r s u b j e c t , was l i n e a r l y r e l a t e d t o v a r i o u s I n t e n s i t i e s o f work lo a d up t o a c r i t i c a l l e v e l ; and (2) t h e r e i s a d e f i n a b l e s t r e s s f a c t o r (o r v a lu e ) w hich , i f exceed ed, a l t a r s t h e l i n e a r r e l a t i o n s h i p o f h e a r t r a t e and work lo a d . 200 150 100 50 _ J _ _ _ _ _ _ _ _ _ _ _ _ I — i_ _ _ _ _ _ _ _ _ _ _ _ I_ _ _ _ _ _ _ _ _ _ _ _ I_ _ _ _ _ _ _ _ _ _ _ I 3 6 9 12 15 18 Word lo ad i n KPM ( x 100) F ig . 7 . —H eart r a t e as a f u n c t i o n o f work lo a d v ji 52 Let us summarize t h e I n f o r m a tio n c o n ta in e d w i t h in t h i s c h a p te r th u s f a r : 1. H e a rt r a t e cannot e q u al z e r o . 2. A ll I n d i v i d u a l s have a maximum h e a r t r a t e . 3. H eart r a t e h as a sim p le l i n e a r r e l a t i o n s h i p w ith work lo a d up t o a d e f i n a b le c r i t i c a l l e v e l . 4 . F or any g iv e n s t r e s s , h e a r t r a t e w i l l i n c r e a s e w ith tim e i f , and o n ly i f , th e system i s c a p a b le o f a t t a i n i n g a s t e a d y - s t a t e o r dynamic e q u ilib r iu m . F ig u r e 6, page 49., d e p i c t s a t y p i c a l h e a r t r a t e re s p o n s e c u rv e . Numerous i n v e s t i g a t o r s have hy p o t h e s iz e d t h a t t h i s grow th cu rve i s b e s t d e s c r i b e d v i a an e x p o n e n tia l f u n c t i o n (1 3 * 1 8 ,2 1 ,2 2 ). R iggs h as su g g e s te d t h a t i f any c u rv e cannot o r does n o t f i t one o f th e f i r s t f o u r e x p o n en ts, t h a t i s , q u a r t i c , c u b ic , sq u a re o r l i n e a r , t h a t c u rv e i s b e s t d e s c r i b e d v i a an e x p o n e n tia l f u n c t i o n (2 2 ). Appendix I c o n ta in s a summary o f d a ta p e r t a i n i n g t o a t r e n d a n a l y s i s on th e mean grow th c u rv e f o r t h i r t y male s u b j e c t s . The a n a l y s i s o f th e d a ta r e v e a l e d a s i g n i f i c a n t t r e n d f o r o n ly one of t h e t h r e e s t r e s s v a lu e s u sed i n t h i s s tu d y . T h e re fo re , i t was co n clu d ed t h a t an e x p o n e n tia l grow th c u rv e was t h e b e s t f i t . 5. The number of c o n s t a n t s r e q u i r e d i n th e model sh o u ld be m inim ized. S e v e ra l d i f f e r e n t ap p ro a ch e s to t h e problem; were u t i l i z e d , each o f which r e s u l t e d i n th e f o r m u la tio n of a m a th e m a tic a l model of h e a r t r a t e re s p o n se t o s t r e s s . A f t e r c a r e f u l e x a m in a tio n of each model a l l b u t two o f t h e m odels' were r e j e c t e d as b e in g i n v a l i d . Of th e two rem a in in g mod e l s , o n ly one was d i r e c t l y u t i l i z e d i n t h i s s tu d y ; f o r t h e r e a d e r ' s i n f o r m a tio n , th e model n o t u t i l i z e d i n th e stu d y i s d e s c r i b e d i n Appendix I I . The c h o ic e of w hich model t o p u rsu e was made on th e b a s i s of t h e f o llo w in g c r i t e r i a : 53 ( l ) which model r e q u ir e d l e s s m an ip u latio n ; and (2) which model had few er number of c o n s ta n ts . W e have p r e v io u s ly s t a t e d : S everal f a c t o r s should be c o n sid e re d when one i s a tte m p tin g to d e s c rib e th e e x p o n e n tia l growth curve. These f a c t o r s a re : th e i n t e n s i t y o f th e s t r e s s and th e d u r a tio n o f s t r e s s . P r io r t o adding th e s e p aram eters o r f a c t o r s to th e model, we w i l l examine th e s im p le s t case of an e x p o n e n tia l curve: e = b a se of a n a t u r a l lo g arith m c = c o n s ta n t = l / x Given t h e s e two p a ra m eters t h e follo w in g ap proxim ation of an e x p o n e n tia l curve i s p l o t t e d on X-Y c o o rd in a te s That thermodynamic laws r e l a t e d to th e c o n se rv a t i o n o f energy a re a r e a l i s t i c e x p re ssio n of man's r e g u l a t i n g mechanisms o r h o m eostatic c o n tr o ls in a c t io n . D - W R i n *to + wo < maximum Energy in p u t i s e q u al to or p r o p o r tio n a l t o h e a r t r a t e f o r g iven tim e and s t r e s s f a c t o r s i f , and only i f , th o se s t r e s s f a c t o r s do not exceed a c r i t i c a l l i m i t . T h ere fo re , 3. H-R*to + wo < maximum Eout -c F ig . 8 . — Course of an ex p o n e n tia l curve y t 54 The p l o t of an e x p o n e n tia l f u n c t i o n on X-Y c o o rd in a te s d e p i c t s a curve t h a t undergoes a r a p i d r i s e which g r a d u a l ly t a p e r s o f f but c o n tin u e s to r i s e to i n f i n i t y . O bviously, th e h e a r t r a t e re sp o n se c u rv e cannot c o n tin u e t o r i s e to i n f i n i t y r e g a r d l e s s of th e s t r e s s o r tim e f a c t o r . W e have p r e v i o u s ly s t a t e d t h a t i n o r d e r f o r any fo rm u la t o be a v a l i d r e f l e c t i o n o f r e a l i t y , i t should c o n ta in in f o r m a tio n r e l a t i n g t o s t r e s s and tim e . We, t h e r e f o r e , modify th e b a s ic form u la to s t a t e : w = s t r e s s t = tim e I n t h i s p a r t i c u l a r case t h e r e s u l t i n g cu rve w i l l be shaped a p p ro x im a te ly l i k e t h e f o llo w in g : c 4. e co+t c e ld + t 1.0 y t F ig . 9 . — E x p o n e n tia l c u rv e a d j u s t e d f o r p l a t e a u 55 The p l o t o f t h i s c u rv e f o llo w s a s i m i l a r c o u rs e a s t h e p r e v io u s c u rv e ; how ever, f o r a l l p r a c t i c a l p u r p o s e s , i t s maximal v a lu e i s re a c h e d a t u n i t y . However, t h i s e q u a tio n a l s o c o n t a i n s a fla w o r an e r r o r ; t h a t i s , r e g a r d l e s s o f t h e v a lu e s o fw o r t , th e f u n c t i o n w i l l c o n tin u e u n t i l some maximum v a lu e ( l = u n i t y ) i s r e a c h e d . I n r e a l i t y , t h i s s i t u a t i o n does n o t e x i s t i f w i s sm a ll o r i f t i s s m a ll, th e c u rv e w i l l r e a c h an asy m p to te b e f o r e maximum h e a r t r a t e v a lu e i s r e a c h e d . Once a g a in , i t i s n e c e s s a r y t o c o r r e c t t h e e q u a t io n . I f we c o r r e c t f o r tim e ( t ) , th e c u rv e s h o u ld t h e n r e a c h an a sy m p to te p r i o r t o maximum v a lu e I f t and w a r e w i t h i n t h e s y s te m ’s c a p a c i t y . c The e q u a tio n e i s a d j u s t e d a s f o llo w s : 4 . (a ) F i r s t , m u l t i p l y t h e e q u a tio n by e t o t h e minus x , o r l / u o r o n e / s t r e s s , w hich I s e q u a l t o t h e b a s i c e q u a tio n p l u s one o v e r s t r e s s . c _ 1 _ c + 1 G O + t U ) U)+t 0 ) { e l e = e (b) We can not add t h i s f a c t o r due t o Im b a la n c e o f t h e d e n o m in a to r. T h e re f o r e , we w i l l m u l t i p l y th e f r a c t i o n by t . (— ) (— ) = — ' w ; v t ; wt (c) Which y i e l d s l t / w t w hich can be summed w i t h t h e o r i g i n a l e q u a t io n and r e s u l t s in . (d) e c+t w+t 56 The e q u a tio n i n 4 (d ) w i l l enable t h e e x p o n e n tia l c u rv e to p l a t e a u , p ro v id in g th e s t r e s s a n d /o r tim e f a c t o r s do n ot exceed th e s y s te m 's c a p a b i l i t y . T his e q u a tio n i s a v a l i d e x p re s s io n o f r e a l i t y f o r h e a r t r a t e re sp o n se which ran g es from z ero t o u n i t y . For example, assum ing t h a t w and t a re f a i r l y la r g e and th e f u n c tio n approached u n i ty a f t e r a s u f f i c i e n t tim e p e rio d , th e curve w i l l approxim ate co u rse A; i f u i s red u ced , th e curve w i l l approx im ate c o u rs e B, as i l l u s t r a t e d i n F ig u re 10. I f we d e fin e u n i t y as th e maximum a t t a i n a b l e h e a r t r a t e , we th e n add t h a t f a c t o r to o u r form ula. c+t B A t F ig . 1 0 .—A pproxim ation o r h e a r t r a t e re s p o n se v ia Model I 5. e w+t + H.R. maximum F ig u re 11 p o r t r a y s s e v e r a l examples o f th e h e a r t r a t e r e sponse to s t r e s s e s of v a r io u s I n t e n s i t i e s and d u r a tio n . 57 As t h e s t r e s s ( l - 5 ) I n c r e a s e s , t h e c o u r s e o f t h e c u rv e s t e e p e n s and a p p ro a c h e s u n i t y a t a more r a p i d r a t e . F o r s t r e s s F iv e , a t r u e a s y m p to tic h e a r t r a t e i s n e v e r a c h ie v e d . t i o n o f a n i n d i c a t o r o f p r e - e x e r c i s e o r r e s t i n g h e a r t r a t e . The p h y s i o l o g i c a l s i g n i f i c a n c e o f r e s t i n g o r p r e - e x e r c i s e h e a r t r a t e h a s n o t been e s t a b l i s h e d ; how ever, t h e p r e e x e r c i s e v a lu e does a f f e c t th e c o u rs e o f t h e c u rv e . T h e re f o r e , i t s h o u ld be in c lu d e d i n t h e e q u a t io n . The f i n a l a l t e r a t i o n t o t h e e q u a t io n i s t h e a d d i - c+ t 6. ( -H . R. j + H. R ) • e W +fc 'maximum7 H.R i n i t i a l = H.R. I 4 3 ■ 1 2 t F ig . 1 1 . —H eart r a t e r e s p o n s e t o s t r e s s 58 I n fin a l- form th e e q u a tio n i s s t a te d th u s : c+t 7 . H .R .fc = ( - H . R . j + H .R .maxlmum) • ^ S im p lify in g th e symbols r e s u l t s in : c+t 8. yk = (-A + B) * e W+^ where yk = Change i n h e a r t r a t e A = I n i t i a l o r p r e - e x e r c i s e h e a r t r a t e B = Maximum h e a r t r a t e c = C onstant t = Time w = S tre s s e = Base of a n a t u r a l lo g arith m CHAPTER IV PROCEDURES The s p e c i f i c prob lem o f t h i s stu d y was t o d e te rm in e i f th e h e a r t r a t e re s p o n s e t o s t r e s s c o u ld be d e s c r ib e d a n d /o r p r e d i c t e d by a m odel. H eart r a t e re s p o n se s were r e c o rd e d w h ile t h e s u b j e c t s ro d e a Monarch b ic y c le e rg o m eter. The d a ta c o l l e c t e d d u r in g t h e e x p e rim e n ta l p e rio d a r e d e s c r i b e d below. The i n f o r m a ti o n p r e s e n t e d In t h i s s e c t i o n re v ie w s : ( l ) a d e s c r i p t i o n o f th e s u b j e c t s who p a r t i c i p a t e d i n t h e stu d y ; (2) th e equipm ent u sed t o measure and r e c o r d th e d a ta d u rin g th e e x p e rim e n t; and (3 ) a d e s c r i p t i o n o f th e e x p e rim e n ta l p r o t o c o l . Subj ects F o rty male s t u d e n ts p a r t i c i p a t e d i n th e d a ta c o l l e c t i o n phase o f t h i s e x p e rim e n t. Due t o a b n o r m a li t ie s i n t h e d a ta or i r r e g u l a r i t i e s w hich o c c u r r e d w hile c o l l e c t i n g t h e d a ta , e . g . , e l e c t r o d e a r t i f a c t , th e d a ta c o l l e c t e d f o r t e n o f th e s u b j e c t s w ere not a n a ly z e d . T able 3 c o n ta in s a summary o f t h e a n th ro p o m e tric d a ta and t h e r e p o r t e d a c t i v i t y l e v e l . The a c t i v i t y l e v e l c l a s s i f i c a t i o n in d e x I s r e view ed i n Appendix I I I . An I n s p e c t i o n o f th e d a t a p r e s e n te d i n th e t a b l e i n d i c a t e s t h a t th e d i f f e r e n c e s betw een 59 TABLE 3 SUMMARY OF SELECTED CHARACTERISTICS FOR ALL SUBJECTS Activity Level Age Height Weight H = 30 N = 10 N = 30 N = 10 H = 30 N = 10 N = 30 M = 10 x = 5.17 x = 3.2 X = 20 X = 2 0 .3 x = 70.63 x = 70.2 X = 168.7 X = 159.8 R = 3-5 R = 3 -4 R = 18-29 R = 18-24 R = 6 if-76 R = 68-74 R = 120-231 R = 137-182 s = .18 s = .816* s = 2.33 s = 1.73* S = 2 .4 2 S = 2 . 19* s = 28.61 s = 14.77* *Wot significant. Appendix I reviews the Activity Level classification system. 0 \ O 61 th e group of s u b j e c t s f o r which th e d a ta were a n a ly z e d and th e group of s u b j e c t s f o r which th e d a ta were n ot a n a ly z e d were m inim al; th e l a r g e s t d i f f e r e n c e o c c u rre d i n t h e su b j e c t s ' body w e ig h t. T here were no s t a t i s t i c a l l y s i g n i f i c a n t d i f f e r e n c e s betw een th e groups on any of th e c h a r a c t e r i s t i c s i n d i c a t e d i n th e t a b l e , p >" .05. F o r th o s e t h i r t y s u b j e c t s f o r whom t h e d a t a were a n a ly z e d , one was a v a r s i t y f o o t b a l l p l a y e r who was a c t i v e l y engaged i n s p rin g p r a c t i c e ; one was a fo rm er c o lle g e f o o t b a l l p l a y e r who had sig n e d a p r o f e s s i o n a l f o o t b a l l con t r a c t ; t h r e e were g r a d u a te p h y s i c a l e d u c a tio n s t u d e n ts ; and t h e rem a in in g t w e n ty - f i v e were v o l u n te e r s from th e " s e r v ic e " c l a s s p h y s i c a l e d u c a tio n program a t th e U n iv e rs ity of S o u th e rn C a l i f o r n i a . A ll s u b j e c t s , w ith t h e e x c e p tio n o f th e g r a d u a te s t u d e n t s , were r e c r u i t e d from one o f t h r e e p h y s i c a l e d u c a tio n c l a s s e s : a fu n d am e n ta ls of p h y s i c a l e f f i c i e n c y c l a s s ; a w eigh t t r a i n i n g c l a s s ; o r a b e g in n in g g y m n astics c l a s s . A ll s u b j e c t s were f r e e o f a p p a re n t i n j u r i e s , were h e a l t h y , and c o n s id e re d t o be a t y p i c a l c r o s s s e c t i o n o f t h e m ale s t u d e n ts a t USC. The i n v e s t i g a t o r met w ith each s u b j e c t e i t h e r i n d i v i d u a l l y o r i n sm a ll gro u p s i n o r d e r t o e x p la in t h e t e s t in g program and t o o r i e n t t h e s u b j e c t s t o t h e e x p e rim e n ta l p ro c e d u re s and t o f a m i l i a r i z e t h e s u b j e c t s w ith ECO t e s t p r o c e d u r e s . A more d e t a i l e d rev iew o f t h e o r i e n t a t i o n w i l l be p r e s e n t e d i n a l a t e r p o r t i o n of t h i s s e c t i o n . 62 E x p e r im e n ta l A p p a ra tu s The f o l lo w in g equipm ent was u s e d i n t h e d a t a c o l l e c t i o n p h a se o f t h i s s tu d y : 1. S c a l e . — A s t a n d a r d p h y s i c i a n 's s c a l e was u se d t o a s s e s s t h e g r o s s w e ig h t and h e i g h t o f t h e s u b j e c t . The b a la n c e o f t h e s c a l e was checked and,, i f n e c e s s a r y , a d j u s t e d a t th e b e g in n in g o f each d a y 's t e s t i n g . 2. S to p w a tc h . —A 1 /10 se co n d s to p w a tc h was u s e d t o "program " t h e t a p e r e c o r d e r . The same w a tch was u se d t o ch eck th e a c c u r a c y o f t h e " p r o g r a m ," p r i o r t o t h e f i r s t e x p e rim e n t and a t t h e c o n c l u s i o n o f t h e l a s t e x p e rim e n t. 3. Monarch b i c y c l e e r g o m e te r . — A s t a n d a r d Monarch e rg o m e te r w hich was m o d if ie d by t h e a d d i t i o n o f r a c i n g p e d a l s and t o e c l i p s was u se d t o p r o v i d e t h e r e s i s t a n c e f o r t h e v a r i o u s e x p e r i m e n t a l c o n d i t i o n s . The u s e o f to e c l i p s a s s i s t s i n s t a n d a r d i z i n g t h e s u b j e c t ' s f o o t p o s i t i o n on t h e p e d a l , t h e r e b y p r e v e n t i n g g r o s s c han ges i n t h e e f f i c i e n c y o f p e rfo rm a n c e . 4 . Tape r e c o r d e r . — A Sony S t e r e o c o r d e r , model num b e r 5 2 1 , was u s e d t o r e c o r d and p l a y a "programmed" t a p e . The t a p e was "programmed" t o i n c l u d e a l l i n s t r u c t i o n s f o r t h e s u b j e c t s , a lo n g w ith a l l tim in g i n d i c a t i o n s . T h is p r o c e d u re h a d t h e f o l lo w in g a d v a n ta g e s : ( l ) each s u b j e c t was g iv e n t h e same s p e c i f i c I n s t r u c t i o n s a t p r e c i s e l y t h e same 63 tim e , (2) any tim in g e r r o r s ( t h e l a r g e s t e r r o r o b s e rv e d was 0 .3 0 o f a second) w ould be c o n s i s t e n t f o r a l l s u b j e c t s , and (3) t h e u se o f t h e t a p e - r e c o r d e d tim in g and s u b j e c t i n s t r u c t i o n s p e r m i t t e d t h e i n v e s t i g a t o r a g r e a t e r freedom o f m otio n and time,* t h i s f l u i d i t y e n a b le d t h e p rim a ry i n v e s t i g a t o r t o t r o u b l e sh o o t and a n n o ta te r e c o r d s when n e c e s s a r y . 5. P h y s lo g ra p h f o u r . —A p h y sio g ra p h f o u r r e c o r d e r and a m p l i f i e r m a n u fa c tu re d by t h e E & M I n s tru m e n t Company was u s e d t o r e c o r d t h e ECG r e s p o n s e . At th e tim e of t h i s e x p e rim e n t, t h e p h y s io g r a p h f o u r had two o p e r a t i o n a l r e c o rd in g c h a n n e ls , i n a d d i t i o n t o a tim e and e v e n t m ark e r. T hroughout t h e e n t i r e c o u r s e o f th e exp erim en t t h e second c h a n n e l was m a in ta in e d i n an o p e r a t i o n a l s t a t e a s a b a ck -u p u n i t . 6. P r e - a m p l i f i e r . —A c a r d i o - p r e - a m p l i f i e r , a l s o m a n u fa c tu re d by E & M I n s tr u m e n t Company, was u s e d t o b o o st th e ECG s i g n a l p r i o r t o i t s c o n d u c ta n c e t o t h e m ain fram e o f t h e p h y s io g ra p h f o u r , 7. E l e c t r o d e s . — E xpendable ( d i s p o s a b l e ) ICU ECG e l e c t r o d e s , p a r t number CSS/15, m a n u fa c tu re d by th e B u rd ic k C o r p o r a tio n were u se d t o t r a n s m i t th e ECG s i g n a l from t h e s t a n d a r d i z e d l e a d t o t h e c a r d i o - p r e - a m p l i f i e r . These e l e c t r o d e s were m a n u fa c tu re d from German s i l v e r , t h e y h a v e a s u r f a c e a r e a of 3 .1 4 trnn and a r e a t t a c h e d t o a 48 in c h 64 s h ie ld e d co n d u ctin g c a b le w ith a s h ie ld e d male p lu g . The e le c t r o d e s were a f f i x e d to th e s u b je c t w ith a s t r i p of m icropore ta p e . A t h r e e e le c t r o d e system was used d u rin g t h i s stu d y : a c t i v e , r e f e r e n c e , and a ground. The e le c t r o d e s were co n nected t o th e c a rd io -p re -a m p by way of a s h ie ld e d t h r e e w ire e x te n s io n , one end of which had a s h ie ld e d f e male r e c e p t a c l e f o r r e c e i v i n g th e e le c t r o d e s c o n n ec to r, and th e o t h e r end was m o d ified to plug i n t o t h e E & M c a rd io p r e - a m p l i f i e r ja c k s . 8. Metronome. A manual metronome was used as an a i d f o r th e c o n t r o l of p e d a l l i n g r a t e . A ll of th e fo re g o in g equipm ent, w ith th e e x c e p tio n of t h e s c a l e , th e ta p e r e c o r d e r , and th e Monarch ergo m eter, was p la c e d i n a m o d ified o f f i c e desk which had been equipped w ith c a s t o r s f o r m o b ility . E x p erim e n tal Design The e x p e rim e n ta l d e s ig n u t i l i z e d i n t h i s stu d y i s an a p p l i c a t i o n o f th e c o r r e l a t e d group d e sig n . As s t a t e d by K e r lin g e r , th e b a s ic p r i n c i p l e u n d e rly in g th e c o r r e l a t e d - g r o u p s 1 p a rad ig im i s t h a t : There i s s y s te m a tic v a r ia n c e i n th e dependent v a r i a b l e m easures due to th e c o r r e l a t i o n betw een th e groups i n some v a r i a b l e r e l a t e d to th e dependent v a r i a b le . T his c o r r e l a t i o n and i t s concom itant v a ria n c e can be in tro d u c e d i n t o th e m easu res—and th e d e s ig n — i n t h r e e ways: ( l ) u se th e same u n i t s , f o r example, s u b j e c t s , i n each o f th e e x p e rim e n ta l gro u p s, . . . ( 1 6 ) 65 The p rim a ry i n t e n t o f t h i s d e s ig n i s t o maximize th e b e tw een g ro u p s v a r i a n c e , i d e n t i f y th e betw een p a i r s v a r i a n c e , and m inim ize t h e e r r o r v a r i a n c e ( 7 # l 6 ) . K e r l i n g e r goes on to e x p la in : The u n i t s u se d do n o t a l t e r t h e v a r ia n c e p r i n c i p l e s i n th e s l i g h t e s t . . . . The im p o rta n t c o n s i d e r a t i o n i s w h e th e r th e u n i t s , w h a te v e r th e y a r e , d i f f e r from each o t h e r . I f t h e y do, v a r i a n c e betw een u n i t s i s i n t r o du ced . I n t h i s s e n s e , t a l k i n g about c o r r e l a t e d g ro u p s o r s u b j e c t s i s t h e same as t a l k i n g ab o u t v a r ia n c e b e tw een g ro u p s o r s u b j e c t s . The n o t io n o f i n d i v i d u a l d i f f e r e n c e s i s e x te n d e d t o u n i t d i f f e r e n c e s . The r e a l v a lu e o f c o r r e l a t e d - g r o u p s d e s ig n s would seem t o be t h a t n o t o n ly do t h e y e n a b le th e i n v e s t i g a t o r t o i s o l a t e and e s t i m a t e th e v a r i a n c e s due t o c o r r e l a t i o n , t h e y a l s o g u id e him t o d e s ig n r e s e a r c h t o c a p i t a l i z e on t h e d i f f e r e n c e s t h a t f r e q u e n t l y e x i s t betw een u n i t s . ( l6 ) The b a s i c w eakness of t h i s p a ra d ig m l i e s i n t h e a s sum ption , f o r t h i s p a r t i c u l a r e x p e rim e n t, t h a t t h e e f f e c t s o f t h e s h o r t d u r a t i o n work lo a d s would n o t a cc u m u late and a f f e c t any o f th e s u b s e q u e n t work lo a d s . T e s ti n g P ro c e d u re s As p r e v i o u s l y s t a t e d , e a c h s u b j e c t r e c e i v e d an o r i e n t a t i o n t o t h e t e s t i n g p r o c e d u r e and t r i a l use o f th e ex p e r i m e n t a l equipm ent a p p ro x im a te ly f i v e d ay s p r i o r to e x p e r im e n ta l t e s t i n g . The f o l lo w in g ite m s w ere p r e s e n t e d a t t h e o r i e n t a t i o n : 1. Each s u b j e c t was r e q u i r e d to r i d e th e Monarch e rg o m e te r a t lo a d s o f v a r i o u s i n t e n s i t i e s . P r i o r t o t h i s r i d e t h e s e a t h e i g h t was a d j u s t e d to t h a t h e ig h t w hich r e s u l t e d i n t o t a l l e g e x te n s i o n when th e p e d a l was a t th e 66 bottom most p o s i t i o n . This h e ig h t was n o ted and reco rd ed f o r use during e x p erim en tal t e s t i n g . Each su b je c t was i n s t r u c t e d to "kick" h i s f e e t f r e e of th e p e d als on th e com mand " s to p ." They were a ls o i n s t r u c t e d t h a t each t e s t i n g c o n d itio n would begin w ith th e l e f t hand p e d al to th e f r o n t w ith th e p edal s h a f t h o r i z o n t a l t o th e ground. Each s u b je c t had an o p p o rtu n ity to p r a c t i c e i n s e r t i n g h i s f e e t i n t o th e to e c l i p s . T his a c tio n was a s s i s t e d by th e i n v e s t i g a t o r who p o s itio n e d th e pedal crank and guided th e s u b j e c t 's f e e t i n t o th e to e c l i p s . O ther in fo rm a tio n r e l a t e d to th e ergom e t e r t e s t were: ( l ) a d is c u s s io n of th e r e s t i n g p o s i t i o n . On th e command "stop" each s u b je c t was to "kick" h i s f e e t f r e e from the p e d a ls and p la c e them on th e fo o t r e s t a t th e f r o n t o f th e ergom eter. With th e e x c e p tio n of th r e e sub j e c t s , a l l of whom were over 6 ' 2" t a l l , th e hands were to rem ain on the h a n d le b a r h a n d g rip s . For th e t h r e e t a l l e s t s u b je c ts th e r e s t i n g p o s i t i o n was ex trem ely uncom fortable and, t h e r e f o r e , was m odified to p e rm it them to r e s t t h e i r forearm s on t h e i r k n ees. (2) During the o r i e n t a t i o n loads each s u b je c t was c o n d itio n e d to th e metronome as a pacing a id j a p p ro x im ately 50 p e rc e n t of t h e s u b je c ts p r e f e r r e d to u t i l i z e th e speedom eter system which i s in c o r p o r a te d on th e bike as an a d ju n c t to th e metronome f o r m a in ta in in g th e p ro p e r r a t e of p e d a llin g . 2. The p r i n c i p l e s and p ro ced u res o f an ECG - t e s t were d is c u s s e d , each s u b je c t was r e q u ir e d t o undergo a 67 t r i a l r u n i n which th e e l e c t r o d e s were a f f i x e d and co n n ected t o t h e a m p l i f i e r . T his was done t o red u c e any a p p re h e n s io n which may d e v elo p a s th e r e s u l t o f th e ECG t e s t . 3. A ll s u b j e c t s were o r i e n t e d t o t h e ty p e of i n s t r u c t i o n s which would be u sed i n th e t e s t s i t u a t i o n a lo n g w ith t h e ap p ro x im ate tim e i n t e r v a l s and t h e t o t a l tim e r e q u ir e d f o r th e e x p e rim e n ta l t e s t . 4 . F i n a l l y , th e s u b j e c t s were g iv e n th e fo llo w in g p r e - t e s t i n s t r u c t i o n s : ( l ) o b t a i n a norm al n i g h t s r e s t ; (2) r e f r a i n from s tr e n u o u s a c t i v i t y p r i o r t o th e t e s t ; (3 ) r e s t r i c t food I n t a k e and smoking f o r a t l e a s t two h ours p r i o r to t h e t e s t ; and (4) n o t i f y th e i n v e s t i g a t o r immedi a t e l y i f i l l n e s s i s f e l t p r i o r t o o r d u rin g any phase of t h e e x p e rim e n ta l t e s t i n g . At t h e b e g in n in g of each t e s t i n g day th e fo llo w in g p ro c e d u re s were e s t a b l i s h e d : 1. S et up t h e t e s t i n g equipm ent. 2. Check and a d j u s t th e b a la n c e o f th e s c a le i f n e c e s s a r y . 3. Check th e f r i c t i o n b e l t and c a l i b r a t i n g w eig ht on th e b i c y c l e erg o m eter and r e p l a c e o r a d j u s t a s r e q u i r e d . 4. Warm up t h e "P hysiograph Four" and C a r d io - p r e - a m p l i f i e r f o r a minimum o f t h i r t y m in u te s. Check th e in k su p p ly , p a p e r su p p ly , c o n t r o l p a n e l s , and o p e r a tio n . 68 5 . Warm up t h e t a p e r e c o r d e r , t h r e a d and a d j u s t t h e program t a p e , c h e c k t h e s t a r t i n g p o s i t i o n . 6. Check t h e e l e c t r o d e c o n n e c tio n s f o r c o r r o s i o n , t h e s u p p ly of c o n d u c tin g p a s t e , t a p e , and c o t t o n M s c u i t s . E x p e r im e n ta l P r o t o c o l When th e s u b j e c t s a r r i v e d a t t h e t e s t i n g s t a t i o n , t h e y w ere im m e d ia te ly r e q u e s t e d t o change i n t o s h o r t s and s h o e s . When th e s u b j e c t s w ere d r e s s e d f o r a c t i v i t y , t h e y w ere w eighed and m easured and t h e n s e a t e d o n t h e Monarch e rg o m e te r i n th e p r e v i o u s l y d e s c r i b e d r e s t i n g p o s i t i o n . A s h o r t i n f o r m a t i o n s h e e t was f i l l e d o u t w h ile t h e s u b j e c t s a t i n t h e r e s t i n g p o s i t i o n . On t h e c o m p le tio n o f t h e i n f o r m a t i o n s h e e t t h e f o l lo w in g s t e p s w ere c a r r i e d o u t: 1. The s a d d l e h e i g h t of t h e b i k e was c h ec k ed o u t , a nd , i f n e c e s s a r y , a d j u s t e d , 2. A b r i e f re v ie w o f th e t e s t p r o c e d u r e was p r e s e n t e d . 3. The e l e c t r o d e s w ere f i x e d t o th e s u b j e c t s . a . The p o s i t i o n s w here t h e e l e c t r o d e s w ere t o be p l a c e d w ere n o te d and m arked. b . The a r e a s o f e l e c t r o d e p la c e m e n t w ere c a r e f u l l y sh a v ed w i t h a s a f e t y r a z o r , i f n e c e s s a r y . c . The a r e a s o f e l e c t r o d e p la c e m e n t w ere c le a n e d w ith a s o l u t i o n o f i s o p r o p y l a l c o h o l . d. The e le c t r o d e a re a s were l i g h t l y a b ra id e d w ith a f i n e g rade of sandpaper. e. A c o t t o n pad a p p ro x im a te ly 1/2 t h e s i z e of th e e l e c t r o d e was s a t u r a t e d w ith co n d u ctin g p a s t e and p la c e d on t h e e l e c t r o d e . The f u n c t i o n of th e pad i s to p re v e n t d i r e c t con t a c t between t h e e l e c t r o d e and th e s u b j e c t . f . The e l e c t r o d e s were f i x e d t o th e s u b je c t u sin g a s t r i p of m ic ro -p o re a d h esiv e ta p e . The e l e c t r o d e placem ent was as f o llo w s : The ground e l e c t r o d e was p la c e d a t t h e base o f t h e sternum j u s t below th e xyphoid p r o c e s s ; t h e a c t i v e e l e c t r o d e was p la c e d on th e su p e r i o r end of th e sternum a t o r n e a r th e manubro- s t e r n a l j u n c t i o n ; and th e r e f e r e n c e e l e c t r o d e was p la c e d on th e f i f t h i n t e r c o s t a l sp ace, a p p ro x im a te ly h a l f way between t h e m i d - a x i l l a r y l i n e and th e m id - c la v i c u l a r l i n e . When n e c e s s a r y , due t o s t r u c t u r e of u n d e rly in g m uscula t u r e , s l i g h t v a r i a t i o n s i n th e p lacem ent o f th e e l e c t r o d e s were made. g. The e l e c t r o d e le a d s were ru n over th e r i g h t s h o u ld e r and down th e back of th e s u b j e c t and ta p e d i n p la c e . The e le c t r o d e s were th e n con n e c te d t o th e e x te n s io n c a b le , th e plug of which was c lip p e d to t h e s u b j e c t 's s h o r t s . was 70 The e x t e n s i o n c a b l e was s e c u r e l y f a s t e n e d to t h e s a d d le o f t h e b i k e . The o t h e r end o f t h e c a b l e was c o n n e c te d t o t h e c a r d i o - p r e - a m p l i f i e r by way o f 3 screw t e r m i n a l s . T h ere was an im m e d ia te c h e c k o f t h e c o n n e c ti o n s and p o l a r i t y ., a d ju s tm e n t s o r r e a p p l i c a t i o n s o f t h e e l e c t r o d e s w ere p e rfo rm e d , i f n e c e s s a r y . 4 . The r e c o r d i n g p a p e r s p e e d o f t h e P h y s io g ra p h P o u r was s e t a t 10 mm/second and t h e tim e m arker a t one p e r se c o n d . P a p e r and i n k s u p p ly w ere a l s o c h e c k e d . 5 . A c a l i b r a t i o n s i g n a l o f 0 .1 mv/mm was r e c o r d e d on t h e r e c o r d , a s was t h e s u b j e c t ' s name and d a t e . The p u rp o s e o f i n t r o d u c i n g t h e c a l i b r a t i o n s i g n a l was t o p e rm it s e c o n d a ry o r su b s e q u e n t u s e o f t h e d a t a . 6 . The t a p e r e c o r d e r was s t a r t e d and t h e t e s t begun; t e s t p r e p a r a t i o n s r e q u i r e d from f i v e t o e i g h t m in u te s ; t h a t i s , once t h e s u b j e c t was d r e s s e d f o r a c t i v i t y t h e t e s t was b egun i n f i v e t o e i g h t m i n u t e s . The d a t a c o l l e c t i o n p r o c e d u r e f o llo w e d ; t h e p r o t o c o l P r e - e x e r c i s e h e a r t r a t e was r e c o r d e d f o r a minimum o f 30 se c o n d s . 71 The f i r s t work bout c o n s is te d of z e ro lo ad a t a p e d a l l i n g r a t e of 60 rpm. The f u n c t i o n o f t h i s zero lo ad was tw o -fo ld : ( l ) i t p re s e n te d th e s u b je c t w ith an o p p o r tu n ity to rev iew th e work pace (60 rpm), and (2) i t a s s i s t e d i n f a m i l i a r i z i n g th e s u b je c t w ith th e work p a t t e r n (3 6 ,4 3 ). T h is a c t i v i t y was c o n tin u e d f o r 30 second s. Two m inutes a f t e r th e c e s s a t i o n o f a c t i v i t y th e f i r s t p o s t t e s t h e a r t r a t e p e rio d was re c o rd e d f o r 15 s e c onds. On th e c o n c lu s io n o f th e p re c e d in g a c t i v i t y , th e s e c ond work bout c o n s i s t i n g o f a 300 KPM/minute work lo a d was i n i t i a t e d , and th e p e d a l l i n g r a t e rem ained a t 60 KPM f o r th e rem ainder of th e work b o u ts. Two m inutes a f t e r th e c e s s a t i o n of a c t i v i t y th e second p o s t t e s t h e a r t r a t e p e rio d was re c o rd e d f o r 15 seconds. At th e end o f th e p re c e d in g i n t e r v a l th e t h i r d work bout began. The work lo ad was 600 KPM/minute; a g a in , th e d u r a t i o n of th e work bout was 30 seconds. Two m inutes a f t e r th e end o f a c t i v i t y th e t h i r d p o s t t e s t h e a r t r a t e p e r io d was re c o rd e d f o r 15 s e c onds . Im m e d ia tely f o llo w in g , th e f o u r t h work bout began, th e work lo a d was 900 KPM/minute and th e d u r a t i o n rem ained th e same. Two m in u tes a f t e r th e work bout ended, t h e t h i r d p o s t t e s t h e a r t r a t e p e rio d was r e c o r d e d f o r 15 seco n d s. The n e x t t e s t c o n s i s t e d o f a PWC-^yQ t e s t . I t was a d m in is te r e d a c c o rd in g t o th e p ro c e d u re s b ased on th e f i n d in g s of s t u d i e s con d u cted by Washlund and S jo s tr a n d ( 4 ,7 1 ) . 1. The f i r s t work lo a d was 300 KPM/minute. The s u b j e c t s were r e q u i r e d t o c o n tin u e p e d a l l i n g th e b i c y c l e e rg o m eter a t a r a t e o f 60 rpm. At th e end of f i v e m inutes th e r e s i s t a n c e was r a i s e d t o 600 KPM/minute. The lo a d was i n c r e a s e d to 900 KPM/minute a t th e c o m p le tio n o f t h e second f i v e m in u tes of r i d i n g . The t e s t was com plete when t h e s u b je c t had f i n i s h e d a t o t a l r i d i n g tim e of 15 m in u te s. S e v e ra l s u b j e c t s were un a b le t o com plete th e e n t i r e t e s t . During t h i s f i n a l t e s t th e h e a r t r a t e r e s p o n s e was r e c o r d e d f o r th e f i r s t m inute of a c t i v i t y f o r each of t h e t h r e e work lo a d s j fo llo w e d by a f i f t e e n second sample f o r each su b seq u e n t m inute of e x e r c i s e f o r each o f th e t h r e e work b o u ts . 73 Recording th e Data The d a ta were re c o rd e d on r e c t i l i n e a r s t r i p c h a rt re c o rd in g p a p er. The p aper was r u le d i n m illim e te r sq u ares w ith f i v e m il l im e te r s u b d iv is io n s . The re c o rd in g u n i t i s an i n t e g r a l p a r t o f th e Physiograph Pour. E xperim ental Schedule A ll t e s t i n g s e s s io n s were scheduled between th e h ours of 9:00 a.m . and 5:00 p.m. The fo llo w in g d a te s were s e l e c te d f o r t e s t i n g : May 2nd, 3rd, 8 th , 9 th , 10th, 15th, and 16th, 1970. F or t h a t p o r t io n of s u b je c ts (n = 10) who were r e t e s t e d , t h e r e t e s t was scheduled at th e same tim e of day as th e f i r s t t e s t . T his was co n sid ere d n e c e s s a ry i n o rd e r to c o n tr o l th e d i u r n a l v a r i a t i o n of h e a r t r a t e (1,5* 1 2 ,1 4 ,2 0 ). f CHAPTER V RESULTS AND DISCUSSION V e r i f i c a t i o n of Model I P r i o r t o i n i t i a t i n g a d i s c u s s i o n o f th e r e s u l t s , i t i s e s s e n t i a l t o rev ie w th e p ro p o se d model and t o v e r i f y th e assu m p tio n s and c o n d it i o n s o f o p e r a t i o n . A summary o f f a c t o r s r e l a t i n g to s e l e c t i n g a ssu m p tio n s as s u g g e s te d by A t- t l n g e r (34) was p r e s e n t e d i n C h a p te r I I . A b r i e f r e s t a t e ment and e x a m in a tio n of t h e s e f a c t o r s w i l l a t t e s t t o th e v e r a c i t y of th e model; i n a d d i t i o n , t h e model sh o u ld be r e view ed f o r v a l i d i t y , g e n e r a l i t y , p r e d i c t a b i l i t y , and comput a b i l i t y . The f a c t o r s l i s t e d by A t t i n g e r w ere: ( l ) t h e model sh o u ld be e s t a b l i s h e d i n term s of p a ra m e te rs which a r e s i g n i f i c a n t and m ea su ra b le ; t h e p a ra m e te rs on which t h e model i s b a sed in c lu d e m easurem ents o f tim e , h e a r t r a t e a t r e s t , h e a r t r a t e d u rin g s t r e s s , and a t h e o r e t i c a l maximum h e a r t r a t e o f two h u n d red b e a t s p e r m in u te. Each o f th e f o r e going p a ra m e te rs a re p h y s i c a l l y r e a l i z a b l e , m e a su ra b le , and t h e v a r i a t i o n s which o c c u r i n a l i v i n g system a r e s i g n i f i c a n t . (2) The model sh o u ld in c lu d e a l l a v a i l a b l e in fo rm a t i o n r e l a f i n g t o th e system u n d e r s tu d y . The p ro p o se d model 74 75 i jis b a se d on t h e c o n s e r v a t i o n o f e n e rg y a s e x p re s s e d v i a therm odynam ic law and each o f th e p a ra m e te rs i n h e r e n t i n t h e law a r e e x p re s s e d i n t h e e q u a t io n w hich d e s c r i b e s th e [model. ( 3 ) The model s h o u ld be r e l a t i v e l y s im p le . I n i t s ’c u r r e n t form , th e model i s a s t r a i g h t f o r w a r d e x p r e s s io n of t h e sy stem u t i l i z i n g v a lu e s f o r f o u r p a ra m e te rs and an e s - Itlm a te o f one c o n s t a n t , (4) The c o n s t r u c t i o n o f t h e model i 's h o u ld p e rm it a l t e r a t i o n s . The model was c o n s t r u c t e d on a sum m ative b a s i s ; t h e r e f o r e , any and a l l components may be m o d if ie d , e l i m i n a t e d o r a d d i t i o n a l com ponents may be added w ith o u t l o s s o f c u r r e n t know ledge and c o n c e p ts . (5) F i - j n a l l y , a good model s h o u ld s u g g e s t a d d i t i o n a l e x p e rim e n ts . U n d o u b te d ly ; t h e e x p l o r a t i o n o f th e p ro p o se d model w i l l r e s u l t i n s u g g e s t i o n s f o r a d d i t i o n a l s tu d y . N ext, i t i s n e c e s s a r y t o re v ie w th e model to d e t e r mine i t s e f f i c i e n c y and e f f e c t i v e n e s s a s i t r e l a t e s t o t h e I f o u r b a s i c c r i t e r i a s t a t e d i n a p r e v i o u s p a r a g r a p h . F i r s t , i v a l i d i t y . A model i s v a l i d i f a l l known c o n seq u en ces o f t h e model a r e c o m p a tib le w ith a l l v e r i f i e d knowledge o f I t h e r e a l s i t u a t i o n . The model s t a t e s : i c+ t t / - q n \ W + t y = (B-AJ • e i ! j. I R e a l i t y d i c t a t e s t h a t y = A when tim e i s e q u a l t o iz e ro o r when s t r e s s i s e q u a l t o z e r o . The p ro o f o f t h i s i s | |s u c h : when s t r e s s i s e q u a l t o z e r o , t i m e - f a c t o r s w i l l be !c a n c e lle d which r e s u l t s i n an exponent of z e ro ; e to a i , ^minus exponent o f z e ro i s equal t o z e ro ; t h e r e f o r e , y = A. |When tim e i s e q u al to z e ro , i t i s ax io m atic t h a t induced i p h y s io lo g ic a l s t r e s s must equal z e ro . Thus, t h e same !co u rse r e s u l t s y = A. At t h i s tim e , th e g e n e r a l i t y of t h e propo sed model cannot be a ff irm e d . I t would ap p ear t h a t because of th e e x p o n e n tia l f u n c t i o n s t a t e d i n th e model, and s in c e i t i s |t h e o p in io n o f numerous i n v e s t i g a t o r s t h a t t h e o p e r a tio n of p h y s i o l o g i c a l system s fo llo w an e x p o n e n tia l c u rv e , g e n e r a l - !i t y to o t h e r b i o l o g i c a l system s may r e s u l t . The t h i r d c r i t e r i o n o f a model r e l a t e s to I t s p r e - i d i c t i v e a b i l i t y . A gain, as of now, t h i s i s an unknown q u a n ti t y ; h o p e f u l l y , t h e model may se rv e as a p r e d i c t o r o r i i n d i c a t o r of h e a r t r a t e f o r any g iv e n tim e f o r a g iv en S i | s t r e s s . j j F i n a l l y , th e model must have a h ig h deg ree o f com- I p u t a b i l i t y , i t must conform to m ath em atical s ta n d a r d s and I be amenable to m ath em atical m a n ip u la tio n . The o p e ra tio n s r e q u i r e d by th e model c o n s i s t of a c om binatio n of sim ple j a r i th m e ti c p ro c e d u re s . This in h e r e n t s i m p l i c i t y should | s a t i s f y th e f o u r t h c r i t e r i a . T r i a l Problem The model s t a t e s : 77 c+t y* = (B-A) - e ~ “ +* y = O bserved h e a r t r a t e f o r a g iv e n tim e B = Maximum h e a r t r a t e w hich h a s been o p e r a t i o n a l l y d e f i n e d as 200 b e a t s p e r m inute f o r t h i s s tu d y A = O bserved r e s t i n g h e a r t r a t e e = Base o f a n a t u r a l lo g a r ith m c = C o n sta n t t = Time u > = S t r e s s A l l of th e a fo re m e n tio n e d q u a n t i t i e s , w ith t h e e x c e p t i o n of Cj a r e knowns o r can be e v a l u a t e d e m p i r i c a l l y . I f one s o lv e s f o r (c) by s u b s t i t u t i o n , one can th e n e v a l u a t e t h e model f o r f i x e d s t r e s s lo a d s and v a r i o u s tim e p e r i o d s . By s u b s t i t u t i o n : W ith th e s u b s t i t u t i o n o f o b se rv e d v a lu e s from a random ly s e l e c t e d s u b j e c t one o b t a i n s ( d a t a v a lu e s from s u b j e c t 2 8 ): D ata V a lu e s c [ l o g ( ^ - ) ] ( t - « ) - t A B y t = 96 A = 66 A = 66 78 Data V a lu es — Continued A B uj = 300 u = 300 o -= 3 " C M I I -P O ■ H I I -P B = 200 B = 200 Set A: c * [lo g(y/B -A )[ ( t - u ) - t = [ l o g ( 96/ 200- 6 6 )] ( 10- 3 0 0) - 10 = [ l o g ( 96/ 13^ )] ( 10- 3 0 0) - 10 = [lo g . 7 16] ( - 2 90) - 10 = (.4868) ( - 290) - 10 = 140.17 - 10 = 130.17 Set B: c = [lo g (y /B -A )] ( t - u ) - t = [ l o g ( 96/ 200- 6 6 )] (240-300) - 240 = [lo g . 7 16] ( - 6 0 ) - 240 = (.4868) ( - 6 0 ) - 240 = 29.21 - 10 = 19.21 cA = 130.17 cB = 1 9 .2 1 79 As one c a n r e a d i l y o b se rv e , c / c . T h is r e s u l t was c o n firm e d on th e d a ta o f a number o f s u b j e c t s who w ere random ly s e l e c t e d . At t h i s p a r t i c u l a r tim e i t i s q u i t e d i f f i c u l t t o a s c e r t a i n why c ^ c . A p o s s i b l e r e a s o n f o r t h i s u n e x p e c te d r e s u l t may be due t o th e a s s u m p tio n t h a t t h e h e a r t r a t e c u rv e i s a smooth c u rv e j i n r e a l i t y , t h i s c o n d i t i o n does n o t e x i s t . The h e a r t r a t e r e s p o n s e c u rv e may be t y p i f i e d a s a s e r i e s o f I r r e g u l a r f l u c t u a t i o n s w hich g r a d u a l l y d i m in is h as th e system " h u n ts" f o r a b a la n c e p o i n t . A more d e s c r i p t i v e a n a lo g y : i t i s a s e r i e s o f "saw to o th " c u rv e s w hich g r a d u a l l y d im in is h i n b o t h p e r i o d i c i t y and a m p litu d e . Due t o t h i s u n e x p e c te d r e s u l t and b e c a u se an a l t e r n a t i v e model had been d e v e lo p e d , i t was d e c id e d to p u rs u e th e e v a l u a t i o n o f t h e a l t e r n a t i v e model r a t h e r t h a n m o d ify in g t h e p rim a ry m odel. The se c o n d a ry model i s p r e s e n t e d i n Appendix I I . V e r i f i c a t i o n o f Model I I P r i o r t o v e r i f i c a t i o n o f Model I I , we s h a l l r e s t a t e t h e e q u a t io n and b r i e f l y d i s c u s s th e method f o r s o l v i n g t h e e q u a tio n f o r th e c o n s t a n t s . Model I I : t - b / t c y - awe 7 = H e art r a t e a t any g iv e n tim e f o r a p a r t i c u l a r s t r e s s 80 a = C onstant to = S tre s s c = Constant t = Time As i n th e p re v io u s model, th e e x p re ss io n of th e v a r i a b le s (y k ,w ,t) a re e m p ir ic a lly observed and th en one so lv e s f o r th e c o n s ta n ts by way of s u b s t i t u t i o n . In t h i s p a r t i c u l a r case an example of th e procedure f o r a s p e c i f i e d s t r e s s i s as fo llo w s: Solving f o r a: - V ( i ) C r " V ( i ) C j . r -*> ^ -■b / 2 ° r - b / 2 ° e /'2' [toae ' - y J + e [toae - y J + e ' [toae ' - y ] , -b/kc r -b/kc t-, + e ' Lwae 1 - J = 0 Solving f o r b: — b>/(^r)° 1 r ~ b / ( h ) C t-, , -b r -b t n e ' - I toae ' - y ] + e [toae - y J 81 S o lv in g f o r c : - 1>/('3')C 1 lr — b/(^r)c t , e • log a[wae M a ' - y ] (2 ) j. Q “' b/ 2° 1 - 1 or -b/2C t, + e ' . — log 2[wae / - y ] 2 + e-b /3 . A. log 3[wae- b/5 C _ yt] 5 * e_t/kC • "c- l0® - y ^ ] = 0 With th e sim u lta n e o u s s o l u t i o n o f t h e s e t h r e e equa t i o n s f o r a g iv e n work lo ad th e v a lu e s o f p a ra m e te rs a , b, and c may be e s tim a te d . The p ro o f of th e s o l u t i o n can be o b ta in e d by s u b s t i t u t i o n o f t h e p a ra m e te rs and m u ltip ly in g th e v a lu e s i n t h e fo rm u la . F o r t u n a t e l y , i t was n o t n e c e s s a ry t o so lv e t h e a fo re m e n tio n e d e q u a tio n s m anu ally. T h is s o l u t i o n s e t f o r a l l c o n d it i o n s and a l l s u b j e c t s would be an enormous t a s k ; however, w ith th e a id o f an e l e c t r o n i c com puter th e s o l u t i o n s e t s may be c a l c u l a t e d i n a s h o r t tim e . The s o l u t i o n s e t s to t h i s problem were so lv e d on an IBM 36O/9 I com puter u t i l i z i n g a program w r i t t e n by Paul Sampson w hich i s in d ex ed i n t h e BMD Manual s e r i e s . T h is p a r t i c u l a r program i s a n o n l i n e a r l e a s t sq u a re s program w hich f i t s a s p e c i f i e d f u n c t i o n w i t h i n th e l i m i t s d e f in e d 82 by th e I n v e s t i g a t o r . The l e a s t sq u a res f i t i s c a lc u la te d v ia ste p w ise i t e r a t i o n s w ith in th e boundary l i m i t a t i o n s imposed by th e i n v e s t i g a t o r . The r e a d e r who d e s i r e s a d d i t i o n a l in fo rm a tio n p e r t a in i n g to t h i s program i s ad v ised to r e f e r to th e BM D Manual p u b lish e d by th e U n iv e rs ity o f C a l i f o r n i a (6 ). The v e r i f i c a t i o n o f th e secondary model fo llo w s: ( l ) th e model i s based on f a c t o r s which a re q u a n t i f i a b l e and which a re c o n sid e re d to be s i g n i f i c a n t ; (2) th e secondary model, as was th e prim ary model, was based on t h e p r i n c i p a l o f energy b a la n c e , and, t h e r e f o r e , conforms to th e s ta n d a rd s and in fo rm a tio n c u r r e n t l y a cc e p te d as r e a l i t y ; (3) t h i s model, as was th e p re v io u s model, i s a r e l a t i v e l y sim ple e x p re s s io n of th e i n v e s t i g a t o r s ' c o n c e p tio n o f r e a l i t y ; (4) th e model was c o n s tr u c te d on an I t e r a t i v e b a s is and m od ified on th e b a s is o f th e s o l u t i o n of numerous p ro b lem s e t s , s u b s t a n t i a l m o d ific a tio n s have been made and ad d i t i o n a l m o d ific a tio n s a re p o s s ib le by expansion or e li m i n a ti o n when d e s ir o u s ; and (5) as i n th e p re v io u s model, the i n v e s t i g a t o r a n t i c i p a t e s th e e x te n s io n of th e model or te c h n iq u e t o o th e r p ro c e ss e s and problem s, i f th e model i s a v ia b le e x p re s s io n of r e a l i t y . I s th e model v a lid ? I n o r d e r f o r th e model to be v a l i d , i t must r e f l e c t th e c u rr e n t knowledge r e l a t e d t o th e t _ h / t c problem . The e q u atio n s t a t e s , y = awe ' . The co n fin e s o f r e a l i t y d i c t a t e t h a t y^ be equal t o zero when e i t h e r t 83 o r w i s e q u a l to z e r o . S ince th e model i s b ased on en erg y b a la n c e , i t i s e s s e n t i a l t h a t y be e q u al t o z e r o , o r i n e s s e n c e , r e f l e c t a b a s a l s t a t e when t a n d /o r w a r e e q u a l to z e r o . The p ro o f i s as f o llo w s : ( l ) when u and t a re e q u al t o z e ro , y i s eq u al t o z e r o ; (2) when e i t h e r w o r t i s v e ry l a r g e , y = c o n s t a n t ; and (3) i t i s a x io m a tic t h a t whenever w o r t i s e q u a l t o z e ro , t h e r e can be no work o u t put and, t h e r e f o r e , becau se = Eou^ y^ must be e q u a l to z e r o . I t sh o u ld be n o te d t h a t t h e p ro o f of t h i s e q u a tio n i s a r e s u l t a n t o f th e p r e v i o u s l y s t a t e d s o l u t i o n s e t s ; he n ce , i t i s an e m p i r ic a l and n o t a t h e o r e t i c a l p r o o f . As p r e v i o u s l y I n d i c a t e d , t h e s o l u t i o n s e t s f o r ob t a i n i n g th e v a lu e s of th e c o n s t a n t s were s o lv e d v i a com p u t e r . A ll of th e s o l u t i o n s e t s in c lu d e d t h e f o llo w in g i n f o r m a tio n : ( l ) p a ra m e te r v a lu e s o f th e c o n s t a n t s ; (2) e r r o r mean s q u a r e — t h i s i s an i n d i c a t i o n a s t o th e q u a l i t y o r goodness of f i t o f th e r e g r e s s i o n e q u a tio n ; (3) a s y m p to tic s ta n d a r d d e v i a t i o n s — an i n d i c a t o r as t o th e a c c u ra c y of f i t o f th e p a ra m e te r e s t i m a t e s ; (4) a sy m p to tic c o r r e l a t i o n of th e p a ra m e te rs t o t h e r e g r e s s i o n e q u a tio n —a g a in , an i n d i c a t o r as to t h e q u a l i t y o f f i t ; (5) v a lu e of th e f u n c t i o n ; (6) th e r e s i d u a l (th e o b se rv ed v a lu e minus t h e e s t i mated v a l u e ) ; (7) th e s ta n d a r d d e v i a t i o n o f t h e p r e d i c t e d v a lu e ; and (8) th e o r i g i n a l d a ta p o i n t s , A rev ie w o f t h e a v a i l a b l e in f o r m a tio n su g g e ste d t h a t a p o r t i o n o f t h e d a t a w ere o f l e s s e r s i g n i f i c a n c e . 84 T a b le 4 I s a re v ie w o f a p o r t i o n o f t h e d a t a f o r S u b je c t 28. The d a t a r e p o r t e d I n c l u d e s : ( l ) t h e s t r e s s o f t h e work l o a d ; ( 2 ) t h e p a ra m e te r e s t i m a t e s ; ( 3 ) t h e a s y m p to tic s t a n d a r d d e v i a t i o n s o f t h e p a r a m e t e r s ; (4) th e a s y m p to tic c o r r e l a t i o n m a t r i x ; and ( 5 ) t h e e r r o r mean s q u a r e . The r e a d e r s h o u ld n o te t h a t t h e a s y m p to tic s t a n d a r d d e v i a t i o n f o r p a r a m e t e r "c" i s e q u a l t o z e r o ; b e c a u se t h e a s y m p to tic s t a n d a r d d e v i a t i o n i s e q u a l t o z e r o , t h e a s y m p to tic c o r r e l a t i o n m a t r i x i s a l s o e q u a l t o z e r o . The z e ro a s y m p to tic s t a n d a r d d e v i a t i o n f o r p a ra m e te r "c" s u g g e s ts t h a t t h i s p a r t i c u l a r e s t i m a t e i s a p e r f e c t f i t . The a s y m p to tic c o r r e l a t i o n m a t r i x and th e a s y m p to tic s t a n d a r d d e v i a t i o n a r e n o t r e p o r t e d i n t h i s s t u d y . A c o m p le te l i s t i n g o f t h e p a r a m e te r e s t i m a t e s f o r each s u b j e c t f o r a l l c o n d i t i o n s o f s t r e s s w i l l be p r e s e n t e d i n a l a t e r s e c t i o n o f t h i s c h a p t e r (T a b le 7j page 121). P r i o r t o a p r e s e n t a t i o n o f t h e p rim a ry d a t a t a b l e s l e t us re v ie w s e v e r a l f a c t o r s w hich s h o u ld be c o n s i d e r e d i n r e a d i n g t h e t a b l e s . Of w hat s i g n i f i c a n c e i s t h e r e p o r t e d e r r o r mean s q u a r e (EMS)? As p r e v i o u s l y s t a t e d , t h e e r r o r mean s q u a re i s one i n d i c a t o r a s t o t h e "g o o d n ess" o f t h e n o n - l i n e a r r e g r e s s i o n f i t o f t h e c u r v e , o r i n more sim p le te r m s , a summary w hich r e l a t e s t o t h e q u a l i t y o f t h e m odel. A f t e r t h e i n i t i a l i n s p e c t i o n o f t h e d a t a , i t a p p e a re d t h a t a d d i t i o n a l i n f o r m a t i o n r e l a t i n g t o t h e m ag n itu d e o f t h e EMS 85 TABLE 4 DATA SUM MARY FOR SUBJECT 28 j-.l . 1 1 1 . . . 300 KPM (Param eter E s tim a te s iAsym ptotic S.D. a .53377 .00616 b 3299.8 75 .199 c 1.9258 0 .0 A sym ptotic C o r r e l a t i o n M a trix a 1 .0 0 0 b 0 .9 8 2 c 0 .0 0 .9 8 2 1 .0 0 0 0 .0 0 .0 0 .0 0 .0 1 E r r o r Mean Square .53354 ' 600 KPM P a ra m e te r E s tim a te s 1 A sym ptotic S.D. a .30121 . OO8638 b 5834.6 336.80 c 2.0044 0 .0 i A sym ptotic C o r r e l a t i o n M a trix a 1 .0 0 0 b 0.953 c 0 .0 0.953 1 .0 0 0 0 .0 0 .0 0 .0 0 .0 E r r o r Mean Square 10.979 900 KPM P a ra m e te r E s tim a te s A sym ptotic S.D. a .22191 .0067681 b 7214.0 474.45 c 2 .0 1 0 8 0 .0 A sym ptotic C o r r e l a t i o n M a trix a 1 .0 0 0 b 0 .9 3 7 c 0 .0 0.937 1.000 0 .0 0 .0 0 .0 0 .0 E r r o r Mean Square 21.391 86 would be of I n t e r e s t . As a p o in t of d e p a rtu r e , a c o e f f i c ie n t o f c o r r e l a t i o n was c a l c u l a t e d between th e EMS and th e d if f e r e n c e between th e observed and p r e d i c te d v a lu e s f o r t of 300 seconds. r = -0.6 47 f o r a s t r e s s of 300 k i l o - xy pound m e ters; r = 0.425 f o r a s t r e s s of 600 kilopound xy m eters; and r r = -0.145 f o r a s t r e s s of 900 k ilopo und xy m eters. Recognizing t h a t c o r r e l a t i o n does not imply c a u s a t i o n , n e v e r th e le s s th e range and m agnitude of th e c o e f f i c i e n ts of c o r r e l a t i o n a re s u r p r i s i n g . One would expect an in v e r s e r e l a t i o n s h i p between mean d e v ia tio n (mean of ob s e r v e d - p r e d ic te d v a lu e s) and the EMS; however, th e p resen ce of a s u b s t a n t i a l p o s i t i v e c o r r e l a t i o n f o r th e 600 k i l o pound m eter s t r e s s f a c to r i s very p u z z lin g and u n e x p la in a b le . The afo rem entio ned c o r r e l a t i o n s su ggested t h a t a d d i t i o n a l c o e f f i c i e n t s of c o r r e l a t i o n be c a l c u l a t e d between EMS and th e range o f th e observed v a lu e s ( in t h i s c a se , th e range of v a lu e s was d e fin e d as th e f i n a l o b s e r v a tio n minus th e i n i t i a l o b s e r v a tio n . T his was tho ugh t n e c e s s a ry b e cause s e v e r a l of th e f i n a l o b s e rv a tio n s were l e s s th a n an observed in te r m e d ia te v a lu e ) . I t should be p o in te d out t h a t th e range of v a lu e s i s i n v e r s e l y p r o p o r t io n a l to th e s t r e s s f a c t o r and t h a t t h i s r e l a t i o n s h i p i s e s s e n t i a l l y l i n e a r i n n a tu r e . C o e f fic ie n t of c o r r e l a t i o n f o r a s t r e s s of 300 KPM was -0 .5 4 8 , f o r 600 KPM 0.367, and f o r 900 KPM 0.00 6. As i n th e p rev io u s c a se , th e m agnitude of th e c o r - 87 r e l a t i o n c o e f f i c i e n t s would a p p e a r to have an in v e r s e r e l a t i o n s h i p ; however., th e r e s u l t s of t h i s a n a l y s i s d id not a s s i s t i n e x p la in in g th e s h i f t which o c c u rre d i n "both s e t s o f c o r r e l a t i o n s . I n summary, i t would ap p ear as though th e EMS i s a pro d u ct of numerous f a c t o r s , s e v e r a l of which may be: ( l ) th e m agnitude o f th e f i n a l o b se rv ed minus p r e d i c te d v a lu e ; (2) th e range of th e o b served v a lu e s ; and ( 3 ) a summation o r p ro d u ct of th e asy m p to tic s ta n d a rd d e v ia ti o n . I f one examines th e v a lu e s i n T ab le 4, th e e f f e c t of th e asymp t o t i c s ta n d a rd d e v ia ti o n i s r e a d i l y a p p a re n t. T ables 5-1 th ro u g h 5-30 d e p ic t t h a t d a ta which were s e l e c t e d f o r p rim ary a n a l y s i s . Each t a b l e i n d i c a t e s : ( l ) th e s u b j e c t ; (2) s e l e c t e d a n th r o p o m e tr ic a l c h a r a c t e r i s t i c s of th e s u b j e c t ; ( 3 ) th e EMS f o r each s t r e s s f a c t o r ; and (4) th e p r e d i c te d and th e observed v a lu e s e x p re sse d as d e l t a A h e a r t r a t e . (A h e a r t r a t e i s e q u al to h e a r t r a t e a t tim e minus h e a r t r a t e a t tim e Q.) I t i s i n t e r e s t i n g t o n o te t h a t th e EMS f o r a s t r e s s of 300 KPM ra n g e s from a h ig h o f 8 9 6 .7 6 (which i s o v e r e le v e n tim e s l a r g e r th a n t h e next l a r g e s t EMS r e g a r d l e s s of s t r e s s f a c t o r ) t o a low of 0.16418; f o r 600 KPM th e rang e was from 27.964 to 1.4584; and f o r 900 KPM th e range was from 72.923 to 7 .7998. A rank d i f f e r e n c e c o r r e l a t i o n was c a l c u l a t e d f o r each p o s s ib l e c om binatio n of EMS v a lu e s . The c o r r e l a t i o n s 88 TABLE 5-1 A* HEART RATE EOR SUBJECT 1-M.H. Predicted (A) and Observed (B) 300 (A) KPM (B) 600 KPM (AT IB) 900 (A) KPM (B) t = 05 19 18 10 10 34 30 t = 10 28 30 20 18 34 30 t = 15 19 18 20 18 19 18 t = 20 19 18 20 18 45 42 t = 25 19 18 31 30 34 30 t = 30 19 18 31 30 45 42 t = 60 24 24 36 36 50 48 t = 120 24 24 36 36 58 60 t — 180 19 18 36 36 58 60 t - 240 19 30 40 42 61 66 t = 300 24 24 40 42 61 66 EMS = 1 .3497 EMS = 4 .3 9 7 EMS = 24.443 A n th ro p o m e tric C h a r a c t e r i s t i c s A ge: 20 H e i g h t : 7 1 .5 W e ig h t: 164 A c t i v i t y In d e x : 5 Body S u r fa c e In d e x : 2 .1 8 *A H.R. = H.R., - H.R., . 89 TABLE 5-2 A* HEART RATE FOR SUBJECT 2-J.F. Predicted (A) and Observed (B) 300 (A) KPM w 600 KPM '(A')” (B) 900 (A) KPM w t = 05 21 20 6 8 21 20 t = 10 33 32 36 32 37 32 t = 15 21 20 22 20 21 20 t = 20 21 20 36 32 37 32 t = 25 21 20 36 32 50 44 t = 30 33 32 22 20 37 32 t = 60 27 26 57 56 79 80 t = 120 38 38 61 62 79 80 t = 180 38 38 61 62 83 86 t = 240 42 44 65 68 86 92 t = 300 42 44 65 68 ** EMS = 3 .0810 EMS = 13.244 EMS = 41.900 A nthropom etric C h a r a c t e r i s t i c s Age: 19 H e ig h t: 67*0 W eight: 222 A c t i v i t y Ind ex : 3 Body S u rfa ce Index: 2.12 *A H.R. = H .R ., - H .R ., . **Did not f i n i s h . 90 TABLE 5-3 A* HEART RATE EOR SUBJECT 3-B.A. Predicted (A) and Observed (B) 300 (A) KPM (B) 600 KPM ( A J W 900 KPM (A) (B) t = 05 29 29 32 29 44 41 t = 10 19 17 18 17 44 41 t = 15 29 29 32 29 33 29 t = 20 29 29 32 29 44 4 l t = 25 29 29 32 29 44 41 t - 30 19 17 42 4 l 44 41 t = 6o 29 29 47 47 58 59 t = 120 29 29 47 47 58 59 t = 180 35 33 47 47 61 65 t = 240 29 29 47 47 61 65 t = 300 29 29 31 35 61 65 EMS - 1 .0956 EMS = 1 3.799 EMS = 19.893 A n th ro p o m e tric C h a r a c t e r i s t i c s A ge: 21 H e i g h t : 6 9 .O W e ig h t: 172 A c t i v i t y In d e x : 3 Body S u rfa c e In d e x : 1 .9 6 *A H.R. = H.R,, - H.R., . 91 TABLE 5-4 A* HEART RATE FOR SUBJECT 4-R.M. Predicted (A) and Observed (B) 300 (A) KPM (B) 600 KPM “ (A) (Bj 900 KPM lA) XBJ- t = 05 13 12 -1 0 ** t = 10 -1 0 -1 0 t = 15 13 12 15 12 o C M I I -P 13 12 -1 0 L T \ O J I I -P 13 12 15 12 t i ll L O O -1 0 -1 0 O V O 1 1 -p 13 12 4o 36 t = 120 24 24 49 48 t = 180 24 24 53 54 o C V J I I -P 24 24 53 54 t = 300 24 24 57 60 EMS = . 45933 EMS = 5.5970 Anthopom etric 'C h a r a c t e r i s t i c s Age: 18 H e ig h t: 7 2 .0 W eight: 152 A c t i v i t y Index: 3 Body S u rface Index: 1.99 *A H.R. = H.R.,. - H .R ., . **Did not f i n i s h . 92 TABLE 5-5 A* HEART RATE FOR SUBJECT 5-S.M. Predicted. (A) and Observed (B) 300 (A) KPM (B) 600 KPM W (B) -HVQ m KPM (B) t = 05 44 42 33 30 34 30 t = 10 33 30 42 42 34 30 t = 15 44 42 42 42 45 42 t = 20 33 30 33 30 34 30 t = 25 33 30 42 42 45 42 t = 30 33 30 33 30 45 42 t = 6o 6o 66 46 48 54 54 t = 120 60 66 46 48 58 60 t = 180 44 42 42 42 58 60 t = 240 26 24 42 42 62 66 t = 300 44 42 42 42 62 66 EMS = 20.781 EMS = 4 .3 5 6 0 EMS = 2 0 .0 9 8 A n th ro p o m etric C h a r a c t e r i s t i c s A ge: 22 H e i g h t : 7 4 .0 W eight: 212 A c t i v i t y In d e x : 4 Body S u rfa c e In d e x : 2.26 *A H.R. = H.R., - H.R., . 93 TABLE 5-6 A* HEART RATE FOR SUBJECT 6-T.F. Predicted (A) and Observed (B) 300 KPM 600 KPM 900 KPM T O '( B y To " T O ' 'T O T O t = 05 23 21 24 21 22 21 t = 10 23 21 24 21 22 21 ct- 1 1 H 1 V Ji 23 21 37 33 53 45 t = 20 23 21 37 33 39 33 t = 25 32 33 37 33 53 45 t a 30 23 21 47 45 53 45 t = 60 32 33 58 63 78 81 t = 120 28 27 55 57 78 81 t = 180 32 33 52 51 80 87 t = 2^0 32 33 58 63 78 81 t = 300 36 39 55 57 78 81 EMS = 4.9492 EMS = 27.964 EMS = 72.923 A nthropom etric C h a r a c t e r i s t i c s Age: 29 H eig h t: 72.0 W eight: 177 A c t i v i t y Index: 3 Body S urface Index: 2.05 *A H.R. = H.R., - H.R.j, . ^n o 94 TABLE 5 - 7 A* HEART RATE FOR SUBJECT 7-A.T. Predicted (A) and Observed (B) 300 KPM W " ( B ) 600 KPM Ta T W 900 KPM (AJ (B) t = 05 10 9 23 21 5 9 t = 10 10 9 9 9 24 21 t = 15 10 9 23 21 24 21 t = 20 21 21 23 21 24 21 t = 25 10 9 9 9 39 33 t = 30 21 21 23 21 39 33 t = 60 21 21 40 39 67 63 t = 120 21 21 40 39 74 7 5 t = l8o 21 21 48 51 74 75 t = 240 26 27 44 45 77 81 t = 300 26 27 44 45 80 87 EMS = . 60901 EMS = 6 .5642 EMS = 36.855 A n th ro pom etric C h a r a c t e r i s t i c s Age: 19 H e i g h t : 6 8 .0 W eig h t: 154 A c t i v i t y In d ex : 3 Body S u rfa c e In d ex : 1.8 2 *A H.R. = H.R., - H.R., . TABLE 5-8 95 A* HEART RATE FOR SUBJECT 8-R.G. Predicted (A) and Observed (B) 300 (A) KPM (B) 600 KPM (A)- (B) 900 KPM '(AT X 'bT t = 05 25 24 27 24 4o 36 t = 10 25 24 38 36 40 36 t = 15 34 36 38 36 4o 36 t = 20 25 24 38 36 51 48 t = 25 34 36 38 36 40 36 t = 30 34 36 38 36 40 36 t = 60 25 24 46 48 59 60 t = 120 25 24 46 48 63 66 t = 180 30 30 46 48 66 72 t = 240 30 30 46 48 63 66 t = 300 25 24 46 48 63 66 EMS = 2 .3747 EMS = 6 .6 6 8 7 EMS = 29.540 Anthropom et r i c C h a r a c t e r i s t i c s Age: 19 H e i g h t : 7 0 .0 W e ig h t: 151 A c t i v i t y In d e x : 3 Body S u rfa c e In d e x : 1 .8 4 *A H.R. = H.R., - H.R,, . v o 9 6 TABLE 5-9 A* HEART RATE FOR SUBJECT 9-T.G. Predicted (A) and Observed (B) 300 KPM 600 KPM 900 KPM Ta T W (a ) (k) TaT T bT t = 05 10 9 23 21 8 9 t = 10 22 21 9 9 8 9 t * 15 10 9 23 21 25 21 t = 20 10 9 9 9 38 33 t = 25 31 33 9 9 25 21 t = 30 22 21 23 21 38 33 t = 60 22 21 39 39 61 63 t = 120 27 27 39 39 61 63 o CO 1 -t 1 1 -p 27 27 39 39 58 57 o OJ 1 1 -p 27 27 43 45 61 63 t = 300 22 21 39 39 65 69 EMS = 1.1241 EMS = 3.8420 EMS = 18.629 A nth ro pom etric C h a r a c t e r i s t i c s Age: 20 H e ig h t: 7 0 .0 W eight: 151 A c t i v i t y Index: 3 Body S u rfa c e In d ex : 1.85 *A H.R. = H.R.. - H.R.. . 97 TABLE 5-10 A* HEART RATE FOR SUBJECT 10-M.C. Predicted (B) and Observed (B) 300 KPM 600 KPM 900 KPM W '(B) T a T W P T W t = 05 23 22 9 10 8 10 t = 10 10 10 24 22 8 10 t = 15 23 22 24 22 26 22 t = 20 23 22 24 22 4 l 34 t = 25 23 22 24 22 26 22 t = 30 10 10 24 22 41 34 t = 6 o 23 22 42 40 73 70 t = 120 28 28 51 52 80 82 t = 180 33 34 51 52 80 82 t = 240 33 34 56 58 80 82 t = 300 38 4o 56 58 83 88 EMS = 1.3237 EMS = 8 .0 7 4 6 EMS = 2 6 .6 6 9 A n th ro p o m e tric C h a r a c t e r i s t i c s A ge: 20 H e ig h t: 6 4 .0 W e ig h t: 120 A c t i v i t y I n d e x : 3 Body S u r f a c e In d e x : 1 .5 6 *A H.R. = H .R ., - H .R ., . un ^o 98 TABLE 5-11 A* HEART RATE FOR SUBJECT 11-T.H. Predicted (A) and Observed (B) 300 (A) KPM (B) 600 KPM (A)-(B) 900 KPM (A) (S) t = 05 16 15 -1 3 17 15 t = 10 16 15 16 15 29 27 t = 15 1 3 16 15 17 15 t = 20 28 27 43 39 38 39 t = 25 16 15 30 37 38 39 t = 30 16 15 30 27 38 39 t = 60 47 51 53 51 ** t = 120 28 27 58 57 t = 180 28 27 58 57 o cu 1 1 -p 28 27 71 75 t = 300 28 27 71 75 EMS = 4 .1450 EMS = 13.589 EMS = 3.7588 A n th ro pom etric C h a r a c t e r i s t i c s Age: 19 H e i g h t : 71 .5 W eig h t: 205 A c t i v i t y In d e x : 3 Body S u rfa c e Index: 2.16 *A H.R. = H .R ., - H .R ., . ‘ 'o **Did n o t f i n i s h . 99 TABLE 5-12 A* HEART RATE FOR SUBJECT 12-R.R. Predicted (A) and Observed (B) 300 (A) KPM (B) 600 (A) KPM (B) 900 (A) KPM (B) t = 05 24 22 9 10 23 22 t = 10 10 10 9 10 23 22 t = 15 10 10 25 22 23 22 t = 20 24 22 25 22 40 34 t = 25 10 10 25 22 54 46 t = 30 24 22 25 22 54 46 t = 60 34 34 52 52 75 70 t = 120 34 34 56 58 87 88 t = 180 39 40 56 58 90 94 t = 24 0 39 4o 52 52 90 94 t = 300 39 4o 56 58 93 100 EMS = 2 .4525 EMS = 9 .0 6 1 0 EMS = ; 5 4 .2 5 1 A n th ro p o m e tric C h a r a c t e r i s t i c s Age: 19 H e i g h t : 7 3 .0 W e ig h t: 177 A c t i v i t y In d e x : 3 Body S u r fa c e In d e x : 2.05 *A H.R. = H.R., - H.R.. . ^n ^o 100 TABLE 5-13 A* HEART RATE FOR SUBJECT 13-M.H. Predicted (A) and Observed (B) 300 KPM 600 KPM 900 KPM P T W W (B) P T W t = 05 26 24 27 24 12 12 t = 10 35 36 13 12 40 36 t = 15 26 24 27 24 28 24 t = 20 26 24 38 36 40 36 t = 25 26 24 38 26 4o 36 t = 30 26 24 47 48 50 48 t = 60 30 30 47 48 57 60 t = 120 35 36 47 48 57 60 t = 180 35 36 47 48 57 60 t - 240 35 36 47 48 57 60 t = 300 35 36 51 54 57 60 EMS = 2.7311 EMS = 8 .1932 EM S = 17.956 A n th ro p o m etric C h a r a c t e r i s t i c s Age s 21 H e ig h t: 7 6 .0 W eight: 231 A c t i v i t y In d e x : 4 Body S u rfa c e In d e x : Not a v a i l a b l e *A H.R. = H.R., - H.R., ^n ^o* 1 0 1 TABLE 5-14 A* HEART RATE FOR SUBJECT 14-R.M. Predicted (A) and Observed (B) 300 KPM '(A) -(B) 6 00 KPM (A) (&)■ 900 (A) KPM t = 05 15 15 14 15 16 15 t = 10 15 15 14 15 16 15 t = 15 15 15 14 15 30 27 t = 20 15 15 14 15 16 15 t = 25 15 27 27 30 27 t = 30 15 15 14 15 30 27 t = 60 15 15 32 33 48 45 t = 120 15 15 32 33 56 57 t = l 8 o 20 21 32 33 56 57 t = 240 9 9 32 33 64 69 t = 300 15 15 32 33 56 57 EMS = . 16418 EMS = 7 .3 2 5 1 EMS = ; 1 5 .1 2 2 A n th ro p o m e tric C h a r a c t e r i s t i c s Age: 18 H e i g h t : 6 8 .0 W e ig h t: 143 A c t i v i t y In d e x : 3 Body S u rfa c e In d e x : I .7 8 *A H.R. = H.R., - H.T., . 1 0 2 TABLE 5-15 A* HEART RATE FOR SUBJECT 15-D. A. Predicted (A) and Observed (B) 300 (Aj KPM (£ ) 600 KPM W VSJ 900 W KPM ( B ) t = 05 13 12 13 12 9 12 t = 10 13 12 13 12 28 24 t = 15 13 12 13 12 28 24 t = 20 -1 0 13 12 28 24 t = 25 13 12 27 24 28 24 t = 30 13 12 27 24 28 24 t = 60 13 12 46 48 65 60 t = 120 23 24 42 42 73 7 2 t = 180 23 24 42 42 76 78 t = 240 23 24 42 42 79 84 t = 300 18 18 46 48 79 84 EMS = .83485 EMS = 5 .3 8 1 3 EMS = ; 3 7 .2 6 6 A n th r o p o m e tr ic C h a r a c t e r i s t i c s A g e : 19 H e i g h t : 7 0 .0 W e ig h t: 164 A c t i v i t y I n d e x : 3 Body S u r f a c e I n d e x : 1 .9 2 *A H.R. = H.R., - H.R.. . 103 TABLE 5-16 A* HEART RATE FOR SUBJECT 16-H.J. Predicted (B) and Observed (B) 300 KPM 600 KPM 900 KPM (A) (B) W l B j W (B; ct- I I 0 vn 23 22 24 22 7 10 0 I — I I I ■ p 11 10 24 22 25 22 t = 15 23 22 24 22 25 22 0 O J I I -p 23 22 36 34 4o 34 t = 25 23 22 36 34 4o 34 t = 30 23 22 36 34 52 46 t = 60 23 22 51 52 71 70 t = 120 32 34 46 46 74 76 t = 180 23 22 51 52 74 76 t = 240 32 34 51 52 78 82 t = 300 28 28 54 58 78 82 EMS = 1 .4 1 9 0 EMS = 9 .0 2 7 0 EMS = V O • I — I 00 A n th ro p o m e tric C h a r a c t e r i s t i c s A ge: 19 H e ig h t: 7 2 .0 W e ig h t: 156 A c t i v i t y In d e x : 4 Body S u r f a c e In d e x : 1 .9 2 *A H.R. = H.R., - H.R., . 104 TABLE 5-17 A* HEART RATE FOR SUBJECT 17-M.S. Predicted (A) and Observed (B) 300 KPM (A) (B) 600 KPM (A) (B) 900 (A) KPM (B) ct I I 0 18 18 33 30 18 18 t = 10 18 18 33 30 18 18 t = 15 18 18 33 30 35 30 0 C V J I I -p 6 6 33 30 49 42 t = 25 18 18 33 30 35 30 t = 30 6 6 ■ 33 30 49 42 t = 60 24 24 49 48 75 72 t = 120 24 24 53 54 82 84 t = 180 24 24 53 54 82 84 0 ■ = t O J I I -P 24 24 57 60 86 90 t = 300 24 24 60 66 89 96 EMS = .28134 EMS = 1 8 .0 2 2 EMS = 5 1 .6 7 3 A n th ro p o m e tric C h a r a c t e r i s t i c s Age: 22 H e ig h t • • 7 0 .0 W eight • f t 136 A c t i v i t y In d e x : 3 Body S u rfa c e In d e x : 1 .7 8 *A H.R. = H.R.. - H.R.. . 105 table 5 - 1 8 A* HEART RATE FOR SUBJECT 18-J.H. Predicted (A) and Observed (B) 300 (A) KPM (B) 600 KPM (A) (B) 900 '(A) KPM (B) t = 05 24 24 12 12 9 12 t = 10 24 24 26 24 27 24 t = 15 24 24 12 12 27 24 t = 20 24 24 26 24 27 24 t = 25 24 24 26 24 4 l 36 t = 30 13 12 38 36 41 36 t = 60 13 12 54 60 64 60 t = 120 24 24 38 36 72 72 t = 180 13 12 43 42 76 78 t = 240 24 24 43 42 76 78 t = 300 24 24 47 48 83 90 EMS = . 37162 EMS = 9 .3 2 2 1 EMS = : 2 8 .6 2 5 A n th ro p o m e tric C h a r a c t e r i s t i c s Age: 18 H e i g h t : 7 1 .0 W e ig h t: 146 A c t i v i t y I n d e x : 3 Body S u r f a c e In d e x : I .8 5 *A H.R. = H.R., - H.R., . io6 TABLE 5-19 A* HEART RATE FOR SUBJECT 19-J.T. Predicted (A) and Observed (B) 300 KPM 600 KPM 900 KPM (A) (B) (A) (B) (A) (B) t = 05 16 15 4o 39 29 27 t = 10 16 15 29 27 29 27 t = 15 16 15 40 39 42 39 t = 20 16 15 29 27 42 39 t = 25 2 3 40 39 42 39 t = 30 16 15 29 27 54 51 t = 60 32 33 45 45 69 69 t = 120 32 33 50 51 73 75 t = 180 27 27 50 51 69 69 o O J I I -p 32 33 50 51 73 75 t = 300 32 33 55 59 77 81 EMS = 1.4405 EMS = 6 .7 7 7 1 EMS = 13*450 A n th ro p o m e tric C h a r a c t e r i s t i c s Age: 18 H e ig h t: 71*0 W e ig h t: 164 A c t i v i t y In d e x : 4 Body S u rfa c e In d e x : 1 .9 4 *A H.R. = H .R ., - H .R ., . 1 3 o 107 TABLE 5-20 A* HEART RATE FOR SUBJECT 20-R.S. Predicted (A) and Observed (B) 300 KPM 600 KPM 900 KPM (A) (B) (A) (B) (A) (B) t = 05 22 22 36 34 39 34 t = 10 22 22 24 22 23 22 t = 15 11 10 36 34 39 34 t = 20 22 22 24 22 39 34 t = 25 22 22 36 34 23 22 t = 30 22 22 24 22 39 34 t = 60 22 22 42 4o 70 70 t = 120 22 22 51 52 74 76 t = 180 27 28 56 58 74 76 t = 240 22 22 56 58 74 76 t = 300 27 28 56 58 77 82 EMS = .44177 EMS = 8 .9 7 8 3 EMS = 3 0 .8 9 7 A n th ro p o m e tric C h a r a c t e r i s t i c s A ge: 21 H e ig h t: 7 1 .0 W e ig h t: 192 A c t i v i t y In d e x : 3 Body S u r f a c e In d e x : 2 .1 0 *A H.R. = H.R., - H.R., . ^n ^o 1 0 8 TABLE 5-21 A* HEART RATE FOR SUBJECT 21-M.W. Predicted (A) and Observed (B) 300 KPM 600 KPM 900 KPM (A) (B) (A) (B) (A) (B) t = 05 10 9 23 21 23 21 t = 10 21 21 9 9 36 33 t = 15 10 9 23 21 23 21 t = 20 21 21 33 33 36 33 t = 25 21 21 23 21 36 33 t = 30 21 21 23 21 ^7 45 t = 60 26 27 33 33 51 51 t = 120 21 21 38 39 51 51 t = 180 26 27 38 39 55 57 t = 240 21 21 38 39 59 63 t = 300 26 27 38 39 59 63 EMS = .48599 EMS = 2 .7 3 1 6 EM S = 13.556 A n th ro p o m etric C h a r a c t e r i s t i c s Age: 19 H e ig h t: 7 2 .0 W eig h t: 173 A c t i v i t y In d e x : 3 Body S u rfa c e In d e x : 2 .0 0 *A H.R. = H.R.. - H.R., . cn ^o TABLE 5-22 A* HEART RATE FOR SUBJECT 22-T .O . P r e d i c t e d (A) and O b serv ed (B) 300 KPM 600 KPM 900 KPM (A) (B) (A) (B) (A) (B) t = 05 28 27 31 27 45 39 t = 10 28 27 31 27 30 27 t = 15 28 27 4 l 39 57 51 t = 20 37 39 4 l 39 45 39 t = 25 28 27 41 39 57 51 o o n II 37 39 4 l 39 45 39 t = 60 37 39 50 51 70 69 t = 120 28 27 50 51 74 75 t = 180 33 33 50 51 81 87 t = 240 33 33 53 57 81 87 t = 300 28 27 53 57 81 87 EMS = 3 .1 8 0 0 EMS = 1 2 .9 0 0 EMS = 6 2 .5 9 2 A n th ro p o m e tric C h a r a c t e r i s t i c s A ge: 19 H e ig h t: 7 0 .0 W e ig h t: 137 A c t i v i t y I n d e x : 4 Body S u r fa c e In d e x : 1 .7 8 *A H .R. = H .R ., - H .R ., . •'n ‘ 'o 1 1 0 TABLE 5-23 A* HEART RATE FOR SUBJECT 23-J.D. Predicted (A) and Observed (B) 300 (A) KPM (B) o^ > o o KPM (B) 900 (A) KPM (B) t = 05 29 28 17 16 32 28 t = 10 29 28 43 40 42 40 t = 15 29 28 31 28 42 40 t = 20 29 28 31 28 50 52 t = 25 17 16 43 40 50 52 t = 30 29 28 43 40 50 52 t = 60 39 4o 57 58 ** t = 120 43 46 57 58 t = l8 o 39 4o 61 64 t = 240 34 34 61 64 t = 300 39 4o 57 58 EMS = 3 .2 1 5 8 EMS = 1 2 .8 4 7 EMS = 8 .8 8 5 4 A n th r o p o m e tr ic C h a r a c t e r i s t i c s A g e : 21 H e i g h t: 7 1 .0 W e ig h t: 139 A c t i v i t y I n d e x : 3 Body S u r f a c e I n d e x : 1 . 8 l *A H.R. = H .R ., - H .R ., . ^n ^o **D Id n o t f i n i s h . Ill TABLE 5-24 A* HEART RATE FOR SUBJECT 24-A.L. Predicted (A) and Observed (B) 300 KPM 600 KPM 900 KPM (A) (B) (A) (B) (A) (B) t = 05 12 12 7 8 7 8 t = 10 18 16 10 10 7 8 t = 15 18 16 12 12 12 12 t = 20 28 24 18 16 12 12 t = 25 35 30 23 20 17 16 t = 30 47 43 25 22 18 18 t = 60 52 50 39 37 22 22 t = 120 53 52 43 42 24 24 t = 180 59 61 48 50 29 28 t = 240 61 65 48 50 32 32 t = 300 61 65 48 50 32 32 EMS = 1 8 .3 2 1 EMS = 7 .6 4 9 4 EMS = 2 .9 8 9 7 A n th ro p o m e tric C h a r a c t e r i s t i c s A ge: 20 H e i g h t : 7 3 .0 W e ig h t: 210 A c t i v i t y I n d e x : 4 Body S u rf a c e In d e x : 2 .2 1 *A H.R. = H.R., - H.R..,. . 1 1 2 TABLE 5-25 A* HEART RATE FOR SUBJECT 25-S.F. Predicted (A) and Observed (B) 300 KPM 600 KPM 900 KPM (A) (B) (A) (B) (A) (B) t = 05 6 6 5 6 19 18 t = 10 6 6 20 18 19 18 t = 15 19 18 20 18 34 30 t = 20 6 6 20 18 34 30 t = 25 19 18 32 30 34 30 t = 30 6 6 32 30 46 42 t = 60 24 24 32 30 55 54 t = 120 24 24 45 48 63 66 t = 180 24 24 4 l 42 63 66 t = 240 19 18 41 42 63 66 t = 300 24 24 4 l 42 63 66 EMS = .3 7 1 1 2 EMS = 5 .4 1 8 4 EMS = 2 0 .8( A n th ro p o m e tric C h a r a c t e r i s t i c s A g e: 21 H e ig h t: 7 1 .0 W e ig h t: 182 A c t i v i t y In d e x : 4 Body S u r f a c e In d e x : 2 .0 3 *A H.R. = H.R., - H.R.. . n ^o 113 TABLE 5-26 A* HEART RATE FOR SUBJECT 26-M.A. Predicted (A) and Observed (B) 300 KPM 600 KPM 900 KPM (A) (B) (A) (B) (A) (B) t = 05 17 18 32 30 33 30 t = 10 28 30 19 18 33 30 t = 15 28 30 32 30 33 30 t = 20 17 18 32 30 33 30 t = 25 28 30 43 42 46 42 0 m I I -p 23 24 32 30 46 42 0 V O I I •p 23 24 43 42 66 66 t = 120 28 30 52 54 70 72 t = 180 28 30 47 48 74 78 t = 240 23 24 47 48 70 72 t = 300 23 24 52 54 74 78 EMS = 8 9 6 .7 6 EMS = 5 .4 0 3 1 EMS = 28.257 A n th ro p o m e tric C h a r a c t e r i s t i c s A ge: 19 H e ig h t: 7 1 .5 W e ig h t: 157 A c t i v i t y In d e x : 3 Body S u rfa c e In d e x : 1 .9 1 *A H.R. = H.R.. - H.R., . un ^o n 4 TABLE 5-27 A* HEART RATE BOR SUBJECT 27-J.M. Predicted (A) and Observed (B) 300 (A) K P M (B) 600 K P M (A) (B) 900 (A) K P M (B) t = 05 17 16 1 4 17 16 t = 10 3 4 18 16 17 16 t = 15 17 16 18 16 46 4o t = 20 17 16 32 28 46 4o t = 25 28 28 32 28 17 16 t = 30 17 16 18 16 55 50 t = 60 23 22 43 4o 66 64 t = 120 23 22 52 52 69 70 t = 180 23 22 47 46 73 76 t = 240 37 40 55 58 76 82 t = 300 28 28 59 64 73 76 EMS = 2 .7725 EMS = 13.914 EMS = : 2 9 .6 5 9 A n th r o p o m e tric C h a r a c t e r i s t i c s A ge: 21 H e i g h t : 74.0 W e ig h t: 178 A c t i v i t y I n d e x : 4 Body S u r f a c e I n d e x : 2 .0 8 *A H.R. = H.R ‘t n “ H .R .. . r o 115 TABLE 5-28 A* HEART RATE FOR SUBJECT 28-R.P. Predicted (A) and Observed (B) 300 KPM 600 KPM 900 KPM (A) (B) (A) (B) (A) (B) t = 05 0 0 -4 0 11 12 t = 10 13 12 13 12 27 24 t = 15 13 12 27 24 27 24 o OJ I I - P 0 0 13 12 40 36 t = 25 13 12 27 24 27 24 t = 30 0 0 13 12 46 42 t = 60 18 18 53 6o 55 54 t = 120 23 24 38 36 55 54 t = 180 23 24 38 36 68 74 ct- I I ro -p o 18 18 42 42 66 70 t = 300 23 24 38 36 60 60 EMS = .53354 EMS = 10.979 EM S = 21.35 A n th ro p o m e tric C h a r a c t e r i s t i c s A ge: 24 H e ig h t: 7 0 .0 W e ig h t: 152 A c t i v i t y In d e x : 3 Body S u r fa c e In d e x : 1 .8 6 *A H.R. = H.R., - H.R., . 1 1 6 TABLE 5-29 A* HEART RATE FOR SUBJECT 29-M.S. Predicted (A) and Observed (B) 300 (A) KPM (B) 600 KPM (A) (B) 900 (A) KPM (B) t = 05 13 13 14 13 14 13 t = 10 13 13 14 13 14 13 t = 15 13 13 25 25 28 25 0 C M 1 ! -P 13 13 14 13 28 25 I T \ C M I I -P 13 13 14 13 28 25 t = 30 13 13 25 25 14 13 t = 60 18 19 25 25 43 43 t = 120 13 13 25 25 47 49 t = 180 13 13 30 31 47 49 t = 240 18 19 30 31 47 49 t = 300 13 13 30 31 47 49 EMS = . 20450 EMS = 1 .4 5 8 4 EMS = 7 .7 9 9 8 A n th ro p o m e tric C h a r a c t e r i s t i c s A ge: 22 H e i g h t : 6 8 .0 W e ig h t: 152 A c t i v i t y I n d e x : 4 Body S u rf a c e In d e x : 1 .8 2 *A H.R. = H.R,. - H.R., . 117 TABLE 5-30 A* HEART RATE FOR SUBJECT 30-R.T. Predicted (A) and Observed (B) 300 KPM 600 KPM 900 KPM (A) (B) (A) (B) (A) (B) t = 05 22 21 23 21 38 33 t = 10 22 21 23 21 21 21 t = 15 10 9 36 33 52 45 t = 20 22 21 36 33 38 33 t = 25 22 21 36 33 52 45 t = 30 10 9 36 33 38 33 o VO I I -p 22 21 4 l 39 67 63 t = 120 22 21 45 45 78 81 t = 180 30 33 49 51 81 87 o O J I I - p 26 27 52 57 81 87 t = 300 22 21 52 57 81 87 EM S = 1 .6 6 7 2 EMS = 16.307 EM S = 6 6 .6 0 2 A n th ro p o m etric C h a r a c t e r i s t i c s A ge: 21 H e ig h t: 6 8 .5 W eig h t: 164 A c t i v i t y In d e x : 3 Body S u rfa c e In d e x : 1 .8 9 *A H.R. = H.R., - H.R., . ^n ^o 1 18 w e re: ( l ) p = 0 .3 5 8 f o r s t r e s s 3 0 0 -6 0 0 ; (2 ) p = 0 .6 4 8 f o r s t r e s s 6 0 0 -9 0 0 ; an d (3 ) P = 0 .4 0 5 f o r s t r e s s 3 0 0 -9 0 0 . One s h o u ld o b s e r v e i n T a b le s 5 -1 th ro u g h 5 -3 0 t h a t t h e l a r g e s t s p r e a d o f v a lu e s b etw een p r e d i c t e d and o b s e rv e d v a lu e s f o r 300 KPM s t r e s s was on t h e o r d e r o f two h e a r t b e a t s p e r m in u te (bpm) f o r t h e I n i t i a l o b s e r v a t i o n and t h r e e b e a t s p e r m in u te f o r t h e f i n a l o b s e r v a t i o n . C o r r e s p o n d in g v a lu e s f o r 6 00 KPM and 900 KPM w ere f o r i n i t i a l v a lu e s 4 bpm and f o r f i n a l v a lu e s 6 bpm and 7 bpm r e s p e c t i v e l y . T a b le 6 d e p i c t s each s u b j e c t 's p h y s i c a l w o rk in g c a p a c i t y e s t im a t e d on t h e b a s i s o f a PW C^q t e s t ( S jo s tr a n d ) a n d a p r e d i c t e d PWC-^q b a se d on th e p ro p o s e d m odel. (Ap p e n d ix IV d e p i c t s t h i s in f o r m a tio n g r a p h i c a l l y . ) An e x a m in a tio n o f th e d a t a p r e s e n t e d In t h e a f o r e m en tio n e d f i g u r e s a p p e a rs t o j u s t i f y t h e f o l lo w in g g e n e r a l i z a t i o n s : 1. The p r e d i c t e d PW C^q (b a s e d on th e m odel) was o f h i g h e r m a g n itu d e i n 27 o f 30 s u b j e c t s . 2. The mean d i f f e r e n c e b etw een th e e s t im a t e d and p r e d i c t e d v a lu e was o n ly 5 9 .2 3 KPM. 3 . By rem o v in g t h r e e s u b j e c t s ( 1 -5 - 2 4 ) w ith t h e g r e a t e s t d i f f e r e n c e b etw een e s t im a t e d and p r e d i c t e d t h e mean d i f f e r e n c e b e tw een e s t i m a t e d and p r e d i c t e d was re d u c e d t o 33*96 KPM. TABLE 6 PREDICTED (A) AND ESTIMATED (B) PWC^q B A 1. 1750 1500 2. 800 758 3. 1455 1365 4. 655 645 5. 1925 1650 6. 1400 1380 7. 1080 1073 8. 1443 1328 9. 1340 1320 10. 773 758 11. 623 608 12. 828 795 13. 1808 1778 14. 1233 1230 15. 1087 1080 1 6 . 933 930 17. 820 840 1 8 . 925 865 19. 885 870 20. 948 908 21. 1275 1230 22. 1110 1050 23. 810 810 24. 1785 1450 25. 1275 1230 2 6 . 945 930 27. 1065 1005 2 8 . 1197 1200 29. 1845 1770 30. 1155 1080 120 4 . The v a l u e s o f b o th t h e e s t i m a t e d a n d p r e d i c t e d PWCi^Q w ere w i t h i n + 5 p e r c e n t o f a c a l c u l a t e d mean f o r 27 o f t h e 30 s u b j e c t s . F u r th e r m o r e , f o r t h e s e same 27 s u b j e c t s ; t h e d i f f e r e n c e b e tw e en t h e e s t i m a t e d and p r e d i c t e d PWC^q was e q u a l t o o r l e s s t h a n 10 p e r c e n t . 5 . The sum o f t h e r e s i d u a l s ( l i n e o f b e s t f i t ) f o r t h e p r e d i c t e d was u s u a l l y l e s s t h a n t h e r e s i d u a l o f t h e e s t i m a t e d v a lu e an d i n no c a s e e x c e e d e d t h e sum o f t h e r e s i d u a l s f o r t h e e s t i m ated pwc1 7 0 . Even th o u g h t h e r e i s an o b v io u s m o n o to n ic t r e n d , a t e s t o f s i g n i f i c a n c e was a p p l i e d t o a s c e r t a i n i f a s t a t i s t i c a l l y s i g n i f i c a n t d i f f e r e n c e b e tw e e n t h e two s e t s o f s c o r e s e x i s t e d . A t t e s t f o r c o r r e l a t e d m eans was u s e d t o t e s t t h e e x p e r im e n ta l h y p o t h e s i s . S t a t e d i n t h e n u l l fo rm , M1 = ^ = ^*46 w hich was n o n - s i g n i f i c a n t a t p > 0 .0 1 l e v e l ; t h e n u l l h y p o t h e s i s o f e q u a l i t y was a c c e p t e d . I n c i d e n t a l l y , f o r t h e r e a d e r ' s i n f o r m a t i o n , t h e c o e f f i c i e n t o f c o r r e l a t i o n b e tw e e n t h e tw o s c o r e s was r = O .9 8 . T a b le 7 p o r t r a y s t h e p a r a m e te r e s t i m a t e s f o r e a c h s u b j e c t f o r a l l c o n d i t i o n s . A g a in , s e v e r a l i n t e r e s t i n g p o i n t s w ere o b s e r v e d : 1. One may r e c a l l t h a t i n t h e d e v elo p m e n t o f M odel I I , i t was assum ed t h a t t h e f u n c t i o n o f p a r a m e te r TABLE 7 PARAMETER ESTIMATES FOR ALL WORK LOADS o 1 o o O n cd ! $ C * 4 C M A J - H C O CJvVOVOCM O H A C O O H H C V J -4" VO A H A 4 AVOCOVO VO C O -4--4" C JV V O H t - O V A H t — C JV O VO C M A C M O O O O CO CTvO VO O O V O O CJVCO 0 \ 0 \ H O O O r l O O n O v O H v o H O Ov H CVIO H O H C V J O V H r l O O CJVCJVO - P O O O O O O O C T \0 O O O O t —4 O O O O O O C J v O O O O «••(/}•••••••#••••• • ■ t 1 • f • • t « i • ■ HHCMOJCMCMCMCMCMCMCMHCMCMCMCMCMHHCMCMCMCMCMCMHCMCMCMCM •p d) .a •p N"\OJ ir\ OJ HMD C M CO ltxE:— I>~vo t— t ^ c o O C — -4* - 4 * • • p • • • • • * • « « • • • * * t * * • 1 » t I I # * » -4" A C V J & A t — A V O A 4 4 -4- OV H OvoQ OV OJ OJ - 4 - - 4 A C — A h A A 4 C O VO CTvCTWO H O CJvVO C - A O - 4 H 4 C —V O if lt\Vo - 4 H 4 v o - 4 OJ A O l A r l O OV v p A A (1 ) C M C M 4 C7\COVOVO M) (M i A r l H 4 H O t—H VO CO O CJv (ACM OJOO t — - 4 CO VO -p V O -4 VO A ACJWO 0 0 A V O CO 0 0 1 4 A t — CO t —VO VO L T V VO CO t —t — 4 L T V -P cd +5 O t— O t — S4-4CO A V O 4 A C JW O vo A C M C M C M CO H A VO OV LCV4 A ( A O H LA CO - 4 l a H , A v o A C O OVO O H - 4 CO - 4 A C O CO H t— H A VOVO A O VO O V O A A C M t - H V O H O OVCM ACJVVO H H A c O A H CO O V O A VO 0 - 0 - 4 A H H A CO A O H O C O C M O V O V O O A O v H H A A A O ACM H HCO H - 4 CM CM CO O h c m c m r c m h c m h c m c m c m c m h c m c m c m c m c m a c m c m c m c m H cmcmcmcmhcm • • • l » l l l l l l l l l » l t l l l l l l * l « l l l o S o ^ o M3 ce H A C O H C O V O cm A r i o v o A O 0 4 C Jv t-V O O A H O VO VO O 4 4 4 A H VO t — OVOCOCM O C O O O V O O O O t-O V C JV C —4 C M O O t v - A O H 0 4 CO C M 0 0 t —VO H O V O V O O V O C JV A O O O 0 0 A C J V 0 V O 4 O O V 0 O O O V O O C O H OvE— A O O V O V O O V O C JV A O O O OVOVCJV CJVCO t— O o O v O O VO O O OV o 1 » I 1 1 I • * * • * • •*••• • 1 •• • 1 I • # • 1 I • h h h c m h h c m h c m h h c m c m c m h h h h h h cm cm h cm cm H cMCMH cm 4 H O V O v o 4 C — O V O A O vC M Ov O A V O V O 0 0 C M t —H C M VO A C O H c o v o A V O OVCM A O O O O O VOVO C M A 4 OVAV OHO VA ACMVOOV OVH VO OO t - C M 4 A C M H C M a H a 4 A C O C M O VO C M t— O 4 C M C M CO C M VO A A OVCM CO 4 Ov A H O A C O O O C O H CO A V O O C M VO VO H A O C T v O v H C M t—4 A A CJVCO E-COVO C M A C M H 0 4 A 4 4 4 C 0 H VO 4 A 4 A A A A C M A 4 V 0 4 4 H A A A 4 Q A 0 V O 4 H O O O CMH Q E — C M O OV A C M CJVCO O V 4 H O V O CJWO CJVH A V O 4 A ACM C M t — A ACM 0 4 H 1 4 O CO t — C M CO Vp t—t — O C M A A L A 4 CM® A A 4 VO CMO A t - t —O t—H VO 4 VO 0 4 t- C O 4 C JV O O O O C M O VO a H C M A 4 v o A t - 0 0 A t —v o h u v r l C M t— A t— C M O O A 4 OV t — A V O CO A O 0 4 A C M A A ACM C M C M C M C M A 4 A C M A C M A A A A ACM C M ACM C M A A ACM C M • • • 1 * • • • • • • • » • ••• • | * * * • » * 1 * « 1 • o 8 H A cd VO V0 4 H O V O V C — C - V 0 4 V 0 A C O A A 4 A A V O t —VO 0 0 VOVO CJWO O O O H O 1 4 1 4 A 1 4 0 OV O A O t —[ 4 0 0 V 4 4 A t 4 A 4 O O CJWO VO OV O V H A l 4 O OV t - t - C O O C JV 4 L A 4 V O C M 4 Ov 4 A A V O C — VO 4 4 Ov C M O C JV 4 OVCM C M H CO t-C JV O V O OVOVCJVOVVO t— C S V o v o v o v a w o 0 0 OV OVCJV C JV O v O OVVO OVOVCJV o • **#••••«••• • i*«i • • I i • i*i #i*i r l r l r l r l C \ j H r l H H r l r l r | r l r l H r | H r | H H H H r | O j H *r-|rHH(\J H VO v o CO K M rv O H O ^ H C O CJ\ H - 4 t>-1>- H CO CO C V J t>* O O O t^ONOO l t \ tr- # » • • • • • • • • ■ • • « • • • * • • • • Lf\ * • • * H CO 4 v O A A 1 4 4 t—H O v t— t 4 H H A A OV A I4CM 1 4 A V O 1 4 CJVCO OVCM C M A O V A A t 4 H OVCOOCOCO H CO A O V CM O V 4 A C M A O 0 0 A C JV • ACJVOVOV H A O C —C JV A O C M C M VO 0 4 OvVOVO A O CO C M CO O v t— v p v o O t —ACM A O C M C M 4 VO ACM A A A H C M 4 A A C M 4 H C M A A A A 4 A 4 O V 4 ACM A O O C M A A 4 A A A 1 4 C 0 A O vo A t — VO OV H ACJVCM O V O A O C O t —CM H C JV H I4VO t - v o A O C O C W 4 I 4 A C JV 4 H C M I4 V O 4 O V A C O O H v o C M t—4 c O VO C M O C M A 4 C O H 0 0 4 C O C M H O v O C M A V O I4 V O 4 A H O C M A Q ACQ OO A A A V O H CO C J V O O H C O O O C JV E — t — A A H A V O CO 4 A O 0 4 A 4 C M 4 VO AVOVO A 4 A A t — VOVO A 4 4 A A A V O A L T V ^ -V 0 4 A • A A 4 4 • • • « * • • • • • « • • •••• • • 1 I # • • # j I I I # • »•••*•••••••••*» • * | I f * I • • » 1 • 1 H C M A 4 A V O t- C O OVO H C M A - ^ A V O t - C O OV O H C M A 4 A V O t - 0 0 CJVO H H H H H H H H H H C M C M C M C M C M C M C M C M C M C M A 121 *Did not finish 1 2 2 ( a ) was t o l i n e a r i z e t h e e f f e c t o f s t r e s s an d t o a d j u s t t h e f u n c t i o n o f s t r e s s i n t h e s o l u t i o n o f t h e problem , s e t s . The d a t a p r e s e n t e d i n T a b le 7 a p p e a r t o s u p p o r t t h i s a ssu m p tio n * a n d , i n a d d i t i o n , i t a p p e a r s a s i f p a r a m e te r ( a ) d o es m o d ify t h e f u n c t i o n o f s t r e s s by a l t e r i n g o r a d j u s t i n g t h e p r o p o r t i o n o f (w ’s ) c o n t r i b u t i o n t o y ^ . F o r ex am p le, l e t u s assum e t h a t (a ) i s e q u a l t o .2 2 0 0 0 , .3 0 0 0 0 , and .5 3 0 0 0 f o r 9 0 0 , 6 0 0 , and 300 ■ f " KPM r e s p e c t i v e l y . I f y = a u , t h e r e s u l t s w ould be 1 9 8* 1 8 0, and 159 (sam e o r d e r a s a b o v e ) ; i n a l l p r o b a b i l i t y , t h e s e v a lu e s a r e r e l a t e d t o t h e a s y m p to te . (P a ra m e te r e s t i m a t e d u s e d above a r e a p p r o x im a tio n s o b t a i n e d on d a t a fro m S u b je c t 2 8 .) A c o m p a ris o n b e tw e e n t h e tim e c u r v e s (A ppendix V) and p a r a m e te r ( a ) a p p e a r s t o s u p p o r t t h i s c o n t e n t i o n . 2. The v a r i a t i o n w h ic h o c c u r r e d i n p a r a m e t e r ( a ) i s i n v e r s e t o s t r e s s ; t h a t i s , t h e s m a l l e s t s t r e s s h a d t h e h i g h e s t a b s o l u t e v a lu e s o f ( a ) and t h e g r e a t e s t am ount o f s u b j e c t - t o - s u b j e c t v a r i a t i o n , w h ile t h e l a r g e s t s t r e s s h a d t h e lo w e s t a b s o l u t e v a lu e s o f p a r a m e te r (a ) and t h e l e a s t am ount o f s u b j e c t - t o - s u b j e c t v a r i a t i o n . F o r e x am p le, f o r s t r e s s o f 300 KPM t h e l a r g e s t v a lu e o f p a r a m e te r ( a ) i s a p p r o x im a te ly 1 .0 3 and t h e s m a l l e s t i s 0 .3 8 . Based on th e s e e s t im a t e s , aw would he e q u a l to 309 and 114 r e s p e c t i v e l y w ith a d i f f e r e n c e betw een v a lu e s o f 195. F o r a s t r e s s o f 600 KPM th e l a r g e s t c a l c u l a t e d v a lu e o f p a ra m e te r (a) was a p p r o x i m a te ly 0 . 4 l and th e s m a lle s t was 0 .2 3 . aw, based on t h e s e - e s t i m a t e s , would be e q u a l to 246 and 138 w ith a d i f f e r e n c e o f 108. F o r a s t r e s s o f 900 KPM th e h ig h and low p a ra m e te r (a ) v a lu e s w ere: .30 and . 1 8; aw was e q u a l to 270 and 108—a g a in a d i f f e r e n c e o f 1 0 8. 3. The e f f e c t of th e e x p o n e n tia l p o r t io n o f th e e q u a t i o n i s p ro b a b ly tw o fo ld : ( l ) t o c o r r e c t a n d /o r a d ju s t th e p r e d ic te d v a lu e ; and (2) t o r e f l e c t t h e shape o f th e h e a r t r a t e re s p o n s e c u rv e . 4 . An e x a m in a tio n o f p a ra m e te r (b) and p a ra m e te r (c ) i n d i c a t e s t h a t (c ) i s o f l i t t l e conseq uence i n d e s c r i b in g t h e r i s e tim e o f th e h e a r t r a t e re s p o n s e c u rv e . On th e o t h e r hand, p a ra m e te r (b) a p p e a rs t o have a d i r e c t r e l a t i o n to s t r e s s . The m agni tu d e o f p a ra m e te r (b) i s p ro b a b ly a compromise betw een an a c c u r a te r e f l e c t i o n o f shape and th e a d ju s tm e n t o f th e p r e d i c t e d v a lu e s . I t a p p e a rs t h a t t h e model d ev elo p ed and e v a lu a te d i n t h i s stu d y i s a c o n s i s t e n t and a d e q u a te r e f l e c t i o n of r e a l i t y . The v e r a c i t y o f t h e p ro p o se d model was s u p p o rte d by: 1 . The s m a ll d i f f e r e n c e b etw een o b s e rv e d and p r e d i c t e d h e a r t r a t e v a l u e s . 2. The a b i l i t y o f t h e m odel t o r e f l e c t t h e v a r i a t i o n s w h ic h o c c u r i n h e a r t r a t e r e s p o n s e t o s t r e s s , p a r t i c u l a r l y f o r l i g h t t o m o d e ra te s t r e s s a n d e a r l y tim e p e r i o d s . 3 . The l a c k o f a s t a t i s t i c a l l y s i g n i f i c a n t d i f f e r e n c e b e tw ee n e s t i m a t e d PWC-^q an d p r e d i c t e d v i a t h e m odel. 4 . W ith t h e e x c e p ti o n o f two o r t h r e e t r i a l s , t h e m a g n itu d e .o f t h e e r r o r mean s q u a r e a p p e a r s t o be w i t h i n a c c e p t a b l e l i m i t s . CHAPTER VI SUM M ARY, CONCLUSIONS AND RECOMMENDATIONS Summary The P urpose The p rim a ry p u rp o se o f t h i s s tu d y was t o d e te rm in e i f an i n d i v i d u a l s u b j e c t 's h e a r t r a t e re s p o n se t o s t r e s s o f v a rio u s i n t e n s i t i e s can be s im u la te d by a m a th e m a tic a l a n a lo g . I t was assum ed t h a t a m a th e m a tic a l model o f t h i s f u n c t i o n c o u ld be d e riv e d , and t h e d e riv e d model would a s s i s t i n d e f in in g a fa m ily o f h e a r t r a t e re s p o n se c u rv e s . The s p e c i f i c problem o f th e I n v e s t i g a t i o n was to a s c e r t a i n t h e " q u a lity " o f th e model and to d e te rm in e i f th e model c o u ld be u sed a s a p r e d i c t o r o f p h y s ic a l w orking c a p a c ity . D e r iv a tio n o f th e Model As a r e s u l t o f t h i s s tu d y , two m odels o f t h e h e a r t r a t e re s p o n s e t o e x e r c is e w ere s y n th e s iz e d . Both models w ere b a sed on th e c o n s e r v a tio n o f e n erg y ; one model was d e sig n e d t o use b a s ic d a ta w hich d id n ot r e q u i r e tr a n s f o r m a t i o n o f any ty p e , w h ile th e o t h e r model r e q u ir e d th e s o l u t i o n o f problem s e t s in o r d e r t o d e riv e t h e p a ra m e te r e s tim a te s which w ere u se d i n t h e a n a lo g . B oth m odels were d e riv e d on a TOTE b a s i s ; t h a t i s , th e m odels were i t e r - 125 126 a t i v e l y f o r m u la te d u n t i l such a tim e when t h e o p e r a t i o n s o f t h e model con form ed t o t h e r e a l i t i e s o f th e f u n c t i o n as d e s c r i b e d i n t h e l i t e r a t u r e . B ecau se o f t h i s t e s t - o p e r a t e - t e s t- e _ x it a p p ro a c h , e ac h m odel conform ed to t h e f o llo w in g s t a n d a r d s : yk = 0 when t = 0 y = 0 when w = 0 y = G when t o r to = °o y^ = C o n tin u o u s v a r i a b l e whose f u n c t i o n i s d e p en d e n t on a c o m b in a tio n o f u > and t E v a l u a t i o n o f t h e M odels I n o r d e r to e v a l u a t e t h e p ro p o se d m o d els, d a t a r e l a t i n g t o th e h e a r t r a t e r e s p o n s e o f 30 s u b j e c t s w ere c o l l e c t e d and a n a ly z e d . Each s u b j e c t was r e q u i r e d t o p e rfo r m a PWC170 t e s t ( S j o s t r a n d te c h n iq u e ) on a M onarch b i c y c l e e rg o m e te r. The h e a r t r a t e d a t a w ere re c o r d e d on an E & M P h y s io g ra p h P o u r. I n a s s e s s i n g t h e v e r a c i t y o f t h e a n a lo g s , t h e e x a m in a tio n o f t h e d a t a was l i m i t e d t o : ( l ) t h e r e l a t i o n s h i p b etw een t h e o b s e rv e d h e a r t r a t e r e s p o n s e c u rv e and t h e p r e d i c t e d h e a r t r a t e r e s p o n s e c u rv e ; and (2 ) th e a s se ssm e n t o f t h e s u b j e c t 's b a se d on t h e p r e d i c t e d h e a r t r a t e re s p o n s e c u rv e . F in d in g s As a r e s u l t o f t h i s i n v e s t i g a t i o n , one model was 127 r e j e c t e d ( i t was n o t fo u n d t o be a n a c c u r a t e r e f l e c t i o n o f r e a l i t y ) and one m odel was a c c e p t e d . The fo rm e r was u n a c c e p t a b l e due t o c h a n g e s w h ic h o c c u r r e d i n t h e r a t e c o n s t a n t . I t s h o u ld be n o te d t h a t a s i m i l a r e r r o r was o b s e rv e d i n t h e m odel w h ich was r e p o r t e d by Suggs ( 6 9 ) . As a r e s u l t o f t h i s d e v e lo p m e n t, t h e e m p h asis o f t h e p r e s e n t i n v e s t i g a t i o n was s h i f t e d t o t h e s e c o n d a ry m o del. An a n a l y s i s o f t h i s m odel i n d i c a t e d t h a t t h e r e was m o n o to n ic t r e n d f o r t h e m odel t o u n d e r e s t i m a t e a s u b j e c t 's t e r m i n a l h e a r t r a t e a t t h e c o n c l u s i o n o f an e x e r c i s e p e r i o d . H ow ever, t h i s u n d e r e s t i m a t i o n was a lw a y s e q u a l t o o r l e s s t h a n 5 p e r c e n t . The s p e c i f i c f i n d i n g s p e r t a i n i n g t o t h i s m odel w e re : 1. The m a th e m a tic a l a n a lo g o f t h e s y s te m was fou n d t o be c o n s i s t e n t an d r e l i a b l e i n d i c a t o r s o f t h e b i o - s y s t e m . T h is f i n d i n g was s u p p o r te d on t h e b a s i s o f t h e f o llo w in g : ( a ) The m ag n itu d e o f d i f f e r e n c e w h ic h was n o te d b e tw ee n th e o b s e rv e d and p r e d i c t e d h e a r t r a t e r e s p o n s e v a l u e s , (b ) The a b i l i t y o f t h e m odel t o d u p l i c a t e t h e r a p i d f l u c t u a t i o n s i n h e a r t r a t e r e s p o n s e , ( c ) The m a g n itu d e o f t h e e r r o r mean s q u a r e . 2. T h e re was no s t a t i s t i c a l l y s i g n i f i c a n t d i f f e r e n c e ( t = 0 .4 6 ) b e tw ee n an e s t i m a t e d PW C^q and b a s e d on t h e p r e d i c t e d v a lu e s a s i n d i c a t e d by t h e m odel. 3. A m a th e m a tic a l a n a lo g o f th e h e a r t re s p o n s e to s t r e s s m ust c o n t a i n an i n d i c a t o r r e l a t i n g t o t h e m ag n itu d e o f th e s t r e s s o r . 4 . Model I I was fou n d t o be a v i a b l e i n d i c a t o r o f th e h e a r t r a t e r e s p o n s e o f th e s u b j e c t s u s e d i n t h i s s tu d y . T h is m odel f o llo w s : y* = au ,e-bA C C o n e lu sio n s H e a rt r a t e re s p o n s e t o s t r e s s may be d e s c r ib e d and p r e d i c t e d w i t h i n an a c c e p ta b le m argin o f e r r o r ( l e s s th a n 5 p e r c e n t u n d e r e s tim a tio n ) by means o f a s im u la te d m ath e m a t i c a l m odel. Recom m endations As a r e s u l t o f t h i s s tu d y and s t u d i e s r e p o r t e d by p re v io u s i n v e s t i g a t o r s , a d d i t i o n a l s t u d i e s p e r t a i n i n g to th e developm ent o f m a th e m a tic a l a n a lo g s In g e n e r a l , and h e a r t r a t e r e s p o n s e m odels s p e c i f i c a l l y , a r e w a r r a n te d . S e v e ra l q u e s tio n s s p e c i f i c a l l y r e l a t e d to h e a r t r a t e r e sp o n se m odels a r e : 1. Can a m odel be f o r m u la te d w hich would r e s u l t i n an I n d i c a t i o n o f an I n d i v i d u a l 's c a p a c i t y to p e rfo rm work? 2. The developm ent o f Model I I sh o u ld be c o n tin u e d i n o r d e r to im prove t h e m o d e l's c a p a b i l i t y f o r __________r e f l e c t i n g r e a l i t y . _____________________________________ 129 3 . M odel I s h o u ld he r e e v a l u a t e d a s a t o o l f o r p r e d i c t i n g when a n i n d i v i d u a l h a s r e a c h e d a s t e a d y s t a t e p r o c e s s . The c o n c e p t o f m o d e lin g s h o u ld he e x te n d e d t o o t h e r p h y s i o l o g i c a l p a r a m e te r s w h ich a r e o f I n t e r e s t t o p h y s i c a l e d u c a t o r s . F o r e x a m p le , a m odel o f s t r e n g t h g a i n , e n d u r a n c e c a p a c i t y , a c c l i m i t i z a t i o n , r e s p i r a t i o n , an d b i o m e c h a n ic s o f movem ents a r e b u t a few o f t h e many p ro b le m s f o r w hich t h i s p r o c e d u r e may be u s e f u l . 130 BIBLIOGRAPHY Books 1. A s tr a n d , P e r - O lo f , and R o d a h l, K a a r e . T e x tb o o k o f Work P h y s io lo g y . New Y ork: M cG raw -H ill, 1970* 2. B ro u h a , L u c ie n . P h y s io lo g y I n I n d u s t r y . O x fo rd : Pergam on P r e s s , I9 6 0 . 3 . Chapman, C a r l e t o n , B ., e d . P h y s io lo g y o f M u s c u la r E x e r c i s e . A m erican H e a r t A s s o c i a t i o n M onograph Number F i f t e e n . New Y ork: The A m erican H e a r t A s s ’n , , 1 9 6 7 . 4 . C o n s o la z io , F ra n k C .j J o h n s o n , R o b e rt E .j and P e c o ra , L o u is J . P h y s i o l o g i c a l M easu rem en ts o f M e ta b o lic F u n c t io n s i n Man. New Y ork: M cG raw -H ill, 1963. 5 . d e V r ie s , H e r b e r t A. P h y s io lo g y o f E x e r c is e f o r P h y s i c a l E d u c a tio n an d A t h l e t i c s . D ubuque, Iow a: Wm. C. Brown, 1966. 6 . D ixon , W. J . , ed. BMP, B io m e d ic a l C om puter P ro g ra m s . Los A n g e le s : U n i v e r s i t y o f C a l i f o r n i a , J a n u a r y , 1964. 7 . E dw ards, A l l e n L. S t a t i s t i c a l M e th o d s. New Y ork: H o l t , R i n e h a r t an d W in sto n , I n c . , 1967- 8 . F e rg u s o n , G eorge A. S t a t i s t i c a l A n a ly s is i n P sy c h o lo g y and E d u c a ti o n . New Y o rk : M cG raw -H ill, 19S 6 . 9 . G r o d in s , F r e d S. C o n tr o l T h eo ry and B i o l o g i c a l S y s tem s . New Y ork: C olum bia U n i v e r s i t y P r e s s , 1963. 10. ' H a m ilto n , W. F . , s e c . e d . Handbook o f P h y s i o lo g y . V o l. I . W ash in g to n , D .C .: A m erican P h y s i o l o g i c a l S o c i e t y , 1 9 6 7 . 11. H a m ilto n , W. F . , s e c . e d . Handbook o f P h y s i o lo g y . V o l. I I I . W ash in g to n , D .C .: A m erican P h y s i o l o g i c a l S o c i e t y , 1 9 6 7 . 131 12. 13. 14. 15. 16. 17. 18. 19. 2 0 . 2 1. 2 2. 23. 24. 25. 132 Jo h n so n , W arren R ., ed. S c ie n c e and M edicine o f Ex e r c i s e and S p o r t s . New Y ork: H arp er & B r o s ., i 9 6 0 . Kalmus, H ., ed. R e g u la tio n and C o n tro l In L iv in g S y stem s. New York: Jo h n W iley and Sons, I n c . , 1 9 6 6. K arp o v ich , P e te r V. P h y sio lo g y o f M uscular A c t i v i t y . P h il a d e l p h ia : W. B. S a n d e rs, 1959. K arvonen, M a r ttI J . , and B a rry , A lan J . , e d s. P h y s i c a l A c t i v i t y a n d 'th e H e a r t . S p r i n g f i e l d , 1 1 1 .: C h a rle s C. Thomas, 19&7. K e r lin g e r , F re d N. F o u n d a tio n s o f B e h a v io ra l R e s e a rc h . New Y ork: H o lt, R in e h a rt”~and" W inston, I n c . , i'9 6 6. Lew is, Don. Q u a n t it a t i v e Methods i n P sy ch o lo g y . New Y ork: M cG raw -H ill, i 9 6 0 . M a rtin , F r a n c is F. Computer M odeling and S im u la tio n . New Y ork: John W iley and Sons, I n c . , I 96B. M e sa ro v ic , M. D ., ed. System s T heory and B io lo g y . New Y ork: S p rln g e r-V e rla g , I n c . , 196b. M orehouse, L aurence E ., and M i l l e r , A ugustus T. P h y sio lo g y o f E x e r c is e . S t. L o u is: C. V. Mosby, N ie ls e n , Kaj L. Methods i n N um erical A n a ly s is . New Y ork: M acm illan, 1956. R ig g s, D ouglas S. The M a th e m a tic a l A pproach to Phys i o l o g i c a l P ro b le m s. B a ltim o re : W illiam s and W ilk in s, I9S3'. S h a p iro , G eorge, and R o g ers, M ilto n . P ro s p e c ts f o r S im u la tio n and S im u la tio n s o f Dynamic S y stem s. New Y ork: S p a rta n Books, 1 9 6 7. Sm ith, J . M aynard. M a th e m a tic a l Id e a s i n B io lo g y . C am bridge: Cambridge U n i v e r s i ty P r e s s , 1 9 6 8. W ilde, D ouglas J . , and B e i g h t l e r , C h a rle s S. Founda t i o n s o f O p tim iz a tio n . Englewood C l i f f s , N . J . : P r e n t i c e - H a l l , I n c . , 1967. 133 P e r i o d i c a l s 26. A lderm anj R ic h a rd B. " I n d i v i d u a l D i f f e r e n c e s i n H e a rt R a te R esponse t o B ic y c le E rg o m e te r W ork." R e se a rc h Q u a r t e r l y , XXXVIII ( 1 9 6 7 ), 3 2 3 -3 2 9 . 27. A ndrew s, R o b e rt B. " I n d ic e s o f H e a rt R a te a s S u b s t i t u t e s f o r R e s p i r a t o r y C a lo r im e tr y ." A m erican I n - d u s t r i a l H ygiene A s s o c ia t io n J o u r n a l , XXVII (1 9 6 6 ), 5 2 6 -5 3 2 . 28. A ndrew s, R o b e rt B. " E s tim a tio n o f V a lu e s o f E nergy E x p e n d itu r e R a te from O bserved V a lu e s o f H e a rt R a te ." Human F a c t o r s , IX ( 1 9 6 7 ), 5 8 1 - 5 8 6 . 2 9 . A n te l, J a c k , and Cumming, Gordon R. " E f f e c t o f Emo t i o n a l S t i m u la ti o n on E x e r c is e H e a rt R a te ," Re s e a r c h Q u a r t e r l y , XXXX ( 1 9 6 9 )* 6 -1 0 . 30. A sm ussen, R ., and Hemmingsen, I . " D e te rm in a tio n o f Maximum W orking C a p a c ity a t D i f f e r e n t Ages I n Work w ith t h e Legs o r w ith t h e A rm s." S c a n d in a v ia n J o u r n a l o f C l i n i c a l and L a b o r a to r y I n v e s t i g a t i o n , X (1 9 5 6 ), 6 7 -7 1 . 31. A sm ussen, E r li n g , and N i e ls e n , M a riu s . " C a rd ia c O u t p u t D u rin g M u sc u la r Work and I t s R e g u l a t i o n s ." P h y s i o l o g i c a l R ev iew s, XXXV (1955)* 7 7 8 -8 0 0 . 32. A s tra n d , P. 0 . "Human P h y s i c a l F i t n e s s w ith S p e c ia l R e fe re n c e t o Sex and A ge." P h y s i o l o g i c a l R eview s, xxxvi (1956), 307-329. 33. A s tr a n d , P. 0 . , and Ryhming, Irm a . "A Nomogram f o r C a l c u l a t i o n o f A e ro b ic C a p a c ity ( P h y s ic a l F i t n e s s ) from P u ls e R a te D u rin g Sub-m axim al W ork." J o u r n a l o f A p p lie d P h y s io lo g y , V II (1 9 5 4 ), 2 18-225. 34. A t t i n g e r , F. 0 . , and Anne', A n th a rv e d i. " S im u la tio n o f t h e C a r d io v a s c u la r S y stem ." A nn als o f t h e New Y ork Academy o f S c i e n c e s , CXXVIII ( I 966')',' 8 1 0 -8 2 9 . 35. B a la n e s c u , F I . ; V o ic u le s c u , A .; and B obocea, A. "The C a r d io - V a s c u la r R espo nse D uring E x e r c is e i n A th l e t e s . " I n t e r n a t i o n a l Z e i t s c h r i f t e Agnew. P h y s io - l o g i e , XXV (1 9 6 8 ), 3 6 1 -3 7 2 . 3 6 . B e a v e r, W illia m L ., and W asserm an, K arlm an. " T ra n s i e n t s i n V e n t i l a t i o n a t t h e S t a r t and End o f E x e r c i s e . " J o u r n a l o f A p p lie d P h y s io lo g y , XXV ( 1 9 6 8 ), 3 9 0 -3 9 9 . 134 37. B erg g ren , G ., and. C h r is te n s e n , E. H. "H eart R ate and Body T em p eratu re as I n d ic e s o f M e ta b o lic R a te Dur in g Work." A r b e its p h y s io lo g ie , V III ( 1 9 6 1), 385- 399. 3 8 . B rodan, V ., and Kuhn, E. "A C o n tr ib u tio n t o th e A naly s i s o f th e P u lse R ate D uring a Load and R eco v ery ." P h y s io lo g ic a B ohem oslovaca, XV ( 1 9 6 6), 6 8 -7 5 . 39. B rouha, L ., e t a l . " P h y s io lo g ic a l R e a c tio n s o f Men and Women D uring M uscu lar A c t i v i t y and R ecovery i n V a rio u s E n v iro n m e n ts." J o u r n a l o f A p p lied P h y s io l ogy, XVI (1 9 6 1 ), 133-140. 40. Brouha, L ., e t a l . "D isc re p a n c y Between H e art R ate and Oxygen Consum ption D uring Work i n th e Warmth." J o u r n a l o f A p p lied P h y sio lo g y , XVIII (1963)> 1095- _ 8 7 41. C ardus, D av is, and V e ig le r , R oyal K. "H eart F re q u en c i e s C urv es. A M a th em atica l M odel." Com puters and B io m ed ical R e se a rc h , I ( 1 9 6 8), 508-526. 42. C ly n es, M anfred, "Use o f Computers f o r P h y s io lo g ic a l D isc o v e ry and f o r D ia g n o sis by Dynamic S im u la tio n ." C i r c u l a t i o n R e s e a rc h , I I ( 1 9 6 2), 515-528. 43. C ra ig , F. N ., and Cummings, E. G. "Slow ing o f th e H e a rt a t th e B e g in n in g o f E x e r c is e ." J o u r n a l o f A p p lied P h y s io lo g y , XVIII (1963)* 353-356. 4 4 . D av ies, C. T. M. " L im ita tio n s t o th e P r e d i c t io n of Maximum Oxygen I n ta k e from C a rd ia c F req u en cy Meas u re m e n ts ." J o u r n a l o f A p p lied P h y sio lo g y , XXIV ( 1 9 6 8), 700-7067 45. D a v ies, C. T. M. "C ard iac F req u en cy i n R e la tio n to A ero bic C a p a c ity f o r W ork." E rgonom ics, XI ( 1 9 6 8), 511- 5 2 6 . 46. D eFares, J . G. "Examples o f C a r d io v a s c u la r M o d els." J o u r n a l o f C hronic D is e a s e s , XIX ( 1 9 6 6), 4 01-410. 47. D eF ares, J. G.j O sborn, J. J . j and H ara, H ir o s h i H. " T h e o r e tic a l S y n th e s is o f th e C a r d io v a s c u la r S ys te m ." A cta P h y s io lo g ic a e t P h arm aco lo g ica N e e r- l a n d l c a , X II (1963)* 189-2657 4 8 . d e V rie s , H e rb e rt A ., and K la fs , Carl. E. " P r e d i c t io n o f Maximal Oxygen I n ta k e from Submaximal T e s t s ." J o u r n a l o f S p o rts M ed icin e, V ( 1 9 6 5)* 207-214. 135 4 9 . D ixon, W. J . "A nnual R e p o rt, H e a lth S c ie n c e s Comput in g F a c i l i t y . " U n i v e r s i t y o f C a l i f o r n i a , Los A n g e le s, 1958. 5 0 . G ro d ln s , F re d S. " G e n e ra l A p p l i c a M l i t y o f C o n tr o l T h eo ry t o B i o l o g i c a l S ystem s w ith Some E x e m p lif ic a t i o n i n t h e R e s p i r a t o r y S y stem ." F e d e r a t io n P r o c e e d in g s , XXVIII ( 1 9 6 9), 4 7 - 5 1 . 5 1. H e lls tr o m , B ., and H olm gren, A. "On th e R e p e a t a b i l i t y o f Subm axim al Work T e s ts and th e I n f l u e n c e o f Body P o s i t i o n on H e a rt R a te D uring E x e r c is e a t Subm axi m al Work L o a d s." S c a n d in a v ia n J o u r n a l o f C l i n i c a l and L a b o r a to r y I n v e s t i g a t i o n , X V III (1 9 5 5 ), 4 7 9 - 455^ 5 2 . H enry, F . M. "A e ro b ic Oxygen C onsum ption and A l a c t i c Debt i n M u sc u la r W ork." J o u r n a l o f A p p lie d P h y s i o lo g y , I I I (1 9 5 1 ), 427-43B5 53* J o n e s , Norman L. " E x e r c is e T e s t i n g ." B r i t i s h J o u r n a l o f D is e a s e s o f t h e C h e s t, LXI ( 1 9 6 7) ~ , 1 5 9 -1 8 9 . 5 4 . K a n to r, Seym our; Ib ra h im , M ic h e l A .; and W in k e ls te in , W arren, J r . "How t o A void Drowning i n y o u r D a ta : L e a rn in g How t o Swim." A m erican J o u r n a l o f P u b lic H e a l t h , LVII ( 1 9 6 7 ), 1 9 7 3 -1 9 7 8 . 55- K a tz , L o u is N. " A n a ly s is o f t h e S e v e r a l F a c to r s Regu l a t i n g t h e P e rfo rm a n c e o f t h e H e a r t ." P h y s i o l o g i - c a l R ev iew s, XXXV ( 1 9 5 5 ) > 9 1 -1 0 6 . 56. K u rc z , R o b e rt L ,; F ox, Edward L . ; and M atth ew s, D onald K. " C o n s tr u c tio n o f a Submaximal C a r d io v a s c u la r S te p T e s t . " R e s e a rc h Q u a r t e r l y , XXXX (1969), 115- 1 2 2. 57. M a lh o tr a , M. S . ; G upta, J . S e n .; and R a l, R. M. " P u ls e Count a s a M easure o f E n erg y E x p e n d itu r e ." J o u r n a l o f A p p lie d P h y s io lo g y , X V III ( 1 9 6 3 )^ 9 9 4 - 5 8 . M a r itz , J . S ., e t a l . "A P r a c t i c a l M ethod o f E s t im a t in g an I n d i v i d u a l 1s M aximal Oxygen I n t a k e . " E rg o n o m ics, IV ( 1 9 6 1 ), 9 7 -1 2 2 . 59. M a s te r, A r th u r M ., and O ppenheim er, E nid T r ib e . "A S im ple T o le ra n c e T e s t f o r C i r c u l a t o r y E f f i c i e n c y w ith S ta n d a rd T a b le s f o r N orm al I n d i v i d u a l s . " The A m erican J o u r n a l o f t h e M e d ic a l S c ie n c e s , CLXXVII ( 1 9 2 1 ), 2 2 3 -2 4 2 . 136 60. M a x fle ld , Mary E . , and B rouha, L u clen . " V a li d i t y o f H e a rt R ate as an I n d i c a t o r o f C ard iac S t r a i n . " J o u r n a l o f A p p lied P h y s io lo g y , XVIII ( 1 9 6 3 )* 1099- 1104. 6 1. M cArdle, W illiam D. ,* Z w iren, L in d a; and M agel, John R. " V a lid ity o f th e P o s te x e r c is e H eart R ate as a Means o f E s tim a tin g H e a rt R ate L u rin g Work o f V a r i ous I n t e n s i t i e s . " R e se a rc h Q u a r te r ly , XXXX ( 1 9 6 9 )* 523-528. 62. Nooney, Grove C. "M ath em atical M odels, R e a l i t y and R e s u l t s ." J o u r n a l o f T h e o r e ti c a l B io lo g y , IX ( 1965)* 239-252. 6 3 . Penman, K enneth A. "A 'F r o n t End' C a rd io v a s c u la r E f f i c i e n c y T e s t." P ro c e e d in g s o f th e A nnual M eeting o f th e N a tio n a l C o lle g e P h y s ic a l E d u c a tio n A s s o c ia t i o n f o r Men, 1966 j pp. 109-114. 64. P ic k e r in g , W . D .; N ik if o r u k , P. N .j and M errim an, J . E. "Analogue Computer Model of th e Human C a rd io v a sc u l a r C o n tro l S ystem ." M ed ical and B i o lo g i c a l E n g i n e e r in g , V II (1969)> 401-41 0. 6 5 . Rushmer, R o b ert F. " P h y s io lo g ic a l A p p lic a tio n o f C om puters." C i r c u l a t i o n R e se a rc h , I I ( 1 9 6 2), 529 - 533. 66. Rushmer, R ob ert F . , and Sm ith, O r v i l le A ., J r . "C ar d ia c C o n tr o l." P h y s io lo g ic a l Review, XXXIX (1959)> 4 1 -6 8 . 6 7 . S c h lip p , R o b e rt W . "A M a th e m a tic a l D e s c r ip tio n o f th e H e art R ate Curve o f Response to E x e r c is e , w ith Some O b s e rv a tio n on t h e E f f e c t s o f Sm oking." Re s e a rc h Q u a r te r l y , XXII (1 9 5 1 ), 439-445. 68. S hephard, R. J . " P r a c t i c a l I n d ic e s o f M e ta b o lic Ac t i v i t y . " I n t e r n a t i o n a l Z e i t s c h r i f t e Agnew. P h y s i- o l o g i e , XXV ( 1 9 6 8), 1 3-24. 6 9 . Suggs, C h a rle s W . "An A n a ly s is of H e art R ate R esponse t o S t r e s s . " R e se a rc h Q u a r te r ly , XXXIX ( 1 9 6 8), 195- 205. 70. Van L in g re n , B .j Seaw ard, P. D .; and O dendaal, W . A. "Work Speed a s a M easure o f an E q u iv a le n t E x e rc is e S t r e s s i n S u b je c ts o f D i f f e r e n t W e ig h ts." C i r c u l a t i o n , x x x i i ( 1 9 6 5 ), 940-947. 137 7 1 . W ahlund, H o lg e r. " D e te r m in a tio n o f t h e P h y s i c a l Work in g C a p a c it y ." A cta M edlca S c a n d in a v ic a * S u p p l. 215 (1 9 4 8 ), 5-1057 138 APPENDIX I TREND ANALYSIS— TIME (G R O W TH 139 TREND ANALYSIS—TIME GROW TH L in e a r Q u a d ra tic Cubic msDi F F MSD1 F 300 KPM - 5f-gjp 1 .5 6 .95 = 1 .6 4 * 3 8 9 .7 700 *56 .95 1 5 6 .2 5 100 1 .5 6 .95 “ 1 .64** 600 KPM 5° i q Q^ 5 .0 2 4 .1 6 = 1 .2 1 * 1 4 .4 4 700 . 02 ^ ~ \ A t - " C I 5 - = K1 2 2 .0 9 100 .2 2 4 .1 6 ~ < 1* 900 KPM 7 2 9 3 .1 6 7 2 .9 3 1 6 .6 4 = 3 .9 1 * 5 6 .2 5 700 .0 8 " <:L 8 5 1 9 .2 9 100 8 5 .1 9 _ 1 8 .6 4 4 .5 7 * * * N o n - s ig n i f l e a n t • * * S i g n i f i c a n t a t p _> .05 l e v e l . O APPENDIX I I DERIVATION OF MODEL I I l 4 l DERIVATION OF MODEL II 142 The s e c o n d a ry m odel w h ic h was s y n t h e s i z e d a s a r e s u l t o f t h i s s tu d y was b a s e d on t h e f u n c t i o n a l r e l a t i o n s h i p w hich e x i s t s b e tw ee n h e a r t r a t e and work lo a d . I t s h o u ld be d u ly n o te d , t h i s r e l a t i o n s h i p e x i s t s , i f , and o n ly i f , t h e sy ste m i s i n a s t e a d y s t a t e o r h o m e o s ta s is c o n d i t i o n . The p a r t i c u l a r m odel w hich w i l l be d e s c r i b e d s h o r t l y , i s a r e s u l t o f a c o m p i l a t io n o f s e v e r a l m odels w h ich w ere d e r i v e d and t h e n a n a ly z e d v i a c o m p u te r. The c u r r e n t s t a t e o f t h e r e s u l t a n t m odel i s due t o t h e i t e r a t i v e ( t r i a l and e r r o r ) p r o c e s s w h ic h was a p p l i e d t o p r i o r c o n c e p t u a l i z a t i o n s . I t was t h e p u rp o s e o f t h i s p a r t i c u l a r m odel t o e x pand t h e f u n c t i o n a l r e l a t i o n s h i p , h e a r t r a t e e q u a l s work lo a d (Y = X ); t h i s r e l a t i o n s h i p was r a m i f i e d t o i n c l u d e o r t o a c c o u n t f o r v a r i a t i o n s i n t h e tim e f a c t o r . The i n c l u s io n o f t h e tim e v a r i a b l e i n t o t h e f u n c t i o n a l r e l a t i o n s h i p (Y = X) w ould t h e o r e t i c a l l y e n a b le one t o p r e d i c t p e ak h e a r t r a t e o r more p r e c i s e l y s t e a d y s t a t e h e a r t r a t e from m ea su re m e n ts o f a r e l a t i v e l y s h o r t tim e p e r i o d . The e x t e n s i o n o f t h i s a s s e s s m e n t t o v a r i o u s s t a t e s o r l e v e l s o f s t r e s s w ould r e s u l t i n a p h y s i c a l w o rk in g c a p a c i t y t e s t w h ich c o u ld be b a s e d on a t r u l y su b -m a x im al e f f o r t ; t h a t i s , a t e s t o f s h o r t d u r a t i o n and m in im al s t r e s s . F o r t h i s p a r t i c u l a r m o d el, a s i n t h e p r e v i o u s l y 143 d e s c r ib e d model., i t was assum ed t h a t t h e h e a r t r a t e r e s p o n s e c u rv e was c a p a b le o f b e in g d e s c r ib e d m ost a c c u r a t e l y v i a a lo g a r ith m ic f u n c t i o n . As i n t h e p re v io u s m odel, th e expo n e n t e -x was s e l e c t e d as t h e b a s e . The r a t i o n a l e f o r t h i s s e l e c t i o n was p r e s e n t e d i n d e s c r i b i n g th e p r e v io u s m odel and w i l l n o t be r e p e a te d i n t h i s s e c t i o n . However, t h e r e i s a m ajo r d i s t i n c t i o n betw een a p rim a ry a ssu m p tio n f o r each o f th e m odels. One may r e c a l l , t h a t i n t h e p r e v io u s c+t model y = (B-A) . e t h i s e q u a tio n e n a b le d th e s te a d y s t a t e h e a r t r a t e t o be l e s s th a n u n i t y and th e maximum h e a r t r a t e t o be e q u a l t o u n i t y . I n t h i s p a r t i c u l a r model t h a t a ssu m p tio n h a s b een n e g a te d . T h is model assum es t h a t u n i t y i s e q u a l t o t h e s te a d y s t a t e c o n d it i o n ; b e c a u se o f t h i s a ssu m p tio n t h e m odel may be s i m p l i f i e d by rem oving from t h e e q u a tio n v a lu e s f o r maximum h e a r t r a t e and v a lu e s f o r t h e s u b j e c t 's i n i t i a l h e a r t r a t e . I t s h o u ld be n o te d t h a t t h e i n v e s t i g a t o r r e c o g n iz e s t h a t t h i s a ssu m p tio n may be u n te n a b le i f a s t r e s s o f lo n g d u r a t i o n o r ex trem e i n t e n s i t y a r e b e in g i n v e s t i g a t e d . D e p le tio n o f e n e rg y s t o r e s o r a s h i f t i n e n e rg y s u p p ly may r e s u l t i n v o id in g th e p r e v i o u s l y s t a t e d a ss u m p tio n . However, s in c e th e fo c u s o f t h i s s tu d y was on work lo a d s o f s h o r t d u r a t i o n and l i g h t t o m o d era te i n t e n s i t y , th e a ssu m p tio n was deemed t e n a b l e . The m odel d e r iv e d i s a s f o llo w s : t b /\j ^ 1. y = e~ ' The e q u a tio n s t a t e s , h e a r t r a t e a t 144 a g iv e n tim e i s e q u a l t o t h e b a s e o f a n a t u r a l lo g m inus a f r a c t i o n * t h e n u m e ra to r o f t h e f r a c t i o n i s a c o n s ta n t* t h e d e n o m in a to r i s tim e t o t h e pow er o f a c o n s t a n t . U n d o u b t edly* t h e r e a d e r h a s o b s e rv e d t h e s i m i l i a r i t y b e tw e e n t h i s e q u a tio n * t h e p r e v i o u s m odel a n d t h e m o d els r e p o r t e d by C ard u s a n d Z e i g l e r and by S u g g s. The p rim e d i s t i n c t i o n b e tw e e n t h i s m odel and t h o s e r e p o r t e d I n t h e l i t e r a t u r e was t h e a d d i t i o n o f t h e n a t u r a l lo g a rith m * e ; t h e m a jo r d i f f e r e n c e b e tw e e n t h i s m odel and t h e one p ro p o s e d i n C h a p te r I I I o f t h i s p a p e r i s t h a t t h e e x p o n e n t i a l f a c t o r now c o n s i s t s o f two c o n s t a n t s and one v a r i a b l e w h e re a s i n t h e p r i o r m o d el, t h e e x p o n e n t i a l f a c t o r c o n s i s t e d o f one c o n s t a n t and two v a r i a b l e s . The p u rp o s e o f t h e e x p o n e n t i a l f a c t o r i s t o d e s c r i b e t h e r i s e tim e o f t h e h e a r t r a t e r e s p o n s e c u r v e . The e x p r e s s io n o f t h e fo rm u la d o e s n o t c o n t a i n a n i n d i c a t i o n of s t r e s s ; one may assu m e, a s d i d Suggs* t h a t t h e s t r e s s f a c t o r w ould be a m o d i f i e r o f r i s e t i m e — and t h i s i s t r u e . However* one may r e c a l l * t h a t t h e m odel p r o p o s e d by Suggs d i d n o t i n d i c a t e o r a c c o u n t f o r t h e a s y m p to te o r s t e a d y s t a t e . I n a n e f f o r t t o p r e d i c t t h i s b en d : 2 . j0 , = u ) • e “ b/ l: The a d d i t i o n o f s t r e s s i n t h e e q u a t i o n r e s u l t s i n a more a c c u r a t e r e f l e c t i o n o f p r e c i s e l y when t h e c u rv e w i l l bend t o a s t e a d y s t a t e c o n d i t i o n ; th u s o v e rc o m in g t h e d i f f i c u l t y e n c o u n te r e d b y S u g g s. The m odel now s t a t e s t h a t y^ i s a f u n c t i o n o f s t r e s s * tim e s t h e b a s e o f a n a t u r a l l o g , m inus an e x p o n e n t. The m odel a s now 145 sta te d ., assum es t h a t th e f u n c t i o n o f s t r e s s i s e q u a l to s t r e s s . One may r e c a l l t h a t C ardus and Z e i g le r s t a t e d t h a t t h i s a ssu m p tio n i n t h e c a se o f t h e i r p a r t i c u l a r model was s u s p e c tj t h e r e f o r e ,' th e p ro p o se d model w i l l undergo one a d d i t i o n a l change. 3. yk = a • w . e T h e f i n a l v e r s io n o f t h i s t model s t a t e s : y e q u a ls a , w hich i s a c o n s t a n t , tim e s s t r e s s , tim e s th e b a se of a n a t u r a l lo g minus an exponent w hich r e f l e c t s th e r i s e tim e o f th e h e a r t r a t e re s p o n se c u rv e . The s p e c i f i c f u n c tio n o f th e f i r s t c o n s ta n t i s to m odify th e e f f e c t o f s t r e s s i n t h e fo rm u la . I n t h i s p a r t i c u l a r c a s e , th e e f f e c t o f a i s th o u g h t t o l i n e a r i z e th e f u n c t i o n of s t r e s s . The r e s u l t a n t i s th e n m u l t i p l i e d by t h a t p o r t i o n o f th e fo rm u la w hich r e f l e c t s t h e h e a r t r a t e re s p o n s e c u rv e . To b r i e f l y r e i t e r a t e , an i n d i v i d u a l 's h e a r t r a t e a t a s p e c i f i c tim e i s a f u n c t i o n o f s t r e s s w hich i s m o d ified by th e a d d it i o n o f a c o n s t a n t , tim e s th e b a se o f a n a t u r a l lo g , w hich p ro v id e s th e c u rv e w ith an a lm o st in s ta n ta n e o u s r i s e tim e w hich i s fo llo w e d by an a sy m p to te , and i s modi f i e d by an exponent w hich r e f l e c t s th e m agnitude o f th e r i s e tim e f o r any g iv e n work lo a d . A gain, s t a t e d i n sym b o l s —yk = awe-13/*' . APPENDIX I I I ACTIVITY CLASSIFICATION INDEX 146 147 ACTIVITY CLASSIFICATION INDEX 1. Sedentary 2. Active 1-2 per month 3. Active 1-2 per week 4. Training once per week 5. Competitive training APPENDIX IV PHYSICAL WORKING CAPACITY ESTIMATED AND PREDICTED 148 H E A R T R A T E 190 S u b je c t: 1-M.H. 175 160 145 130 115 100 (x 100) 6 18 3 9 12 15 1750 KPM P r e d i c t e d 1500 KPM E s tim a te d Work i n k ilo p o u n d m e te rs F ig . 1 2 . 1 . — P h y s i c a l w o rk in g c a p a c i t y e s tim a te d and p r e d i c t e d -t= - y o 190' 175 i 6o 145 130 115 100 85 70 2 — f i n . S u b je c t: 2 - J .F __________________________ i_________________________________i________________________________ l___________________________________l_________________________________ ' ■ 3 6 9 12 15 18 (x 100) KPM P r e d i c t e d Work l n k l l °P °und m e te rs KPM E s tim a te d P ig . 1 2 . 2 . — P h y s i c a l w o rk in g c a p a c i t y e s t im a t e d and p r e d i c t e d 150 190 S u b je c t: 3-B .A . 175 160 145 130' 115 100 8 5 - 18 (x 100) 6 15 9 12 3 _1 4 5 5 KPM P r e d i c t e d Work l n ^ ilo p o u n d m e te rs - 1365 KPM E s tim a te d F ig . 1 2 . 3 . — P h y s i c a l w o rking c a p a c i t y e s tim a te d and p r e d i c t e d 151 190 S u b je c t: 4-R.M 175 160 145 130 115 100 12 Work i n k ilo p o u n d m e te rs 655 KPM P r e d i c t e d 645 KPM E s tim a te d P ig . 1 2 . 4 . — P h y s i c a l w o rk in g c a p a c i t y e s t im a t e d and p r e d i c t e d 152 190 S u b je c t: 5-S.M . 175 160 145 130 115 100 9 12 Work i n k ilo p o u n d m e te rs - 1925 KPM P r e d i c t e d — 1650 KPM E s tim a te d F ig . 1 2 . 5 . — P h y s i c a l w o rk in g c a p a c i t y e s t im a t e d and p r e d i c t e d 153 H E A R T R A T E 190 S u b j e c t : 6 -T .F 175 160 145 130 115 100 85 70 6 18 (x 100) 9 3 15 12 1400 KPM P r e d i c t e d Work i n k ilo p o u n d m e te r s 1380 KPM E s tim a te d F i g . 1 2 . 6 . — P h y s i c a l w o rk in g c a p a c i t y e s t im a t e d an d p r e d i c t e d H ui 4 = ^ H E A R T R A T E 1 9 0 S u b j e c t : 7 -A .T . 175 160 145 130 115 100 18 (x 100) 6 3 15 9 12 - 1080 KPM P r e d i c t e d Work l n ^ ilo p o u n d m e te rs — 1073 KPM E s tim a te d Rig, 1 2, 7 .— Physical working capacity estimated and predicted 155 190 S u b j e c t : 8 -R .G . 175 160 145 130 115 100 85 70 18 (x 100) 6 15 12 3 9 -1443 KPM P r e d i c t e d Work l n k H o p o u n d m e te r s - 1328 KPM E stim a te d . P i g . 1 2 . 8 . — P h y s i c a l w o rk in g c a p a c i t y e s tim a te d an d p r e d i c t e d m C T \ 190 S u b je c t: 9 -T .G . 175 160 145 130 115 100 9 12 Work i n k ilo p o u n d m e te rs — 1340 KPM P r e d i c t e d — 1320 KPM E s tim a te d F ig . 1 2 . 9 . — P h y s i c a l w o rk in g c a p a c i t y e s t im a t e d and p r e d i c t e d 157 190 S u b j e c t : 10-M .C. 175 160 145 130 115 100 12 Work i n k ilo p o u n d m e te rs - 773 KPM P r e d i c t e d — 758 KPM E s tim a te d P ig . 1 2 .1 0 .— P h y s i c a l w o rk in g c a p a c i t y e s t im a t e d an d p r e d i c t e d 00 190 S u b je c t: 1 1 -T .H . 175 160 145 130 115 100 18 (x 100) 9 12 Work I n k ilo p o u n d . m e te rs — 623 KPM P r e d i c t e d — 608 KPM E s tim a te d F i g . 1 2 .1 1 .— P h y s i c a l w o rk in g c a p a c i t y e s tim a te d an d p r e d i c t e d 159 190 S u b je c t: 12-R .R 175 160 130 115 100 85 70 18 (x 100 6 15 12 3 9 - 828 KPM P r e d i c t e d Work i n k ilo p o u n d — 795 KPM E s tim a te d Pig. 12.12.— Physical working capacity estimated and predicted 190 S u b j e c t : 13-M .H . 175 160 145 130 115 100 12 Work i n k ilo p o u n d m e te rs - 1808 KPM P r e d i c t e d — 1778 KPM E s tim a te d F i g . 1 2 .1 3 .— P h y s i c a l w o rk in g c a p a c i t y e s t im a t e d an d p r e d i c t e d 191 190 S u b j e c t : 14-R.M 175 160 145 130 115 100 18 (x 100) 12 Work i n k ilo p o u n d m e te rs - 1233 KPM P r e d i c t e d — 1230 KPM E s tim a te d F i g . 1 2 .l 4 . — P h y s i c a l w o rk in g c a p a c i t y e s t im a t e d and p r e d i c t e d 162 1 9 0 S u b je c t: 1 5 -D .A . 175 160 130 115 100 85 70 18 (x 100 6 3 12 15 9 - 1087 KPM P r e d i c t e d Work i n m e te rs — 1080 KPM E s tim a te d Fig. 12.15.— Physical working capacity estimated and predicted 190 S u b je c t: 1 6 -J .H . 175 160 145 130 115 100 85 70 6 18 (x 1 0 0 ) 3 9 12 15 - 933 kpm P r e d i c t e d Work i n k il° p o u n d m e te rs - 930 KPM E s tim a te d Fig. 12.16.— Physical working capacity estimated and predicted 164 190 S u b j e c t : 1 7 -M .S . 175 160 145 130 115 100 85 70 6 18 (x 100 3 9 15 12 - - 840 KPM Predicted Work in kil°P°tmd meters 820 KPM Estimated Pig. 12.17.— Physical working capacity estimated and predicted 1 9 0 S u b je c t: 1 8 -J .H 175 160 145 130 115 100 85 70 6 18 (x 100 3 9 15 12 ---------------- g 2 5 kpm P r e d i c t e d WoI>k l n k l l ° P ° ™ d m e t e r s ------------- 865 KPM E s tim a te d Pig. 12.18.— Physical working capacity estimated and predicted 166 1 9 0 S u b j e c t : 1 9 - J .T . 175 160 145 130 115 100 85 70 18 (x 100 6 3 12 15 9 - 885 KPM P r e d i c t e d Work l n k i l° P ° « n d m e te rs 870 KPM E s tim a te d Pig. 12.19.— Physical working capacity estimated and predicted H E A R T R A T E 1 9 0 175 160 145 130 115 100 85 70 948 KPM P r e d i c t e d 908 KPM E s tim a te d J - x S u b j e c t : 20- 9 12 15 Work i n k ilo p o u n d m e te rs 18 (x Pig. 12.20.— Physical working capacity estimated and predicted ■ R . S. 100) ! 168 190 Subject: 21-M.W 175 160 145 130 115 100 85 70 i I 6 18 (x 100 3 12 9 15 - 1275 KPM Predicted Work in kil°pound meters j 1230 KPM Estimated j j Fig. 12.21.— Physical working capacity estimated and predicted ! ___ . ... vo j 190 Subject: 22-T.O. 175 160 130 115 100 85 70 6 18 12 x 100 - 1110 KPM Predicted Work in *il°P°™d meters — 1050 KPM Estimated Fig. 12.22.— Physical working capacity estimated and predicted 1 9 0 S u b j e c t : 2 3 - J .D . 175 160 130 115 100 85 70 18 (x 100) 6 3 9 12 15 - 810 KPM P r e d i c t e d Work l n k i l o Pound rae te r s 810 KPM E s tim a te d Pig. 12.23.— Physical working capacity estimated and predicted 190 S u b j e c t : 2 4 -A .L . 175 160 145 130 115 100 85 70 18 (x 100) 6 15 9 12 3 - 1785 KPM P r e d i c t e d Work i n ^ ilo p o u n d m e te rs — 1450 KPM E s tim a te d Fig. 12.24.— Physical working capacity estimated and predicted 1 9 0 S u b je c t: 2 5 - S .F . 175 l6 o 145 130 115 100 85 70 6 18 (x 100 3 9 12 15 - 1275 KPM P r e d i c t e d Work i n k * l° P ° ™ d m e te rs 1230 KPM E s tim a te d Fig. 12.25.— Physical working capacity estimated and predicted uj; 190 175 160 145 130 115 100 85 70 _ Q S u b j e c t : 3 6 9 12 15 18 KPM P r e d i c t e d Work i n k i 1° P ° '« id met eI,s KPM P r e d i c t e d 26-M .A . r (x 100) 1 2.2 6,— Physical working capacity estimated and predicted 190 S u b je c t: 2 7 -J.M . 175 160 145 130 1 1 5 100 85 70 18 6 9 x 100 3 12 15 - . n C r . . , , Work i n k ilo p o u n d m e te rs — 1065 KPM P r e d i c t e d * 1005 KPM E s tim a te d Fig. 12.27.— Physical working capacity estimated and predicted H E A R T R A T E 190 S u b je c t: 2 8-R .P 175 160 145 130 115 100 85 70 6 18 (x 100 3 9 12 15 1197 KPM P re d ic te d . 1200 KPM E s tim a te d Work i n k ilo p o u n d m e te rs Pig. 1 2.2 8.— Physical working capacity estimated and predicted 1 i — ^ j CT\ 190 S u b je c t: 29-M .S. 175 160 130 115 100 85 70 6 3 18 (x 100) 9 12 15 - 1845 KPM P r e d i c t e d Work l n k U o p o u n d m e te rs 1770 KPM E s tim a te d Fig. 12.29.— Physical working capacity estimated and predicted 177 190 Subject: 30-R.T 175 160 145 130 115 100 70 6 3 18 (x 100 9 12 15 - 1155 KPM Predicted Work ln kilopound meters — 1080 KPM Estimated Fig. 12.30.— Physical working capacity estimated and predicted APPENDIX V A HEART RATE FOR THREE LEVELS OF STRESS 179 S u b je c t: 1-M.H 20 m in u te s R /T im e f o r 3 l e v e l s o f w ork: A = 300 KPMj B = 600 KPMj C = 900 KPM P i g . 1 3 . 1 . — A H e a r t r a t e f o r t h r e e l e v e l s o f s t r e s s 081 H E A R T R A T E S u b je c t: 2 - J .F 80 60 4o 20 0 4 1 2 3 5 m inutes R/Time f o r 3 l e v e l s o f work: A = 300 KPMj B = 600 KPMj C = 900 KPM F ig . 1 3 . 2 . —A H eart r a t e f o r t h r e e l e v e l s o f s t r e s s 181 S u b je c t: 3-B.A 20 m inutes R/Time f o r 3 levels of work: A = 300 KPMj B = 600 KPM; C = 900 KPM Fig. 1 3 . 3 . —A H e a rt r a t e f o r t h r e e l e v e l s of s t r e s s 182 S u b je c t: 4-R.M. 20 m inu tes ; R/Time f o r 3 l e v e l s of work: ; A = 300 KPM; B = 600 KPM; C = 900 KPM ! CO I 1 3 . 4 . —A H e art r a t e f o r t h r e e l e v e l s of s t r e s S u b je c t: 5-S.M. 20 m inutes R/Time f o r 3 l e v e l s of work: A = O D v u l i r n ; v j = y \ j \ j m m F ig . 1 3 . 5 . —A H eart r a t e f o r t h r e e l e v e l s o f s t r e s s Subject: 6-T.F. 80 - 20 m inutes R/Time f o r 3 l e v e l s o f work: A = 300 KPMj B = 600 KPMj C = 900 KPM F ig . 1 3 . 6 . —A H eart r a t e f o r t h r e e l e v e l s o f s t r e s s 00 VJl S u b je c t: 7-A.T. 20 minutes R/Time f o r 3 l e v e l s o f work: A = 300 KPM; B = 600 KPM; G = 900 KPM Fig. 1 3 . 7 . —A H e art r a t e f o r t h r e e l e v e l s o f s t r e s s 186 S u b je c t: 8-R.G. H E A R T R A T E 80 - 60 - 4o ~ 20 » 0 R/Time for 3 levels of work: A = 300 KPM; B = 600 KPMj 0 = 900 KPM Fig. 13.8.— A Heart rate for three levels of stress m inutes oo S u b je c t: 9-T.G 20 5 m in utes R/Time f o r 3 l e v e l s of work: A = 300 KPM; B = 600 KPM; C = 900 KPM ___________ F ig . 1 3 . 9 . —A . H eart r a t e f o r t h r e e l e v e l s o f s t r e s s 188 S u b je c t: 10-M.C 20 R/Time f o r 3 l e v e l s of work: A = 300 KPMj B = 600 KPMj C = 900 KPM ______________________ P ig . 13>10.— A H eart r a t e f o r t h r e e l e v e l s o f s t r e s s oo vo S u b je c t: 11-T.H 20 m inutes R/Time for 3 levels of work: A = 300 KPMj B = 600 KPM; C = 900 KPM __________________ Fig. 13.11.— A Heart rate for three levels of stress i — 1 vo S u b je c t: 12-R.R. 80 20 m inutes R/Time f o r 3 l e v e l s o f work: A = 300 KPMj B = 600 KPMj C = 900 KPM F ig . 1 3 . 1 2 . —A H eart r a t e f o r t h r e e l e v e l s of s t r e s s i-1 vo S u b je c t: 13-M.H 20 m inutes R/Time f o r 3 l e v e l s o f work: A = 300 KPMj B = 600 KPMj C = 900 KPM F ig . 1 3 . 1 3 . —A H e a rt r a t e f o r t h r e e l e v e l s o f s t r e s s 192 S u b j e c t : H E A R T R A T E 1 2 R/TIme f o r 3 l e v e l s o f work: A = 300 KPMj B = 600 KPMj C = 900 KPM Pig. 13.14.— A Heart rate for three levels of stress m inu tes vo u> Subject: 15-D.A. 40 20 5 m inutes R/Time f o r 3 levels o f work: A = 300 KPMj B = 600 KPM; C = 900 KPM Pig. 1 3 .1 5 .—A H eart r a t e f o r t h r e e l e v e l s of s t r e s s vo H E A R T R A T E S u b je c t: 1 6-J.H . 20 4 1 2 3 5 m inutes R/Time for 3 levels of work: A = 300 KPM; B = 600 KPMj C = 900 KPM Fig. 13.16.— A Heart rate for three levels of stress 195 S u b je c t: 17-M.S 20 5 m inutes R/Time for* 3 l e v e l s of work: A = 300 KPMj B = 600 KPM; C = 900 KPM P ig . 1 3 .1 7 .—A H eart r a t e f o r t h r e e l e v e l s of s t r e s s S u b je c t: 18 -J.H . 20 m inutes R/Time f o r 3 l e v e l s o f work: A = 300 KPMj B = 600 KPMj C = 900 KPM P ig . 1 3 .1 8 . —A . H eart r a t e f o r t h r e e l e v e l s of s t r e s s S u b je c t: 1 9 -J.T . 20 5 m inu tes R/Time f o r 3 l e v e l s of work: A = 300 KPMj B = 600 KPMj C = 900 KPM F ig . 1 3 . 1 9 . —A H e art r a t e f o r t h r e e l e v e l s of s t r e s s S u b je c t: 20-R.S 20 1 2 3 ^ 5 m inutes R/Time f o r 3 l e v e l s of work: A = 300 KPM; B = 600 KPM; C = 900 KPM P ig . 1 3 . 2 0 . — A H e art r a t e f o r t h r e e l e v e l s of s t r e s s 199 S u b je c t: 21-M.W. 20 5 m inutes R/Time f o r 3 l e v e l s o f work: A = 300 KPMj B = 600 KPMj C = 900 KPM P ig . 1 3 .2 1 .— A H eart r a t e f o r t h r e e l e v e l s of s t r e s s 200 S u b je c t: 22-T.O 80 20 5 m inutes R/Time for* 3 l e v e l s of work: A = 300 KPMj B = 600 KPMj C = 900 KPM F ig . 1 3 . 2 2 . —A H e art r a t e f o r t h r e e l e v e l s of s t r e s s 201 S u b j e c t : 2 3-J.D 20 m inutes R/Time f o r 3 l e v e l s o f work: A = 300 KPMj B = 600 KPM; G = 900 KPM F ig . 1 3 . 2 3 . — A H e art r a t e f o r t h r e e l e v e l s of s t r e s s 202 S u b je c t: 24-A.L. 20 5 m inutes R/Time for 3 levels of work: A = 300 KPMj B = 600 KPMj C = 900 KPM Fig. 13.24.— A Heart rate for three levels of stress 203 Subject: 25-S.F. 20 5 minutes R/Time f o r 3 l e v e l s o f work: A = 300 KPMj B = 600 KPMj C = 900 KPM Fig. 13.25.— A . Heart rate for three levels of stress S u b je c t: 26-M.A. 60 40 20 1 2 3 4 5 m inutes f o r 3 l e v e l s o f work: 300 KPM; B = 600 KPMj C = 900 KPM P ig . 1 3 .2 6 . — A H e art r a t e f o r t h r e e l e v e l s of s t r e s s R/Time A = ui A H E A R T R A T E S u b je c t: 27-J.M 80 60 4o 20 0 4 l 5 m inutes 2 3 R/Time f o r 3 l e v e l s o f work: A = 300 KPM; B = 600 KPM; C = 900 KPM Pig. 13.27.— A Heart rate for three levels of stress 206 S u b je c t: 28-R.P. 40 20 5 m inutes R/Time f o r 3 l e v e l s of work: A = 300 KPM; B = 600 KPM; C = 900 KPM F ig . 1 3 . 2 8 . — A H eart r a t e f o r t h r e e l e v e l s of s t r e s s S u b je c t: 29-M.S. 8 0 H E A R T R A T E 60 20 5 m inutes R/Time for 3 levels of work: A = 300 KPM; B = 600 KPM; C = 900 KPM Pig. 13.29.— A Heart rate for three levels of stress ro 0 1 00 S u b j e c t : 30-R.T. H E A R T R A T E 80 60 40 20 0 4 1 2 3 5 m inutes ; R/Time for 3 levels of work: | A = 300 KPMj B = 600 KPMj C = 900 KPM Pig. 13.30.— A Heart rate for three levels of stress 60S

Linked assets

University of Southern California Dissertations and Theses

Conceptually similar

PDF

Effects Of Light And Heavy Equipment On The Acquisition Of Sports-Type Skills By Young Children

PDF

Concepts Derived From Observed Movement Patterns Represented By Visual Forms

PDF

The Effect Of Body Position On The Development Of Isometric And Isotonic Strength

PDF

The Rate Of Learning Motor Tasks

PDF

Economy Of Learning At Beginning Levels Of Gross Motor Performance

PDF

The Effect Of Participation In A Girls Inter-University Athletic Program Upon Selected Physiological Variables

PDF

The Differential Effects Of Viewing Selected Moving Visual Figure Patterns On The Performance Of A Dynamic Balance Task

PDF

The Effect Of Practice On Three Dynamic Components Of Kinesthetic Perception

PDF

The Relationship Between Work Capacity And Motor Learning

PDF

Transfer And Retention Of Selected Balance Skills

PDF

Flexibility Changes As A Result Of Isometric And Isotonic Exercise Over Limited Ranges Of Motion

PDF

A Comparison Of The Ability To Control Single Motor Units In Selected Human Skeletal Muscles

PDF

A Comparison Of Interference And Decay During Short-Term Motor Memory

PDF

Differential Applications Of Resistance And Resulting Strength Measured At Varying Degrees Of Knee Extension

PDF

The Effects Of General And Specific Warm-Up On Subsequent Motor Performance

PDF

A Cross-Sectional Study Of The Personality Factors Of Girls And Women In Competitive Lacrosse

PDF

Identification Of Some Philosophic Beliefs Of Influential Contemporary Authorities In Recreation

PDF

The Retroactive Effect Of Strenuous And Exhaustive Exercise On Maze Task Learning

PDF

Effect Of Training With Ankle Weights On Running Skill

PDF

Sex Differences In Learning A Complex Motor Task

Asset Metadata

Creator Freeland, Thomas Edward (author)

Core Title Heart Rate Response To Stress: A Mathematical Model

Degree Doctor of Philosophy

Degree Program Physical Education

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Education, Physical,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Lersten, Kenneth C. (committee chair), Hall, J. Tillman (committee member), Lockhart, Aileene (committee member), Metheny, Eleanor (committee member), Morris, Royce (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-540277

Unique identifier UC11362126

Identifier 7203775.pdf (filename),usctheses-c18-540277 (legacy record id)

Legacy Identifier 7203775.pdf

Dmrecord 540277

Document Type Dissertation

Rights Freeland, Thomas Edward

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA