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Construction and Validation of Multivariable Population
Pharmacokinetic Models — Utility of Allometric Forms to
Predict the Pharmacokinetic Disposition of Gentamicin in
Pediatric Patients With Appendicitis
by
Thomas M. Gilman
A Thesis Presented to the
Faculty Of The Graduate School
University Of Southern California
In Partial Fulfillment of the
Requirements for the Degree
Master Of Science in Applied Biometry
August 1996
Copyright 1996 Thomas M. Gilman
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UMI Number: 1381586
UMI Microform 1381586
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UNIVERSITY OF SOUTHERN CALIFORNIA
THE GRADUATE SCHOOL
U N IV E R S ITY PARK
LOS ANGELES. C A LIF O R N IA 9 0 0 0 7
This thesis, written by
Thomas M . Gilman
under the direction of hXs~JThesis Corfimittee,
and approved by a ll its members, has been pre
sented to and accepted by the Dean of The
Graduate School, in p artia l fulfillm ent of the
requirements fo r the degree of
Applied Biometry__________________
Dean
Da^„A^gHSt.„20,M .1996
THESIS COMMITTEE
Chairman
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Dedicated to m y wife and to my family
Acknowledgments
I thank Dr. Irving Steinberg, Dr. Tracey Yoshida, and Dr. Annet Arakelian for
providing the patient data that made this work possible. I am particularly
grateful to Dr. Steinberg for the time he spent increasing my aware ness of the
need to characterize the disposition of drugs in pediatric patients.
I thank the members of my thesis committee: Dr. Leslie Bernstein, Chair; Dr.
Eugene Sobel, and Dr. David D’Argenio. I express my appreciation to the
many members of the Preventive Medicine Faculty who stimulated my interest
in the analysis and interpretation of data and especially to Dr. Stan Azen for
his encouragement and support.
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Table of Contents
Dedication and Acknowledgments ii
List of Tables iv
List of Figures vi
Summary viii
1 1ntroduction 1
2 Methods 3
3 Results 12
4 Discussion 47
5 Appendix: Biometric Methods 56
6 References 61
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List of Tables
I: Formulas for Estimating Creatinine Clearance from Post-natal Age in
Years, Height in cm, and Serum Creatinine in mg/dL.
6
II: Predictor Variables in Maximum Models.
10
Ilia: Characteristics of Group I: Model Building and Internal Validation
Group.
14
lllb: Characteristics of Group II: External Validation Group.
15
IV: Models from All-subsets-regression Procedures Using the Model
Building Group (Group 1).
17
Va: Models for Predicting CL in Pediatric Appendicitis. Model Variables,
Coefficients, and Goodness of Fit.
20
Vb: Internal Validation of Models for Predicting CL in Pediatric Appendici
tis. Model Forms, Goodness of Prediction Statistics, and Percent Coef
ficient of Variation.
21
Via: Models for Predicting Vd in Pediatric Appendicitis. Model Variables,
Coefficients, and Goodness of Fit.
22
Vlb: Internal Validation of Models for Predicting Vd in Pediatric Appendici
tis. Model Forms, Goodness of Prediction Statistics, and Percent Coef
ficient of Variation.
25
Vila: External Bias of Models for Gentamicin CL in Pediatric Patients with
Appendicitis.
43
Vllb: External Bias of Models for Gentamicin Vd in Patients with Appendici
tis.
44
iv
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Villa: External Precision of Models for Gentamicin CL in Pediatric Patients
with Appendicitis.
45
Vlllb: External Precision of Models for Gentamicin Vd in Pediatric Patients
with Appendicitis.
46
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List of Figures
1 Gentamicin CL
28
2 Gentamicin CL
29
3 Gentamicin CL
30
4 Gentamicin CL
31
5 Gentamicin CL
32
6 Gentamicin CL
33
7 Gentamicin Vd
34
8 Gentamicin Vd
35
9 Gentamicin Vd
36
10 Gentamicin Vd
37
1 1 Gentamicin Vd
38
12 Gentamicin Vd
39
13 Gentamicin Vd
40
14 Gentamicin CL
52
15 Log-iog transform of gentamicin CL vs. total body weight (Group 1)
53
16 Gentamicin Vd vs total body weight (Group 1).
54
VI
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17 Log-log transform of gentamicin Vd vs. total body weight (Group 1).
55
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SUMMARY
BACKGROUND. Better methods for precise prediction of initial dosage needs
of aminoglycoside antibiotics in pediatric patient populations are needed.
Multivariable models for predicting gentamicin pharmacokinetics and serum
concentrations in pediatric patients with appendicitis were developed and
validated.
METHODS. Population pharmacokinetic models for gentamicin were con
structed using data from 79 pediatric patients without renal insufficiency and
ranging in age from two to 17 years. Serum clearance (CL) and apparent vol
ume of distribution (Vd) were determined for each patient using measured se
rum concentrations obtained at steady-state.
Multivariable models for predicting CL and Vd were developed using all
subsets regression. Several naive models were tested as well. Patient age,
gender, height, weight, presence of obesity, surgical condition of the appen
dix, number of days after surgery that serum samples were taken, serum
creatinine and albumin concentrations were considered as potential predictor
covariates and factors. Candidate models were selected using the Mallow Cp
statistic and the adjusted R2 . External validation of candidate models was
performed using data from an additional 53 patients.
RESULTS. The best most parsimonious models with acceptable predictive
behavior had the following allometric forms: CL (L/h) = 0.7417 (weight,
kg)0-5236 and Vd (L) = 1.035 (weight, kg)0-6239. Corresponding mean CV%
were 25% and 12%. The external validation demonstrated that predictions of
pharmacokinetic parameters were unbiased and precise with median absolute
prediction errors of 14.1% for CL and 17.7% for Vd.
viii
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CONCLUSION. Gentamicin pharmacokinetic parameters and serum concen
trations in pediatric appendicitis patients were well-explained using allometric
models incorporating weight as the sole predictor variable. Other models were
identified as well. These models should prove useful for selection of initial
dosage regimens and for Bayesian dosage adjustment in similar patients.
These population pharmacokinetic models appear well suited for predicting
the pharmacokinetic disposition of aminoglycosides in pediatric patient groups
where size ranges broadly. This is not unexpected since allometric models
are well accepted for predicting pharmacokinetic behavior in different species
by scaling variates of interest to body size.
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INTRODUCTION
1.1 Appendicitis and Target Gentamicin Concentrations.
Appendicitis is a common cause of hospitalization and morbidity in the pedi
atric age group. Gentamicin, a naturally occurring mixture of three aminogly
coside antibiotics, is commonly administered in combination with other antimi
crobial agents to avoid the post-operative complications of gangrenous or
perforated appendicitis (Gamal eta!.; 1990). Gentamicin, like other amino
glycoside antibiotics, has concentration-dependent killing against susceptible
bacteria. Peak serum gentamicin concentrations are therefore used as indi
rect markers of therapeutic response (Hyatt et a/.; 1995). Inadequate gen
tamicin dosages that result in peak serum concentrations below 6.0 mg/L
have been associated with an increased risk of infectious complications in
adults with perforated or gangrenous appendicitis (Gill et al.; 1986).
1.2 Variation in Serum Concentrations.
In the past, pediatric appendicitis patients were usually dosed empirically with
five mg/kg body weight/day of gentamicin. This has been observed to result in
a five-fold range of peak serum concentrations in these patients at the Pedi
atric Wards of the Los Angeles County — University of Southern California
(LAC-USC) Medical Center (I. Steinberg, unpublished data). Some of this
variation may be the result of mistiming of samples, mistiming of infusions, in
accurate measurement of doses, or inaccurate measurement of weight but
much of it is probably the result of interpatient differences in pharmacokinetic
disposition characteristics. These differences may relate to differences in
body composition and in elimination organ function that occur with maturation.
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1.3 Gentamicin Pharmacokinetics.
In the case of aminoglycoside antibiotics like gentamicin, distribution occurs
into the extracellular fluid space which averages 20 to 30% of body weight. A
portion distributes from the extracellular fluid into tissues where it is bound
and slowly released after termination of treatment. Elimination of the amin o-
glycosides from the body depends primarily upon kidney function. Using a
one-compartment structure as a pharmacokinetic model, the size of the distri
bution space is denoted as the volume of distribution (Vd) and the amount of
volume that is cleared of drug per unit time is denoted as clea ranee (CL).
1.4 Population Pharmacokinetic Models.
Dosing methods that result in less variation in peak serum concentrations
than those seen with the empirical 5 mg/kg/day dosage are needed for s e-
lecting initial gentamicin regimens. In this manuscript, conventional biometric
methods were used to develop and validate multivariate population models for
predicting individual patient pharmacokinetics. Two novel measures of goo d-
ness of predictive behavior were considered and used along with conve n-
tional measures. The resulting models are expected to permit more prudent
individualization of initial dosage regimens for gentamicin in pediatric patients
with appendicitis and probably for other infected pediatric patients as well.
1.5 Specific Aims.
The objective of this study was to develop and validate multivariable models
for predicting gentamicin CL and Vd in pediatric patients with appendicits.
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METHODS
2.1 Patients.
All patients were identified from pharmacokinetic records of the Division of
Pediatric Pharmacotherapy, Department of Pediatrics at the Los Angeles
County-University of Southern California Medical Center (I. Steinberg, Direc
tor). The following criteria were met for inclusion of patient data.
1) Age between one and 18 years.
2) Surgical diagnosis of appendicitis.
3) Treatment with intermittent, intravenous infusions of gentamicin.
4) Steady-state, peak concentration measured at 30 to 60 minutes after a 30
to 60 minute intravenous infusion. The existence of steady-state was i n-
ferred if a fixed dosing regimen had been administered for more than 16
hours by the time serum samples were taken.
5) Steady-state, mid-interval concentration measured at three to four hours
after the same infusion
6) Complete or partial records of gentamicin doses and administration times.
7) Demographic information including age and gender and anthropometric
information such as height and weight.
8) Absence of indicators of clinical shock — such as diminished urine ou t-
put, low blood pressure, or obtundation — during the period of data acqu i-
sition.
9) Serum creatinine and albumin concentrations obtained within 24 hours of
pharmacokinetic sampling.
3
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The records of 135 pediatric patients were screened for model building. Of
these, 56 had missing values for serum creatinine or serum albumin or both.
Model building was performed using the remaining 79 records and these p a-
tients were denoted as Group I. Information from an additional 53 patients,
who were treated subsequent to the original 135, was acquired for external
validation of all-subsets-regression models and these patients were denoted
as Group II.
2.2 Pharmacokinetic Analysis.
For the original sample of 135 patient records and for the first 27 validation
patient records, partial dosing records were available. Gentamicin elimination
rate constant (Kel, hr "1), half-life {Wz, hr), clearance (CL, L/hr) and apparent
volume of distribution (Vd, L) for a one-compartment model were calculated
from steady-state gentamicin concentrations using Eqns. 1-6 (Gibaldi and
Perrier; 1975). For the final 26 validation patients, complete records of all
doses were available. For these patients, CL and Vd were estimated by fitting
timed doses and serum concentrations for each patient to a one-compartment
model using nonlinear, least-squares regression with unity-weighting. Kel was
then calculated as CL + ■ Vd and V/z as log(2) + Kel.
Kel = [ log(Cpeak) - log(Cmid) ] - 1 Eqn. 1
where Cpeak and Cmid are the steady-state peak and mid
interval serum concentrations, t is the time between Cpeak and
Cmid in hrs, and Kel is the elimination rate constant in reciprocal
hrs. NB: log refers to the natural logarithm operator throughout.
Cmax = Cpeak + exp(-Kel x tpeak) Eqn. 2
where Cpeak is the measured peak concentration, tpeak is the
time in hrs between the end of the infusion dose and the time
Cpeak was obtained, exp is the exponentiation operator, and
4
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Cmax is the theoretical value of the steady-state serum concen
tration at the end of the infusion.
Cmin s Cmax x exp(-Kel x (T - 1 ’) ) Eqn. 3
where Cmax is the value from Eqn. 2, Kel is the elimination rate
constant in hrs'1 , T is the dosing interval in hrs, t’ is the infusion
interval in hrs, and Cmin is the theoretical steady-state value of
the serum concentration at the beginning of the infusion.
Vd = Dose + ( f x Kel) * (Cmax - Cmin x exp(-Kel x t’) ) Eqn. 4
where Dose is the gentamicin dose in mg administered prior to
the measured serum concentrations, t’ is the intravenous infu
sion interval in hrs, Kel is the elimination rate constant from Eqn.
1, Cmax and Cmin are the calculated theoretical steady-state
serum gentamicin concentrations in mg/L at the start and finish
of each steady-state intravenous infusion from Eqns. 2 and 3,
and Vd is the volume of distribution in L.
CL = Kel * Vd Eqn. 5
where CL is in L/hr and represents the volume of serum being
cleared of gentamicin per unit time,
t = log(2) - Kel Eqn. 6
where t Y z is the elimination half-life in hrs. This is a constant
value and represents the time during which the serum conce n-
tration falls to one half of its starting value.
2.3 Anthropometric and Renal Function Indices.
Four indices were examined for their possible value in predicting pharmacok i-
netic parameters:
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1. Body surface area (BSA) was calculated by the following formula:
BSA (M2 ) = weight (kg)0-3378 height (cm)0-3964 (Haycroft et a/.; 1978).
2. Body mass index (BMI) was calculated in the conventional way:
BMI = weight (kg) / height (cm2 ) (Keys et al.; 1971)(Simopoulos and Van
Itallie; 1984).
3. Obesity was considered present if the patient’s weight was above the 95th
percentile for BMI according to age using published tables of standardized
percentiles (Hammer et al.; 1991).
4. Creatinine clearance (CLcr) was calculated in mL/min/1.73 M2 according to
age group and gender using the formulas in Table I (Schwartz et al.;
1987). Values of CLcr were expressed in L/h by multiplying first by
BSA (M2 ) 1.73 (M2 ) and then multiplying by 60 (min/h) - s - 1000 (mL/L).
Table I: Formulas for estimating creatinine clearance from postnatal age
in years, height in cm, and serum creatinine in mg/dL.
Postnatal Age Gender Formula for CLcr
1.0 to <1.5 y F, M 0.44 height (cm) / Scr (mg/dL)
1.5 to <12.5 y F, M 0.56 height (cm) / Scr (mg/dL)
12.5 to 21 y F 0.59 height (cm) / Scr (mg/dL)
12.5 to 21 y M 0.73 height (cm) / Scr (mg/dL)
2.4 Regression Models.
Model Forms. Multiple linear regression models were constructed for predict
ing gentamicin CL and Vd in pediatric patients with appendicitis. Three model
forms were fitted to the data by conventional least-squares techniques for k
predictor variables:
6
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1. Log-log models of the form log(Y) = log(Bo) + 01 x !og(Xi) + ... + x
log(Xk) + E. For the log-log models, exponentiation gives an allometric
form which can be generalized as: Y = exp(Bg) x X i x ... x X ^ k x
exp(E).
2. Linear models of the form Y = Bo + B-j x X-j + ... + Bk x *k + E-
3. Log-linear models of the form log(Y) = Bq + B-| x X-j + ... + Bk x Xk + E.
For the log-linear models, exponentiation gives a product of exponential
terms.
NB all log operations are natural logarithms. For all three model forms: Y
stands for CL or Vd; B q , B-|,..., Bk are the regression coefficients which are
estimated; X-j, X2, .... Xk are predictor variables which may represent demo
graphic or anthropometric characteristics; E represents the error in the pre
diction of Y for each individual using the model equation and it is as sumed
that E ~ f^O.G2). Solving each of these equations with E = 0 gives the pre
dicted value of Y which is designated as Y hat-
2.5 Model Building And Internal Validation.
Maximum models, which utilize all possible covariates and factors, were con
structed and examined together with all possible restricted models using all
subsets multiple-linear regression (SAS Institute Inc.; 1987). Promising all-
subsets-regression predictive models were identified by examining R2 adj. ancl
the Mallows Cp statistic (Glantz and Stinker; 1990, pp. 249-50), (Hocking;
1976), (Mallows; 1973). Residuals were plotted against predictions and
against all ordinal predictor variables to look for outliers and to ensure lack of
heteroscedasticity, bias, and trend. Studentized residuals and the Cook D
statistic were used to screen for influential observations (Kleinbaum et al,
7
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1988, pp. 181-227). The variance inflation factor (VIF) was used to check for
muiticollinearity (Glantz and Stinker, 1990, pp. 191-3). These model selec
tion statistics are detailed in the appe ndix.
Comparison of promising models for the same dependent variable was ac
complished using the PRESS(.j) or predicted residual sums of squares statis
tic (Glantz and Stinker, 1990, pp. 249-55). Models which give more precise
predictions for patients with larger values of the dependent variable tend to
have smaller values of PRESS(-j). Two alternative comparative statistics,
RPRESS(.j) or Relative PRESS(.j)) and APPE(.j) or Absolute Percent Predic
tion Error, were calculated in an attempt to compensate for this limitation of
PRESS(-j). These are detailed in the appendix. To ensure comparable stati s-
tics, the predicted residuals from the log-log and the log-linear models were
transformed to the original units of the dependent variates before calculating
PRESS(.j), RPRESS(.j), and APPE(.j). The goodness of prediction measures
(PRESS(.j), RPRESS(.j), and APPE(.j)) are detailed in the appendix. The
percent coefficient of variation was calculated for each model as 100 times
the ratio of the square root of the mean squared error to the mean value of the
variate (note all values of the variates are positive). When the log of the var i-
ate was used as the dependent variable then the ratio was taken after exp o-
nentiation.
2.6 Centering And Scaling.
For all subsets building of linear models, interval predictor variables (also r e-
ferred to as covariates) were rescaled to have a mean of zero and a standard
deviation of 1.0 to minimize muiticollinearity. This was only done prior to ge n-
erating quadratic predictor variables for accommodating nonlinearity in the
linear and the log-linear models. (Kleinbaum etal, 1988, pp. 217-8). For all-
subsets-regression models that included such quadratic terms, final hypoth e-
8
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sis testing, goodness of fit statistics, residual analyses, and computation of
adjusted R2 , studentized residuals, and Cook’s D were reported for the scaled
and centered transform of the data. This would have been done for any model
with one or more parameters having a VIF > 10 as well but none of the se
lected models met that criterion.
2.7 Maximum Models.
The potential predictor variables in the maximum models for CL and Vd are
listed in Table II along with the units or coding used for model building.
Demographic and anthropometric variables were age, gender, height, weight.
For the log-log models, BMI was not included since log(BMI) is a linear com
bination of log(weight) and log(height). Laboratory variables were serum a I-
bumin and creatinine concentrations. Clinical variables were the number of
days after surgery at which serum sampling occurred and whether the appe n-
dix was found to be perforated or gangrenous at surgery.
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Table II: Predictor Variables in Maximum Models.
Loa-loa Models
Covariates Factors
Age, log(years) Gender (Male 0, Female 1)
Height, log(cm) Perforation Status
Body Weight, log(kg) (gangrenous 0)
Serum Albumin, log(g/dL) (perforated 1)
Serum Creatinine, log(mg/dL) Obesity (0,1)
Time After Surgery, log(days) BMI >25 (0, 1)
Linear and Loo-Linear Models
Covariates Factors
Height (ht, cm), ht*
Gender (male 0, female 1)
Weight (wt, kg), wt2
Perforation Status
2
Age (years), age
(gangrenous 0)
Serum Albumin (g/dL) (perforated 1)
Serum Creatinine (mg/dL) Obesity (0,1)
Time After Surgery (days) BMI >25 (0,1)
2
Body Mass Index (kg/M )
2.8 External Validation.
Model equations developed in Group I were validated by predicting CL and
Vd for Group II patients. Predictive bias was evaluated by calculating percent
prediction error (PPE). The formula for PPE is ((Yhat - Y) x 100) / Y, where Y
and Y hat are the actual and the predicted values respectively. Predictive pr e-
10
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cision was studied using APPE, PRESS, and RPRESS as noted above and
detailed in the appendix.
2.9 Statistics.
Statistical procedures were performed with SAS Version 6.04 for personal
computers (SAS Institute Inc.; 1987). Kruskal-Wallis tests were used to
compare interval variables among more than two groups. Mann-Whitney rank
sum tests were used to compare interval variables between two groups. Like-
2
lihood ratio X tests were used to compare categorical variables among more
than three groups. Wilcoxon signed rank tests were used to detect overall
predictive bias of models and to compare predictive precision between mod
els. All tests were two-tailed and the significance level was set at 0.05 unless
otherwise indicated. The Bonferonni inequality was used to adjust the signif i-
cance level when an overall significance level of 0.05 was desired for a r e-
lated series of tests.
1 1
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RESULTS
3.1 Patients.
Patient characteristics are summarized in Tables Ilia and lllb for Group I and
Group II respectively. Characteristics of Groups I and II were compared using
Wilcoxon-Mann-Whitney rank sum tests and Likelihood ratio X2 tests
(o c = 0.05) showed statistically significant differences in median serum cre
atinine between Groups I and II (p=0.01). Despite this difference for serum
creatinine, estimated creatinine clearance did not differ between the two
groups. There were no statistically significant differences in CL, L/h; Vd, L;
2
Vd, L/kg; t-|/2, h; age, y; ht, cm; wt, kg; BMI, kg/M ; serum albumin, g/dL; days
after surgery; or distribution of gender, obesity, or perforation status.
For both groups combined, 63.6% were males and 36.4% were females and
this is very consistent with the general predominance of males among pediat
ric patients with acute appendicitis (Gamal and Moore; 1990) (Janik and Fi-
ror; 1979) (Marchildon and Dudgeon; 1977) (Shandling et al.; 1974). A
gangrenous appendix was diagnosed in 90.2% and a perforated appendix
was found in the other 9.8%. Pharmacokinetic studies were performed 1 to 7
days after surgery and the average number of postoperative days to sampling
was 2.1. A broad range was observed for the pharmacokinetic variables as
expected for samples taken from a population whose members range greatly
in size. Mean CL was 5.3 with a range of 1.4 to 10.4 L/h. Mean Vd was 11.4
with a range of 3.5 to 39.4 L and, expressed as a fraction of body weight, av
eraged 0.30 with a range of 0.11 to 0.73 L/kg.
Age ranged from 2 to 17 years and averaged 10.3 years. Height ranged from
83.0 to 184.2 and averaged 140.0 cm. Weight ranged from 10.4 to 114.0 and
averaged 41.9 kg. Body mass index ranged from 10.5 to 41.8 and averaged
12
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20.0 kg. Obesity was present among 24 of 132 (18.2%). Body surface area
2
ranged from 0.49 to 2.34 and averaged 1.26 M . Serum creatinine measure
ments ranged from 0.2 to 1.2 and averaged 0.7 mg/dL. None of the patients in
this sample had clinical evidence of renal insufficiency. Serum albumin
measurements ranged from 2.0 to 4.3 and averaged 3.4 g/dL.
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Table Ilia: Characteristics of Group I: Model Building and Internal Valida
tion.
N Mean ± SD Range
Pharmacokinetic Parameters
Clearance, L/h 79 5.4 ±1.9 1.9-10.4
Distribution Volume, L 79 11.6 + 5.9 3.6 - 39.4
Distribution Volume, L/kg 79 0.28 + 0.11 0.12-0.73
Half-Life, h 79 1.4 ± 0 .4 0.9 -2.7
Demoaraohic Variables
Age, y 79 10.8 ±4.3 2 -17
Male 52/79 (65.8%)
Female 27/79 (34.2%)
AnthroDometric Variables
Height, cm 79 143.5 ±23.0 90.0-184.2
Weight, kg 79 45.3 ±22.0 11.0-114.0
Body Mass Index, kg/sqM 79 20.6 ± 5.6 10.5-41.8
2
Body Surface Area, M
79 1.33 ±0.42 0.52 - 2.34
Obesity 15/79(19.0%)
Laboratory Variables
Serum Creatinine, mg/dL 79 0.7 ± 0.2 0.4 -1.2
Creatinine Clearance, L/h 79 6.1 ±2.1 2.1 -11.1
Serum Albumin, g/dL 79 3.5 ± 0.4 2.2 - 4.3
Clinical Variables
Time After Surgery, d 79 2.2 ±1.0 1 -7
Perforated Appendix 67 / 79(84.8%)
Gangrenous Appendix 12/79(15.2% )
14
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Table lllb: Characteristics of Group II: External Validation.
N Mean ± SD Ranae
Pharmacokinetic Parameters
Clearance, L/h 53 5.1 ±1.9 1.4 -8.7
Distribution Volume, L 53 11.0 ± 5 .2 3.5 -29.2
Distribution Volume, L/kg 53 0.32 ±0.13 o
•
o
0 0
Half-Life 53 1.5 ± 0 .4
C \ J
C O
1
0 0
o
Demoaraohic Variables
Age, y 53 9.6 ±4.0 2-17
Male: 32/53 (60.4%)
Female: 21/53 (39.6%
Anthrooometric Variables
Height, cm 53 134.8 ±21.5 83.0-175.3
Weight, kg 53 36.9 ±17.0 10.4-83.0
Body Mass Index, kg/sqM 53 19.0 ±4.1 12.0-29.6
Body Surface Area, M
53 1.16 ±0.36 0.49 - 2.02
Obesity 9/53 (17.0%)
Laboratory Variables
Serum Creatinine, mg/dL 53 0.6 ± 0.2 0.2 - 0.9
Creatinine Clearance, L/h 53 6.2 ± 3.0 1.6-14.7
Serum Albumin, g/dL 53 3.4 ± 0.5 2.0-4.1
Clinical Variables
Time After Surgery, d 54 2.0 ± 1 .0 1 -5
Perforated Appendix 52/53 (98.1%)
Gangrenous Appendix 1 /5 3 (1.9%)
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3.2 Naive Models.
Qentamicin is primarily eliminated by renal glomerular filtration and renal
function is highly dependent on body size in this population. As a result, two
naive models for predicting CL from either BSA (designated as CL model 3)
or estimated CLcr (designated as CL model 4) were studied. Each specifies
CL as a linear function of the predictor variable with the intercept forced
through the origin. This has the intuitively appealing property of constraining
CL to approach zero as estimated CLcr approaches zero. Since gentamicin
distributes into extracellular fluid and the volume of this fluid is highly de
pendent on body size, two naive models for predicting Vd from either BSA
(designated as Vd model 3) or weight (designated as Vd model 4) were also
studied. These were formed in the same manner as the naive models for CL.
3.3 All-subsets-regression Models.
The best models found with the all-subsets-regression procedures are shown
in Table IV. The best Cp models are shown in bold typeface for each of the
three possible model forms: log-log, log-linear, and linear-linear. The log-log
and log-linear models with log(weight) or weight as the sole predictor varia -
bles were retained for testing despite their high Mallow C P values and are
shown in regular typeface. This was done because weight is the conventional
predictor variable for Vd and therapy is often initiated without benefit of info r-
mation other than weight. Each of the model forms are separated from the
others in the table. The dependent variable, the predictor variables included
in each model, R2 , adjusted R2 , and Mallows Cp are shown and these can only
be compared for models of the same form because these models have identi
cal maximum models. The final column of the table is the arbitrary model
designation number.
1 6
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Table IV: Models From All-subsets-regression Procedures Using Model
Building Group (Group 1).
Dependent Predictor Mallows
Variable Variable(s) R2 R2 adj
cP
Model #
CL Models
Log-log
log(CL) log(weight) 0.58 0.57 0.331 1
Linear-linear
CL weight 0.63 0.62 2.035 2
Log-linear
loa(CL) weight 0.56 0.55 8.420 5
log(CL) height, weight 0.60 0.59 1.534 6
Vd Models
Log-log
loa(Vd) iog(weight) 0.51 0.51 8.171 1
log(Vd) log(albumin),
log(height)
0.57 0.56 0.735 2
Log-linear
log(Vd) weight 0.53 0.52 15.594 5
log(Vd)
2
height, height >
weight2 , albumin
0.62 0.60 4.450 6
Linear-linear
Vd albumin, height,
height2 , cre
atinine,
weight2 , postop
days,
BMI > 25 (0,1)
0.76 0.73 6.821 7
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3.4 Internal Validation.
The fitted parameters and associated statistical tests of the all-subsets-
regression and naive predictive models for CL are summarized in Table Va
(coefficients, inferences, VIF, goodness of fit) and Table Vb (PRESS(-i),
RPRESS(-i), APPE(-i), studentized residuals, Cook D, percent coefficient of
variation). For all CL models, the predictor variables are statistically signif i-
cantly associated with their respective dependent var iables (p < 0.05).
Clearance model 1 (CL = exp(B(j) x weight®1 )) with RPRESS(.j) = 5.13, model
3 (CL - 3 qx BSA) with RPRESS(.j) = 4.43 and model 6 (CL = exp(B(j +
Bi height + B^weight)) with RPRESS(.j) = 4.88 appear most promising. Clear
ance Model 4 (CL = I3 q x CLcr) appears least promising. It has absolute stu
dentized residuals which equal or exceed 3.0 indicating the presence of ou t-
liers that are not well explained by the model. The PRESS (.j) and RPRESS(.j)
statistics are about twice as high as for any other CL models and R 2 acjj is only
0.293. The median APPE(.j) was slightly larger than the others but not large
enough to dismiss this model by itself.
The fitted parameters and associated statistical tests of the predictive models
for Vd are summarized in Table Via (coefficients, inferences, VIF, goodness
of fit) and Table Vlb (PRESS(.j), RPRESS(.j), APPE(.j), studentized residuals,
Cook D, percent coefficient of variation). There was a statistically sig nificant
association between dependent and predictor varia bles in all Vd models ex-
2
cept model 6 (p < 0.05). One predictor variable in Vd model 6, height , was
not statistically significantly associated with the dependent varia ble (p = 0.15).
Volume of distribution models 2 (Vd = Bq x height1 ^ x A lbum in^) with
RPRESS(.j) = 8.07 and model 6 (Vd = exp(Bo + B *| weight + B2height +
18
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fr e ig h t2 + B4albumin)) with RPRESS(.j) = 8.07 appear most promising.
They are followed closely by Vd models 1 (Vd = exp(Bo)x w e ig h t^ ) with
RPRESS(.j) = 10.23 and model 5 (Vd = exp(Bg) x exp(B-| weight)) with
RPRESS(.j) = 10.10. Only Vd model 3 (Vd = Bq x BSA) has absolute studen
tized residuals greater than 3.0 and its maximum Cook D was also the largest
at 0.79. This model appears the least promising.
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Table Va: Models for Predicting CL in Pediatric Appendicitis. Model Variables, Coefficients, and Goodness of
Fit.
Dependent
Variable
Predictor
Variables
Model
Coefficients
p-Value *
VIF t Goodness of Fit f R2 adj Comment
1. log(CL) Intercept
log(Weight)
-0.313242
+0.527730
.1048
< .0001
0.0
1.0
F=106.1, pc.0001 .574 All
Subsets
2. CL Intercept
Weight
+2.436596
+0.066385
< .0001
< .0001
0.0
1.0
F=118.7, p<.0001 .601 All
Subsets
3. CL BSA +4.043113 < .0001 0.1 .588 Naive
4. CL CLcr +0.844242 < .0001 0.1 .293 Naive
5. log(CL) Intercept
Weight
+1.083053
+0.012156
< .0001
< .0001
0.0
1.0
F=96.0, p<.0001 .549 All
Subsets
6. log(CL) Intercept
Height
Weight
-0.445713
+0.006114
+0.006853
.0462
.0035
.0018
0.0
3.2
3.2
F=57.6, p<.0001 .592 All
Subsets
* t test, Hq: Parameter estimate is zero,
t Variance inflation factor
t F test, Hq: No association between dependent and predictor variables.
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Table Vb: Internal Validation of Models for Predicting CL in Pediatric Appendicitis. Model Forms, Goodness of
Prediction Statistics, and Percent Coefficient of Variation.
Model Form N
Median
APPE^.j j PRESS(.j) RPRESS(_j)
Studentized
Residual
Cook
D
Mean
CV%
1. CLh a i = e^O Wt^1
79 18.6% 117.8 5.13 -2.2 to 2.2 .13 25%
2. CLh a i = 3q + &1 Wt
79 17.4% 113.2 6.29 -2.1 to 2.3 .09 22%
3. CLhal - BqSSA 79 17.6% 116.1
4.43*
-2.1 to 2.4 .09 22%
4. CLhat= BoCLcr 79 19.7% 202.0 11.80 -3.9 to 3.0 .67 29%
5. CLh a , = e®0 e^tW t
79 18.5% 119.2 5.61 -2.6 to 2.0 .14 25%
6. C Lhat = e^O e ^lH t eB2w t
79
16.2% * 111.5*
4.88 -2.2 to 2.2 .12 24%
* Indicates the best values for goodness of prediction statistics.
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Table Via: Models for Predicting Vd in Pediatric Appendicitis. Model Variables, Coefficients, and Goodness of Fit.
Dependent
Variable
Predictor
Variables
Model
Coefficients
p-Value *
VIF f
Goodness of Fit ♦
R2 adj Comment
1 . log(Vd) Intercept
log(Weight)
-0.069391
+0.654034
.7975
< .0001
0.0
1.0
F=81.7, pc.0001 .509 All
Subsets
2. log(Vd) Intercept
log(Albumin)
log(Height)
-6.028172
-1.211088
+1.994456
< .0001
< .0001
< .0001
0.0
1.0
1.0
F=50.4, p<.0001 .559 All
Subsets
3. Vd BSA +8.881623 < .0001 0.1 .510 Naive
4. Vd Weight +0.246202 < .0001 0.2 .524 Naive
5. log(Vd) Intercept
Weight
+1.635339
+0.015634
< .0001
< .0001
0.0
1.0
F=87.2, p<.0001 .545 All
Subsets
* t test, Hq: Parameter estimate is zero,
t Variance inflation factor
t F test, Hq: No association between dependent and predictor variables.
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Table Via (Continued).
Dependent
Variable
Predictor
Variables
Model
Coefficients
p-Value *
VIF t Goodness of Fit % R2 adj Comment
6. log(Vd)§
Intercept -0.525904 < .0001 0.0 F=30.1, pc.0001 .599 All
Weight2
+0.000063213 .0029 1.4 Subsets
Height +0.025662 .0001 1.2
Height2
-0.000056617 .1487 1.4
Albumin +0.237411 .0118 1.2
* t test, Ho: Parameter estimate is zero,
t Variance inflation factor
$ F test, Hq: No association between dependent and predictor variables.
§ Model coefficients are fitted from original data whereas p-va(ues, VIF, goodness of fit and adjusted R 2 are from the
centered and scaled transform of the data.
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Table Via (Continued).
Dependent
Variable
Predictor
Variables
Model
Coefficients
p-Value*
VIFf
Goodness of Fit$ Rz adj Comment
7. Vd§
Intercept -0.154531 <.0001 0.0 F=35.2, p<.0001 .734 All
Albumin -2.872784 .0058 1.3 Subsets
Height +0.174010 .0010 2.8
Height2
-0.000610 .0030 1.8
Creatinine +6.715366 .0029 2.6
Weight2
+0.001730 < .0001 3.6
Postop Days +0.604383 .0469 1.0
BMI>25 (0,1) -2.387555 .0134 2.8
* t test, Hq: Parameter estimate is zero,
t Variance inflation factor
$ F test, Hq: No association between dependent and predictor variables.
§ Model coefficients are fitted from original data whereas p-values, VIF, goodness of fit and adjusted R 2 are from the
centered and scaled transform of the data.
Table Vlb: Internal Validation of Models for Predicting Vd in Pediatric Appendicitis. Model Forms, Goodness of
Prediction Statistics, and Percent Coefficient of Variation.
Model Form N
Median
APPE(.j) PRESS(.j) RPRESS(.})
Studentized
Residual CookD
Mean
CV%
1. Vdh a , = e&0Wt&1
79 24.7% 1437.9 10.23 -2.2 to 2.2 .17 12%
2. V dhat = e&0 Ht&1 Alb®2
79 23.9% 1449.3
8.07 f
-1.6 to 2.5 .21 12%
3. Vdhat = B()BSA 79 22.4% f 1385.8 11.96 -1.7 to 4.6 .79 35%
4. V dhat = B0Wt
79 22.4% f 1342.1 11.36 -2.0 to 2.9 .58 35%
5. V dhat = e^o e&lW t
79 24.2% 1104.9 10.10 -2.0 to 2.2 .06 13%
6. Vdhat* = eB0 eB1w t e62Ht
e&3Ht2 e ^ A lb
79 22.9% 1086.6 f 8.071
-1.8 to 2.6 .12 13%
7. Vdha* = 60 + 01 Alb + B2Ht
+B3Ht2 + B4Scr + B5Wt2
+ BgPostop Days
+ B7[BMI>25 (0,1)]
79 27.2% 1182.2 11.58 -2.0 to 3.0 .15 30%
*The studentized residuals, and Cook D statistic were calculated from the centered and scaled transform of the data,
t Indicates the best values for goodness of prediction statistics.
3.5 External Validation: Scatterplots.
The predictive performance is depicted for the six CL models in Figs. 1
through 6 and for the seven Vd models in Figs. 7 through 13. Each figure is a
scatterplot of model predictions on the ordinate versus meas ured pharma
cokinetic parameter values on the abscissa. Group I (the original model
building and internal validation sample) data are indicated by open circles
and Group II (the external validation sample) data are indicated by filled ci r-
cles. The line of identity is plotted on each graph. The model equations are
also given for each figure.
When examining these figures, it should be bome in mind that the group I
data were used to develop the model equations whereas the group II data had
no influence on the predictions. Therefore the distribution of the filled circles
about the line of identity is of greater interest.
For the CL models, inspection reveals that the open and filled circles distri b-
ute near the identity line with a tendency to overpredict low measurements
and to underpredict high measurements for models 1, 2, 3, 5, and 6. For
model 4 on the other hand, the open and filled points scatter more regularly
above and below the identity line but the deviations from that line are greater.
For the Vd models, inspection reveals a tendency for the open and filled data
points to scatter about the line of identity and to overpredict low values and
underpredict high values for models 1,2,3, 5, 6, and 7. Careful inspection of
Fig. 4 shows that the data points tend to congregate below the identity line for
model 4 indicating a tendency to underpredict all Vd values. There were three
patients (two in Group 1 and one in Group II) with very high measured Vd va I-
ues whose data points fell below the identity line indicating that they were not
well accommodated and were underpredicted by all of the models. Models 4,
26
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5, 6, and 7 accommodated the highest of these Vd values better than models
1, 2, or 3.
27
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Gentamidn CL, Model 1
12-
1 0 -
s
8<
2 6
| 4
£ 4
2
^r. W r n Z Z O .
0 2 4 6 8 10 12
Measured (L/h)
Groups: 1, 2 Identity line: 3 «
Fig. 1: Gentamicin CL (L/h) model 1 predictions vs. observations
(G roups 1 and 2). Predictions of CL are depicted on the ordinate and o b-
served measurements are shown on the abscissa. Group 1 model building
data are the open circles, Group 2 validation data are the filled circles, and
the dotted line is the identity line. Log(CL) = >0.313 + 0.528 log(weight) or
CL = 0.731 weight0528.
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Gentamicin CL, Model 2
12 -
10:
1 8
1 6
u
■ g 4
£
2
0 2 4 6 8 10 12
Measured (M i)
Groups: 1 , 2 Identity Line: 3 »»« l . . . 2 — 3
Fig. 2: Gentam icin CL (L/h) model 2 predictions vs. observations
(Groups 1 and 2). Predictions of CL are depicted on the ordinate and o b-
served measurements are shown on the abscissa. Group 1 model building
data are the open circles, Group 2 validation data are the filled circles, and
the dotted line is the identity line. CL = +2.44 + 0.0664 weight.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Gentamicin CL, Model 3
1 2
t
i
1 0 j
§
i
*i
T 3
i
■ 8
6-
U
© «i
a
2
0
0 2 4 6 8 10 12
Measured (IVh)
Groups: 1 , 2 Identity Line: 3 ° ° « i • • • 2 — 3
Fig. 3: Gentamicin CL (L/h) model 3 predictions vs. observations
(Groups 1 and 2). Predictions of CL are depicted on the ordinate and o b-
served measurements are shown on the abscissa. Group 1 model building
data are the open circles, Group 2 validation data are the filled circles, and
the dotted line is the identity line. CL = 4.04 BSA.
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Gentamicin CL, Model 4
12
10'
t
i
i
&
1 6
£
£ 4
2
0
• •
0 2 4 6 8 10 12
Measured [ L / h ]
Groups: 1 , 2 Identity Line: 3 <
1 • • • 2 ------ 3
Fig. 4: Gentamicin CL (L/h) model 4 predictions vs. observations
(Groups 1 and 2). Predictions of CL are depicted on the ordinate and o fa -
served measurements are shown on the abscissa. Group 1 model building
data are the open circles, Group 2 validation data are the filled circles, and
the dotted line is the identity line. CL = 0.844 CLcr.
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Gentamicin CL, Model 5
1 2
10]
§ 8 j
t S j
nd
0 3 6 J
.a
i 4
2
0
° •
o
/ - o
®wii§'£KcP*
° /
0 2 4 6 8 10 12
M easured (l/h )
Groups: 1 . 2 Identity Line: 3 »oo 1 m 2 — 3
Fig. 5: Gentamicin CL (L/h) model 5 predictions vs. observations
(G roups 1 and 2). Predictions of CL are depicted on the ordinate and o b-
served measurements are shown on the abscissa. Group 1 model building
data are the open circles, Group 2 validation data are the filled circles, and
the dotted line is the identity line. Log(CL) = +1.08 + 0.0122 w eight o r CL
2.94 exp(0.012 weight).
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Gentamicin CL, Model 6
1 2
10
§ 8
3
6
4
£
2
0
0 2 4 6 8 10 12
M easured (iVh)
Groups: 1 . 2 Identity Line: 3
O O O 1 • • • 2 — 3
Fig. 6: Gentamicin CL (L/h) model 6 predictions vs. observations
(Groups 1 and 2). Predictions of CL are depicted on the ordinate and o b-
served measurements are shown on the abscissa. Group 1 model building
data are the open circles, Group 2 validation data are the filled circles, and
the dotted line is the identity line. Log(CL) = -0.446 + 0.00611 height +
0.00685 weight or CL = 0.96 exp(0.0061 height) exp(0.0068 weight).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Gentamicin Vd: Model 1
40f
I
O t . --------- — ,------------- ----------------
0 10 20 30 40
M easured (L)
Groups: 1. 2 Identity Line: 3 oooi«*«2 — 3
Fig. 7: Gentamicin Vd (L) model 1 predictions vs. observations (Groups
1 and 2). Predictions of Vd are depicted on the ordinate and observed mea s-
urements are shown on the abscissa. Group 1 model building data are the
open circles, Group 2 validation data are the filled circles, and the dotted line
is the identity line. Log(Vd) = -0.0694 + 0.654 log(weight) or Vd = 0.933
w eight0,654.
34
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Gentamidn Vd: Model 2
40f
i
_ 3 o |
j i
"d
O-'L , --------- . --------- .
0 10 20 30 40
Measured (L)
Groups: 1, 2 Identity lin e: 3 oool . •. 2 — 3
Fig. 8: Gentamicin Vd (L) model 2 predictions vs. observations (Groups
1 and 2). Predictions of Vd are depicted on the ordinate and observed meas
urements are shown on the abscissa. Group 1 model building data are the
open circles, Group 2 validation data are the filled circles, and the dotted line
is the identity line. Log(Vd) = -6.03 -1.21 log(albumin) +1.99 log(height) or
Vd = 0.0024 albumin'1 * 2 1 height*1* 9 9 .
35
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Gentamicin Vd: Model 3
40f
t
!
t
_ 3 0 l
0 ^ --------------
0 10 20 30 40
M easured (L)
Groups: 1 , 2 Identity Use: 3 ==■«! • • • 2 — 3
Fig. 9: Gentamicin Vd (L) model 3 predictions vs. observations (Groups
1 and 2). Predictions of Vd are depicted on the ordinate and observed mea s-
urements are shown on the abscissa. Group 1 model building data are the
open circles, Group 2 validation data are the filled circles, and the dotted line
is the identity line. Vd = 8.88 BSA.
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Gentamicin Vd: Model 4
40:
i
Measured (L)
Groups: % 2 Identity line: 3 1 • • • 2 — 3
Fig. 10: Gentamicin Vd (L) model 4 predictions vs. observations (Groups
1 and 2). Predictions of Vd are depicted on the ordinate and observed mea s-
urements are shown on the abscissa. Group 1 model building data are the
open circles, Group 2 validation data are the filled circles, and the dotted line
is the identity line. Vd s 0.246 w eight.
37
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Gentamicin Vd: Model 5
40-
30i
TJ
%20
^ 10
P o
0 10 20 30
M easured (L)
Groups: 1. 2 Identity Line: 3
40
2 — 3
Fig. 11: Gentamicin Vd (L) model 5 predictions vs. observations (Groups
1 and 2). Predictions of Vd are depicted on the ordinate and observed mea s-
urements are shown on the abscissa. Group 1 model building data are the
open circles, Group 2 validation data are the filled circles, and the dotted line
is the identity line. Log(Vd) = +1.64 + 0.0156 w eight or Vd =
5.16 exp(0.0156 weight).
38
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Gentamicin Vd: Model 6
40r
Oci----------— — , --------- -------------
0 10 20 30 40
Measured (L)
Groups: 1 , 2 Identity line: 3 »«• 1 • • • 2 — 3
Fig. 12: Gentamicin Vd (L) model 6 predictions vs. observations (Groups
1 and 2). Predictions of Vd are depicted on the ordinate and observed mea s-
urements are shown on the abscissa. Group 1 model building data are the
open circles, Group 2 validation data are the filled circles, and the dotted line
is the identity line. Log(Vd) = -0.526 + 0.0000632 weight2 + 0.0257 height -
0.0000566 height2 + 0.237 albumin or Vd = 0.591 exp(0.0000632 weight2 )
exp(0.0257 height) exp(0.0000566 height2 ) exp(0.237 albumin).
39
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Gentamicin Vd: Model 7
40
r i
^ 30;
J !
T3 i
*§ 20
• S 3
d :
^ 10
o
0 10 20 30 40
Measured (L)
Groups: 1, 2 Identity Line: 3 oooi • • • 2 — 3
Fig. 13: Gentamicin Vd (L) model 7 predictions vs. observations (Groups
1 and 2). Predictions of Vd are depicted on the ordinate and observed mea s-
urements are shown on the abscissa. Group 1 model building data are the
open circles, Group 2 validation data are the filled circles, and the dotted line
is the identity line. Vd = *0.154 - 2.87 albumin + 0.174 height - 0.000610
height2 + 6.72 creatinine + 0.00173 weight2 + 0.604 postop days - 2.39
BMI > 25 (0,1).
40
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3.6 External Validation: Tables.
The results of the external validation using Group II are summarized in Table
Vila (bias of CL predictions), Table Vllb (bias of Vd predictions), Table Villa
(precision of CL predictions), and Table V llib (precision of Vd predictions).
None of the predictive models for CL were statistically significantly biased (p
> 0.20). Volume of distribution model 4 (Vd = 0.246 weight) was statistically
significantly biased with a median prediction error of >19.4 percent (p=.002).
The external bias of Vd model 4 is evident in the disposition of group II data
points in Fig. 10 whereas there is no internal bias evident in th e group I data
points. The other Vd models were statistically signifi cantly unbiased (p >
0.37).
Clearance model 1 (CL = l3ow6'9 ^ 1 . RPRESS 5.13, PRESS 103.1, median
APPE 14.1%), CL model 3 (CL = I3 0BSA, RPRESS 4.43, PRESS 103.9, and
median APPE 19.7%) and CL model 6 (CL = exp(f3o + & 1 height + f32we'9ht) >
RPRESS 5.15, PRESS 105.6, median APPE 16.1%) appear the most precise.
Examination of the corresponding Figs. 1, 3, and 6 suggests there is little to
choose between these three models. The principle of parsimony would favor
CL model 1. Clearance model 4 (CL = SgCLcr, RPRESS 11.80, PRESS
290.9, and median APPE 22.3%) was the least precise and this is quite ev i-
dent in Fig. 4.
Volume of distribution model 1 (Vd = SgweightBl, RPRESS 6.54, PRESS
897.8, median APPE 17.7%), Vd model 5 (Vd = exp(&o + Bi weight), RPRESS
6.45, PRESS 924.9, median APPE 21.8%), and Vd model 6 (Vd = exp((30 +
weight + 02height - S3height2 - S 4albumin), RPRESS 6.45, PRESS 853.6,
median APPE 25.0%) appear most precise. Examination of the correspond
ing Figs. 7, 11, and 12 indicates that Vd model 1 works very well with ind i-
41
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viduals with lower values for Vd but underpredicts for individuals with very
high values for Vd. Model 5 and 6 so slightly better but Model 1 is more pa r-
simonious and appears to perform best overall. The least precise was Vd
2 2
model 7 (Vd * 3q + I3i albumin + B2height + ^h e ig h t + B4Scr + Bsweight +
Bgpostop days + B7(BMI > 25, RPRESS 10.03, PRESS 1026.1, median APPE
30.2%).
42
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Table Vila: External Bias of Models for Gentamicin CL in Pediatric Patients with Appendicitis.
Model Form N
Median
PPE
25,h
%ile
75t h
%ile
P
value* Comment
1. C Lha, = e^O Wt®
53 -2.3 -18.7 8.3% 0.45 Unbiased
2. CLh a t = Go + B-jWt 53 +3.2% -19.2% 14.9% 0.80 Unbiased
3. C Lhat = BqBSA
53 -5.6% -21.1% 9.9% 0.20 Unbiased
4. CLh a t = BgCLcr 53 -6.0% -20.3% 28.9% 0.80 Unbiased
5. C Lha, = efy) eR1Wt
53 -1.7% -23.9% 12.7% 0.52 Unbiased
6. C Lhat = e^O eR1Ht efi2w t
53 -1.3% -22.9% 10.6% 0.41 Unbiased
* Wilcoxon signed rank, H q'. Median % prediction error is zero.
6
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Table Vllb: External Bias of Models for Gentamicin Vd in Patients with Appendicitis.
Model Form N
Median
PPE
25t h
%i!e
75,h
%ile
P
value*
Comment
1 .V d h a , = e&0Wt&1
53 -1.0% -24.0% 10.4% 0.37 Unbiased
2. Vdha , = e^O Ht^1 Alb&2
53 -1.3% -22.0% 29.6% 0.91 Unbiased
3. V dhat = BqBSA 53 4.4% -16.7% 15.6% 0.54 Unbiased
4. V dh at = B0Wt 53 -19.4% -33.4% 3.0% 0.002 Biased
5. V dha, = e^O e^1 Wt
53 -4.3% -23.7% 21.0% 0.44 Unbiased
6. V dha, = e&0 e&1w t e ^ H t
x e ^ H t2 eB4Alb
53 -4.7 -25.3% 22.7% 0.66 Unbiased
7. V dha, = B0 + B1 Alb+ B2Ht
+ B3Ht2 + B4Scr + B5Wt2
+ BgPostop Days
+ By[BMI>25 (0,1)]
53 -4.0% -28.9% 32.2% 0.91 Unbiased
* Wilcoxon signed rank, H q: Median % prediction error is zero.
s
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Table Villa: External Precision of Models for Gentamicin CL in Pediatric Patients with Appendicitis.
Model Form N
Median
APPE PRESS RPRESS
P
value*
Comment
1. CLh a t = eB0W t6
53 14.1% + 103.1 f 5.13
—
Best overall
2. CLh a « = Bo + B ^ t 53 17.3% 104.3 6.29 0.74
3. CLhat= BoBSA
53 19.7% 103.9 4.43 f 0.23 Best by RPRESS
4. CLhat = BgCLcr 53 22.3% 290.9 11.80 0.01 t Worst
5. CLhat = e^O e&1w t
53 18.4% 116.8 6.57 0.48
6. CLhat = e&0 e ^ l Ht eBiW t
53 16.1% 105.6 5.15 0.54
* Wilcoxon signed rank, Hq: Median difference from model 1 is zero,
t Indicates the best values for goodness of prediction statistics.
t Significance Level: a =.05/5=.01 by Bonferonni inequality.
Si
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Table Vlllb: External Precision of Models for Gentamicin Vd in Pediatric Patients with Appendicitis.
Model Form N
Median
APPE PRESS RPRESS
P
value*
Comment
1. Vdh a I = eB0 WtB1
53 17.7% 897.8 6.54
—
Best overall
2. Vdh a t = eB 0 HtB1 AlbB2
53 24.2% 907.5 8.26 .12
3. Vdhat = BqBSA 53 16.7% t 985.6 7.42 .13 Best by APPE
4. V d h a « = B0Wt 53 26.0% 1530.6 6.90 .02
5. Vdhat = eB0 eB1w t
53 21.8% 924.9 6.45 f .88 Best by RPRESS
6. Vdhat = eB 0 eB1w t eB2Ht
x eB3^ 2 eB4^lb
53 25.0% 853.6 f
6.451
Best by PRESS and
RPRESS
7. Vdhat = B0 + B1A,b + B2 Ht
+ B3Ht2 + B4S cr+B 5Wt2
+ B0Postop Days
+ B7[BMI>25 (0,1)1
53 30.2% 1026.1 10.03
•O H
Worst
4*
O )
* Wilcoxon signed rank, Hq'. Median difference from model 1 is zero,
t Indicates the best values for goodness of prediction statistics.
$ Significance Level: « =.05/6=.008333 by Bonferonni inequality.
DISCUSSION
4.1 Aminoglycoside Dosing Considerations.
Over much of the past two decades, the prevailing point of view among phar-
macokineticists has been that the variation in gentamicin pharmacokinetics
precludes meaningful prediction of pharmacokinetic behavior for an individual
patient based on average population models. From this premise, it was sug
gested that dosage regimens must be individualized based on timed drug
administrations and two or more measured serum samples (Sawchuk et at.;
1977).
A recent innovation in the approach to dosing aminoglycosides is to focus on
attaining a therapeutic peak serum concentration in the range of 10 to 15
mg/L and then wait 24 hrs between doses. With this strategy, no attempt is
made to control the entire concentration-time profile. Instead reliance is
placed on obtaining a serum concentration measurement at the middle of the
dosing interval and comparing it with a desired target range. Population
pharmacokinetic models can be used to determine initial dosing regimens
using either of these two dosing strategies.
It should be noted that optimal target serum concentration profiles for gen
tamicin have probably not been completely defined for pediatric or adult ap
pendectomy patients. In addition, it is not clear that all appendectomy patients
have the same dosage needs.
4.2 Internal and External Sources of Pharmacokinetic Variation.
Clinical researchers have observed extensive variation in measured amino
glycoside pharmacokinetic parameters in certain patient populations. The
most prominent sources of variation are internal to the patient: 1) abnormal or
47
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changing renal function which alters CL and 2) variation in body composition
which effects Vd.
Any attempt to obtain serum concentrations to estimate individual pharma
cokinetic parameters is compromised to some extent by sources of variation
which are external to the patient: 1) variation from stated content of dosage
admixtures, 2) error in assayed concentrations, 3) inexact timing of dosage
administration, 4) inexact timing of serum sampling, 5) poorly designed
sampling strategy (D'Argenio et a/.; 1981).
4.3 Bayesian Regression Analysis of Serum Concentrations.
Serum sampling has associated costs in the form of capital expenditures, al
location of clinician time from other patient-related activities, and patient dis
com fort In order to help minimize these costs and reduce the effect of exter
nal sources of variation, nonlinear regression programs have been written
with Bayesian convergence criteria which provide meaningful posterior pa
rameter estimates regardless of the availability of measured concentrations.
By incorporating additive terms in the convergence criterion which square
deviations from the expected values of pharmacokinetic parameters and then
divide by estimates of their variances, these programs attempt to provide a
balance between what is known and what is unknown about each patient
(Sheiner et al.; 1979) (Katz et ah; 1981). To utilize the model equations pro
vided here for Bayesian estimation, the model prediction gives the expected
value and the percent coefficient of variation may be used to calculate the
variance. When no serum concentrations are available, the posterior model is
the same as the prior. When many serum concentrations are available, the
posterior may be quite different from the prior.
48
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4.4 Multiplicative Models and Allometry.
Simple multiplicative models have found applications in allometry
(Boxenbaum; 1984) (Rowland M; 1985). The simplest alio metric equation is
given by Y = «xG, where « and 3 are parameters, X is the sole covariate, and
Y is the variate of interest. For pharmacokinetic variates where X is a mea s-
ure of body size, « is quite variable from drug to drug whereas 3 tends to be
more consistent. The value of 3 is typically near 0.75 for CL and near 1.0 for
Vd according to one author (Ings; 1990). The data presented here are some
what but not entirely consistent with this idea since p was 0.53 for the allomet-
ric CL model and 0.62 for the allometric Vd model.
A previous study of pooled gentamicin Vd data from various research groups
indicated that Vd can be explained using an allometric equation (Keller;
1989). The equation was developed from 183 Vd observations in human i n-
o 79
fants, children, and adults. The model equation was Vd = 0.57 weight ' and
it explained the data better than a simple proportion or a linear-linear model.
This equation overpredicted Vd in the subjects who were over 100 Kg and
probably obese and underpredicted Vd in two subjects with very large Vd
measurements.
4.5 Managing Outlying Data.
One patient included in the model building group had a Vd of 39L. This p a-
tient was a 14 year old male who had sepsis and a bilateral pleural effusion.
He was intubated in the emergency room and admitted directly to an intensive
care bed. An ultrasound study of the abdomen showed the presence of an
abdominal fluid collection in the lower right quadrant. After surgical explor a-
tion, he was diagnosed with a perforated appendix and peritonitis. Although
he appeared to be an outlier in the models for predicting Vd, the Cook D was
49
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always less than 0.3 indicating that there was no excessive degree of infl u-
ence on the final parameter estimates for any Vd model. Despite having a Vd
which was poorly accommodated by the fitted models, this patient’s CL was
easily explained using models incorporating body weight or body surface
area. He was, however, an outlier for the CL model incorpo rating CLcr. His Vd
in L/Kg was high and the serum creatinine may have been diluted by an i n-
crease in extracellular fluid which generally accompanies sepsis and is co n-
sistent with the clinical findings of a pleural effusion and abdominal fluid. This
would result in a gross overestimate of CLcr. Because of the lack of influence
of his data on the models, the patient was retained in the model building
group.
4.6 Best Models.
The best models were selected on the basis of their relative lack of bias, their
goodness of precision, and their simplicity. That is to say that when more than
one model appeared to perform well, the best model was selected based on
the principle of parsimony. For the two variates of interest, simple allometric
forms that incorporate information about body weight alone appeared best.
These two models are depicted graphically in Figs. 14 through 17. Examin a-
tion of Figs. 14 and 16 shows that the models fit the data well for patients
weighing about 15 to 80 Kg. Above this weight, the models tend to underpre-
dict the measured values. Figs. 15 and 17 show the linearization of the data
after application of the log-log transform.
The allometric log-log models and the log-linear models required less predi c-
tors to explain Vd when compared to the linear-linear models. This may be an
artifact due to the presence of two individuals in the model building group with
very high measured volumes of distribution (Fig. 16).
50
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Although albumin was a statistically significant predictor for the C p best log-
log, log-linear, and linear-linear models of Vd, it did not appear to lend any
advantage when studied in the external validation group. Based on the inte r-
nal validation, however, it does appear to lend slight advantage to Vd models.
Since the excess body water observed with sepsis is expected to dilute serum
albumin concentrations, it may be prudent to incorporate albumin information
when available to predict Vd in patients with moderate to severe sepsis. Most
of the patients in this series do not fit this profile. Neither obesity nor gender,
both of which are expected to affect body composition and body water, had
any influence on models for CL or Vd.
Conclusion
Gentamicin pharmacokinetic parameters and serum concentrations in pediat
ric appendicitis patients were well-explained using allometric models incorpo -
rating weight as the sole predictor variable. Other models were identified as
well. These models should prove useful for selection of initial dosage reg i-
mens and for Bayesian dosage adjustment in similar patients. These popul a-
tion pharmacokinetic models appear well suited for predicting the pharm a-
cokinetic disposition of aminoglycosides in pediatric patient groups where
size ranges broadly. This is not unexpected since allometric models are well
accepted for predicting pharmacokinetic behavior in different species by
scaling variates of interest to body size.
5 1
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Gentami cin CL: Model 1
1 2 -r
I
i
i
0 20 40 60 80 100 120
Total Body Weight, K g
Measured 1, Predicted 2, 95% □ 3,4 • • • 1 2 3 4
Fig. 14: Gentam icin CL vs. total body weight (Group 1). Clearance is de
picted on the ordinate in L/h and total body weight is given on the abscissa in
Kg. The solid line is the line of best fit and the dotted lines on either side are
the 95% confidence range for the mean.
52
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Gentami ci n CL: Model 1
100 *
CL = 0.73 Weight
10 100 1000
Total Body Weight, Kg
Measured 3, Predicted 2, 95% Q 3,4 • • • 1 2 3 4
Fig. 15: Log-log transform of gentamicin CL vs. total body weight (Group
1). Clearance is given on the ordinate in log(L/h) and total body weight is
given on the abscissa in log(Kg). The solid line is the line of best fit and the
dotted lines on either side are the 95% confidence range for the mean. Co m-
parison of this figure with Fig. 14 shows how the log-log transform linearizes
the relationship between CL and weight.
53
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Gentami dn Vd: M odel 1
N * 79
Q
1 0
0.65
Vd = 0.93 Weight
80 100 120 60 20 40 0
Total Body Weight K g
Measured 1, Predicted 2, 95% Q 3,4 • • • 1 2 3 4
Fig. 16: Gentamicin Vd vs. total body weight (Group 1). Volume of distri
bution is depicted on the ordinate in L and total body weight is given on the
abscissa in Kg. The solid line is the line of best fit and the dotted lines on e i-
ther side are the 95% confidence range for the mean. The model appears to
accommodate the relationship between Vd and weight except where Vd is
very large.
54
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Gentamicin Vd: Model 1
100
£
N = 79
0.65
Vd = 0.93 Weight
10 100 1000
Total Body Weight, Kg
Measured X Predicted 2, 9596 Cl 3,4 • • • 1 2 3 4
Fig. 17: Log-log transform of gentamicin Vd vs. total body weight (Group
1). Volume of distribution is given on the ordinate in log(L) and total body
weight is given on the abscissa in log(Kg). The solid line is the line of best fit
and the dotted lines on either side are the 95% confidence range for the
mean. Comparison of this figure with Fig. 16 shows how the log-log transform
linearizes the relationship between Vd and weight.
55
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Appendix: Biometric Methods
5.1 Correlation Coefficient, R 2 .
R2 measures the degree to which a regression model fits a data set. More
precisely, it is the amount of variation in the dependent variable which is e x-
plained by the model.
R2=1 - (SSE + SST).
SSE is the sum of squared errors and SST is the corrected total sum of
squares. The addition of predictor variables always increases R 2 and this
makes it difficult to compare models with different numbers of predictor vari
ables (Glantz and Slinker; pp 245-254).
5.2 Adjusted Correlation Coefficient, tfadj.
R2 adj attempts to compensate for this effect of additional predictors by impos -
ing a penalty for each added variable.
R2 adj = 1 ■ MSE + MStotal = 1 - { [ SSE + (n-k-1) ] [ SST (n-1) J}.
The value of n is the number of independent observations and k is the nu m-
ber of independent variables in the model. Addition of an independent var i-
able will decrease SSE but R2 adj W 'H not increase unless this more than off
sets the effect of the lost degree of freedom which causes n-k-1 to de -
crease. The best model using this criterion is the one with the largest R 2 a(jj.
(Glantz and Slinker; pp 245-254).
56
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5.3 Mallows Cp Criterion.
The Mallows Cp criterion may also be used to help select the best model,
Cp = [ SSE(p) + MSE(k) ] - n + 2p + 2.
The symbol SSE(p) is the sum of squared errors with p predictor variables,
MSE(k) is the mean squared error for the maximum model with k predictor
variables, and n is the number of independent observations. The Mallows C p
approaches p + 1 when MSE(p) is close to MSE(k). The minimum value of Cp
is 2p - k + 1 where p is the number of independent variables in the restricted
model. The best model is indicated when Cp is near p +1 (Glantz and
Slinker; pp 245-254).
5.4 Studentized Residual, r(.
Studentized residuals can be used to detect outlier data points. The ith st u-
dentized residual (rj) is found by dividing the residual (e j) by its standard er
ror,
r\ = e ; + ( s x S Q R T ( 1 - h j ) ) ,
h j = X j ( X ’ X ) “ 1 x j \
The symbol s is the root mean squared error, SQRT is the square root o p-
erator, and hj is the ith leverage specified in matrix notation. To calculate h j, X j
is the ith row of regressors from the data matrix, X is the complete data matrix,
and xj and X are their respective transposed matrices. Outliers have stude n-
tized residuals with absolute values of 2 or more (Glantz and Slinker; pp
136-141).
5 7
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5.5 Cook Distance Statistic, D.
Outliers may or may not have undue influence on parameter estimates de
pending upon how far they are from the center of the independent data space.
The Cook distance studies influence by calculating the effect on the regres
sion parameter estimates when a point is omitted. The ith Cook distance (Dj)
is found by
Di = I R*i s- (k+1) ] x [ hi * (1-hj) J,
where k is the number of independent variables, r; is the ith studentized re
sidual given above, and hj is the ith leverage given above. Values of Dj
greater than 1 are considered troublesome and values above 4 are consid
ered overly influential. (Glantz and Slinker; pp 136-141) (Kleinbaum et al.;
1988, pp. 200-1).
5.6 Variance Inflation Factor, VIF.
The VIF is a simple transform of the multiple correlation coefficient (R2 j) for a
regression of the independent variable of interest against all the other inde
pendent variables in the model.
VIFj = 1 -= -(1 -R *j).
Either VIF or Rj may be used to look for multicollinearity. The VIF indicates
whether the variance of the regression parameter for the excluded independ
ent variable is inflated due to redundance with the other independent vari
ables. A VIF of one means R2 j = 0 and indicates no redundance. A VIF of 10
or more means R2 j = 0.9 or more and indicates great uncertainty in the pa
rameter estimate. Hypothesis tests on parameter estimates with multicolline
arity are unreliable (Glantz and Slinker; pp 191-193).
58
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5.7 Predicted Residual Error Sum of Squares, PRESS.
The PRESS(.j) statistic has been proposed to internally validate a pre dictive
regression equation in cases where testing in an independent data set is i m >
possible. It is calculated as
PRESS(.i) = E (ei+ ( 1 - hj) f s j (e(.j))a
where ej is the ith residual error and h j is the ith leverage as defined above.
The ratio of ej to 1 - hj conveniently gives the predicted residual for the ith o b-
servation where the ith observation is dropped from the regression e (.j). The
model with the smallest PRESS(.j) may be considered best for ability to pre
dict (Glantz and Slinker; pp 249-250).
In this paper, PRESS without the (-i) subscript is used to denote the predicted
residual error sum of squares when a predictive model developed from me m-
bers of a group is used to estimate the dependent variable of interest for
members of another group. The following formula applies:
PRESS = Z (ej)2
5.8 Relative Predicted Residual Error Sum of Squares,
RPRESS.
RPRESS is a modification of the PRESS statistic devised for this paper. The
author does not know whether it has been previously used and described. It
expresses the signed, predicted residual error as a fraction of the measured
value of the dependent variable. Thus in the formula above for PRESS, ej is
replaced by e \ + Yj, where Yj is the value of the dependent variable for the ith
individual and e j = Yj - Yhatj. RPRESS(_j) is completely analogous to
PRESS(.j). Choosing models that minimize this squared error criterion gives
59
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no advantage to models which predict large Y j better than small Yj as is the
case for PRESS. Since the models in this paper are designed to predict var i-
ates which range broadly, RPRESS should be a better model selection crit e-
rion than PRESS.
5.9 Absolute Percent Prediction Error, APPE.
APPE is the absolute value of the prediction error e j expressed as a percent
age of the measured value of Yj.
APPE = I e j x 100 * Y j I ,
APPE(.j) = I e ( . j ) x 100 - s - Y j I.
The value of APPE(-i) may be used for internal validation and APPE may be
used for external validation. This statistic should also be a better model s e-
lection criterion than PRESS for the reasons given in section 5.8.
60
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Asset Metadata

Creator Gilman, Thomas Marvin (author)

Core Title Construction and validation of multivariable population pharmacokinetic models: Utility of allometric forms to predict the pharmacokinetic disposition of gentamicin in pediatric patients with app...

Contributor Digitized by ProQuest (provenance)

School Graduate School

Degree Master of Science

Degree Program Applied Biometry

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

Tag engineering, biomedical,Health Sciences, Pharmacy,OAI-PMH Harvest

Language English

Advisor Bernstein, Leslie (committee chair), D'Argenio, David (committee member), Sobel, Eugene (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-4690

Unique identifier UC11341305

Identifier 1381586.pdf (filename),usctheses-c16-4690 (legacy record id)

Legacy Identifier 1381586.pdf

Dmrecord 4690

Document Type Thesis

Rights Gilman, Thomas Marvin

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