Cytotechnology 12: 315-324, 1993. 9 Kluwer Academic Publishers. Printed in the Netherlands.

Mathematical models for multidrug resistance and its reversal Seth Michelson Department of Biomathematics, Syntex Discovery Research, MS S3-1B, 3401 Hillview Avenue, Palo Alto, CA 94303, USA

Key words: facilitated diffusion, mathematical models, MDR

Abstract

Mathematical models describing drug resistance are briefly reviewed. One model which describes the molecular function of the P-glycoprotein pump in multidrug resistant (MDR) cell lines has been developed and is presented in detail. The pump is modeled as an energy dependent facilitated diffusion process. A partial differential equation linked to a pair of ordinary differential equations forms the core of the model. To describe MDR reversal, the model is extended to add an inhibitor. Equations for competitive, one-site noncompetitive, and two-site noncompetitive inhibition are derived. Numerical simulations have been run to describe P-glycoprotein dynamics both in the presence and absence of these kinds of inhibition. These results are briefly reviewed. The character of the pump and its response to inhibition are discussed within the context of the models. All discussions, descriptions, and conclusions are presented in nonmathematical terms. The paper is aimed at a scientifically sophisticated but mathematically innocent audience.

I. Introduction

Experiments in vitro with cultured tumor cell lines have revealed a series of very complicated mechanisms, both at the genetic and biochemical levels, that can account for drug resistance (Warr and Atkinson, 1988; Marx, 1986; Curt et al., 1984; Schimke, 1984; Harris, 1984). Many are reviewed in this special issue. The most common forms of resistance are: 1) decreased drug uptake, 2) increased drug efflux, 3) increased degradation/metabolism of drug, 4) increased drug-target concentration, and 5) alteration in drug-target properties. Cells can become resistant to only a single selective agent (single drug resistance), or, in some cases, to agents which are structurally and mecha-

nistically diverse (multiple or pleiotropic drug resistance) (Pastan and Gottesman, 1987; Moscow and Cowan, 1988; Gottesman and Pastan, 1988; Croop et al., 1988). Classical multidrug resistance (MDR) falls into the second category outlined above. The mechanism underlying MDR involves the overexpression of an energy-dependent efflux pump in the plasma membrane (Ling and Thompson, 1974; Jul~iano and Ling, 1976; Kartner et al., 1983; Inaba et al., 1979; Tsuruo et al., 1982). The pump is a transmembrane glycoprotein referred to as p170 or P-glycoprotein. Often, one finds that stepwise selection of cells with one agent, say doxorubicin, leads to the generation of a cell line that is also resistant to other natural product agents including anthracy-

316 clines, vinca alkaloids, podophyllotoxins, and colchicine (reviewed by Pastan and Gottesman, 1987; Gottesman and Pastan, 1988). Coincident with the development of these experimental models has been the evolution of a mathematical theory to describe it. And while the aim of this paper is to describe in detail a particular model of the P-glycoprotein pump (see Sections III.C and IV below), I recognize that mathematical models are no different from the experiments they mimic in that they must be viewed in the appropriate context. Therefore, I have decided to "zoom in" on the appropriate models of MDR by moving from the general case of models for the emergence of resistance (Section II.A) to models of tumorwide resistance (Section II.B) to models of cellular resistance (Section II.C) to models of MDR (Section III) and its reversal (Section IV). Recognizing that the audience for this special issue probably spans a spectrum of mathematical sophistication, I have decided to present the mathematics in as straightforward a manner as possible. Where necessary, mathematical equations are presented. However, the thrust of this paper is to describe in plain English the assumptions underlying the theory, the results of those theoretical speculations, and the limitations one must expect of the theory.

II. General models of drug resistance A. Stochastic models of emergence

By stochastic, the mathematician means random. In these models, the emergence of a drug resistant subclone is considered a fortuitous result of random mutations expressed during tumor evolution. Given the complexities in both definition and characterization of drug resistance, it has been difficult to develop manageable mathematical models to describe its emergence. Goldie and Coldman have proposed some of the most notable examples (Goldie and Coldman, 1979, 1984; Goldie et al., 1982; Coldman and Goldie, 1983; Coldman et al., 1985). Their models are phenomenological, predicting that single agent resistance

depends upon an unspecified cellular mutation. They predict that the frequency of cross-resistance to several agents should be the product of the individual underlying single mutation frequencies, and account for multiple resistance on the tumor level in this manner. Furthermore, in their earlier work, they assumed that "resistant" meant completely resistant, i.e., that no clinically acceptable level of drug could kill the resistant subclone. This assumption has been relaxed in subsequent models. Simple pencil and paper analyses point out the limitations of this model. Even if the viable mutation rate is small (i.e. one mutation in 10 6 mitoses), tumors presenting at an early clinical stage (approximately 1 cm 3) have negligible chance of being homogeneously chemosensitive. In fact, there are probably multiple subpopulations already present in a tumor of that size. If one also assumes, as Goldie and Coldman have, that resistant cells are totally resistant, one must conclude that upon presentation, the chances of curing any moderately sized tumor are negligible. Even if caught early, massive levels of a vast array of drugs would be needed to cure the cancer. Goldie and Coldman address multiple drug resistance on the cellular level incidentally. They assume that individual subpopulations emerge independently, and that these subclones are resistant to different drugs. MDR, as we understand it, must then be quite rare, the product of two mutations, yielding a cell that is doubly mutated. Therefore, the odds of achieving resistance on the cellular level to three or more drugs are astronomical. Thus resistance to an array of drugs is a tumorwide, not cellular, phenomenon. Day (1986) has extended their work to include asymmetry in growth, mutation and death rates to show that, in multiple resistant tumors, optimal treatment is always achieved with multiple drug regimens, but that the sequencing strategy employed may depend upon the underlying transition rates established for each subpopulation independently. However, Day states quite clearly that this type of model may be insufficient when considering MDR as expressed by the P-glycoprotein pump.

317 B. Deterministic models of therapy

Deterministic models are mathematical models, usually described by differential equations, which are fairly robust to small population variances. The biological effects being modeled act on total populations, and are strictly defined by dynamic equations which predict the mean behavior of the population. Fluctuations about the mean (i.e., what any single individual might add to the effect) are considered negligible in these models. Like the stochastic models outlined above, deterministic models have been developed which portray drug resistance as a classifier. These models usually define a resistant subpopulation in generic terms; e.g., varying transition rates for death, growth, etc. Overall tumor growth is modeled as a process in which two populations vie for survival in a hostile (drug treated) environment. Invariably, the loss rates for the resistant subpopulation are constants, fixed at smaller magnitudes than those defined for sensitive population. None of these models directly describes resistance mechanistically. For example, in three related papers (Gregory et al., 1988; Hokanson et al., 1986; Birkhead et al., 1986) tumor growth-rate and overall tumor volume at the time of presentation are used as predictors of response in the presence of single drug therapy. However, the authors state explicitly that they only meant to describe resistance as a phenomenon, and that no attempt was made to model it mechanistically. As a case in point, they assumed that sensitivity and resistance are nonacquirable traits, fixed in an altering environment. This kind of assumption would totally ignore the effects of gene amplification and DHFR upregulation in methotrexate treated tumors (Schimke, 1984). No explicit definition or attempt to model MDR on a cellular basis is tried either. In a more realistic model, Duc and Nickolls (1987) linked the pharmacokinetic profile of a single course of drug therapy to a tumor growth model. The distribution of drug was determined theoretically in three physiological compartments (the plasma, over the sensitive tumor, and over the resistant tumor). They assumed that each tissue

compartment exhibited first order kinetics and that the sensitive and resistant tumor compamnents were completely segregated and did not communicate biophysically. A standard set of ordinary differential equations was derived and solved. They modeled cell growth within each compartment independently. The presence of one population did not affect the growth or loss of the other. The level of drug at each site was determined separately. A general growth-death model for cycle specific drugs took the following form: d N = F(N) N(t) - kf(t)mxF(N) N(t) dt

(1)

For cycle nonspecific drugs the model looked like: dN -_F(N) N(t) - kf(t)m N(t) dt

(2)

In these general models, F(N) may be used to represent a logistic growth term. If one were to define F(N) appropriately, as the size of the tumor, N, grows, F(N) would shrink. This would force the rate of growth, dN/dt, to decelerate and eventually slow to zero. kf(t) represents the time-dependent distribution of drug in a particular tissue compartment. The terms on the right of the minus sign in these equations represent loss, and the difference in the cell death rates is due entirely to the term m x, which is large if the population is sensitive and small if it is resistant. The model could be modified to implicitly represent MDR by representing the pharmacokinetics of the two (segregated) tissue compartments as asymmetric, e.g., by allowing for in= creased outflow from the resistant tissue compartment. Then kf(t), representing whole tissue and not intracellular concentrations, would be different for the two populations. However, the authors do not address this aspect of resistance explicitly, and one should note that this type of model does not distinguish between decreased drug accumulation in the tissue and decreased drug accumulation in the cell.

318 C. The hybrid

Michelson and Slate (1989, 1991) developed a mathematical model which describes drug resistance in a more mechanistic manner, by defining it as one or all of the physiologic pathways listed in the Introduction above. Transitions (e.g., growth, death, acquisition of resistance) are expressed stochastically for individual cells. However, the probability rates defining the transitions are dynamic, determined at each time step by the level of active drug at the target site of the average tumor cell. The model assumes that: 1) Each cell experiences a transition (or none at all!) with a given probability during a given time step independently, and does not depend on its neighbors for signals or controls. 2) The lifelength of each individual cell is a random variable that is identically and independently distributed throughout the cell population. 3) Drug is uniformly distributed throughout the cell population; no spatial hindrance in access of drug to any cell is encountered. 4) Cell death is due strictly to the cytotoxic effects of the drug, and the risk of cell death is proportional to the concentration of drug at the target site in the cell. The distribution of drug moieties throughout the average cell is modeled using standard concentration-dependent first order compartment kinetics to describe the interstitial spaces, the cytoplasm, and the target site (usually the nucleus). While the variable expression of any of the resistance mediators listed in the Introduction may be due to any of a number of underlying processes (e.g, gene amplification, alterations in transcriptional or translational efficiency, etc.), the model only deals with their functional consequences (e.g, decreased uptake, increased efflux0 differential target sensitivity, etc.) explicitly. The very simple assumption is made that cell death is strictly proportional to the concentration of drug at the target site: If the concentration of an active agent at its target site is high enough, the cell will very likely die, and the more drug present, the more likely the death. Now cell death is a probabilistic event, based entirely upon the ability of a drug to penetrate, distribute, and accumulate in the cell. Using these

models, we have shown that any cell which pumps out enough drug such that its concentration at the target site remains "low enough", significantly enhances its chances for survival (Michelson and Slate, 1989; Slate and Michelson, 1991). And though the molecular structure of the pump is not addressed explicitly in this model, a simple pump is an inherent component in the calculations of the drug distribution equations. Thus, this type of hybrid model brings the definition of drug resistance from the tumor-wide level down to the cellular level. One should recognize the inherent limitations of this model. Clearly, in large, poorly vascularized tumors, cell-cell interactions (including "competition" for the drug) cannot be ignored. Any claim that drug is equally distributed across a large tumor mass is self-evidently erroneous. However, on the micro-pharmacological level, the level at which the P-glycoprotein pump works, the distribution of drug within the tumor cell may yield a useful insight into the dynamics of drug resistance and its reversal. It is this simple fact that motivates the development of the types of models described in the next section.

III. Specific models for multidrug resistance A. Intracellular micropharmacology

Demant et al. (I990) developed a model of drug transport very similar to the one Michelson and Slate (1992) developed to describe the P-glycoprotein pump on the molecular level. Their questions were slightly different, however. Demant and colleagues asked: "Could endosomal transport of drug under varying levels of pH account for a major portion of drug efflux in MDR cell lines"? Their model described three compartments: the extracellular medium, the cytoplasm, and the endosomal vesicles. Within the cytoplasm the drug could be in three states: free, bound to low affinity membrane binding sites, or bound to high affinity nuclear binding sites. Based upon the theory of mass action, assuming that the amount of drug bound to the membrane sites is significantly small-

319 er than the number of possible sites (i.e., ignoring saturation), and assuming that equilibrium is achieved rapidly (i.e., instantaneously) in the cytoplasm, they derive an equation for the dissociation constant for membrane binding. Active transport across the membrane is then defined by Michaelis-Menten kinetics. From their model they concluded that active transport is the primary efflux mechanism in MDR cell lines, and that diffusion and exocytosis are not fast enough to account for the rapid efflux observed experimentally. Michelson and Slate (1992, 1993) have expanded upon this initial model to include diffusion and the energy dependence of the pump, and have modeled active transport as a facilitated diffusion process (see Section III.C below). B. Calculation o f p u m p K m

All mathematical models used to describe MDR transport are designed within an experimental context to explain an observed dynamic. Today most researchers (all?) believe that the pump binds to a cytotoxic target drug before actively transporting it out of the cytoplasm. The binding and facilitation of the transport are reminiscent of enzyme kinetics. Therefore, most (all?) theoreticians have described the pump and its dynamics using a Michaelis-Menten rate equation. Horio et al. (1990) set up their model to mimic an experiment in which apical-to-basal and basal-to-apical flux across MDCK epithelial cells was measured. Based upon differential flux characteristics and relative diffusion rates, they derived a final equation for the apparent K m algebraically. Transport observed experimentally for each direction was used to get a handle upon the intracellular concentration of drug so that the Eadie-Hoffstra plots could be employed. In a similar study Spoelstra et al. (1992) based their model on an experimental flow through system. Flux, defined as a time derivative, d/dt, is the result of net diffusion and Michaelis-Menten transport. They derived net flux estimates from intracellular and extracellular concentrations, and by assuming equal membrane diffusability in both directions. Using Scatchard plots they derived

estimates for the number of binding sites available to the target drug, and their affinity. However, they also observed that the Hill slopes of their Scatchard lines were greater than one, making strict interpretation of their data difficult. C. Energy-dependent facilitated diffusion

Michelson and Slate (1992) developed a model of the MDR pump based upon a washout experiment: tumor cells are preloaded with radioactively tagged drug, removed from the drug-loaded medium, washed, and restored to a new drug-free medium. The intracellular drug concentrations are then monitored over time. The P-glycoprotein associated with MDR is an energy-dependent pump which can be described by the following enzyme kinetics: E +2 A T P o E ' 2 A T P E ' 2 A T P +S(inside) ~ E ' 2 A T P . S E ' 2 A T P ' S ,, E + 2 A D P +2P +S(outside)

(3)

where E is the concentration of p170, and S is the concentration of the substrate (drug). The rate of the first reaction is given by the typical MichaelisMenten formula V 1 = VAre'ATP

(4)

K A +ATP

The rate of the second reaction is given by

V~= ~

Vs'S

(5)

K~+S

Because the total flux of substrate out of the cell is limited by the slower of the two rates, V 1 and V v we define VMAX,the minimum efflux rate, as the minimum of the V 1 and V 2. Thus, the ove.rall flux is given by FLUX =

VMAx'ATP'S (Ka + A T e ) ( K g +S)

(6)

320 The level of ATP in the resting cell is dynamically maintained by conversion to and from ADP and free phosphate. For the purposes of the model, Michelson and Slate assumed that the cell is a homogeneously-mixed compartment packed with enzyme processes which maintain this homeostatic condition. They further assumed (for numerical purposes) that these enzyme reactions could be conglomerated into a single energy maintenance process which follows Michaelis-Menten kinetics. Mathematically, then, the energy pools of the cell can be described by dA T P

VIA TP

V *ADP

dt

K I +ATP

K *+ADP

dADP

V IATP

V *ADP

dt

K / +ATP

K *+ADP

(7)

where (V*,K*) and (V',K') are the Michaelis-Menten constants for the ADP-ATP conversion process. When a cell is challenged with a cytotoxic agent, the efflux pump begins operating, and the ATP pool is depleted by two molecules for each molecule of drug pumped from the cell. Consequently, for each molecule of drug pumped from the cell two molecules of ADP are produced. Therefore, Eq. 7 was modified to be: dATP =-2FLUX WATP + V*ADP dt K I +ATP K* +ADP dADP V I A TP V *ADP = 2FLUX + dt K I +ATP K *+ADP

(8)

To represent the entire pump and its machinery, one can generalize the formalism developed by Joshi (1985) as follows: OS

Ot

~S

-~

bx 2

-F--0

VMAx'ATP'S

F=

(Ka +ATP) ( K M +S) dATP

WATP = -2F

dt dADP

~ = dt

-

(9) V *ADP

+

K I +ATP V I A TP

K* + A D P V *ADP

K I +ATP

K *+ADP

2F+ .

where ~ is the diffusion rate constant for drug through the membrane (from the cytoplasm to the interstitial space) and x represents the perpendicular radial distance through the membrane from inside to out. The spatial characteristics of the model assume that substrate is distributed only within the pump (i.e., within the transmembrane channel of the pump), and that the concentration of substrate in the cytoplasm is derived from a well mixed compartment. Therefore, the concentration of substrate at the internal membrane surface is representative of the entire cytoplasmic concentration. A similar assumption is made about the external cell surface. What this model says, is that drug would normally leave a preloaded cell via diffusion. This assumption is similar to that made by Spoelstra et al. (1992). Mathematically, simple diffusion over a spatial distance, x, (in our case, radially through a membrane) is described by the partial differential equation given in System 9 when F = 0. To facilitate this diffusion, something extra must be added. In this model, the flux term, F. The derivation of the flux term is crux of the model. It is this term which explicitly accounts for the energy dependence of the pump and its binding characteristics for the target drug. Suspected inhibitors would also be accounted for by the flux term. Therefore, characterization of flux is necessary if one is to describe MDR reversal in a mechanistic way (see Section IV, below). Numerical simulations were run to confirm the reasonableness of the model. A drug washout experiment was mimicked. The preloaded cells were allowed to equilibrate in fresh medium, and the cytoplasmic concentration profiles for the target drug were plotted over time. The results were as expected. Without the pump, drug halflife in the cytoplasm was 12 h (defined by pure diffusion). With the pump, the halflife was about 30 min (similar to that observed in the laboratory experiments). Rate constants for the different pumping processes were derived from both laboratory measurements and from standard literature estimates.

321 IV. Model for reversal of MDR

One way to reverse MDR is to take advantage of the energy dependence of the pump. However, based upon their model, Michelson and Slate concluded that that strategy was fraught with peril. Mathematical analysis of the basic model (System 9) shows that since the ATP-ADP predrug steady state is stable, dynamic, and nonoscillatory, one could, in theory, exhaust the ATP pool in the local milieu of the pump. This could be achieved by increasing flux (a self-defeating strategy), blocking the efficiency of ADP to ATP synthesis, or increasing ATP hydrolysis. The fact that the energy specific parameters represent potential targets in resistance reversal completely ignores the fact that manipulating the energy producing systems in a cell would have to be exquisitely precise for one to minimize toxicity in vivo. However, Michelson and Slate's objection to this means of MDR reversal is based upon the theoretical fact that, with respect to ATP levels, the pump is, in fact, a self-regulating mechanism. Once a cytotoxic challenge begins, and if sufficient levels of ATP and target drug are around, the pump will saturate, forcing flux to some positive constant, F*, less than or equal to Vmax. During the transition phase from a completely inactive pump to one which is totally saturated, the steady state concentrations of ATP and ADP shift in their phase space towards the ADP axis. When the pump is fully saturated and flux equals F*, a new steady state is established. If the level of drug in the cytoplasm is such that the energy control mechanisms can maintain high enough ATP levels for the pump to function, all the drug will be pumped out, flux will tend to zero, and the original steady state will be re-established, If the levels of drug in the cytoplasm are such that one begins to exhaust the ATP pool, then flux decreases, maybe even being forced to zero, i.e., the pump stops, until the ATP levels are re-established, and a new steady state (at some nonzero ATP level) is achieved. When that occurs, the flux is re-established at a new lower level. When all the drug is pumped out of the cell, the flux again tends to zero, and the original steady state is re-established.

Simulation studies (Michelson and Slate, 1992) have shown that under physiologically reasonable conditions, even in the presence of energy disruption, the ATP pool is replenished much more quickly than the pump diminishes it. At best, the effiux of drug is slowed only slightly. If a compound could be created which only attacks the energy conversion process for the pump !in tumor cells, and thus remain nontoxic to other ceils in vivo, it would, at best, act as an adjunct to one of the other therapeutic strategies. Given that the energy pool is probably a suboptimal target for MDR reversal, how would one inhibit the P-glycoprotein pump? The inhibitor could: 1. Attack the outward diffusion kinetics through the membrane. 2. Attack the transport activity of the pump at the inner surface of the membrane (e.g., its ability to bind drug). 3. Attack the efflux efficiency of the pump by "clogging" the transmembrane pore complex extracellularly. Strategy number 2 is the one presently being pursued most actively in the clinic. Compounds are sought which block the efflux action at the drug binding site(s) of the pump. Essentially, this is the motivation for using the calcium channel blocker, verapamil, and other P-glycoprotein inhibitors as MDR reversal agents. Michelson and Slate (1993) have extended their original model to include such theoretical reversal agents. Suppose one introduces a competitive inhibitor, I, into the system. Then the V z equation above (Eq. 5), should be altered as follows: Vs.S V2 = KM(1 + (I/K1)) + S

(10)

where K I is the dissociation constant of the Pglycoprotein-inhibitor complex. Efflux becomes

F=

VMax'ATP'S

(G +ATe) (K~(1 + (I/K,)) +S)

(11)

322 Adding inhibitor diffusion and effiux to the original system results in:

3s __

3t OI -a7 F 1=

32s -

_

o',

-F~=0

bx 2 B 321 -~x 2 G=~ VMAx'ATP'S -

(KA +ATP) (KM(1 +(I/KI)) +S) VMAx"A TP "I

(12)

G= (XA +ATP) (K/(1 + (S/Ks)) +I dA TP _ = dt dADP= dt

_

V *ADP K* +ADP (FI+F2) _ . V*ADP K * +ADP

- ( F 1 + F 2)

+

where 13 is the diffusion constant for the inhibitor, F 1 is the effiux of the drug, and F 2 is the effiux of the inhibitor. Since the same fixed number of ATP molecules is required to transport each molecule out of the cytoplasm (whether S or I), and since the two substrates are competitive inhibitors, a total effiux term, (F 1 + F2), must be added to the energy conversion equations. A similar analysis can be developed for a noncompetitive inhibitor. By replacing V 2 with

v2 =

V 1*S

(13)

(KM+S) (1 +(I/X,)) one rederives the diffusion equation with new flux terms

VMAx'ATP'S FI*

,

(Ka +ATP) ((KM+S)(1 +(I/g~))) VMax'ATP'I

(14)

G= (KA +ATP) ((K~+I) (1 + (S/Ks)))

Simple inspection of Eqs. 10 and 13 shows that a competitive inhibitor increases K m by a factor of (1 + I/KI), a normalized measure of the inhibitor

activity. The noncompetitive inhibitor, on the other hand, decreases the transport (reaction) rate by a factor of 1/(1 + I/K~), which again has I normalized to its own binding affinity. The meaning here is clear: In the first instance, as the inhibitor competes with the substrate, more substrate will eventually overcome any inhibition, while in the second, no amount of substrate can overcome the blockade, and the reaction is slowed as long as inhibitor is present. The question Michelson and Slate asked is: "What should the initial loading of a test compound be given its 1) inhibitive character, and 2) its affinity for the binding site? Is any one kind of inhibitor better than any other? Does one need to worry about the timing and strategy for administration"? Their simulation studies showed that over a fixed range of initial loading and inhibitor affinities, noncompetitive inhibitors are more efficacious than competitive inhibitors. But these advantages are only evident at the lower loading concentrations (suggesting an answer to their second question about the pharmacokinetics of the inhibitor). If enough competitive inhibitor can get into the tumor cells prior to drug therapy, and if it can be maintained there at high enough levels despite the pump's activity, then competitive inhibitors with proper pharmacokinetic profiles may be acceptable as clinically relevant MDR reversal agents. Michelson and Slate also looked at the question of whether a one site noncompetitive inhibitor, one which is pumped out of cell along with the target drug, is significantly different in its effectivity from a two-site noncompetitive or allosteric hindering inhibitor which is not. They showed that if an inhibitor is not extruded by the pump, then its advantage over a similar compound which is forced from the cytoplasm is exerted only at high affinitylow concentration combinations. If both inhibitors are low affinity, then no real advantage was observed.

323

V. Implications of the theory A cautionary word is in order before one ventures to interpret the theoretical results described above. First, note that the models of Spoelstra, Horio, and Michelson and Slate are all variations upon the Michaelis-Menten transport theme. Each, in its own way, describes transport as a saturable rate phenomenon depending upon the mass-action chemistry of the pump molecule. The differences between the three models reside in the detailed descriptions of diffusion, energy dependence, etc. Each is based upon the experimental design of choice used within their own laboratories to investigate transport. Second, these models are in a state of evolutionary flux. For example, Michelson and Slate's earlier results (1990, 1992) did not adequately mimic the inhibition of p170 activity. When modeling inhibition, the stereochemistry of the ATP binding sites, the nature of the inhibitor (competitive, noncompetitive, or allosteric), the stoichiometry of the drug binding site(s), etc., were not accounted for. It was these added layers of complexity that required the development of a more sophisticated model like that presented above. Therefore, one must accept these initial results with some degree of caution. For example, in the Michelson and Slate model (1993) outlined above in Section IV, they modeled diffusion as the only other mechanism of inhibitor loss. This is especially important when considering the case of a twosite allosteric noncompetitive compound. For example, they completely ignored the possibility that the inhibitor is the target of biochemical degradation, metabolism etc. But suppose it is. Then one should expect that the target drug effiux will accelerate with inhibitor degradation, and might even approach that observed when a one-site noncompetitive inhibitor is used. In a worst case scenario, if the degradation rate of the two-site inhibitor is faster than the effiux rate of the one-site inhibitor, the effectivity of the stable one-site inhibitor could even surpass that of the two-site one. The models that have been developed thus far can be used to make simple predictions about how MDR reversal agents could be optimally employed to block pumping activity. However, in order to

create a more realistic model, one must consider other complexities of P-glycoprotein function. For example, how many binding sites are there and what are their structures? How does binding affect ATPase activity? Are all P-glycoprotein molecules identical? How does posttranslational modification of the protein affect is transport and binding characteristics? It is the evolution of these questions, derived from the experimentalist, that drives the theorist to develop new mathematical models of MDR and its reversal.

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Address ,for offprints: S. Michelson, Department of Biomathematics, Syntex Discovery Research, MS A4-100, 3401 Hillview Avenue, Palo Alto, CA 94303, USA.