Predicted and Observed Effects of Antibody Affinity and Antigen Density on Monoclonal Antibody Uptake in Solid Tumors Cynthia Sung, Ty R. Shockley, Paul F. Morrison, et al. Cancer Res 1992;52:377-384. Published online January 1, 1992.
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[CANCER RESEARCH 52, 377-384, January 15, 1992]
Predicted and Observed Effects of Antibody Affinity and Antigen Density on Monoclonal Antibody Uptake in Solid Tumors Cynthia Sung,1 Ty R. Shockley,2 Paul F. Morrison, Harold F. Dvorak, Martin L. Yarmush, and Robert L. Dedrick Chemical Engineering Section, BiomédicalEngineering and instrumentation Program, National Center for Research Resources, NIH, Bethesda, Maryland 20892 ¡C.S., P. F. M., R. L. D.]; Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 [T. R. S.¡;Department of Pathology, Beth Israel Hospital and Harvard Medical School and the Charles A. Dana Research institute, Boston, Massachusetts 02215 [H. F. D.]; and Department of Chemical and Biochemical Engineering and the Center for Advanced Biotechnology and Medicine, Rutgers University, Piscataway, New Jersey 08854 and Massachusetts General Hospital and Harvard Medical School, Boston, Massachusetts 02215 [M. L, Y.J
ABSTRACT The uptake and binding of monoclonal antibodies (MAbs) in solid tumors after a bolus i.v. injection are described using a compartmental pharmacokinetic model. The model assumes that M Ah permeates into tumor unidirectional!}1 from plasma across capillaries and clears from tumor by interstitial fluid flow and that interstitial antibody-antigen interactions are characterized by the Langmuir isotherm for reversible, saturable binding. Typical values for plasma clearance and tumor capil lary permeability of a MAb and for interstitial fluid flow and interstitial volume fraction of a solid tumor were used to simulate the uptake of MAbs at various values of the binding affinity or antigen density for a range of MAb doses. The model indicates that at low doses, an increase in binding affinity may lead to an increase in MAb uptake. On the other hand, at doses approaching saturation of antigen or when uptake is permeation limited, an increase in the binding affinity from moderate to high affinity will have only a small effect on increasing MAb uptake. The model also predicts that an increase in antigen density will greatly increase MAb uptake when uptake is not permeation limited. Our experiments on MAb uptake in melanoma tumors in athymic mice after injection of 20 ¿tgMAb (initial plasma concentration, about 120 MM)are consistent with these model-based conclusions. Two MAbs differing in affinity by more than 2 orders of magnitude (3.8 x 10" M ' and 5 x 10'" M ') but with similar In vivo antigen densities in M21 melanoma attained similar concentrations in the tumor. Two MAbs of similar affinity but having a 3-fold difference in in vivo antigen density in SK-MEL-2 melanoma showed that the MAb targeted to the more highly antigen attained a higher MAb concentration. We also discuss predictions in relation to other experiments reported in the The theoretical and experimental findings suggest that, for applications, efforts to increase MAb uptake in a tumor should the identification of an abundantly expressed antigen on tumor than the selection of a very high affinity MAb.
expressed the model literature. high dose emphasize cells more
INTRODUCTION Uptake of MAbs3 by solid tumors depends on factors govern ing the antibody-antigen interaction as well as on host-related factors such as plasma clearance and tumor physiology. Phar macokinetic models offer theoretical descriptions of the inter dependence of these various factors (1-6). The models can suggest experiments to test the relevance of those factors as well as provide ideas for strategies to improve MAb-based therapies. In this paper, we present a physiologically based, compartmental pharmacokinetic model which relates tumor uptake of a MAb to its plasma kinetics, capillary permeabilityarea product, and binding affinity and to the interstitial fluid flow rate, interstitial volume fraction, and concentration of Received 6/19/91 ; accepted 11/1/91. The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked advertisement in accordance with 18 U.S.C. Section 1734 solely to indicate this fact. 1To whom requests for reprints should be addressed, at Building 13, Room 3W13. 2 Present address: Baxter Healthcare Corp., Round Lake, IL 60073. 3The abbreviations used are: MAb, monoclonal antibody: AUC, area under the concentration-time curve; CEA, carcinoembryonic antigen.
antigen in the tumor. The purpose of developing this model was 2-fold. One was to simulate the effects on MAb uptake in solid tumors produced by changing the value of parameters believed to play an important role in transport and binding. By relating the transport and binding events to physiological proc esses and antibody-antigen interactions, we could examine the oretically the effect of parameters within a range of values representative of MAbs and test experimentally whether param eter changes would lead to model-predicted changes in MAb uptake. The other was to apply the model to analysis of exper imental data in order to obtain quantitative estimates of the transport and binding constants in in vivo systems. Fitting data to a mathematical model requires that the model be parsimon ious in the number of unknown constants. Therefore, our approach was to develop as simple a model as possible, includ ing only the predominant processes governing transport and binding while recognizing that some quantitative conclusions drawn from data fitting are limited by approximations in the model. Our pharmacokinetic model differs from previously published models (1, 3-5) in that the others include a more comprehensive range of processes that may affect uptake of MAbs in tumors. For the experimental conditions considered by us, however, the larger number of adjustable parameters and the complexity of the equations in those models makes them more cumbersome for data analysis. The model presented in this paper provides a useful exploratory tool for the experimen talist and aids in understanding how certain MAb and tumor properties affect the pharmacokinetics of MAbs. Recently, we described the special case when MAb doses are low enough such that the binding sites in the tumor are far from being saturated (6). The low dose case allowed us to represent the ratio of free and bound drug in the interstitium as a constant. This has the advantage that the relevant differ ential equation is linear, making it easier to understand the mathematical relationship among the various parameters. Often, however, doses of interest are sufficiently high that the binding sites approach saturation (7-10). In the present paper, we extend our earlier work over a wider dose range by repre senting binding with the Langmuir adsorption isotherm for reversible, saturable binding. The differential equation which describes the interstitial MAb concentration no longer has an exact analytical solution but must be solved numerically. Below we show several model simulations on the effects of varying the affinity constant, K,, or the concentration of binding sites, B0. We also present experimental data on MAb uptake at high doses in solid melanoma xenografts using MAbs which vary in K, or Bo. The experimental results support predictions of the model that a very high affinity MAb does not necessarily attain measurably higher uptake in tumors compared to a moderate affinity MAb whereas increases in B0 are more likely to yield higher MAb uptake. 377
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AND EXPERIMENTAL
ANTIBODY PHARMACOKINETICS
MATERIALS AND METHODS
Plums
Model Development. The plasma and total tumor concentrations of a MAb are experimentally measurable quantities in tumor uptake studies. The total interstitial tumor concentration (e.g. mol/g interstitial fluid) can be calculated from the total tumor concentration (mol/g tumor) by subtracting the amount of MAb which exists in the plasma space of the tumor and dividing the remainder by the fractional inter stitial volume of the tumor, 0 (6, 11). The model represents the plasma and interstitial tumor spaces as compartments of uniform concentration (Fig. 1) and assumes that: (a) after a bolus i.v. injection, the plasma MAb concentration, CP, decreases biexponentially from an initial plasma concentration CPO(Equation A); (¿>) MAb is transported from the plasma into the tumor unidirectionally at a rate equal to the product of the plasma-to-tissue transport constant, k, and the coexisting plasma concentration; (c) MAb in the tumor interstitium rapidly equilibrates between free and bound forms according to the Langmuir adsorption isotherm (Equation B), where K, is the MAb affinity constant and B0 is the interstitial concentration of binding sites; and (d) loss of free MAb from the tumor occurs at a rate governed by a loss constant, L, and the concentration of free antibody in the tumor. Physiological interpretations of k and L are that k represents the capillary permea bility times the capillary surface area per g of tumor and that L represents the interstitial fluid flow per g of tumor, the mechanism for clearance of free interstitial proteins from the tumor. Assumptions b and d are combined in an ordinary differential equation that describes the overall rate of change of antibody concentration in the tumor interstitium (Equation D). Equation C is a mass balance which states that the sum of the bound and free interstitial concentrations, Cb and Ci, respectively, is equal to the total interstitial concentration, C,.
Tissue Interstitium C,
bound
Fig. 1. Pharmacokinetic model of MAb transport from plasma to tissue. The transport of MAb from plasma to tissue is assumed to be unidirectional and governed by the capillary permeability-area product, k, and the coexisting plasma concentration. In the tissue compartment, the MAb is assumed to rapidly distrib ute between free and bound compartments in accordance with the Langmuir equilibrium binding isotherm where K, is the MAb affinity and A0 is the concen tration of binding sites. Unbound MAb is assumed to be transported out of the tissue at a rate equal to the convective flow of interstitial fluid in tumors or lymph flow in normal tissues, L, times the coexisting interstitial unbound MAb concen tration. This figure is similar to one appearing in Ref. 6 except that K. and B0 now are treatable as separate parameters.
Mm,fairly uniform concentrations are obtained with MAb affinities less than IO8 M~' but that nonuniformities become more pronounced as MAb affinity is increased. Thus for high affinity antibodies and large intercapillary distances, the number of binding sites that is functionally accessible to MAb may be only a fraction of the total. Dose also affects the homogeneity of MAb distribution: Matzku et al. (8) and Fenwick et al. (9) have shown that as dose is increased, the spatial distribution of MAb becomes increasingly uniform. The assumption of rapid equilibrium was tested in Fujimori's model
Plasma compartment: CP
(3) which allows for nonequilibrium binding. It was found that for a MAb with 10' M~' affinity, when the forward rate constant is greater than 3 x IO4ivr1 s~' as is the case for most MAbs (15, 16), the average tumor concentration is not affected.4 Thus, the equilibrium assumption
(A)
Tissue compartment: (B) C, = Cb + Cf
Cr - LC, dt
fre«
(C) (D)
We have modeled transcapillary transport as a unidirectional process because macromolecules the size of albumin and larger are transported into normal tissues mainly by hydrostatically driven convection and are cleared mainly by lymphatic drainage (12) rather than being transported back across the capillaries (13). Tumors lack a lymphatic network, but interstitial fluid similar in protein content to normal lymph is observed to exude from tumors (14). Thus, interstitial fluid flow in tumor probably serves a role similar to lymph flow in normal tissue as the main clearance mechanism for proteins. The assumption of unidirec tional transcapillary transport, however, limits application of the model to large macromolecules; the model equations as written would not be appropriate for MAb fragments such as Fab or Fv. This assumption can easily be relaxed by including transcapillary tumor-to-plasma trans port in Equation D, although at the cost of introducing another parameter. The validity of the assumption that the tumor may be represented as a compartment of uniform concentration depends on many factors such as the homogeneity of tumor structure, intercapillary distances, MAb affinity, and dose. For example, the model is probably appropriate for small tumors but not for large tumors which have substantial pressure gradients or necrotic areas. The distributed model of MAb transport of Fujimori et al. (3) predicts that for an intercapillary distance of 170
is expected to apply for most MAbs. Measurement of binding will also depend upon the extent of internalization and metabolism of the MAb. For example, if a radiolabeled MAb is rapidly internalized and metab olized and the radioactive label is lost from the tumor as a small diffusible compound, or if the MAb binds to a shed antigen and the immune complex is a mobile species which can be cleared by interstitial fluid flow, less radioactivity would be found in the tumor than if such events were absent. Therefore, the model may underestimate the extent of binding when the MAb is directed against a secreted or rapidly metabolized antigen. Because of the difficulty in obtaining both binding and metabolic constants from fitting a model to time-varying average tumor concentration data, we chose not to include an explicit metabolic term. It is important to bear in mind, however, that interpretation of the model-fitted value for B0 may be influenced by internalization and metabolic effects as well as the degree to which MAb is nonuniformly distributed. Hence, B0 represents only an apparent binding site density. Computer Calculations. Simulations were generated by solving the differential equation (Equation D) numerically with the condition that the tumor concentration is initially zero. A fourth-order Runge-Kutta algorithm in Mathematica (Wolfram Research, Inc., Champaign, IL) was used on a Macintosh lici. Fit of experimental data to the model was carried out using Gear's method (17) for numerical solution of the differential equation and a finite difference Levenberg-Marquardt al gorithm (18) for performing a nonlinear weighted least-squares regres sion; the weighting factors were the variances in concentration at each time point. The computer program uses subroutines from the IMSL statistical and mathematical libraries (IMSL, Inc., Houston, TX). In terstitial concentrations for binding and nonbinding MAbs at various times after i.v. injection are input values for the computer program. Animal Model. The animal model was the athymic mouse with s.c. solid human melanoma tumor M21 or SK-MEL-2 receiving an i.v. injection of 20 Mgof '25I-labeled specific MAb and 20 ng of '"I-labeled nonspecific MAb. Details of the animal experiments and characteriza-
378
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THEORETICAL
AND EXPERIMENTAL
ANTIBODY PHARMACOKINET1CS
B = 100
tion of the antibodies and tumor antigens may be found in the accom panying paper (11).
150p C, (nM)
RESULTS
100 •¿
Model Simulations. The pharmacokinetic equation (Equation D) can be solved numerically for given parameter values of the plasma kinetics (Cpo, R, Xi, X2),tumor physiology (k, L, 0, and MAb binding (K,, B0). To carry out the simulations, we used values that are characteristic of MAbs and tumor-associated antigens. For the plasma kinetics, we used data on the plasma clearance of an IgG monoclonal antibody (MAb 436) after i.v. injection into athymic mice (11): R was set equal to 0.459; Xi to 1.11 x 10~2 min'1 and X2to 1.38 x 1(T4 min'1. We set k equal to 0.5 n\/m'm/g based on experiments reported by Colapinto et al. for a MAb given to mice bearing s.c. glioma xenografts (19). L was set equal to 0.8 n\/mm/g based on our experiments with s.c. TE671 rhabdomyosarcoma xenografts (6), a value which also is consistent with experiments by Nakagawa et al. (20) with intracranial gliomas. A value of 0.243 was used for B0 (see "Appendix").
C
po
= 1 nM ,10' IO8 •¿10s
C (nM)
3000 300 30
3.0
For therapeutic applications, MAb doses typically are in the range approaching saturation of the binding sites in a tumor, whereas for imaging applications, nonsaturating doses usually provide better contrast between tumor and normal tissue. The response of the pharmacokinetic model to changes in K„ de pends very much on the administered dose. Table 1 shows the maximum MAb concentration (C,.ma«) attained in the interstitium when Ka = IO8, IO9, 10'°,and infinity for initial plasma concentrations ranging from 1 to 300 nM. For a 20-g mouse, these values of CPOrepresent MAb doses ranging from approx imately 0.15 to 45 /ig; for a 70-kg human, they represent doses from approximately 0.45 to 130 mg. Values of B0 from 10 to 1000 nM were examined. When the maximum interstitial MAb concentration at infinite affinity is less than A0, the maximum is called the permeation limit (see "Appendix," Equation J). In
2.0
1.0
0.0
24
48
72
120
96
Time (hours) C
po
= 100 nM
C (nM)
_.
10
300
Table 1, it can be seen that at the lower doses, an increase in affinity from IO9 M~' to 10'°M~' can lead to a significant, although much less than order-of-magnitude, increase in the maximal interstitial MAb concentration. This occurs only when uptake is not permeation limited. In contrast, increasing A'., over this range of affinities leads to very small increases in C, at the higher doses as antigen saturation is approached (C,.max > 50% Bo) or as the permeation limit is approached (C,.max> 50% permeation limit). From these calculations, it appears that
200
100 30 3
24
48
72
96
120
Table I Maximum interstitial MAb concentration after i.v. injection (nM) Time (hours) M'10.50.91.93.45.2«1.62.85.710.115.6«5.08.517.732.551.5«13.221.244.988.9150.3°36.850.697.2213.1448.3°98.2115.01 10« 10'M-'1.83.35.2°6.5°7.4°4.58.915.0°19.5°22.3°9.721.344.8°63.4°74.0°17.434.187.6172 = 10'°M-'4.5°6.3°7.4°7.8°8.0°8.717.221.9°23.4°24.1°12.230.167.8°77.6°80.2°18.437.4 109 M~') greatly slows MAb penetra
SK-MEL-2 (nM)10050B/"
•¿i-• i/1•i^ir-\ $—¿
-ì
•¿ 436 n IND1
24
Time
48
72
(hours)
Fig. 6. Experimental data on uptake of MAbs in human melanomas xenografled in athymic mice. (A) M21 tumor. MAbs 436 and 9.2.27 differ in K. by a factor of 130 (A', of 3.8 x 10' and 5 x 10'°NT1, respectively) but have nearly the same /(„. A dose of 20 nu was administered; the average CM for animals receiving MAb 436 was 127 nM, for MAb 9.2.27, 115 nM. The pharmacokinetic model predicts that at this dose and these values of A*,and assuming that the plasma kinetics of the antibodies are identical, an increase in binding affinity will result in only a marginally higher uptake in the tumor (see Fig. 3). The experimental data show that the higher affinity antibody MAb 9.2.27 attained a slightly lower tumor concentration than MAb 436. This can be largely attributed to the slightly lower plasma concentrations of MAb 9.2.27 compared to MAb 436 during the 72 h after injection [11]. (A) SK-MEL-2 tumor. MAbs 436 and INDI have identical plasma kinetics and nearly identical affinities (3.9 x 10" and 3.3 x 10" M"1, respectively), but B0 for MAb 436 is 3 times that of MAb INDI, based on measurements on tumor cells obtained directly from s.c. tumors. The experimen tal data support the model prediction that higher antibody uptake will be attained in the tumor with the higher antigen density (see Fig. 5). The curves in both A and B represent least-squares fits of the experimental data to the pharmacokinetic model (see "Materials and Methods"). Model Tits yielded quantitative estimates of the capillary permeability-area product, the apparent binding site concentra tion, and the interstitial fluid flow (11).
our recently published pharmacokinetic model (6) to encompass a broad range of doses that might be used experimentally and clinically. The transport and binding parameters in the model are known for antibodies within a fairly well-defined range, and therefore simulations with numbers within this range can pro vide insight into which parameters will affect tissue concentra tions. This paper examines some of the general features of the model and shows that the effects of binding affinity and antigen density on tumor uptake predicted by the model were observed in animal experiments. The pharmacokinetic model predicts that the effect of in creasing antibody affinity on MAb uptake in tumors is very dependent on dose. At doses which do not approach saturation of binding sites, the model predicts that increases in MAb affinity as high as 10" M ' can lead to substantial increases in
tion and leads to spatial nonuniformity. The pharmacokinetic model predicts that tumor interstitial MAb concentrations are very sensitive to changes in binding site concentrations in a range characteristic of tumor-associated antigens when doses are high enough such that uptake is not permeation limited. Our experiments with two MAbs of similar affinity but differing in antigen density in SK-MEL-2 tumor showed that a significantly higher tumor uptake was attained with the MAb which has approximately a 3-fold higher in vivo antigen density (Fig. 6Ä). Goldenberg et al. (25) have found similar results with a MAb targeted to an antigen (epidermal growth factor receptor) expressed at different densities on three different tumor cell lines. The percentages of injected dose per g of tumor for tumors expressing 5 x IO3, 3 x 10s, and 2x10* receptors/cell were 0.9 (day 7), 4.2 (day 7), and 12.4 (day 8), respectively. While Goldenberg does not describe how the tu mors differed in vascularity, capillary permeability, and frac tional interstitial volume, factors which can also affect tumor uptake of MAb, the sizable difference in antigen expression among the different tumor cells likely dominates the results of their experiments. Philben et al. (26) carried out experiments with a MAb to the CEA on four different tumors with different CEA concentrations. MAb uptake increased with increasing antigen content except for the highest CEA content tumor; 48 h after injection of MAb, the percentages of injected dose/g of tumor for tumors containing 0, 1 x IO2, 2 x IO3, and 1.8 x 10" ng CEA/g were 1.4, 16.4, 51.1, and 29.5, respectively. The authors had evidence suggesting that the tumor containing the highest amount of CEA shed the CEA antigen more rapidly than the other tumors: CEA was secreted more rapidly by cultured tumor cells with the highest CEA content compared to the second highest; and the liver, the organ which catabolizes
382
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THEORETICAL AND EXPERIMENTAL ANTIBODY PHARMACOKINETICS
shed immune complexes, had the highest radioactivity in the animals with the tumor producing the highest levels of CEA. Boerman et al. conducted a study similar to ours wherein comparison was made between two MAbs of similar affinity but targeted to different antigens on the same cell. The in vitro antigen density differed by a factor of 3, but in contrast to our results, they found that MAb uptake was lower for the MAb to the more highly expressed antigen (27). They were not able to detect shedding of the antigen, but a possible explanation for their experiments is that the MAb directed to the more highly expressed antigen was more rapidly catabolized in the tumor compared to the other MAb, resulting in loss of the radioactive label from the tumor cells. Another possible explanation for their results is that antigen expression was different on cells in vivo compared to cells from cultures. We have observed differ ences in antigen expression on M21 and SK-MEL-2 cells in culture and on cells directly taken from in vivo tumors (11). Antigen expression was sometimes increased and sometimes decreased, by up to a factor of 3 in either direction, and we could not discern any factor which correlated with the magni tude or direction of the changes. While our experiments and many in the literature support the pharmacokinetic model prediction that higher antigen density will lead to higher tumor interstitial concentrations of MAb, it is evident that further experiments are needed to examine in vivo-in vitro antigen differences and the effect of MAb catabolism by the target cells on tumor uptake of MAbs. The model clearly is a simplification of the many complex factors which govern MAb localization to a target tissue. For example, it does not contain any information on the spatial distribution of MAb within the tumor tissue. If there are regions of tumor which are far from a capillary or are unperfused or necrotic, then the volume that is accessible to the MAb will be lower than the total interstitial volume of the tumor. Developing an accurate spatially distributed model, unfortunately, is con founded by the need to incorporate information on the distri bution of capillaries, pressure gradients, existence of poorly perfused regions, etc., information which generally is not avail able or well characterized, although progress is being made in this area by several groups (5, 28). The advantage of the simple compartmental model presented in this paper and its limited number of adjustable parameters is that it allows the use of the model for data fitting when only the average MAb concentra tion in the interstitium is known at various times after injection of MAb. In the accompanying paper, we have demonstrated the application of the model to analysis of data on uptake of MAbs in solid melanoma tumors. Model-fitted values of B0 were consistently lower than values from direct measurements on cells removed from in vivo tumors, pointing to the possible existence of heterogeneous MAb distribution in the tumor tissue or MAb internalization and metabolism. On the other hand, values obtained for the capillary permeability-area prod uct and interstitial fluid flow were consistent with histológica! characteristics of the tumors and with other published experi ments on tumor physiology, demonstrating the utility of the model in characterizing quantitatively these transport parame ters for MAbs in solid tumors.
kinetic model. We also would like to acknowledge Dr. Richard J. Youle for critiquing the manuscript.
APPENDIX Equations B and C may be combined to obtain an expression for (', in terms of C,. The result is Equation E. C, = 0.5[(-fi0 + C, - l /K.) + V(fi„-C,+
i/K.)2 + 4CJK.]
(E)
In the limit as K,—»oc, Equation E shows that when C, < fio, then CV-» 0. The differential equation which describes the time-varying total interstitial concentration, Equation D, simplifies to for 0 < t < to
at
(F)
where Iois the time required to attain complete saturation of the binding sites, i.e., C, = fio, and is given by Equation G. **
=
k
f"
Jo
Cp(t)dt
(G)
For times greater than to, i.e., C, > Bo, Equation E shows that in the limit as Kt— *»,C|— KC, - fi0). Equation D becomes Equation H with initial condition that
