Chemical Engineering in Medicine

4 Pharmacokinetics and Cell Population Growth Models in Cancer Chemotherapy

Downloaded by UNIV OF CINCINNATI on May 27, 2016 | http://pubs.acs.org Publication Date: June 1, 1973 | doi: 10.1021/ba-1973-0118.ch004

K E N N E T H B. BISCHOFF, K E N N E T H J. H I M M E L S T E I N , a R O B E R T L . D E D R I C K , and D A N I E L S. Z A H A R K O School of Chemical Engineering, Cornell University, Ithaca, Ν. Y. 14850, De partment of Chemical Engineering, University of Maryland, College Park, M d . 20742, Biomedical Engineering and Instrumentation Branch, DRS, and Labora tory of Chemical Pharmacology, N C I , National Institutes of Health, Bethesda, M d . 20014

The development of mathematical models to simulate cancer chemotherapy for ultimate use in optimizing treatment has two main requirements: prediction of the time course of drug distribution in the body, or pharmacokinetics, and quantitative description of the cancer cell growth and death as a function of the drug concentration in the local environ ment. The first has been determined for certain important cancer chemotherapeutic agents, yielding predictive, physio logically consistent models for several mammalian species. The second aspect is beginning; the basis for the models is a description of the cell population maturation and death where the loss term is a function of the drug concentration. Some available data for L1210 leukemia in mice will be used to illustrate the model predictions and compare these with experimental data.

'he rational and reliable prediction of drug effects based on dosage regimen and concomitant biochemical events has been a long soughtafter goal. Two necessary steps are involved: (a) prediction of the drug distribution, or pharmacokinetics, throughout the body, including the local concentration at the site of action; and ( b ) prediction of the actual drug action as a function of the local concentration. There is a great a Present address: Dow Chemical Co., Midland, Michigan 48640. 47 Reneau; Chemical Engineering in Medicine Advances in Chemistry; American Chemical Society: Washington, DC, 1973.

48

C H E M I C A L ENGINEERING I N M E D I C I N E


similarity to chemical reactor analysis and design where the overall mass and energy balances and the local kinetics are used. There are no known examples where the complete task has been done in a truly quantitative and a priori fashion although this general type of reasoning is used by pharmacologists and clinicians to design therapeutic procedures. Because of the experimental and analytical difficulties involved in each phase, there may be some doubt that definitive predictions are feasible. H o w ever, the potential benefits to be obtained, especially for individuals, justify every reasonable effort that can help to simplify and reduce the typically extensive and somewhat risky clinical experimentation. W e w i l l consider here this problem i n the context of cancer chemo therapy. Since antineoplastic drugs by necessity are quite toxic, it would seem that meaningful mathematical models could be important guides in the optimization of various types of experimental and clinical trials. The mode of action of many of these drugs, at least semiempirically, is qualitatively and sometimes quantitatively known, and the pharmaco kinetics of several types have been studied. More specific information seems to be available concerning the action on tumor cells than toxicity i n critical normal cells such as bone marrow and gastrointestinal tract. However, to be useful i n a clinical sense, both effects must be predictable since successful therapy is based on the balance between the cancer cell death and the toxicity. This paper summarizes some recent work in each of the two areas of pharmacokinetics and local drug action and then illustrates an initial attempt at combining them to predict drug actions on mouse L1210 tumors. Cell

Kinetics

First consider the mathematical descriptions of cell kinetics which define the drug effects for cancer chemotherapy. Quantitative models to describe mammalian (or other) cell growth and death are still rela tively crude. Although the complicated biochemical events leading to D N A and other chemical production, cell mitosis, and population growth have been studied qualitatively in some depth and quantitative aspects of some reactions are known, synthesis of this with a comprehensive description has not yet been completely accomplished. The formal structure of various types of mathematical models has been summarized by Frederickson, Ramkrishna, and Tsuchiya (J) and Weiss (2). The various biological assumptions that are implicit i n many simple models, such as the Michaelis-Menten formula, are discussed, and generalized approaches are suggested. F e w specific actual examples

Reneau; Chemical Engineering in Medicine Advances in Chemistry; American Chemical Society: Washington, DC, 1973.

4.

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ET AL.

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Cancer Chemotherapy

are known, however, where an analysis has been carried to completion. W e w i l l discuss here a fairly simple model, stating only the most impor tant assumptions involved. For many of the drugs of interest, action occurs presumably only during some fraction of the cell mitotic cycle. Most often this is con sidered to be the D N A production or S-phase of the total G i - S - G - M cycle. A useful model, therefore, should at least be able to incorporate this type of information—e.g., the structured models of Tsuchiya, Fredrickson, and Aris ( 3 ) . This means that simple gross descriptions, such as M i c h a e l i s - M e n t e n - M o n o d models are not sufficient, and some atten tion must be paid to the cells during growth. W e assume here for a first approximation that a single cell age or maturation variable is sufficient.


2

Many investigators have studied models of the von Foerster type (see Trucco (4) for several references and properties), which belong to the general population balance category used in kinetic theory, particle agglomerization, crystallization, etc. (see Himmelblau and Bischoff (5) for a textbook treatment). W e w i l l use the model of Rubinow (6) for reasons discussed below. Rubinow defined a normalized cell maturation variable, μ such that a cell w i l l divide when μ = 1. N o w μ is related to the biochemical events occurring during the cell cycle in some undetermined fashion. It is, there fore, a semiempirical variable and is operationally defined i n terms of measured cell cycle times. This implies that all cells divide after the cell cycle time of τ hours. It is observed, however, that cell division times are scattered about a mean value. This randomness must be accounted for suitably. Most models account for this with an explicit operation i n the mathematical solution by averaging cell division times over the entire population. This scheme leads to the solution of a rather difficult integral equation (see, e.g., Trucco (4)). Recently Subramarian et al. (8) have considered weighted-residual methods for more easily solving these problems. Rubinow uses a simpler mathematical approach which assumes that groups of cells behave i n a deterministic manner, a l l dividing when their μ = 1, and then averages over a distribution of maturation rates for the entire population. Both approaches lead to similar final results although the fundamental biological assumptions are somewhat different. Our purpose i n this first attempt is to assess the feasibility of the fundamental approach, so we would prefer the simplest mathematical description. Ultimately, more involved models should prove useful, such as the comprehensive numerical model of Shackney ( 7 ) ; however, here we discuss only simple models.


50


W i t h this description, then, the standard population balance equa tion describing the cell kinetics is:

S + çw

-

-

».

(D

where: η (μ, t) = distribution function of cell maturation at time, t Downloaded by UNIV OF CINCINNATI on May 27, 2016 | http://pubs.acs.org Publication Date: June 1, 1973 | doi: 10.1021/ba-1973-0118.ch004

υ (μ, t) = maturation growth rate λ (μ, t) = death or loss rate of cells The pertinent boundary conditions for solving this partial differential equation are: (a) initial state of system: η ( μ , 0 ) = Ν. g

(2)

fa)

where N = total cell density (b) mitosis boundary condition for binary fission: 0

υ (0, t) η (0, 0 = 2 υ (1, t) η (1, t)

(3)

In addition, the final solution can be averaged over the distribution function of maturation rates, W(v). It is not obvious exactly how the drug effect should enter the model since the maturation rate, υ, and the direct loss function, λ, could be affected. It is simplest to use only the second approach, and this w i l l be done for the present. Also, if a constant maturation growth rate, ν = 1/T, is used as a crude approximation to a more realistic function, Equation (1) reduces to: θη , 1 θη d t - , ^ +

.

( =

~

λ

(

Μ

)

η

( Λ

.

( l a )

and Equation (3) to: η (0, t) = 2 η (1, t)

(3a)

Relatively simple analytical solutions can now be obtained, and these are considered in more detail below. The explicit variable usually observed is the total cell density since separating the cells into their cell cycle fractions would be difficult to


4.

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Cancer Chemotherapy

51

achieve. I n terms of the model, this corresponds to adding a l l the /i-fractions, so that the total cell density is: Ν (t)

j\

η (μ, t) d μ

(4)


Finally, possible functional forms for λ (/*,*) should be discussed. This is how the drug effect is introduced into the model and is, therefore, the crux of the issue. One simple form is a direct first-order dependency on the local drug concentration, C(t): λ (μ, t) = Κ (μ, t) C (t)

(5)

where the rate constant, K, could depend on the cell cycle and also could be a function of time through deactivation or acclimatization effects. A s an example of maturation dependency, the drug might only be effective during a certain fraction of the cell cycle, a ^ μ ^ b, so that: i 0, 0 < μ < a Κ fat) = \K, a < μ < 6 / 0, b < μ < 1 The local drug concentration is given by the pharmacokinetics. A more realistic form takes into account commonly observed drug saturation effects:

Loss functions defined b y Equation (6) can be substituted into Equation ( l a ) , and the solution obtained. Before this is done, however, the types of functions, C(t), obtained from pharmacokinetics must be described. Pharmacokinetics If the intent is to develop a complete a priori description, the pharma cokinetics must be included. Several relevant studies have been done for some of the commonly used cancer chemotherapeutic agents; two are described here. T h e first is the folic acid antagonist, methotrexate ( M T X ) ; the second is cytosine arabinoside ( A R A - C ) . Both are active against leukemia, which is of primary interest here. A comprehensive pharmacokinetic model for M T X has been devel oped for wide ranges of doses, and for mammalian species from mouse



to man—see the papers of Bischoff, Dedrick, and Zaharko (9,10,11,12) for detailed description of the models and references to prior work. Figure 1 shows a flow diagram that was adequate to represent many of the complicated phenomena occurring during the distribution of M T X i n the body. It consists of discrete compartments representing basic functional plasma, highly perfused viscera, and poorly perfused tissues with the second of these further differentiated into the portal circulation because of the great importance of biliary secretion.


Transient mass balances for the various compartments can be written: plasma V

p

I F

^J

=

-

e c t i o n

) +

(QL +

Q

τα +

l

QK +

Q

k

rÎ

+

Q

m

û

(7a)

QM) CP

muscle (7b) kidney

ν V

« - a ~dT ~

(r

dC

k

Q

\

K

C C

k

\ -

TK)

~

P

t, ^ ** RK

(7c)

liver V ^ = ( Q

L

- Q o ) ( c

P

where

-

C

^ )

k

KL

0

(7d)

C /R

L

To =

Q o ( % - ^ ) - r

+

L

+C

L

L/RL

bile ducts τ gut tissue

= r

M

- r „ « = 1, 2, 3)

(7e)

^ - ^ ( ' - ë ) Sï[sîfc; '] c

+

gut lumen dCo dt

L

_ 1V

+ K7

1

-ΪΣΪ

dd

(7f)

(7g)

(7h)


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53

Cancer Chemotherapy

PLASMA Q -Q L

G


jGut Absorption

Ci-4c +c fc 2

3

• Fecet

4

Gut Lumen

KIDNEY Urine

MUSCLE Figure 1. VOL

4

Compartmental model for MTX distribution

dÇi = kV dt F

(C,

GL

« = 2, 3, 4) These equations contain the usual input-output terms of compartmental mass balances and also a simple first-order renal clearance, which is close to inulin clearance for M T X . T h e R i are tissue-to-plasma distribution ratios to account for protein binding. The volumes V , and flows, Q, are known from recorded anatomy and physiology. Other parameters are defined as follows: k , renal clearance; k , saturable rate of drug trans port into bile; K , saturation constant for bile transport; k , saturable rate of intestinal absorption; K , saturation constant for intestinal absorption; b, nonsaturable rate of intestinal absorption; k , reciprocal of nominal transit time in small intestine. Several complicating factors occur i n the portal system balances. Equation ( 7 d ) contains a term, r , that represents the biliary secretion and could also model saturation effects. Equation (7e) approximates K

L

L

Q

G

F

0


54

C H E M I C A L ENGINEERING I N MEDICINE

distributed (vs. lumped) behavior by using three compartments-in-series. A similar scheme is used for the gut lumen, as indicated i n the remaining equations. T h e latter also account for absorption through the gut wall by a saturable process. Further details of this aspect of pharmacokinetic modeling are given b y Bischoff et al (13).


ΙΟΟρ-

001 « 0 υ

1

60

1

120

1

180

— 240 1

TIME, Minutes

Figure 2. Comparison between curves predicted by model and observed data in mice at 3 mg/kg MTX As described i n detail i n the above articles, the basic premise of the models is to utilize as much physiologic information as possible to ap proach an a priori predictive result. Therefore, a l l the flows, organ volumes, tissue binding, renal clearances, etc., are taken from standard physiological sources or are independently measured. These are incor-


4.

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55

Cancer Chemotherapy


porated into the model to attempt prediction of the experimental results obtained with intact animals. This approach deliberately focuses atten tion on the important physiologic and pharmacologic functions, thus leading to a more meaningful interpretation of experimental results and improved predictive models. Figure 2 compares the model predictions with data from mice for each of the body regions, and excellent agreement is obtained. Figure 3 shows clinical plasma data i n man. Although the time scale of events is quite different, the same pharmacokinetic model successfully predicts them. 100 c-

0

90

180

270

360

TIME, Minutes

Figure 3. Comparison between curves predicted by model and observed data in man at 1 mg/kg (data from Reference 14)



56


Figure 4. Model predictions in mouse vs. rat for observed LD doses 50

The potential importance of pharmacokinetic information is illus trated i n Figure 4 where known L D (lethal dose) doses of M T X were simulated i n the mouse and the rat. Despite the 6+-fold difference i n L D doses, the model predicts quite similar time histories i n the plasma at longer times. Therefore, searching for a biological reason for the dif ferent L D should certainly take into account the pharmacokinetics. The extrapolation of these ideas into the necessity for pharmacokinetic information i n developing cancer chemotherapeutic drug dosage regimens should be clear. The distribution of M T X is dominated by rather complicated biliary secretion phenomena with ultimate urinary or fecal excretion; it requires a detailed compartmental model for its description. The drug A R A - C is quite different since its distribution is dominated b y rapid metabolism 5 0

5 0

5 0


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Cancer Chemotherapy

57

i n the liver and other organs of man. Dedrick, Forrester, and H o (15) have shown how rather simple considerations can lead to a predictive model for A R A - C pharmacokinetics. Figure 5 shows a model that was devised to elaborate the detailed disposition of A R A - C .


Mass balances comparable with Equations (7) were used, but omit ting the complications of biliary secretion and adding certain enzyme reaction rate terms. As an example the kidney balance [Equation ( 7 c ) ] becomes VR —ΤΓ =

QK (Cblood ~

ν

j7-

J^m, Κ

r

, π "Τ*

CR) — k'RCh\ooa V K, tissue

(8)

where F K

m a x

M

= enzyme activity at saturation, yjg/g min = Michaelis constant, μ&/πι1

The enzyme (pyrimidine nucleoside deaminase) levels i n various body regions have been measured, and in vitro reaction rate studies give numerical values for V and Km. Using all of this information together with physiological values for the organ weights and flows, Dedrick, Forrester, and H o (15) were able to predict concentrations of A R A - C and its metabolite i n the blood of human cancer patients; urinary excre tion of the drug and its metabolite were also simulated. m a x

A sample of the results is shown i n Figure 6 where the lower curve is for A R A - C ; the upper curve represents A R A - C plus metabolite, and the points are clinical data. Figure 6 shows that there is a very rapid drop i n the drug concentration i n the blood followed b y a slower expo nential phase. This behavior represents a complex interaction of physical and metabolic processes as discussed in the original source. The model shown i n Figure 5 has not yet been applied definitively to the mouse; however, Skipper, Schabel, and Wilcox (16), Mellett (17), Neil, Moxley, and Manak ( I S ) , and Borsa et al. (19) have presented data on plasma levels. The pharmacokinetics can be approximated by an exponential decay with a time constant of about 1/2 hour. These examples show how description of the local drug concentration, C(t), can be obtained for substitution into the loss function, λ(μ,ί). F o r leukemias the plasma values may be what is required. F o r solid tumors, however, the local drug concentration would be of primary concern and would require a more comprehensive pharmacokinetics model.


58



Application

to Cancer

Chemotherapy

Since the necessary cell kinetics and the pharmacokinetics have been described, they can be combined i n an attempt to show the results of drug treatment. Several general solutions to Equations ( l a - 3 a ) have been developed by Himmelstein (20); only one w i l l be described here. A search for experimental data with which to test the models revealed that too little was available to utilize the full cell cycle specific provi sions i n the model. Computational comparisons (20) indicated that a time-dependent rate constant, would give results similar to the cell cycle effects although much biological input might be lost with this semiempirical approach. For this case the loss function is a function only of time, λ = λ ( ί ) , and the solutions to Equations ( l a - 3 a ) are par-

Lean

Figure 5.

Compartmental model for ARA-C distribution



I 0

Figure 6.

I

I

20

40

I

I

60 80 Tl ME ( minutes )

I

100

I

120

I

140

Comparison between curves predicted by model and observed data in man at 1.2 mg/kg ARA-C

ticularly simple. However, more general solutions are available for use when more complete data become available. Our application is to mouse L-1210 leukemia cells where it is ob served experimentally that essentially exponential growth occurs during the main middle portion of cell growth (see Reference 16). A n asymp totic solution for this case can be developed by assuming a functional form: η (μ, 0 = N e* h (μ) exp [ - Λ (t)] 0

1


60


where Η(μ) = function of μ to be determined.

Λ (t) m

f

λ ( 0 dt'

Substituting this into Equation ( l a ) , and using the mitosis boundary condition, Equation (3a), yields the final solution: η (μ, t) = ΛΓ (2 In 2) 2 k " * ) e'

A ζ

L1210 CELLS 25mg/kg ARA-C DAILY

102

I ι I

θ

ι

12

-L 16 20

. ι

J-

24

-L 28

TIME , DAYS Figure 8. Cell density predicted by model and comparison with experimental mean death day for dosage regimen indicated Figure 8 shows the model results for one such regimen. The heavy zig-zag lines are the model predictions for the drug effects from Equation (10). T h e same parameters as given above were used; the scalloped nature of the curves is not apparent on the graph scale. The table above the figure compares the model predictions for various doses, found from the dashed lines on the graph, with the experimental mean death days. Rather remarkable agreement is obtained. Because of the simplicity of the model and the various approximations used, this agreement is prob ably somewhat fortuitous. However, Figure 9 shows a similar comparison for an alternative dosage regimen with different behavior. It appears, therefore, that models


4.

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Cancer Chemotherapy

63

of this type, even when oversimplified, may offer the ability to describe some of the overall events that occur when cancer chemotherapeutic agents are administered. Similar predictions of toxic effects on normal cells are also required before a true dosage regimen optimization can be attempted. NO. OF INJECTIONS 0 EXPERIMENTAL 8.5 RESULTS Downloaded by UNIV OF CINCINNATI on May 27, 2016 | http://pubs.acs.org Publication Date: June 1, 1973 | doi: 10.1021/ba-1973-0118.ch004

1

2

3

4

9

10

13

14 14.5 15

7

8

16

17

9.0 10.5 11.7 12.9 14.1 15.0 15.9 16.8 17.7

MODEL RESULTS

• I

5

6

1

I

I ' I ' I '—I

1

' I

l/i 'III!

10'

'!!

L12I0 CELLS 15mg/kg ARA-C Every 3 hours for one day

10* £ 1 0

ι

I

ι I

8

.

SURVIVALS-CLINICAL RESULT I ι I ι I ι I ι I

12

16

20

24

28

TIME , DAYS Figure 9. Cell density predicted by model and comparison with experimental mean death day for dosage regimen indicated It seems justified to continue along this line with future work. A s discussed above, more realistic loss functions already have simple mathe matical solutions available and could be easily utilized if the specific data required to evaluate the model parameters were available. T h e direction of some interesting experimental work is indicated since a framework within which to place the results can be formulated. Although


64


much remains to be done before precise clinical results can be predicted, the simple models suggested here can offer significant guidance.


Literature Cited 1. Fredrickson, A. G., Ramkrishna, D., Tsuchiya, H . M., Math. Biosci. (1967) 1, 327. 2. Weiss, G. H . , Bull. Math. Biophys. (1968) 30, 427. 3. Tsuchiya, H . M., Fredrickson, A. G., Aris, R., Adv. Chem. Eng. (1966) 6, 125. 4. Trucco, E., Bull. Math. Biophys. (1965) 27, 285, 449. 5. Himmelblau, D. M . , Bischoff, Κ. B., "Process Analysis and Simulation," Wiley, 1968. 6. Rubinow, S., Biophysical J. (1968) 8, 1055. 7. Shackney, S., Cancer Chemother. Repts. (1970) 54, 399. 8. Subramanian, G., Ramkrishna, D., Fredrickson, A. G., Tsuchiya, H . M., Bull. Math. Biophys. (1970) 32, 521. 9. Bischoff, Κ. B., Dedrick, R. L., Zaharko, D. S., J. Pharm. Sci. (1970) 59, 149. 10. Dedrick, R. L., Bischoff, Κ. B., Zaharko, D. S., Cancer Chemother. Repts. (1970) 54, 95. 11. Zaharko, D. S., Dedrick, R. L., Bischoff, K. B., Longstreth, J. Α., Oliverio, V. T., J. Natl. Cancer Inst. (1971) 46, 775. 12. Bischoff, Κ. B., Dedrick, R. L., Zaharko, D. S., Longstreth, J. Α., J. Pharm. Sci. (1971) 60, 1128. 13. Bischoff, Κ. B., Dedrick, R. L., Zaharko, D. S., Slater, S. M., Proc. Ann. Conf. Eng. Med. Biol. (1970) 23, 89. 14. Henderson, E. S., Adamson, R. H . , Oliverio, V. T., Cancer Res. (1965) 25, 1018. 15. Dedrick, R. L., Forrester, D. D., Ho, D. H . W., Biochem. Pharma. (1972) 21, 1. 16. Skipper, H . E., Schabel, F. M . Jr., Wilcox, W. S., Cancer Chemother. Repts. (1967) 51, 125. 17. Mellett, L . B., "Progress in Drug Research," E. Jucker, Ed., Vol. 13, Birkhauser Verlag, Basel, 1969. 18. Neil, G. L., Moxley, T. E., Manak, R. C., Cancer Res. (1970) 30, 2166. 19. Borsa, J., Whitmore, A. F., Valeriote, F. Α., Collins, D., Bruce, W. R., J. Natl. Cancer Inst. (1969) 42, 235. 20. Himmelstein, K. J., Ph.D. Thesis, University of Maryland (1971). RECEIVED November 22, 1971.


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