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A physiologic model of granulopoiesis
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A physiologic model of granulopoiesis
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A PHYSIOLOGIC MODEL OF GRANULOPOIESIS by Karyn Alise Redekopp 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 (Biomedical Engineering) May 1995 Copyright 1995 Karyn A. Redekopp This thesis, written by !> . t o y O £ I! S< € k < ? p £ .................. under the guidance of Faculty Committee and approved by all its members, has been presented to and accepted by the School of Engineering in partial fulfillment of the re quirements for the degree of M A M C . ^ S C I t f S C . t . ....................................................... . & c (f.fr i.A . .fyr. D ate ___............................................ Faculty (Committee ......... ^ C h a irm a n M V • \ < o £ e J F * ~ ~ - Acknowledgments I would like to thank Dr. David D’Argenio for his quality advice, support, and encouragement, not only in relation to this research but throughout my college education, Dr John Rodman for proposing this area of research; Dr Merrill Egorin for the use of his experimental data; Dr. Larry Redekopp for the many morning discussions over a cup of coffee, and Mark Redekopp for his assistance with the preparation of this document Dedication This work is dedicated to my family for the support and encouragement they have provided me over the past 24 years , and in memory of my aunt, Cheryl Redekopp, whose fight with leukemia initiated my interest in this subject Contents Acknowledgments ii Dedications iii List of Tables v List of Figures vi 1 Introduction I 1.1 Basic Physiology.................................................................................................. 1 1.2 Experimental S tu d ies........................................................................................... 4 1.3 M o d e ls................................................................................................................... 7 14 Specific Aims.......................................................................................................... 12 2 Methods 17 2.1 Design of the M o d e l........................................................................................... 17 2 2 Steady State Sim ulations.................................................................................... 20 2.3 Sensitivity S tu d y .................................................................................................. 22 2.4 Simulation o f Drug T h e ra p y ............................................................................. 23 3 Discussion of Results 27 3.1 Steady State Sim ulations.......................... 27 3 .2 Sensitivity S tu d y .................................................................................................. 30 3 .3 Simulation o f Drug T h e ra p y ............................................................................. 33 4 Conclusion 48 4.1 Model S u m m a ry ................................................................................................. 48 4.2 Future W o r k ........................................................................................................ 49 Reference List 54 Appendix 57 iv List of Tables 1 1 Mitotic Index Data taken from Killmann, 1962 ..................................................... 14 1.2 Compartment Transit Times (CTT) taken from Rondanelli, 1967 14 2 1 Model Parameter V a lu e s.......................................................................................... 26 3.1 Compartment Size Ratios for Steady State Simulation with Varying CTTs . . . 36 3 .2 Comparison o f Steady State Simulation Values with Literature Values . . 36 v List of Figures 1.1 Granulocyte Production from Stem Cell to Blood Pool 14 1.2 Models o f Proliferative C om partm ents................................................................... 15 1.3 Cronkite and Vincent M o d e l.................................................................................... 16 1 4 Model o f Proliferative Compartments by Fokas e t a l . ............................................. 16 2 1 Physiologic Model of Granulopoiesis....................................................................... 25 3.1 Emergence T im e s ...................................................................................................... 37 3.2 Leukophoresis o f 50% Circulating N eutrophils..................................................... 38 3 .3 Response of Circulating Neutrophils to Variations in Production R a t e .. 39 3 .4 Response o f Circulating Neutrophils to Additional Cell Divisions 39 3 .5 Response o f Circulating Neutrophils to Variations in Proliferation Fractions 40 3.6 Response o f Circulating Neutrophils to Variations in Compartment Transit T im e s ........................................................................................................................... 41 3 7 Response o f Circulating Neutrophils to Variations in Maturing Band Cell Fraction 43 3 8 Response o f Circulating Neutrophils to Variations in Demargination Rate 43 3.9 Simulation o f Drug Therapy: Removal o f Active C e lls ......................................... 44 3.10 Simulation of Drug Therapy: Removal of Maturing C e l l s .................................. 45 3 1 1 Simulation of Drug Therapy: Removal of Non-proliferating C e l l s .................... 46 3 .12 Myelosuppression in a Patient Treated with Cyclophosphamide and GM-CSF 47 VI Chapter 1 Introduction 1.1 Basic Physiology Granulo'ytopoiesis is the process by which granulocytes are produced, proliferate, mature and are released to the circulating blood pool. This process occurs in the bone marrow and is believed to originate from a pluripotent stem cell capable o f producing erythrocytes, granulocytes, and thrombocytes. The predominant role of granulocytes is the phagocytosis o f microorganisms, especially in acute inflammation [1] Because neutrophils are the most abundant form of granulocytes, approximately 90% of circulating granulocytes, the two terms are often used interchangeably Once the granulocytic pathway is initiated, the cell matures through six developmental stages until it is released into the blood pool (Figure 11). These maturation stages consist of myeloblasts, promyelocytes, myelocytes, metamyelocytes, band, and segmented neutrophils. The first three cell types are capable o f proliferating although it has been suggested that metamyelocytes may also divide under certain conditions [2]. The last two stages are often referred to as the marrow storage pool from which large numbers of mature granulocytes can be released upon demand. 1 The blood granulocyte pool can be thought of as containing two sub- compartments, circulating blood granulocytes and marginated granulocytes, which rapidly equilibrate in healthy individuals. Marginated granulocytes are those cells that cannot be directly measured from the circulating blood, a portion of which adhere to the vessel walls. This was first discovered when only 50% of radio labeled neutrophils that had been injected into the blood stream could be detected [3] Exercise or administration of epinephrine will lead to an increased blood granulocyte concentration due to rapid release of granulocytes from the marginated pool [4], This pool o f marginated cells is considered to be slightly larger than the circulating granulocyte pool [S] and may provide the first source o f granulocytes to the circulating blood pool during infection. There is some speculation whether the marginated pool fills preferentially before the circulating pool in times of decreased neutrophil levels known as nuetropenia [3]. It may be possible that under certain conditions such as nuetropenia or some other system stress the circulating blood pool and marginated pool do not equilibrate rapidly and must be considered as two separate kinetic units [5] Cell death from the blood pool is a random process with a half life of 6.8 hours The proliferation scheme of immature granulocytes in the bone marrow has been an area of much disagreement. This is due, in part, to the anatomical difficulty of studying a process that occurs in the bone marrow However, there are data that lead to several possible conclusions. One fact that is generally agreed upon is a DNA synthesis time of 13-16 hours. When a cell culture is pulsed with tritiated thymidine, those cells undergoing DNA synthesis will incorporate the radioactive label into their DNA which allows the number of 2 cycling cells to be quantified. The labeling index is the ratio o f labeled cells to the total number of cells present A high labeling index indicates a majority o f cells in DNA synthesis It has been reported that both myeloblasts and promyelocytes have high labeling indices while the myelocyte has a much lower labeling index [6,7] Mitotic indices have been used to give an estimate o f the proliferation o f specific cell types By definition, the mitotic index is the ratio of dividing cells to the total number of cells Results from this technique have been varied due to the difficulty of counting mitoses and categorizing each cell class The data excerpted from KJllmann et al and reported in Table I I are the values generally accepted for mitotic activity [7,8] Some researchers use the frequency of mitosis to propose a proliferation scheme in which myelocytes have the largest number of mitoses. However, the use of frequency o f mitosis data can be misleading because it does not take the compartment size o f each cell class into account Others have proposed proliferation models based on the specific mitotic index which is the mitotic index of a particular precursor class normalized to the number of cells in that cell type Use of these data lead to a proliferation scheme with the greatest number of mitoses occurring at tne most immature cell type, the myeloblast This approach appears to be correct as it implements mitotic index with respect to the number of granulocytes of each specific cell class The kinetic properties of granulopoiesis are also important and have been studied using flash labeling with tritiated thymidine One parameter of interest is the emergence time defined as the time from uptake of label to the time o f emergence into the circulating blood pool. Often this reflects the time from the last myelocyte division until entrance to the blood A range for emergence times in healthy patients has been reported to be 3 between 96 and 144 hours, although values o f 273.6 , 204, and 151 hours have also been recorded [5,10,11,12], One research group reported the mean time from myelocyte DNA labeling to emergence in the blood to be 102 hours with a standard deviation of 13.8 hours [13]. As indicated, there exists a large discrepancy in emergence times which may reflect variations in mature granulocyte storage time in the bone marrow. During infection the emergence time is reduced and can be as low as 48 hours [2,14]. Another kinetic parameter is the compartment transit time originally defined by Killmann et al. and reported by Rondanelli et al. [9,15], The compartment transit time (CTT) is defined as the time from a cell’s entrance into a specific cell compartment to the time it, or its progeny, leave the cell compartment. The compartment transit times reflect an increase in mitotic time as the cell increases in maturity (Table 1 2) These researchers do not report CTTs for the non-proliferative compartments, although cell turnover times have been recorded for non-proliferative cells [14]. 1.2 Experimental Studies Much more can be learned about granulopoiesis by perturbing the system and observing the resulting dynamics, uncovering inherent control mechanisms within the system that are undetected in steady state behavior. The most common experimental methods used to study the dynamics o f granulopoiesis are x-radiation of the animal under study or leukophoresis, the removal o f circulating neutrophils. Other possibilities include inducing infection or injecting bacterial endotoxin [16,17,18]. 4 One experiment found an increasing number of non-segmented ceils were being released into the blood in response to leukophoresis [17], The appearance o f non segmented cells, mainly band cells, represents a possible primary recovery mechanism in which the body attempts to restore the normal circulating granulocyte concentration by releasing cells from the marrow reserve pool. The degree of non-segmented cell release was found to be related to the percentage decrease in blood pool granulocytes It was also noted that mitotic activity increased in the precursor cells, possibly as an attempt to repopulate a depleted marrow store Another experiment designed to study the dynamic state of granulopoiesis used irradiation to induce a state o f neutropenia [16]. A hind leg o f irradiated mice was shielded from the radiation and the subsequent recovery activity in the healthy bone marrow was studied. The basic premise of this experiment was that the radiation did not alter the normal granulopoietic regulatory mechanisms in the shielded marrow In their results, Morley and Stohlman conclude that the granulocyte production rate and the release rate from the mature marrow store are increased in neutropenia. They suggest that the increased production rate is a result o f increased stem cell differentiation into the myeloblast compartment and not of increased mitotic activity in subsequent granulocyte compartments. It should be noted that this conclusion was not based on measurement of mitotic activity but as a hypothesis for the four-fold increase in the number of myeloblasts and promyelocytes over a two to three day period. Previously, it had been thought that the granulocyte concentration in circulating blood followed an oscillating time course [19] This had been reported in a disease state known as cyclic neutropenia, but was also believed to occur in normal, healthy individuals. 5 The oscillations were reproduced by a computer model that incorporated two nonlinear feedback loops dependent on the circulating blood granulocyte concentration [20]. More recently the data was reassessed and found to produce oscillations in only 2 o f the 11 subjects [21], This leads to the conclusion that, although steady oscillations may occur in certain disease states, a stable neutrophil oscillation is not present in healthy individuals. Unlike erythropoiesis or thrombopoiesis, no basal granulopoietic regulator has been isolated. In the process o f red blood cell formation, erythropoietin is the molecule used in communication between red blood cell concentration and production of new erythrocytes. This communication molecule in granulopoeisis is still only hypothetical. A breakthrough in hematopoietic regulation occurred with the discovery o f colony stimulating factors or CSFs. The four stimulating factors, macrophage-CSF (M-CSF), granulocyte/macrophage-CSF (GM-CSF), granulocyte-CSF (G-CSF), and muhi-CSF or Interleukin-3 (LL-3), have stimulating effects on overlapping cell types The CSFs cause inactive cells to enter into DNA synthesis and decrease the cell cycle time. As a result, the number o f progeny of a granulocyte precursor cell can be directly related to the concentration o f CSFs administered in-vitro [22]. Besides a proliferative effect, CSFs also contribute to the differentiation of cells to committed cell lines, maturation of cells, and stimulate the functional activity of mature cells The CSF serum concentration in normal healthy individuals is usually very low, but does increase during infection. This indicates that CSFs may play a regulatory role in times o f system stress such as infection, but they are not proven to act as the granulopoietic regulator under normal conditions. However, these molecules, which have 6 been cloned and produced for therapy, are proving to be useful in patients that experience myelosuppression as a side effect of their treatment Although a regulatory pathway has not been explicitly defined for granulopoiesis, it is evident from the data reviewed above that some feedback communication is occurring. This is most noticeable in disease state when the demand for more granulocytes is increased During infection, when an increased number of granulocytes is needed, the emergence time for a granulocyte to pass from myeloblast to the blood is reduced to approximately half the normal value [2,14], This has been explained to be a result of decreased maturation times in the non-proliferating cell pools [2], As well, mitotic activity of granulocyte precursors can increase with the possibility of metamyelocyte proliferation [2,17] The proliferation fraction of myelocytes may increase during infection [2,6]. Granulocyte production rate has been shown to increase when the circulating neutrophil count is reduced, and the mature marrow store may release cells more quickly under the same condition [16] Band cells appear in the blood in large amounts during times of high demand [17]. The kinetic interaction o f marginated and circulating cell populations may also become important in the case of infection [3], 13 Models All of these experimental findings suggest many possible control mechanisms which may be acting. In order to investigate the effects o f these mechanisms, mathematical models have been proposed that describe the process o f granulopoiesis. 7 Perhaps the most diversity exists in the models of the proliferative compartments, due to the difficulty in obtaining information from the bone marrow. Other investigators have lumped compartments together in comprehensive granulopoeisis models in order to simplify the description o f the process. Two models o f the proliferative compartments were proposed in order to explain blood granulocyte specific activity (BGSA) curves in dogs that were injected with diisopropyl fluorophosphate, DFP32 (Figure 12) [18] In the first model, stem cell, myeloblast, promyelocyte, and myelocyte were all considered to have a self perpetuating stem cell quality. That is to say, upon division, a fraction of the progeny would mature to the next compartment and the rest would stay in the compartment and divide again. This hypothesis is contrasted with a model that has a self perpetuating stem cell and one division in each o f the myeloblast and promyelocyte compartments in which all the progeny mature to the next compartment. Myelocytes are considered to divide three times before maturing to become metamyelocytes Both these proliferation schemes were compatible with the BGSA curves. One o f the most widely accepted models o f granulopoiesis was proposed by Cronkite and Vincent (Figure 1 3) [14] Their model postulates an influx of cells into the myeloblast compartment and does not describe the action of the stem cell. The myeloblast and promyelocyte compartments contain one mitosis each while the myelocyte compartment has two divisions each requiring 54 hours. The total granulocyte transit time predicted by this model is greater than 250 hours. In a study on the growth fraction o f myelocytes, the Cronkite and Vincent model was unable to explain kinetic data because of the long myelocyte cycle times [6]. The 8 labeling index (LI) of myelocytes, as measured by Dresch and his colleagues, never rose above 50% o f the LI for myeloblasts and promyelocytes This can be interpreted in two ways. Either the myelocyte population has a prolonged Gl rest phase of the cell cycle or only a fraction o f the myelocytes are cycling. Dresch suggests that the long myelocyte cycle time is similar to a non-proliferating myelocyte population and proposes a myelocyte compartment transit time o f approximately one to two days with a proliferating myelocyte fraction near 0 5 This model predicts a peak tritiated thymidine specific activity at six days which corresponds to in-vivo data o f healthy individuals In an attempt to explain observed oscillations in blood granulocyte concentration, King-Smith and Morley proposed a model for granulopoiesis utilizing two feedback loops [20]. Their purpose for this model was not to simulate a granulocyte production scheme, but rather to demonstrate that the nature o f the feedback loops could produce stable oscillations such as those thought to occur in blood granulocyte concentrations. Although it has been shown that these oscillations rarely exist in healthy individuals, the influence of feedback loops remains an important area o f investigation Under normal neutrophil conditions, the King-Smith and Morley model reproduced stable low amplitude oscillations The two nonlinear feedback loops, dependent on the circulating granulocyte concentration, controlled the granulocyte production rate and a variable time delay representing marrow storage time of mature granulocytes. When the variable time delay loop was removed, the system produced unstable oscillations with increasing amplitude suggesting a damping influence by the variable time delay. The nature of the granulocyte production rate curve also influenced the output. If a less steep curve was chosen, the model resulted in damped oscillations. 9 No experimental basis was given for selecting the specific properties of the feedback relationship. It appears that the characteristics of the feedback production loop determine the oscillatory behavior Rubinow and Lebowitz propose a comprehensive mathematical model consisting o f five compartments [5]. They suggest that insufficient data exists to model each granulocyte cell class separately and have therefore lumped the proliferative cells together in active and resting compartments. From there, the cells must pass through maturation and resting compartments before reaching the blood pool. Unlike King-Smith and Morley, the release rate o f granulocytes to the active and maturation compartments is dependent on the total number of granulocytes in the system They postulate that increased production would occur if marrow cell counts as well as circulating neutrophil counts were depressed. The reserve compartment release rate is dependent only on the blood neutrophil count. Simulations of the BGSA curve led to several conclusions By choosing different values for the time of maturation and reserve compartment release rates the transit time of the non-proliferative pool was calculated to be 8, 9, or 10 days. The total transit time of the simulation was 10.25 days which compared favorably with the larger estimate o f 11 25 days reported in the literature [12], They found that increasing the maturation time lengthened the transit time, but no significant effect was seen by increasing the release rate. Rubinow and Lebowitz conclude that release from the reserve compartment has both “first in, first out’ ' and random characteristics, with the random characteristics being more dominant under system stress. 10 The dynamic response of the system to leukophoresis resulted in small amplitude damped oscillations in the blood pool and reserve compartment. The amplitude and period o f the response could be modified by the parameters describing the reserve store release rate. When increased leukophoresis was simulated, the degree o f initial overshoot seen in the system response was larger. Rubinow and Lebowitz point out that this oscillation or “ringing” is most likely due to the control elements dependent on the total neutrophil count Total body irradiation simulations also produced damped oscillations although the initial overshoot in response to the radiation was larger than that o f the leukophoresis simulation Oscillations o f this nature have not been reported in leukophoresis experiments, but the authors suggest that the total neutrophil population as a function o f time has not been amply investigated. Because these oscillations are of low amplitude and damp out slowly, cyclic neutrophil counts would rarely be measured in healthy individuals. O f course, in response to a significant neutropenia greater amplitudes o f oscillation occur and are more easily detected. Another reason for the absence o f cyclic neutrophil counts in patients may be due to the dynamic interaction between the circulating and marginated pool. The Rubinow/Lebowitz model neglects this interaction on the basis that little qualitative information exists on the subject. Another model o f the proliferative granulocyte compartments was developed in order to describe granulopoiesis in healthy individuals and in patients with Chronic Mylogenous Leukemia (CML) (Figure 1.4) [23], This model implements a proliferation scheme based on the specific mitotic index data. That is to say, the blast cell has the most divisions while the myelocyte has only one. The metamyelocyte is also capable of dividing 11 in this representation A constant production rate of myeloblasts from committed stem cells was assumed for the initial model, although cycling o f the stem cell was included later The unique characteristic of this model is the inclusion of cell populations within the proliferative compartment that do not divide but only mature. By choosing the proliferative fraction of myelocytes to be 0 5, the myelocyte compartment reflects the labeling index data reported by Dresch [6] This model was applied to CML in order to postulate how leukemic cells eventually outnumber healthy granulocytes The results show that by increasing the proliferation fraction of the dividing precursors a greater number of cells are produced No kinetic data was studied, only steady state values for cell populations were reported This model provides a possible mechanism for increased leukemic cell production and will be useful in designing further experiments in this area of research 1.4 Specific Aims The models mentioned above provide information about specific parts of the granulopoiesis process, most notably the proliferative stages of cell development However, a physiologic model representing the complete granulocyte production and maturation process does not exist. This type of model would be helpful in defining important parameters within the system and understanding their contribution to the overall process With this insight, the model may be used to simulate the granulopoietic 12 response to external factors such as chemotherapy in order to determine whether this model offers a viable means of understanding physiologic applications The model proposed by Fokas et al. offers a good starting point for the design of a physiologic model o f granulopoiesis because it describes the proliferative process in more detail The following points provide an outline of the approach that was taken to develop and understand this model 1 Expand the model of granulopoiesis proposed by Fokas et al. to include non proliferative and blood pool compartments 2. Compare steady state simulations o f the model with varying compartment transit times and several possible representations of the blood pool 3 Quantify the effects of parameter variations on the circulating neutrophil count in a sensitivity study. 4 Simulate circulating neutrophil response to possible mechanisms of drug action as seen in chemotherapy 13 Figure 1.1 Granulocyte Production from Stem Cell to Blood Pool myelobteet promyetocyto myelocyte metamyelocyte bend - O - O ^ o - c T - O Table 1.1: Mitotic Index data taken from Killmamv 1962 Cell Type miooo nucleated cells in bone marrow particle smear relative compt size relative frequency of mitoses it of cells in mitosis / 1000 nucleated cells specific mitotic index* myloblast 10.0 1.00 1.0 0249 0.0249 promyelocyte 33.7 3.37 2 0 0498 0.0148 myelocyte 163.0 16.30 7 1 1.778 0.0109 non-dividing * * 361.4 36 14 0 0 0 * specific mitotic index = fraction of specific cell type in mitosis at any time *• non-dividing marrow granulocytes - metamyelocytes to marrow segmented neutrophils Table 1.2: Compartment Transit Times (CTT) taken from Rondanelli, 1967 Minimum CTT (in hours) Maximum CTT (in hours) myeloblasts 31 promyelocytes 33 66 myelocytes 63 143 all mitotable granulocytopoietic cells 127 240 segmented 14 Figure 1,2: Models o f Proliferative Compartments cocnmtod stem cell n-prollferating cells myeloblast promyelocyte myelocyte commrted stem myeloblast promyelocyte myelocyte non-proliferating cell cells 15 Figure 1.3: Cronkite and Vincent Model committed myelobiaat promyelocyte myelocyte non-proiiferating item cell cells Figure 1.4: Model o f Proliferative Compartments by Fokas et al. Stem Myeloblast Promyelocyte .Myelocyte O ^ O r ^ O 16 I Chapter 2 Methods 2.1 Design of the Model In an attempt to obtain a more realistic model o f granulopoiesis we have expanded the model proposed by Fokas and his colleagues. A schematic representation of the model is shown in Figure 2.1. The proliferative precursor classes are comprised of parallel active and maturation compartments. A fraction of the cells that are produced by the stem cell will have proliferative capability and are passed to the first active myeloblast compartment, while the remaining cells are placed in the first myeloblast maturation compartment. After a specified time the active cells will divide but the progeny remain blast cells Again, a fraction of the newly divided cells will continue to have proliferative capablitiy and are passed to the next active blast compartment. The remaining cells are placed in the second blast maturation compartment along with the cells from the first maturation compartment. After the last blast division the cells become promyelocytes and are passed to the first promyelocyte active and maturation compartments. This process 17 continues throughout the proliferative granulocyte precursors. The number o f divisions assigned to each cell class was determined by specific mitotic index data. Although the metamyelocyte compartment is generally considered non proliferative, in the model it was allowed proliferative capability which could be useful in times o f system stress. The band cell population is non-proliferative and is used for further cell maturation. However, in this model band cells are capable of flowing directly to the circulating blood pool, bypassing the segmented cell compartment and offering a quick means o f boosting circulating neutrophil levels. The kinetic interaction of the circulating and marginated cell pools is an important consideration and is incorporated into the model. The rate constants for flow between the two compartments are assumed to be equal for simulation o f normal granulopoiesis. Granulocytes are eliminated from the blood pool at a rate calculated from the 6.8 hour neutrophil half-life. Although the action o f the stem cell can be modeled in detail, this representation assumes a constant stem cell production rate. It should be noted that physiologically there exists a pluripotent stem cell capable of producing erythrocytes, granulocytes, and thrombocytes. The stem cell shown in this model is the committed stem cell compartment which produces only granulocytes. Theoretically, the fraction of cells entering each successive active compartment should decrease from a value near one for the first myeloblast stage to approximately 0 5 at the myelocyte division [5]. In an attempt to simplify the model, the active fraction was assumed to be piecewise constant for each precursor. Ideally it should be possible for a fraction o f cells from the maturation compartments to become active again, but for simplicity this possibility was excluded. 18 The time allowed for transit through each proliferative precursor class (e.g. from blast to promyelocyte) was taken from compartment transit time data reported by Rondanelli [IS]. Mitotic times o f the proliferative cells are known to increase with increasing maturity and this is reflected in the compartment transit time. In the model, the time required for each division was calculated by dividing the compartment transit time of each cell class by the number o f cell divisions that occur within that cell class. The time calculated for one division was also assumed to be the same time for maturing cells to pass through the corresponding maturation compartment. The transit times for non- proliferating cells were inferred from the cell turnover times reported by Cronkite [14]. The mathematical representation of the previous description consists o f 18 first order differential equations. Each compartment follows first order kinetics. For example, the rate of change o f the number o f cells in a compartment is equal to the rate of cell entrance into that compartment minus the rate at which cells leave the compartment. The active and maturation compartments of the proliferative cells can be described as follows where the value of j represents the compartment number (e g. l-3=myeloblast, 4- 5=promyelocyte, etc.). < u , d t d t T„, ' T , d M j 2(1 M j ~ * ~ = t~> +^ 7 ' T 19 A, and Mj represent the number of cells in active and maturing compartment, fj represents the fraction o f cells which continue to be active, and Tj represents the division/maturation time. Multiplication of the A^i term by a factor o f two represents proliferation o f the cells in the Aj.i compartment. The number o f cells within each proliferative class was determined by summing the number o f cells in each compartment comprising that cell class. The non-proliferative cell classes also followed first order kinetics Because these cell types did not undergo any divisions, the number of cells within the compartment represents the number of cells o f that cell class. The model was simulated on a SUN SPARCstation2 The differential equations were solved by the Livermore Solver for Ordinary Differential Equations with Automatic Method Switching for Stiff and Nonstiff Problems (LSODA). A time step o f 0.1 hours was used with relative and absolute error tolerances of 10"6 Output data were analyzed using MATLAB. A complete set o f equations for the model is contained in the appendix 2.2 Steady State Simulations The model described above was used to determine the steady state values of each granulocyte compartment. The steady state response was then compared with values reported in the literature to determine if the model parameters were correctly chosen. The proliferation fractions were the same as those used by Fokas et al. In the case o f the metamyelocyte compartment the proliferation fraction was considered to by very small in 20 order to approximate this compartment as mostly non-proliferative. The contribution of cells to the blood pool from the band cell compartment was also considered very small under normal granulopoiesis conditions In the simulation performed by Fokas and his colleagues the division/maturation time, T, was considered to be constant and equal to 20. We investigate the steady state response of two sets o f compartment transit times The first set of CTTs represent the use o f the low end of the CTT spectrum reported in Table 12 while the second set of CTTs indicate the use o f the midpoint o f the CTT range. The parameter values given in Table 2 1 were used in these simulations. The number o f granulocytes in the blood pool was simulated by three different models. As mentioned previously, the blood pool can be thought o f as two separate compartments that equilibrate rapidly approximated as one kinetic unit. Although not enough is known about the rate o f margination and demargination of cells in the blood pool, it is thought that this process may be an active component in the system response to increased granulocyte demand. The first model assumed the blood pool was one kinetic unit and did not address the subject o f distinct marginated and circulating blood compartments The other 2 models included marginated and circulating blood compartments with equivalent rate constants allowing for transfer o f cells between the two compartments. These last two models differ from one another by the choice of which compartment the granulocytes are eliminated from the body. The time necessary for a cell to travel from the original production site to release into the blood is a parameter o f granulopoiesis that is not clearly determined. Many estimates exist. To investigate the total transit time o f the model an input o f cells, 21 analogous to an injection, was given at the level of the stem cell The simulation was begun at steady state Varying levels of increased cell production were allowed for the first hour of the simulation and then returned to the normal input value The transit time was then determined as the time at which the peak value o f cells within the blood compartment occurred 2.3 Sensitivity Study The circulating neutrophil count is the variable that is most easily measured experimentally and is used to describe the physiologic state of the system such as neutropenia Therefore, it is important to understand the effect of each parameter on the circulating nuetrophil count. Each parameter was varied by 20 and 50 percent of the benchmark value used in the steady state simulations while the other parameters were held at their original value A dynamic disturbance was simulated by removing 50% of the steady state value of circulating blood cells over a period of one hour The system response due to variations in each parameter was observed for 500 hours (20 8 days) The model was also evaluated for its response to increased proliferation by the addition of cell divisions Mathematically this was accomplished by allowing the proliferation constant, usually assumed to be two, to be increased to four This is not an ideal simulation of additional proliferation in that there is no additional time allowed for this division In other words, in the time normally allotted for one division two divisions actually occur Increased proliferation by one extra division in the blast, promyelocyte. 22 and myelocyte compartments individually was simulated The final simulation allowed three extra divisions to occur at the same time, one for each compartment. 2.4 Simulation of Drug Therapy One important application of granulopoiesis modeling involves chemotherapy. Often therapy must be discontinued due to the myelosuppressive effects of the therapeutic agents. An understanding of the induction o f myelosuppression by the chemotherapeutic agents and the recovery mechanisms inherent to the granulopoiesis process during and after chemotherapy may lead to the design o f more efficacious treatments In order to model this situation, the mechanism of drug action, which is different for specific drugs, must be known. Some agents are known as cycle specific drugs which suggests that they kill ceils in certain phases of cell division. This may be simulated by removal o f cells from proliferating cell compartments Other drugs are not specific for cell cycling and have another means of inducing cell death. The simulations that were implemented considered three broad cell classes - active, maturing, and non-proliferating - which would be affected by different drug mechanisms The active cell class contained those cell compartments that experienced cell divisions. The maturing cell class was considered to contain those cell compartments that belonged to proliferative cell types but were not undergoing cell division. These two broad cell categories would experience cell loss due to those drugs which acted on immature cell types or non-cycle specific drugs. The active compartment would also experience cell loss for drugs that were cycle specific and depended on the proliferating capabiltiy o f those 23 cells to produce cell death. The last broad cell category was considered to be the non- proliferating cells. These cells experience cell death by some other drug induced mechanism which might act on specific cell receptors displayed by the mature cells. The effect o f cell death in each o f the categories was simulated by a constant rate o f cell removal for 24 hours. This period o f time was chosen to correspond with experimental pharmacokinetic data from patients treated with Cyclophosphamide and GM-CSF [23]. The time period for a designated drug action can be modified depending on the dosing strategy implemented. Cells were removed over the 24 hour period from each compartment within the designated cell class at a specified rate for each simulation 24 Figure Z 1: Physiologic Model o f Granulopoiesis committed myeloblast promyelocyte myelocyte mete stem cell f f c f h V f p W *0-*— O— O-L-'O-*— C L b a n d • * 8 m e n t * d v r r n circu latin g , O i marginated Q ^ blood pool 25 Table Z I : Model Parameter Values Symbol Description Value < 1 stem cell production rate 4 0 xlO 6 ceUs/kg BW /hour h dividing blast cell fraction 0.8 f. dividing promyelocyte cell fraction 0.6 f - dividing myelocyte cell fraction 0 5 fm dividing metamyelocyte cell fraction 0 1 fbd maturing band cell fraction 0 9 T „ time for 1 blast cell division varied T P tune for 1 promyelocyte cell division varied T m time for 1 myelocyte cell division varied T m time for 1 metamyelocyte ceil division 30.3 hours T m time in band cell compt 49.9 hours tune in segmented cell compt 71.8 hours rate constant from circulating to marginated compartments 1.0 hours 1 k m c rate constant from marginated to circulating compartments 1.0 hours 1 k ' i rate constant for granulocyte elimination 0.102 hours 1 nb number of blast divisions 3 * * > number of promyelocyte divisions 2 » m number of myelocyte divisions 1 H m t \ number of metamyelocyte divisions 1 26 Chapter 3 Discussion of Results 3.1 Steady State Simulations The results from the steady state simulations show that the extended model of granulopoiesis compares favorably with several values reported in the literature. These results are displayed in Tables 3.1 and 3 2 Because this is a physiologic model of granulocyte production and maturation, the model variables that are incorporated may provide some insight into the biologic process that is occurring. A comparison o f the simulation results with values reported in the literature shows that a better approximation of compartment size is gained from using CTT data rather than a constant T,. The simulation produces a promeylocyteblast ratio closer to the literature value when the low end of the CTT range is used. However, the myelocyte:blast ratio is a much better approximation when the mid-point o f the CTT range is used, presumably because the longer myelocyte CTT allows an accumulation o f myelocytes within the compartment and therefore increases the ratio. This may suggest that the best choice for the promyelocyte CTT lies on the low end o f the range while the myelocyte 27 CTT is closer to the mid-point. These CTTs were used during the sensitivity study simulations The hypothesis that more cell divisions take place in the early proliferative compartments is justified by the steady state simulations results. Other models have assigned only one division to the blast and promyleocyte compartments while the myelocyte compartment generally has several divisions These assumptions were based on frequency of mitosis for each compartment However, use of the specific mitotic index which is normalized to compartment size seems to be a more appropriate choice This leads to the assignment of 3 blast divisions, 2 promyelocyte divisions, and 1 myelocyte division In fact, the steady state simulation results for compartment size coincide nicely with literature values when the number of cell divisions for each class follow the specific mitotic index Previously, Cronkite had proposed a long transit time for the myelocyte compartment which included two myelocyte divisions On the other hand, Dresch modeled this compartment as containing only one division with a proliferation fraction of 0 5 based on labeling index data The model that we have proposed includes aspects from both of the previous models The myelocyte compartment transit time of 103 hours is necessary to accommodate the low myelocyte proliferation fraction in order to achieve the correct compartment size A physiologic explanation of the long myelocyte transit time is not immediately apparent It is unclear if the myelocyte begins to increase the maturation process in order to exhibit more functional capability, but this seems to be a reasonable assumption 28 The modeling approach for the blood cell compartment has an effect on the asymptotic values o f blood pool granulocytes. When the blood compartment is treated as one kinetic unit, the simulation results in a value near that reported in the literature. However, when the blood pool is treated as two distinct compartments the number of granulocytes in the blood effectively doubles. This is related to the transfer rate constants between marginated and circulating compartments which brings these two compartments into equilibrium with one another. A slight difference in number o f circulating granulocytes is evident when the elimination rate constant is switched from the marginated to circulating compartment Based on physiology, it appears that the granulocytes should be eliminated via the marginated compartment as this would more accurately represent cell removal by the liver. More work, both experimental and theoretical, will need to be done in the area of demargination in order to provide a more realistic physiologic result The emergence time, defined as the time to circulating neutrophil peak after an increased production of cells by the committed stem cell, was approximately 260 hours or 10.8 days (Figure 3.1). This can be viewed as an approximation o f the time required by a cell to traverse the maturation steps in the bone marrow The estimate o f 10 8 days is reasonably good compared to the longer value o f 11 25 days reported in the literature, demonstrating that the transit times for each cell class are reasonable choices. 29 3.2 Sensitivity Study The response o f the model to a removal of 50% o f the initial circulating blood neutrophils for one hour was simulated with the parameter values given in Table 2.1. The blood neutrophil count had recovered to greater than 95% o f the steady state value within 50 hours (Figure 3 .2). The Idnetics o f the asymptotic circulating nuetrophil response were slow, lasting about 500 hours (20 8 days). When the stem cell production rate was varied the initial Idnetics of the blood pool neutrophils remained unchanged until 100 hours into the simulation (Figure 3.3). However, the steady state values were determined by the amount o f variation. The greatest amplification o f circulating neutrophils resulted from an increased number of divisions within the proliferating cell classes (Figure 3 .4) By adding one extra cell division to the model the blood neutrophil count was increased from 1.5 to 2 times the original count, depending on which cell class held the extra division. When each cell class was allowed an extra cell division (3 divisions total) the circulating neutrophil count was increased to greater than 5 times the initial steady state count after a period o f 500 hours. The amplification did not affect the early kinetic response of the model to the leukophoretic perturbation. Variations of the proliferation fraction of cells for the first four cell stages also resulted in different steady state values o f the circulating neutrophil count (Figure 3 5). Each precursor class required the same amount of time, approximately 75 hours, to produce deviations in circulating neutrophil counts from the original simulation values. The greatest deviations from the original steady state values occurred when the blast and 30 promyelocyte stages were allowed to have increased active cell fractions. The metamyelocyte proliferation fraction had little effect on the circulating nuetrophil count The stem cell production rate, number o f divisions, and the proliferation fractions offer a means o f increasing circulating neutrophil counts during times of circulating granulocyte depletion such as infection. A cooperative action between these three parameters may be the most effective way to boost declining blood pool neutrophil counts. It is important to remember that none of these parameters will provide a quick response to the need for neutrophils. Changes in the early time course must result from another parameter The model produced interesting results when the compartment transit times were varied (Figure 3 .6). The blast and promyelocyte times had only a slight effect on the level o f circulating neutrophils which eventually returned to the original steady state value The variations in the early proliferative compartments did not affect the initial ldnetic response to the perturbation However, as compartment transit times were varied for increasingly mature cell types, the early kinetic response differed. The myelocyte compartment produced the greatest overshoot in circulating nuetrophil counts with a reduced time of recovery (50 hours). The time o f recovery decreased as the compartment transit times for more mature cells types were varied, with a recovery time of less than five hours due to the segmented celt transit time. Presumably this is due to the distance between the precursor cell type and the circulating blood compartment. Although variations in compartment transit times did not affect the steady state value, the kinetics o f the asymptotic response was not constant across every cell class. 31 The early ldnetic response o f the model could also be changed by variations in the maturing band cell fraction and the demargination rate. Decreasing the band cell maturation fraction allowed more cells to be shunted directly to the circulating blood compartment and thereby cause a more rapid increase in the circulating nuetrophil count with an increased overshoot o f the steady state value. No change in the asymptotic value of the circulating neutrophil count was produced (Figure 3 7) Variation of the demargination rate had an effect on both the early kinetic response and asymptotic value o f the circulating neutrophil count (Figure 3.8). This contribution to the asymptotic value and early kinetic response by the demargination rate suggest that it may play an important role in regulation o f circulating neutrophil counts It seems likely that a combination o f the parameters would be able to produce an early kinetic response to the dynamic disturbance as welt as an increased production of granulocytes in the case of prolonged disturbance The choice o f parameters to produce a combined effect is not immediately apparent. Decreasing compartment transit times of the mature cell classes along with shunting immature band cells to the blood pool may be the first choice for producing the early kinetic response. The proliferation parameters and stem cell production rate may all be involved in providing increased numbers of granulocytes to the blood pool, but the degree o f their interaction must be determined. 32 3 3 Simulation of Drug Therapy Myelosuppression is a toxic response in many chemotherapy treatments leading to the discontinuation o f therapy because the patient no longer has the ability to fight off ordinary infection. The detailed physiologic model for granulopoiesis is used to provide insight into granulocyte suppression occurring during treatment. Cell loss due to drug therapy was simulated by a constant rate o f cell removal from compartments within each category of affected cells for 24 hours. The rate o f removal was limited by the stipulation that the number of cells within each cell category must remain above zero The response o f circulating neutrophils to removal o f cells from active and maturing compartments results in a delayed suppression (Figures 3.9 and 3.10) This delay is more noticeable in the case o f cell removal from the acitve compartments The minimum value o f granulocytes, commonly termed the nadir value, occurs approximately two days earlier when cells are removed from the maturing compartments than when cells are removed from the active compartments. Recovery takes place gradually although it is slightly faster in the simulation with cell removal from maturing compartments. By comparison, the response o f circulating neutrophils to removal of cells from the non-proliferating compartments shows an immediate decline with a very early nadir (Figure 3 11). Recovery of circulating neutrophils occurs twice as fast as recovery from removal of active or maturing cells. This quick decline in circulating neutrophils is presumably due to the close proximity o f the non-proliferating cells to the blood pool compartment. 33 The shape and time course o f each response is useful for understanding possible mechanisms of drug action that may produce myelosuppression. Those drugs that cause cell death by interacting with DNA synthesis, protein synthesis, or other factors of cell division may produce a slow response to therapy. As is evident from the physiologic model, many steps are involved in granulocyte production and maturation, and time is required for the transit o f cells from the cell division phase to the blood pool. If the immature, proliferative cells are affected by a drug, the immediate effect may not be seen in the circulating blood pool because the non-proliferating cells are still being released from the marrow. However, after time the number of non-proliferating cells will also decrease as there are fewer cells upstream to feed into the remaining compartments This is the cause of the delayed suppression which can be easily seen with the physiologic model. The time course o f myelosuppression in a patient treated with Cyclophosphamide and GM-CSF is shown in Figure 3.12. Cyclophosphamide is metabolized to several active compounds that produce their therapuetic effect by alkyiation or crosslinking of the DNA [24], This suggests that immature cells are being killed although it is known that Cyclophosphamide spares stem cells. A comparison of the actual time course data with the simulation results show that the nadir is reached at approximately the same time when cells are removed from the active and maturing compartments. The simulations also produce the characteristic delay in suppression of granulocytes. It is important to note that the patient was also being treated with GM-CSF, and this effect is not explicitly included in the simulations. 34 There are still many unanswered questions in the approach to modeling of myelosuppression. One question to answer is whether the normal mechanisms of granulopoiesis are overwhelmed or ineffective during chemotherapy. This would affect the model o f granulopoiesis during the granulocyte recovery period after therapy is ended. More data on the time course of myelosuppression produced by therapy is necessary to validate any model results. This is difficult to obtain, especially when treating patients with high doses o f chemotherapy, because drugs that counteract the toxic side effect must be given concurrently to prevent patient death. 35 Table 3.1: Compartment Size Ratios for Steady State Simulation with Varying CTTs myeloblast promyelocyte myelocyte nondividing* from literature 1 3.4 1 6 3 36.1 CTT: with constant T b^60,p-40.m-20 hours 1 2.2 1.8 t7.0 CTT: low end o f range b^31,p=34,m=63 hours 1 3 6 10 8 33.3 CTT: midpoints o f range b=3I.p^50.m = 103 hours 1 5 4 17 9 33.3 CTT: combined b=3I,p=34,m = 103 hours 1 3 6 1 7 9 33.3 * meta, band, and segmented compartments Table 3.2: Comparison o f Steady State Simulation Values with Literature Values literature values simulation values CTT: b=31 ,p=34,m= 103 hrs size o f marrow storage pool 9 0 6 4 (band * seg kg BW) x 10 v 5.0 marrow storage circulating pool 13 116 * 11.6 ** 10.5 ••• circulating f marginated pools 7 5.5 * (cells kg BW) xlO 3 1 1 ** 11.6 **• total number granulocytes in body 1 14 1.39 * (cells/kg BW) xlO 1 0 1.45 •• 1 45 *** total transit time 11.25 108 (days) * model with lumped blood pool compartment ** model with granulocyte elimination from circulating compartment *•* model with granulocyte elimination from marginated compartment 36 C irc u la tin g N eutrophils Figure 3.1: Emergence Time Emergence Times For Verted 1 Hour C sM Infusions 1.35 1.3 1.25 1.2 1.15 1.05 0.95. 150 200 250 300 350 Tims (hours) 100 400 450 500 = 10 times normal production rate = 100 times normal production rate * Circulating neutrophils normalized to steady state value (6 1 x 10 8 cells/kg BW) Figure 3 .2: Leukophoresis o f 50% Circulating Neutrophils flsspon— of Clroutattng NsutrapMIs to Lsukophorssis 1.5 1.4 1.3 0.9 o.a 0.7 0.6 0.5, S O 100 150 200 250 300 360 Tima (hours) 400 450 500 * Circulating neutrophils normalized to steady state value (61x10* cells/kg BW) 38 Figure 3.3: Response o f Circulating Neutrophils to Variations in Production Rate Stem Cell Production Rate — — — * * ^ 100 200 300 400 Time (hours) = steady state value 50% decrease 20% decrease = 50% increase = 20% increase Figure 3.4: Response o f Circulating Neutrophils to Additional Cell Divisions Extra Cell Divisions 100 Time (hours) * Circulating neutrophils normalized to steady state value (6 1 x 1 0 1 cells/kg) 39 F igure 3.5: Response o f Circulating Neutrophils to Variations in Proliferation Fractions Myeloblast Promyelocyte 1.5 gO.S 100 200 300 400 1.5 ? 1 1 g0.5 O Myelocyte me mm 0 100 200 300 400 Time (hours) 1.5 0.5 Metamyelocyte 1.5 0.5 Time (hours) = steady state value 50% decrease 20% decrease = 50% increase - 20% increase * Circulating neutrophils normalized to steady state value (6 1 x 10 g cells/kg ) 40 Circulating Neutrophils Figure 3.6: Response of Circulating Neutrophils to Variations in Compartment Transit Times 2 •1.5 1 P I 0.5 2 1.5 1 0.5 0 0 100 200 300 400 Myelocyte — I 0 100 200 300 400 Time (hours) 1.5 0.5 100 200 2 1.5 1 0.5 0. Metamyelocyte J-.J 0 100 200 300 400 Time (hour*) 41 Circulating Neutrophil* Figure 3.6 (cont) Band Sagmantad 2 1.5 1 0.5 0 0 100 200 300 400 Tima (hours) M 1.5 0.5 100 Tima (hours) s steady state value 50% decrease 20% decrease = 50% increase = 20% increase * Circulating neutrophils normalized to steady state value (6.1x10* cells/kg ) 42 Figure 3.7: Response of Circulating Neutrophils to Variations in Maturing Band Cell Fraction Maturing Band Fraction J S 1.5 z 0 . 5 O Time (hours) Figure 3.8: Response of Circulating Neutrophils to Variations in Demargination Rate Demargi nation Rate § 0.5 100 200 300 400 Time (hours) = steady state value = 50% decrease = 50% increase . . . = 20% decrease - 20% increase * Circulating neutrophils normalized to steady state value (6 1 x 10 * cells/kg BW) 43 Figure 3.9: Simulation o f Dm g Therapy: Removal o f Active Cells Reeponaa to Removal of AcBv Ceie lor 24 Home 0.9 0.5 15 20 25 10 Response of Cell Types 0.8 0.6 0.4 0.2 20 Rate o f removal = 7 .5 x 10 6 cells/hr — = active compartment = maturing compartment - - = non-proliferating compartment * Neutrophils normalized to steady state values 44 Figure 3.10: Simulation o f Drug Therapy: Removal o f Maturing Cells naaponas 1 0 Removal ol Maturing CeAe tor 24 Hour* 0.96 0.9 0.85 0.75 20 25 Tima {days) Response of Call Types 0.8 0.6 0.4 0.2 15 20 25 Tima (days) Rate of removal = 3 .0 x 10 7 cells/hour - = active compartment - - = maturing compartment - - = non-proliferating compartment * Neutrophils normalized to steady state values 45 Figure 3.11: Simulation o f Drug Therapy; Removal o f Non-proliferating Cells nesponee to Removal of Non-preMmttnB C olt for 24 Hours 0.9 0.4 20 25 15 Time (daps) Response of Cell Types ..........-.......... ■ ! |M ----- n..................... l— ------ I l ✓ V ______________ 1 ______________ ---------------------- 1 — ...... ........... 10 15 Time (days) 20 25 Rate o f removal = 1 Ox 10 8 celts/hour . . . = active compartment - = maturing compartment — = non-proliferating compartment * Neutrophils normalized to steady state values 46 W h i t e B l o o d C e l t s {*1000) Figure 3.12: Myelosuppression in a Patient Treated with Cyclophosphamide and GM-CSF 10 8 6 4 2 0 White Blood Cel Reeponee — M L . « ....................................... 1 ......................... ...... ' ~lr— — — -------- nr“ ---------------------------------- ---------------------------------------- 1 K . , , , ...................... jri “ ’ .....................................* H .................................... K — 1 * 1 I 1 X * * ................. . ... 10 15 Time (days) 20 25 47 Chapter 4 Conclusion 4.1 Model Summary The initial steady state simulation of the model with no feedback communication produces compartment sizes similar to those reported in the literature. This model assumes that there are 3 cell divisions within the myeloblast compartment, 2 divisions within the promyelocyte compartment, and only 1 division within the myelocyte compartment. This representation is supported by specific mitotic index values which suggest that the most immature cell type experiences the greatest proliferation The compartment transit times used in the sensitivity study simulations were based on the steady state compartment size they produced Granulocyte elimination from the marginated blood pool was considered to be the best choice as it approximated the actual physiology Although the simulation produces acceptable steady state results, the model is unable to reproduce the dynamic response to leukophoresis or x-radiation as seen experimentally, which suggests that feedback loops are necessary to accomplish such a response by the model. From the literature there are many possible mechanisms that may 48 be investigated. The following conditions may be active during times of system stress such as leukophoresis or radiation. • decreased CTTs or maturation time [2] • possibility o f metamyelocyte proliferation [2,17] • increased fraction o f blasts, promyelocytes, and myelocytes entering mitosis [2,16,17] • increased granulocyte production rate by stem cells [16] • increased release rate o f mature marrow storage [16] • increased release o f band neutrophils released to the blood pool [ 17] • interaction o f circulating and marginal blood pools [3, 5] 4.2 Future Work It is clear that this model represents a good beginning for a detailed physiologic model o f granulopoiesis, but there are several areas where more work is needed. First, reduction o f the model to a more concise and flexible form is desired The kinetic interaction between circulating and marginated blood pools requires further investigation. Finally, feedback loops that will provide the model with the capability of regulating granulocyte counts must be determined. The model as presented in Figure 2.1 consists o f 18 first order differential equations and is not capable o f handling changes in the number o f mitoses. It would be appropriate to reduce the model to include the necessary parameters and allow more 49 flexibility for variations in the number o f cell divisions. A physiologic approach would be to reduce the model to contain one active and one maturation compartment for each proliferative cell type while adding a parameter that defines the number of cell divisions within the active compartment. This would keep the physiologic aspect o f the model in tact. However, assuming first order kinetics, a greater number o f cells would be passed to each successive compartment and the compartment sizes would increase This occurs because the driving force or cell population that determines the number of cells flowing to the next cell compartment would be the total number o f cells within a cell class and not simply those cells undergoing the last division within that cell class. Another approach to model reduction is by cluster analysis [25]. This method reduces the model by lumping groups o f states that behave similarly It is possible that this approach may be applied to the current model o f granulopoiesis Several possible models for the blood pool compartment have been simulated The best model choice based on resulting steady state compartment size is a lumped circulating and marginated granulocyte compartment. However, the dynamic interaction between these two compartments during system stress may represent a very important recovery mechanism and therefore deserves further attention. The rates for margination and demargination were chosen arbitrarily and a more detailed investigation of the effects of these two parameters is suggested. The most important area for continued work on this model is the design of feedback loops. It is this feedback communication that will allow the model to be useful in dynamic applications. Although the lack o f experimental data precludes validation of the 50 feedback loops it is necessary to investigate their possible existence and function so that a theoretical framework is available for future experimental results. The difficulty of investigating the effect of feedback mechanisms begins with the large number o f parameters within the model which may be involved in feedback. In addition to speculating which parameters are most likely to respond to feedback communication, the controlling parameter is also unknown. Finally, the problem of quantifying the relationship between controlling and responding parameters must be addressed Each o f these issues is difficult to resolve without experimental data to use as validation. The list of possible feedback mechanisms mentioned above encompasses all the parameters presented in the model. The sensitivity study has defined the effect o f each parameter on the circulating neutrophil count which provides some directions in choosing parameters that produce a specific result It is difficult to say if these parameters may be acting cooperatively or independent from each other Determination o f the controlling parameter o f feedback communication presents another problem. Because no physiologic data exist for this control mechanism it is difficult to determine which parameters should be used If the control mechanism of erythropoiesis is used as an example of hematopoietic control we may assume that it is the concentration o f granulocytes in the circulating blood pool that controls granulocyte production. If in Act GM-CSF and G-CSF are the communication molecules in granulopoiesis and are produced in cells lining the blood vessels, the blood granulocyte concentration may be the controlling parameter Rubinow and Lebowitz propose that the total granulocyte concentration (blood and marrow) controls the proliferation and 51 maturation o f immature marrow granulocytes while the blood granulocyte concentration controls the release o f mature granulocytes to the circulating blood pool [5], As is evident, there appears to be several good choices for controlling parameters. Based on qualitative physiologic data several possibilities of controlling and responding parameters may be chosen. In studies on the effects o f G-CSF it was found that G-CSF stimulates increased mitosis in early proliferating cells, decreased transit time of post-mitotic cells, and increased demargination of cells from the marginal pool to the circulating pool. These three sets of parameters may be initially chosen as the parameters regulated by blood granulocyte concentration. The next step is to determine the quantitative relationship between controlling and responding parameters Although the basic relationship between the parameters is apparent the exact definition is not known Another issue to address in defining feedback loops is the possibility o f a time delay, which adds an extra variable to the already complicated process of choosing feedback parameters. It is feasible to expect that a molecular form o f communication may take some amount of time to reach the site where the desired effect is produced, which can be viewed as a time delay The existence o f a time delay or the degree o f delay is not known; it remains speculation until experimental data is available to validate simulation results. The underlying problem for designing feedback mechanisms is the lack of experimental time course data which is necessary to provide validation for each feedback hypothesis. It is difficult to obtain this information due to the location o f granulocyte formation in the bone marrow. Human granulocyte time course data is further complicated by the willingness o f patients to subject themselves to constant granulocyte 52 monitoring. Until this information is available detailed mathematical models will only be hypotheses o f the physiologic process. However, these models may be useful in defining important parameters and suggesting direction for future experimental work. 53 Reference List [1] Ludyard, P., Grossi, C Cells Involved in Immune Responses In Roitt et al. (eds), Immunology. Mosby, St. Louis, p. 2.17. (1993). [2] Cronkite, E.P. Analytical Review of Structure and Regulation of Hemopoiesis. Blood Cells 14:313 (1988). [3] Boggs, D R. The Kinetics of NeurophilJc Leukocytes in Health and in Disease Sem. Hemat. 4:359 (1967) [4] Mary, J Y Normal Human Granulopoiesis Revisited I. Blood Data Biomedicine and Pharmacotherapy. 38:33 (1984) [5] Rubinow, S. I., Lebowitz, J L. A Mathematical Model o f Neutrophil Production and Control in Normal Man. Journal o f Mathematical Biology. 1:187 (1975). [6] Dresch, C. Troccoli, G , Mary, J Y Growth fraction o f myelocytes in normal human granulopoiesis. Cell Tissue Kinet. 19:11 (1986). [7] Mary, J Y Normal Human Granulopoiesis Revisited II. Bone Marrow Data Biomedicine and Pharmacotherapy 38:66 (1984) [8] Killmann, S., Cronkite, E.P., Fliedner, T M., Bond, V P Mitotic Indices of Human Bone Marrow Cells. I. Number and Cytologic Distribution o f Mitoses. Blood 19:743 (1962) [9] Killmann, S., Cronkite, E.P, Fliedner, TM., Bond, V P , Brecher, G Mitotic Indices o f Human Bone Marrow Cells. II. The Use of Mitotic Indices for Estimation of Time Parameters o f Proliferation in Serially Connected Multiplicative Cellular Compartments. Blood 21:141 (1963) [1 0 ]F lied n er, T.M., Cronkite, E.P., and Robertson, J S Granulocytopoiesis. I. Senescence and Random Loss o f Neutrophilic Granulocytes in Human Beings Blood 24:402 (1964). [1 lJFliedner, T.M., Cronkite, E.P., Killmann, S.A., Bond, V.P. Granulocytopoiesis. II. Emergence and Pattern o f Labeling o f Neutrophilic Granulocytes in Humans. Blood 24:683 (1964). 54 [12] Warner, H R , Athens, J.W An Analysis of Granulocyte Kinetics In Blood and Bone Marrow Ann. N.Y. Acad Sci. 113:523 (1964). [13]Maloney, M.A., Patt, H.M Granulocyte Transit from Bone Marrow to Blood. Blood. 31:195 (1968). [14]Cronkite, E.P., Burlington, H., Chanana, A.D., Joel, D.D Regulation of Granulopoiesis. In Cronkite et al (eds ), Hematopoietic Stem Cell Physiology. Liss, New York, NY p. 129 (1985) [15]Rondanelli, E G , Magluilo, E., GiraJdi, A., Carco, F P. The Chronology of the Mitotic Cycle of Human Granulocytopoietic Cells Phase Contrast Studies on Living Cells in Vitro. Blood. 30:557 (1967) [16]Morley, A., Stohlman, F Jr. Studies on the Regulation o f Granulopoiesis I. The Response to Neutropenia. Blood. 35:312 (1970). [17]Patt, H.M., Maloney, M.A., Jackson, E.M. Recovery of Blood Neutrophils After Acute Peripheral Depletion. Amer. J. Physiol. 188:585 (1957) [18]Boggs, D R., Athens, J W , Cartwright, G.E., Wintrobe, M.M Leukokinetic Studies. IX. Experimental Evaluation of a Model o f Granulopoiesis J. o f Clin. Invest. 44: 643 (1965). [19]Morley, A. A. A Neutrophil Cycle in Healthy Individuals Lancet. 2:1220 (1966) [20]King-Smith, E.A., Morley, A Computer Simulation of Granulopoiesis: Normal and Impaired Granulopoiesis Blood. 36:254 (1970) [21]Dale, D C., Ailing, D.W., Wolff, S.M. Application o f Time Series Analysis to Serial Blood Neutrophil Counts in Normal Individuals and Patients Receiving Cyclophosphamide British Jounal of Haematology. 24:57 (1973) [22]Lord, B I , Bronchud, M.H, Owens, S., Chang, J , Howell, A , Souza, L., Dexter, T.M The Kinetics o f Human Granulopoiesis Following Treatment with Granulocyte Colony-Stimulating Factor in vivo. Proc. Natl. Acad. Sci. USA. 86:9499 (1989). [23]Lichtman, S.M., Ratain, M.J, Van Echo, D A , Rosner, G., Egorin, M.J., Budman, D R., Vogelzang, N.J., Norton, L., Schilsky, R.L. Phase I Trial of Granulocyte- Macrophage Colony-Stimulating Factor Plus High-Dose Cyclophosphamide Given Every 2 Weeks: a Cancer and Leukemia Group B Study. 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IEEE Transactions on Automatic Control. 30:478 (1985). 56 Appendix Active Com partm ents M aturation Com partm ents - f ya ^ dym b \ ym b i dt f bq' T „ dt * ^ *tya » 2 2 f h y \ , y \ t dym bi 2(1 - f h)y* * + y m „ x y " b2 dt T b T b dt T b T b <ty\> 2 f b y \2 y \ 3 d y m b i 2 ( 1 - / b ) y a » z + y m b2 y m b i dt Tb T b dt T b T b < fya» 2 f p y \ , y % t <fym 2 ( i - / , ) y ‘ » + y m » y m p i dt T b tp dt T b T P <iy\.x 2 / / , . y % i d y m P2 2 ( i - f p ) y * P \ + y v y m p2 dt T, tp dt T P T P 4ya. 2 / ^ V y*- dym m 2 ( 1 ~/m )ym # + y m » y \ dt T , T m dt T P T m dya~ 2fmy am y aim dym m 2 d - f m)ya~ + y m ~ y m ~ dt T T dt T T m * m ( ‘ m 1 m l 57 Non-Proliferative Compartments d y M y * m + y m m t y b J d t T m , T ^ j d y * t _ f^y* ym g Blood Pool Compartments INFORMATION TO USERS This manuscript has been reproduced from the m icrofilm master. U M I films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the qualify of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margin*, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note wiS indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. 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University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Redekopp, Karyn Alise
(author)
Core Title
A physiologic model of granulopoiesis
School
Graduate School
Degree
Master of Science
Degree Program
Biomedical Engineering
Degree Conferral Date
1995-05
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
biophysics, general,health sciences, immunology,health sciences, oncology,OAI-PMH Harvest
Language
English
Contributor
Digitized by ProQuest
(provenance)
Advisor
D'Argenio, David B. (
committee chair
), Kalaba, Robert (
committee member
), Khoo, Michael Chee-Kuan. (
committee member
)
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c18-4083
Unique identifier
UC11357747
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1376504.pdf (filename),usctheses-c18-4083 (legacy record id)
Legacy Identifier
1376504-0.pdf
Dmrecord
4083
Document Type
Thesis
Rights
Redekopp, Karyn Alise
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
Tags
biophysics, general
health sciences, immunology
health sciences, oncology