Journal o f Pharmacokinetics and Biopharmaceutics, Vol. 3, No. 2, 1975

The Kinetics of Methotrexate Distribution in Spontaneous Canine Lymphosarcoma Robert J. Lutz, 1 Robert L. Dedrick, 1 James A. Straw, 2 Michael M. Hart, 2 Philip Klubes, 2 and Daniel S. Zaharko 3 Received Nov. 22, 1974--Final Jan. 28, 1975

A mathematical model is presented to simulate the time-dependent uptake of methotrexate in spontaneous canine lymphosarcomas in vivo. Blood flow rates in these tumors are high so that transporl ~to the tumor is limited by cell membrane resistance. A significant amount of rapidly exchangeable methotrexate appears to exist in extracellular space loosely bound to proteins or cell membranes. Transmembrane drug transport follows Michaelis-Memen kinetics, with the maximum facilitated transport ranging from 0.002 to O.O071~g/min/ml for the separate tumors studied and a Michaelis constant for transport equal to 0.2 ~g/ml. This is in the range of Michaelis constants reported for normal tissues in rats in vivo and in several cell lines in vitro. KEY W O R D S : methotrexate; spontaneous lymphosarcoma; pharmacokinetic model; transport in tumors.

INTRODUCTION Methotrexate is an effective agent for treatment of several kinds of cancers, including acute lymphoblastic leukemia, choriocarcinoma, Burkett's lymphoma, and some sarcomas (1). By strongly binding to the intracellular enzyme, dihydrofolate reductase, methotrexate inhibits the conversion of dihydrofolate to tetrahydrofolate, a cofactor in DNA synthesis. The efficacy of methotrexate depends on the ability to maintain intracellular drug concentrations in the target tumor sufficient to inhibit dihydrofolate reductase while maintaining tolerable inhibition of the host's normal tissues. Many data have been reported regarding the transport of methotrexate in several tumor cells lines in vitro, as reviewed previously (3); however, few data are tBiomedical Engineering and Instrumentation Branch, National Institutes of Health, Bethesda, Maryland 20014. 2Department of Pharmacology, George Washington University, Washington, D.C. 20037. 3Laboratory of Chemical Pharmacology, National Institutes of Health, Bethesda, Maryland 20014. 77 9 1975 Plenum PublishingCorporation, 227 West 17th Street, New York, N.Y. I0011. No part of this publication may be reproduced,stored in a retrievalsystem, or transmilted,in any form or by any means, electronic, mechanical, photocopying, microfilming,recording, or otherwise, without written permission of the publisher.


Lutz, Dedrick, Straw, Hart, Klubes, and Zaharko

available on the kinetics of methotrexate distribution in solid tumors in vivo. Indeed, most solid tumors have been particularly resistant to standard chemotherapeutic modalities (2), resulting, perhaps in part, from an inability to maintain effective drug levels at the tumor site. A better understanding of the drug transport mechanism in solid tumors could lead to improved treatment. Recently, a mathematical model has been reported which describes the transport of methotrexate in vivo in several normal tissues in rats (3). The principles of that model have been utilized in this work to describe" the timedependent uptake of methotrexate, in vivo, in spontaneous canine lymphosarcomas. Factors considered which are relevant to the transport of drug include (a) tumor blood flow rates, (b) cell permeability, (c) enzyme content, and (d) both intracellular and rapid-equilibrium extracellular binding. Experimentally measured parameters are used in the model wherever possible, e.g., blood flow rates and enzyme concentrations ; other parameters are inferred from the data, e.g., Michaelis constants. MATERIALS AND M E T H O D S A detailed description of the experimental methods has been published (4); only a brief summary will be repeated here. Dogs with a clinical diagnosis of lymphosarcoma were utilized in this study. Multiple tumor sites were usually present in each animal. Regional blood flow of involved lymph nodes was determined by a thermal dilution technique described by Tuttle and Saddler (5). The method was checked by comparing measured blood flow with published values for several normal tissues of dogs, and in some cases by direct measure of blood flow by venous cannulation. They were found to be in good agreement. Chromatographically purified 3H-methotrexate sodium salt was administered by rapid intravenous injection at doses of 0.01, 0.03, 0.1, 0.3, and 3 mg/kg to pentobarbital-anesthetized dogs. Chromatographically purified 14C-inulin was administered simultaneously with methotrexate in order to measure extraceUular space. Blood samples were withdrawn through a venous cannula at 2-min intervals up to 20 min followed by 5-min samples up to 90 min after injection of the drug. Peripheral tumors from among the multiple tumor sites in each dog were excised with a minimum of bleeding at 789 15, 30, 60, and 90 min after administration of the drug. The excised tumors were frozen in liquid nitrogen, sliced into cubes representing various areas of the tumor, and, after drying and combustion to 3H20 and 14CO2, assayed for radioactivity by liquid scintillation spectrometry. Total tissue concentrations of methotrexate and inulin were determined from the specific activity of the administered drug.

The Kinetics of Methotrexate Distribution in Spontaneous Canine Lymphosarcoma


Some tumors were excised prior to drug administration, frozen, and homogenized for assay of dihydrofolate reductase activity. In addition, the homogenate was titrated with methotrexate in the presence of dihydrofolate and NADPH to determine the concentration of methotrexate required for 100 % inhibition of dihydrofolate reductase. DEVELOPMENT OF THE MODEL A complete model for the prediction of the time course of plasma and tissue concentrations of methotrexate in the dog is available (6). However, since the primary focus of attention was on the tumor uptake of the drug in these studies, actual experimental plasma concentration curves were more convenient to use as model inputs. This also obviated the problem of accounting for all the variabilities among dogs of different breeds and various states of health. A triexponential equation of the form Cp = a l e -b'~ + a2 e -b~t + a3 e -b~'


was fitted to the plasma data with a nonlinear, least-squares routine (13). The earliest experimental plasma concentration taken was at 2 min. A zero time point for the triexponential curve fit, representing the peak in the plasma concentration, was derived from a simulation using the full dog model (6) rather than merely extrapolating the 2-rain point to zero time. This method should yield reliable estimates of the area under the plasma concentrationtime curve at early times. Simultaneous tumor and plasma concentrations are plotted in Fig. 1. At early times and at high doses, drug is mostly extracellular and serves to determine extracellular methotrexate space. Figure 1 shows that no unique relationship exists for the methotrexate tissue-to-plasma ratio. At a given plasma concentration, for example, 0.01/zg/ml, there are different tumor concentrations depending on the dose administered and the time the tumor sample was taken. This indicates that the tumor is not a flow-limited, equilibrium compartment, and suggests a membrane-limited transport similar to that reported for bone marrow, spleen, and small intestine in rats (3). Figure 2 is a scheme of the tumor model. Exchange across the capillary membrane is so rapid that the plasma and interstitial fluid are assumed to form one equilibrium compartment. That is, the concentration of free drug in the plasma is the same as the concentration of free drug in the interstitial fluid, denoted by Co in equation 2. The total concentration of drug in the extracellular space, Ce (which includes plasma and interstitial space), may be greater than the free concentration if binding occurs in extracellular space, perhaps as adsorption of drug to cell surfaces or to interstitial proteins.


Lutz, Dedrick, Straw, Hart, Kiubes, and Zabarko







z Z

n,, Iz (j= Z O r LM

//=oQ 9







of 9


/ 0 0





X [,J I0













I I0

Fig. 1. Distribution of methotrexate between plasma and lymphosarcoma in dogs following intravenous doses of 3.0 ( ~ , t ) , 0.3 (IS], II), 0.1 (V, V), 0.03 (A,&), and 0.01 ( 9 mg/kg at times from 789 to 90 min. Open and shaded symbols represent two different dogs at each dose. Dashed line represents tissue-to-plasma ratio of 1 : 1.










Vi Ci


Fig. 2. Diagram of tumor model. Q is plasma flow rate to tumor, ,Cp is methotrexate plasma concentration entering tumor compartment, Co is free methotrexate plasma concentration leaving in equilibrium with the compartment, Ce is the extracellular space concentration which includes a saturable fraction in rapid exchange with the plasma, (jA) is the transmembrane flux, and Ci is the free intracellular concentration.


The Kinetics of Methotrexate Distribution in Spontaneous Canine Lymphosarcoma


If the binding is in rapid equilibrium, then the total concentration of drug in the extracellular space is a direct function of the plasma concentration, and the plasma and interstitial fluid can be lumped together as one equilibrium compartment with concentration Ce. In the model, transport across the cell membrane occurs by a saturable, facilitated mechanism demonstrating Michaelis-Menten kinetics. The mass-balance equations on the individual compartments are presented below. A more complete description of the derivation has been published (3). For the extracellular compartment, the mass balance is dCe Ve-d~ = Q(Cp - Co) -. (jA)


where V~is the tissue extracellular volume (ml), Q is the plasma flow rate to the tumor (ml/min), Cp is the plasma methotrexate concentration (#g/ml), Co is the concentration of free methotrexate exiting in equilibrium with the extracellular compartment (#g/ml), C~ is the total extraceltular methotrexate concentration (kLg/ml), and ( j A ) is the net flux of drug to the intraceltutar compartment (#g/rain). The mass balance on the intracellular compartment results in the following equation in terms of the free intracellular drug : dC i (jA) v~ a--i- = 1 + ~a'/(~ + 6 ) 2


where Ci is the free intracellular methotrexate concentration (/~g/ml), a' is the intracellular strong binding capacity of dihydrofotate reductase (flg/ml), and e is the dissociation constant of the drug-enzyme complex (~g/ml) (7). The total intracellular drug concentration is described as the sum of the free and strongly bound drug by qi = Ci +



~+ Ci


Previous experience with bone marrow and gastrointestinal tissue in rat indicated that methotrexate distributes uniformly in extracellular water, so that the equilibrium extracellular concentration, C~, is equal to the equilibrium exiting concentration in the tissue, Co. As we describe in more detail in the next section, use of this assumption would not allow adequate modeling of the lymphosarcoma data. Rather, it appears that there is a sizable fraction of drug in the extracellular space which is rapidly and reversibly bound by a saturable process. Perhaps this fraction exists as drug bound to cell membranes or to interstitial proteins. The nature of the data did not permit us to differentiate between these possibilities. Nevertheless,


Lutz, Dedrick, Straw, Hart, Klubes, and Zaharko

an equilibrium distribution ratio for methotrexate in the extracellular compartment, Ce/Co, can be written as Re =

f B+C

1.0 + -




where R e is the methotrexate distribution ratio (CdCo), f is the maximum bound concentration 0tg/ml), and B is the dissociation constant for adsorption, or binding. If the rapidly exchanging phase is not present, then f = 0 and Ce = Co ; i.e., the total extracellular concentration is simply equal to the free equilibrium concentration exiting from the extracellular compartment. Noting that (6)

Co = C d R e

equation 2 can be written as

dce e dt =

Q(Cp - Ce/Re)




The flux across the cell membranes is a saturable process described by the following equation : (jA) -

kV(Ce/Re) K + (Ce/R~)

kVC~ K + C~


where k is the maximum facilitated transport rate (#g/min/ml) based on total tissue volume, V is total tissue volume (ml), and K is the Michaelis constant for transport (#g/ml). Equations 3, 7, and 8 are solved simultaneously to derive the total extracellular concentration and the free intracellular concentration. Equation 4 gives the total intracellular concentration. The tissue average concentration is the volume weighted average of the extracellular and intracellular concentrations. DETERMINATION


Values of the parameters used in the mathematical model were obtained by direct measurement wherever possible; others were inferred from the tissue and plaSma data. Plasma Concentration Curve


Plasma data were fitted to a triexponential curve as shown in equation 1 by a nonlinear least-squaresregression (13). The plasma curves along with the data a r e shown in Fig. 3 for five individual dogs, one at each of five different doses. The resulting analytical expression described by equation 1 served as the input forcing function to the model.

The Kinetics of Methotrexate Distributionin SpontaneousCanine Lymphosarcoma



I01 DOSE (mg/kg)



(LS2 ,~}




(LS3, 'el




(LS?, e)

-6 ;r o



(=: =o ,~





t ~o

[ 20

I 30

I 40

I 5o

I 60

_L ?0

] 80

I _1 90 ~00


Fig. 3. Plasma concentration of methotrexate vs. time following several intravenous doses. Points represent experimental data. Solid line represents triexponential curve fitted as described in the text.

Flow Rates (Q) Flow rates were obtained by a thermal dilution technique which allowed measurement of spatial variations of flow through the tumor from its periphery to its center. Blood flow was greatest immediately below the surface of the tumor and decreased progressively toward the center, Still the minimum flow rate encountered in any region of the tumor was greater than in some

Lutz, Dedriek, Straw, Hart, Klubes, and Zaharko


Table I. Model Parameters of Lymphosarcomas in Five Dogs ~ Dog No.

Dose, mg/kg Ve, % ECF V~, % ICF a', strong binding, #g/ml k, maximum facilitated transport, #g/ml/min K, Michaelis constant, ,ug/ml f, maximum extracellular binding, /~g/ml B, Michaelis constant for extracellular binding, ~tg/ml






0.01 14 86 0.21 0.007

0.03 15 85 0.10 0.0064

0.1 27.5 72.5 0.21 0.007

0.3 22 78 0.06 0.002

3 19 81 (0.2)b 0.005

0.2 0.85

0.2 0.85

0.2 0.85

0.2 0.85

0.2 0.85






%: 10-5 #g/ml, Q = 0.4 ml/min/g tumor (plasma) for all dogs. bEstimated. =

well-perfused tissues, such as brain (4). The model was not flow limited even with use of the m i n i m u m flow rate in the tumor. Increased values of Q did not increase simulated drug uptake in the tumor. Blood flow, therefore, does not a p p e a r to limit methotrexate delivery to any area of these tumors.

lntraeellular Binding Constant (a') The a m o u n t of methotrexate required to p r o d u c e 100% inhibition of dihydrofolate reductase was determined by in vitro titration of excised, h o m o g e n i z e d t u m o r samples prior to methotrexate exposure of some dogs. The value is the methotrexate-equivalent enzyme content, a', in Table I. W h e n no data were available, a' was set equal to 0.2 #g/ml. This assumption is reasonable since it has been shown that the predicted concentrations in membrane-limited tissue are independent of a' if the dihydrofolate reductase is not saturated (low doses) and also are not very sensitive to a' at high doses (3).

Membrane Transport Parameters (k, K) At the two lowest doses, the methotrexate t u m o r uptake was approximately p r o p o r t i o n a l to dose, implying a linear transport mechanism. Until all the dihydrofolate reductase is saturated with drug, virtually no free drug exists intracellularly, and only influx occurs. Since (Ce/Re)~ Cp

The kinetics of methotrexate distribution in spontaneous canine lymphosarcoma.

A mathematical model is presented to simulate the time-dependent uptake of methotrexate in spontaneous canine lymphosarcomas in vivo. Blood flow ratew...
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