Antibody-Directed Effector Cell Therapy of Tumors: Analysis and Optimization Using a Physiologically Based Pharmacokinetic Model

RESEARCH ARTICLE

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Antibody-Directed Effector Cell Therapy of Tumors: Analysis and Optimization Using a Physiologically Based Pharmacokinetic Model1 Stuart W. Friedrich 2,3, Stephany C. Lin 2, Brian R. Stoll, Laurence T. Baxter 4, Lance L. Munn and Rakesh K. Jain Department of Radiation Oncology, Massachusetts General Hospital and Harvard Medical School, Boston, MA 02114, USA Abstract The failure of the cellular immune response to stop solid tumor growth has been the subject of much research. Although the mechanisms for tumor evasion of immune response are poorly understood, one viable explanation is that tumor - killing lymphocytes cannot reach the tumor cells in sufficient quantity to keep the tumor in check. Recently, the use of bifunctional antibodies ( BFAs ) has been proposed as a way to direct immune cells to the tumor: one arm of the antibody is specific for a known tumor - associated antigen and the other for a lymphocyte marker such as CD3. Injecting this BFA should presumably result in cross - linking of lymphocytes ( either endogenous or adoptively transferred ) with tumor cells, thereby enhancing therapy. Results from such an approach, however, are often disappointing — frequently there is no benefit gained by using the BFA. We have analyzed the retargeting of endogenous effector cells by BFA using a physiologically based whole - body pharmacokinetic model that accounts for interactions between all relevant species in the various organs and tumor. Our results suggest that the design of the BFA is critical and the binding constants of the antigen and lymphocyte binding epitopes need to be optimized for successful therapy. Neoplasia ( 2002 ) 4, 449 – 463 doi:10.1038/sj.neo.7900260 Keywords: bifunctional antibody, lymphocyte, trafficking, mathematical model, tumor localization.

Introduction Adoptive effector cell therapy relies on the delivery of lymphokine - activated and expanded effector cells or specific tumor - infiltrating lymphocytes and activated natural killer cells. Administration of interleukin - 2 – activated effector cells has been shown to create significant clinical toxicity [ 1,2 ], and the isolation of specific lymphocytes that target tumor tissues is difficult and only possible when antigens have been isolated from the specific tumor cell line. For clinical application, an ideal immunotherapy would direct the body’s endogenous effector cells to a tumor and only activates the effector cells in the presence of tumor antigens. Such therapies are being explored using bifunctional antibodies ( BFAs ) that have specificity for both a

tumor antigen and the T - cell receptor – CD3 complex on T cells [ 3 – 14 ]. These BFAs have been engineered with the expectation that they will induce the T - cell repertoire to generate lytic and inflammatory responses at the tumor site. F( ab0 )2 BFAs, which lack functional Fc portions, have been shown to induce lymphocyte proliferation only in the presence of tumor cells and produce no in vivo systemic activation [ 8,15 ]. The availability of many antitumor antibodies, including several that demonstrate broad antitumor reactivity against a variety of human adenocarcinoma cell lines, and the ability of F( ab0 )2 antibodies to penetrate tumors more efficiently than full - size antibodies has generated considerable interest in this form of therapy [ 3 – 5,10 ]. The application of retargeting endogenous effector cells using BFAs will require the optimization of several BFA properties and treatment conditions. The systematic evaluation of the effects of changing these variables is perfectly suited to analysis using a physiologically based pharmacokinetic model. Such a model can be used to study BFA and lymphocyte biodistribution by examining the significance of transport mechanisms and BFA properties with respect to the localization of effector cells at a tumor site. The results from these modeling studies can be used to more effectively guide preclinical experiments and bridge the gap between the optimal preclinical conditions and therapy in humans. The objective of the research presented in this paper is to develop a pharmacokinetic model that can be used in the design of BFA - directed effector cell therapies. Two previously developed models that were used to individually study the distribution of antibodies [ 16 ] and effector cells [ 17,18 ] form the basis for this work. Both models are physiologically based, using measurable physiological parameters such as organ volumes and blood flow rates and incorporating the

Address all correspondence to: Rakesh K. Jain, Department of Radiation Oncology, Massachusetts General Hospital, 100 Blossom Street, Cox - 7, Boston, MA 02114, USA. E - mail: [email protected] 1 This work was supported by National Institutes of Health grants PO1 CA80124 ( R.J.K., L.L.M. ), and R01 HL64240 ( L.L.M. ). 2 These authors contributed equally to this work. 3 Current address: Pharsight Corporation, Cary, NC 27511, USA. 4 Current address: Advanced Process Combinatorics, West Lafayette, IN, USA. Received 16 January 2002; Accepted 27 February 2002. Copyright # 2002 Nature Publishing Group All rights reserved 1522-8002/02/$25.00

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mechanism - based transport processes that determine how antibodies and effector cells distribute in the body. Data collected by Bakacs et al. [ 19 ] was used to estimate unknown parameters for use in the model in the absence of experimental values for these parameters. Bakacs et al. followed the biodistribution of a BFA after IV administration in a syngeneic mammary adenocarcinoma BALB / c mouse model. A lung metastasis model and a subcutaneous solid tumor model were considered. In the former, 64PT cells were IV injected into the tail vein of BALB / c mice. BFA treatment was found to prolong the survival of these mice. In contrast, the BFA was unable to prolong the survival of mice with subcutaneous 64PT tumor grafts. A biodistribution study showed that the BFA was accumulating preferentially in the spleen and lymph nodes, as it was designed to bind to the T cells abundant in these tissues, rather than in the subcutaneous tumor. The authors suggested that poor vascularity or physiological barriers presented by the solid tumors may have prevented the BFA and T cells from infiltrating the tumor. Smaller tumors, such as those in the lung metastases model, may not present such barriers. Our research focuses on the poor homing of BFA to solid tumors. Our model supports the observations of Bakacs et al. and further examines how the interplay between BFA, lymphocytes, and tumor antigen may be exploited to optimize antibody - mediated adoptive immunotherapy.

Methods Model Development The model we developed to study BFA distribution and retargeting of lymphocytes is based on two previously developed models. A whole - body physiologically based model is used in which the major organs of the body are represented as compartments that are connected in an anatomical manner by the systemic and lymphatic circulation ( Figure 1 ). The organ compartments are then further subdivided into vascular and extravascular subcompartments. The detailed description of the antibody and effector

Figure 1. Whole - body pharmacokinetic model. All the major organs are included in the model, and the various compartments are interconnected by the blood ( black arrows ) and lymphatic ( gray arrows ) systems.

cell transport processes is adapted from previous models [ 18,20,21 ]. Briefly, the main transport processes in the antibody model include: convective and diffusive transport across capillary walls; reversible, nonsaturable, nonspecific binding in the extravascular compartments; reversible, saturable, specific binding to tumor antigens; and elimination through the kidneys. The main transport processes in the effector cell model include: capture and arrest at the endothelial wall through cell adhesion molecules; extravasation of arrested lymphocytes into the extravascular compartment; and depletion of cells through either a catabolic process or apoptosis. In addition to these processes, the model developed in the present study accounts for saturable binding between BFAs and lymphocytes and the normal production of endogenous lymphocytes. Functional, immunohistochemical, and morphologic studies indicate that the vascular endothelium of tumor vessels may be partially composed of tumor cells exposed to the blood flow [ 22 – 24 ]. Therefore, the model also accounts for binding of BFA and BFA – lymphocyte complexes to antigens in the vascular space of the tumor compartment. Figure 2 shows a schematic of the two subcompartments of the tumor compartment and the associated BFA, lymphocyte, and BFA – lymphocyte complex species. The model nomenclature is given in Appendix A and the mass balance and other model equations can be found in Appendices B and C, respectively. Model Parameters To facilitate comparison between model predictions and the experimental data collected by Bakacs et al. [ 19 ], the physiological parameters for mice were used in the simulations ( see Appendix A ). The plasma flow rates for the tumor, muscle, and skin were determined experimentally; other plasma flow rates were obtained from the literature [ 25 ]. Lymphatic flow rates and interstitial fluid recirculation rates ( J R,i ) were determined by fitting model predictions to experimental data [ 21 ]. Total organ volumes were determined experimentally by weighing each organ and assuming a density of 1 g ml  1 except for the bone where a density of 1.5 g ml  1 was assumed. Vascular and interstitial volumes of organs were calculated as percentages of total organ volume as given in the literature [ 21,26,27 ]. These physiological parameters are summarized in Appendix A.1. The BFA osmotic reflection coefficients ( sS, sL ) for small and large pores have been determined previously to be 0.96 and 0.11, respectively [ 21 ] and the BFA permeability – surface area products ( PSS, PSL ) have been determined to have values of 2.310  5 and 7.910  6 ml min  1 g  1, respectively [ 21 ]. The binding affinity of the BFA to antigen has been determined by Bakacs et al. [ 19 ] to be 3.0106 M  1. The reverse binding rate constant of BFA to antigen has been determined to be approximately 1.010  2 min  1 [ 21 ], which results in a forward binding constant of 3.0107 ml mol  1 min  1. The binding affinity of BFA to CD3 molecules on lymphocytes was determined by Bakacs et al. [ 19 ] to have a value of 1.0107 M  1. The reverse binding rate constant of BFA to CD3 is not known; therefore, as an

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Figure 2. The various species considered in the model. Effector cells ( large circles ) can be bound to BFA ( ), which can bind to tumor antigen ( green diamonds ) or CD3 ( blue squares ). Antibody can also bind nonspecifically in the interstitial space ( small circles ). -

=-

== 

initial estimate, the value for the reverse binding rate constant between BFA and antigen was used ( 1.010  2 min  1 ), which resulted in a forward binding constant of 1.0108 ml mol  1 min  1. The density of CD3 molecules on lymphocytes was estimated to be 7.5104 molecules per cell, and a tumor antigen density of 1.210  11 mol ml  1 was used as reported in the literature [ 28 ]. Nonspecific binding of the BFA in the extravascular space was assumed to be negligible due to the absence of an Fc domain that normally accounts for the majority of nonspecific binding. The final parameter related to BFA distribution is the rate of elimination in the kidneys; this parameter was determined by fitting the model to the experimental data. The extravasation of lymphocytes from the vascular to the extravascular space of organs requires ( i ) contact and adhesion of the lymphocytes with the endothelial walls ( cell capture ), ( ii ) stable adhesion ( cell spreading ), and ( iii ) extravasation into the interstitium. The relevant model parameters are the density of endothelial adhesion sites in each organ, the binding affinity between the lymphocytes and adhesion sites, and the migration rate of a bound lymphocyte across the endothelium. The transcapillary migration rate used in the simulations was 2.910  4 min  1 as estimated previously [ 17 ]. The rate of capture, release, and arrest is expected to vary between organs. Zhu et al. [ 17 ] achieved varying rates by fixing the adhesion site density in each organ and calculating a varying capture and arrest rate constant for each organ by fitting model predictions to experimental data. In the simulations for this study the capture, release, and arrest rate constants were assumed to be equal in all organs with values of 9.01017 ml mol  1 min  1, 200 min  1, and 110  3 min  1, respectively, which are the average values determined by Zhu et al. for nonactivated T cells [ 17 ]. This assumption was made as a first approximation due to a lack of experimental organ specific values. However, this is a reasonable assumption because capture and arrest rates are expected to vary between organs primarily because of differences in

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adhesion site densities, not differences in the rate constants. Adhesion site densities were allowed to vary between organs and were estimated by fitting model predictions to experimental data. The rate of lymphocyte recirculation in the extravascular space through the lymphatic circulation was previously determined by Zhu et al. [ 17 ] to have values of 0.018, 0.01, and 1.0 for the lymph node, tumor, and all other tissues, respectively. The lymphocyte depletion rate constant was set at 4.210  3 min  1 in the lungs and zero in other tissues as determined by Zhu et al. [ 17 ]. The rate of lymphocyte generation in the body was determined by fitting the model calculated steady - state concentration of lymphocytes in the plasma to the known concentration in mice, 9.010  18 mol ml  1 [ 29 ]. Appendix A.2 summarizes the kinetic parameters used in the model simulations. Model Simulations A FORTRAN program that incorporated the ordinary differential equation solver LSODE was written to perform the simulations. The differential equations that describe the mass balance of each species in each tissue and subcompartment are given in Appendix B. As mentioned previously, the unknown parameters that were estimated by fitting experimental data were the elimination rate of BFA in the kidneys, the normal rate of generation of lymphocytes needed to maintain a constant level in the body, and the density of adhesion sites on the vascular endothelium in different organs. Data collected by Bakacs et al. [ 19 ] was used in the parameter - estimation process. Bakacs et al. studied the distribution of BFA in BALB / c mice bearing subcutaneous tumors. Mice were given subcutaneous injections of 64PT cells, which express high levels of MMTV gp52 envelope glycoprotein, 6 days before BFA treatments. Radiolabeled BFA, with specificity for gp52 and CD3, was given through tail vein injection, and organ radioactivity was measured at various times after injection to determine the BFA biodistribution. This data was used in the estimation of the vascular adhesion site densities and the elimination rate of BFA in the kidneys. The first step in the model simulations was to determine the generation rate of lymphocytes required to maintain the known normal steady - state concentration of lymphocytes in the blood in the absence of BFA. This was accomplished by setting the adhesion site density in each tissue to an estimated initial value and then determining the generation rate of lymphocytes required to maintain the correct concentration of lymphocytes in the blood. The second step in the simulations was to fit the organ concentrations of BFA ( as determined by Bakacs et al. ). Because BFAs are able to bind to lymphocytes, the concentration of BFA in the tissues is in part regulated by the concentration of lymphocytes in each tissue, which is in turn controlled by the adhesion site density in the vascular space of the tissue. Therefore, the BFA concentrations calculated by the model were fit to the experimental data of Bakacs et al. by varying the adhesion site density in each tissue in addition to the elimination rate of

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Figure 3. BFA does not alter the effector cell distribution. Black bars represent the steady - state effector cell AUC in the various organs before BFA injection. The dark gray bars are the predicted AUC values after BFA administration. The parameters associated with these runs are shown in Appendices A.1 – 3. In each organ, lymphocyte concentrations varied by less than 1% after injection, indicating that the BFA is not effective in directing the effector cells to the tumor under these conditions. The light gray bars show the effect of increasing the capture site density in the tumor ( CV,tumor ) by one order of magnitude on the AUC. Adjusting this parameter has a significant effect on the level of lymphocyte exposure in the tumor.

step fitting process was repeated until the model produced good fits using the same parameter values in both fitting steps. Nonlinear regression fitting routines were used to fit the experimental data. Two different routines were used, and the resulting parameters were compared as a check of consistency. The two routines produced parameter values with the same order of magnitude. In addition, the estimated parameters resulting from the two different routines varied by less than 15%, and the majority of the parameters varied by less than 5%. These slight differences can be attributed to differences in the convergence algorithm. After determining the model parameters that gave the best fit to the experimental data, we performed a sensitivity analysis. Each parameter of the model was varied and the percent change in exposure ( area under the curve [ AUC ] ) to lymphocytes in the plasma, tumor, and lung was determined.

Results BFA in the kidneys. The fitted values of adhesion site density were then substituted for the initial values used in the first step to calculate steady - state levels of lymphocytes, and a new lymphocyte - generation rate was calculated. This two -

Lymphocyte Distribution We first analyzed the lymphocyte distributions before and after administration of the BFA using the baseline parameters given in Appendices A.1 – 3. The lymphocyte profiles at

Figure 4. BFA kinetics are determined by the lymphocyte distribution. ( A ) BFA clearance curves normalized to injected dose for the various organs. Solid lines represent the case where lymphocytes are present, and the dotted lines are runs without lymphocytes. Note that the presence of lymphocytes affects the pharmacokinetics of the BFA, especially in lymphocyte - rich tissue. ( B ) BFA clearance curves normalized to the plasma levels. Although there appears to be some relative accumulation in the tumor, the endogenous trafficking of the lymphocytes still dominates the system.

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48 hours after injection of BFA were also analyzed for the cases where the model parameters were varied from the baseline values. AUC values for the lymphocyte profiles were calculated for each organ as a measure of the cumulative exposure of the tissue to lymphocytes after 48 hours. Figure 3 shows the predicted lymphocyte AUC values for the baseline case before and after administration of the BFA ( black and dark gray bars, respectively ) and for the case where C V,i in the tumor was increased by one order of magnitude ( light gray bars ). The AUC values are based on organ concentrations that include the lymphocytes in the vascular and interstitial space of the organ as well as those captured or arrested at the vascular endothelium. In general, the highest level of exposure to lymphocytes is observed in the tumor, followed by the spleen, lymph node, and skin. The lowest concentrations are in the liver. BFA Kinetics The model predicts a lack of effector cell homing to the tumor in the baseline case. The model - calculated levels of BFA in mice are shown in Figure 4, A and B. In each panel the solid line represents the best - fit level of BFA in each tissue when lymphocytes are present, and the dotted line represents the BFA levels in each organ using the same model parameters but without BFA binding to lymphocytes ( lymphocytes are not present ). The plots in Figure 4A show the amount of BFA in each tissue normalized with respect to the total dose of BFA. Figure 4B shows the same data except with the amount of BFA in each tissue normalized to plasma BFA levels. As a result, the trends in Figure 4B indicate the rate of elimination from each organ relative to plasma; increasing values ( positive slope ) indicate that BFA is accumulating in the tissue relative to plasma, and decreasing values indicate that the BFA is being eliminated from the tissue at a higher rate than from the plasma. Also shown in Figure 4B is the experimental data of Bakacs et al., which compares reasonably well with the model calculated data. The model parameters estimated by fitting the model predictions to experimental data are shown in Appendix A.3. Sensitivity Analysis A sensitivity analysis was performed to assess the effect of each parameter on the model solution. The information obtained from this analysis is important in determining the effect of unknown or estimated parameter values on the results and in predicting how the system responds to variations from the baseline case, which can be useful in optimizing the treatment design. Figure 5 shows the sensitivity of the therapeutic index ( TI ) to the model parameters. The TI is defined as AUCtumor / AUCplasma where the AUC represents the cumulative exposure to lymphocytes after 48 hours. The sensitivity of TI to changes in the model parameters was calculated as a percent change from the baseline case. The effects of increasing and decreasing each parameter by an order of magnitude are shown in Figure 5 as dark and light bars, respectively. Positive values signify an increase in TI as a result of

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the parameter change, and negative values indicate a decrease. The TI is the most sensitive to the adhesion site density in the tumor vasculature ( C V,i ) followed by the association and dissociation rate constants for the capture c ) and the of the effector cell – BFA complex ( K f,ic and K r,i association rate constant for the arrest of the complex ( K f,ia ).

Discussion The predictions of the model clearly show that physiological conditions must be carefully considered for successful BFA - directed effector cell therapy. Figure 3 indicates that rather than guiding the T cells to the target tissue, the BFA is being dragged around the system by the cells. Lymphocyte concentrations in each organ varied by less than 1% after injection of BFA, indicating that the BFA is not effective in redirecting the effector cells under these conditions. Figure 4A shows the specificity of the localization of the BFA, as all data are relative to plasma levels. Note that compartments with higher baseline concentrations of lymphocytes ( e.g., lymph and spleen ) show the largest differences between runs with and without effector cell – BFA binding. The normalized BFA levels in these compartments are significantly higher with effector cell – BFA binding, suggesting that the effector cells are changing the distribution of the BFA, not vice versa, as was originally hoped.

Figure 5. Sensitivity of the therapeutic index ( TI ) to selected model parameters. Values represent the percent change compared to the baseline CD values for K f,i ( association constant for binding between BFA and CD3 ); ag CD K r,i ( dissociation constant for binding between BFA and CD3 ); K f,i ( assoag ciation constant for binding between BFA and antigen ); K r,i ( dissociation c constant for binding between BFA and antigen ); K f,i ( association constant for c capture of EC – BFA complex ); K r,i ( dissociation constant for capture of EC – a BFA complex ); K f,i ( rate constant for arrest of EC – BFA complex ); CD ( number of CD binding sites per cell ); PS ( permeability – surface area product ); Li ( lymph flow out of tumor ); Q i ( blood flow into tumor ); JR,i ( fluid recirculation flow rate of tumor ). Tumor size; CV,i ( density of capture sites in the vascular space of tumor ); A EV,i ( density of antigen binding sites in the fB extravascular space of tumor ); E kidney ( elimination rate constant for free BFA in kidney ); Fi ( fraction of EC – BFA complex in extravascular space of tumor that recirculates ). Dark and light bars correspond to an order of magnitude increase or decrease in the parameter, respectively. The left - hand panel shows parameters that affect all organs; those on the right are specific to the fB tumor, except for E kidney .

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It is also important to consider the nonnormalized data in Figure 4B because, in the absence of toxic side effects, it is the absolute area under these curves that determine therapeutic benefit. Even larger differences are apparent here between the runs with and without BFA – effector cell binding, and the differences are again largest in compartments rich in lymphocytes. Figure 5 shows the sensitivity of the TI ( TI = AUCtumor / AUCplasma ) to changes in various model parameters. The sensitivity analysis indicates that the greatest positive impact on the effectiveness of therapy is achieved by increasing the number of adhesion sites available for lymphocyte capture in the tumor. Varying the rate constants for lymphocyte capture and arrest also have a significant effect on the TI. This implies that selectively modifying the tumor vasculature by increasing ( C V,i ) or by changing the rate constants in favor of capture and arrest will have a favorable effect on the therapy. Figure 3 shows that increasing C V,i by one order of magnitude in the tumor is capable of increasing the effector cells AUC in the tumor by more than six times the baseline case. Though these parameters have the greatest impact on the TI, they may be difficult to manipulate. Some consideration, therefore, should be given to other parameters that may be easier to control. For example, Figure 5 shows that the TI is also sensitive to the tumor lymph flow rate. Decreasing this flow rate encourages the accumulation of effector cells in the tumor; therefore, this parameter should also be considered in the optimization of this therapy approach. Figure 5 also shows that the TI is relatively insensitive to changes in the BFA. Varying BFA binding to CD3 and to tumor antigens appear to have little effect on the TI. This implies that under the model conditions, BFA is not able to home effector cells to the tumor. Changing the BFA binding parameters by one order of magnitude is not sufficient to significantly alter lymphocyte accumulation in the tumor, and more drastic changes are necessary for successful BFA directed effector cell therapy.

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Appendix A Nomenclature 1. Concentrations C fB V,i C fB EV,i C bB V,i C bB EV,i C EB V,i C EB EV,i agB C V,i agB C EV,i C cEB V,i C aEB V,i C agEB V,i C agEB EV,i C nB EV,i C nEB EV,i C total V,i C total EV,i

free BFA in vascular space of compartment i ( mol cm  3 ) free BFA in extravascular space of compartment i ( mol cm  3 ) BFA bound to effector cell ( EC ) in vascular space of compartment i ( mol cm  3 ) BFA bound to EC extravascular space of compartment i ( mol cm  3 ) EC – BFA complex in vascular space of compartment i ( mol cm  3 ) EC – BFA complex in extravascular space of compartment i ( mol cm  3 ) BFA bound to antigen in vascular space of compartment i ( mol cm  3 ) BFA bound to antigen in extravascular space of compartment i ( mol cm  3 ) captured EC – BFA complex in vascular space of compartment i ( mol cm  3 ) arrested EC – BFA complex in vascular space of compartment i ( mol cm  3 ) EC – BFA complex bound to antigen in vascular space of compartment i ( mol cm  3 ) lymphocyte – BFA complex bound to antigen in extravascular space of compartment i ( mol cm  3 ) nonspecifically bound BFA in vascular space of compartment i ( mol cm  3 ) nonspecifically bound EC – BFA complex in extravascular space of compartment i ( mol cm  3 ) total concentration of cells in vascular space of compartment i ( mol cm  3 ) total concentration of cells in extravascular space of compartment i ( mol cm  3 )

2. Kinetic Parameters K f,iCD association constant for binding between BFA and CD3 on EC in compartment i ( cm3 mol  1 min  1 ) CD K r,i dissociation constant for binding between BFA and CD3 on EC in compartment i ( min  1 ) ag K f,i association constant for binding between BFA and antigen in compartment i ( cm3 mol  1 min  1 ) ag K r,i dissociation constant for binding between BFA and antigen in compartment i ( min  1 ) c K f,i association constant for capture of EC – BFA complex in compartment i ( cm3 mol  1 min  1 ) c dissociation constant for capture of EC – BFA complex in compartment i ( min  1 ) K r,i a K f,i rate constant for arrest of EC – BFA complex in compartment i ( min  1 ) n K f,i association constant for nonspecific binding of BFA in compartment i ( min  1 ) n K r,i dissociation constant for nonspecific binding of BFA in compartment i ( min  1 ) fB E kidney elimination rate constant for free BFA in kidney ( cm3 min  1 ) cEB Ei elimination rate constant for captured EC – BFA complex in compartment i ( min  1 ) aEB Ei elimination rate constant for arrested EC – BFA complex in compartment i ( min  1 ) aEB Ji transmigration rate constant for arrested EC – BFA complex in compartment i ( min  1 ) fB transcapillary exchange rate of free BFA in compartment i ( mol min  1 ) Ji J S,i transcapillary fluid flow rate through small pores in compartment i ( cm3 min  1 ) J L,i transcapillary fluid flow rate through large pores in compartment i ( cm3 min  1 ) J R,i fluid recirculation flow rate of compartment i ( cm3 min  1 ) P baseline production rate of EC in plasma ( mol min  1 ) 3. Flow Rates Qi Li Q hepatic

blood flow into compartment i ( cm3 min  1 ) lymph flow out of compartment i ( cm3 min  1 ) blood flow into liver before the addition of blood flow from GI and spleen ( cm3 min  1 )

4. Volumes V V,i V EV,i

vascular volume of compartment i ( cm3 ) extravascular volume of compartment i ( cm3 )

5. Other Parameters Fi CD A V,i A EV,i C V,i

fraction of EC – BFA complex in extravascular space of compartment i that recirculates number of CD binding sites per cell ( mol mol  1 ) density of antigen binding sites in the vascular space of compartment i ( mol cm  3 ) density of antigen binding sites in the extravascular space of compartment i ( mol cm  3 ) density of capture sites in the vascular space of compartment i ( mol cm  3 )

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density of free CD3 sites in the vascular space of compartment i ( mol cm  3 ) density of free CD3 sites in the extravascular space of compartment i ( mol cm  3 ) density of occupied CD3 sites in the vascular space of compartment i ( mol cm  3 ) density of occupied CD3 sites in the extravascular space of compartment i ( mol cm  3 ) density of free capture sites in the vascular space of compartment i ( mol cm  3 ) density of free antigen sites in the vascular space of compartment i ( mol cm  3 ) density of free antigen sites in the extravascular space of compartment i ( mol cm  3 ) Pecle`t number for flow through small pores in compartment i Pecle`t number for flow through large pores in compartment i partition coefficient for free BFA between vascular and extra vascular space in compartment i permeability – surface area product ( cm3 min  1 g  1 )

fCDV,i fCDEV,i oCDV,i oCDEV,i fCV,i fAV,i fAEV,i PeS,i PeL,i ki PS

Appendix B Mass Balance Equations

fB þðQGI LGI ÞCV;GI fB ðQliver Lliver ÞCV;liver

1. Plasma

fB Jliver

1.1 Free BFA VV;plasma 

fB KfCD ;liver CV;liver fCDV;liver VV;liver

fB dCV;plasma

dt

bB þKrCD ;liver CV;liver VV;liver

fB ¼ ðQlung Llung ÞCV;lung

Qhepatic þ Qlnode þ Qtumor þ Qbone þ QGI

! fB CV;plasma

þQheart þ Qkidney þ Qmuscle þ Qskin þ Qspleen

2.2 Free EC – BFA complex in vascular space VV;liver

fB KfCD ;plasma CV;plasma fCDV;plasma VV;plasma

EB dCV;liver dt

EB þðQspleen Lspleen ÞCV;spleen

bB þKrCD ;plasma CV;plasma VV;plasma

EB þðQGI LGI ÞCV;GI EB ðQliver Lliver ÞCV;liver

1.2 Free BFA – effector cell complex dC EB VV;plasma V;plasma dt



¼

EB fB Kfc;liver CV;liver fCV;liver VV;liver

EB ðQlung Llung ÞCV;lung

Qhepatic þ Qlnode þ Qtumor þ Qbone þ QGI þQheart þ Qkidney þ Qmuscle þ Qskin þ Qspleen

EB ¼ Qhepatic CV;plasma

cEB þKrc;liver CV;liver VV;liver

! EB CV;plasma

2.3 Captured EC – BFA complex in vascular space VV;liver

þP

cEB dCV;liver dt

EB ¼ Kfc;liver CV;liver fCV;liver VV;liver cEB Krc;liver CV;liver VV;liver

1.3 BFA bound to effector cells VV;plasma 

bB dCV;plasma dt

cEB Kfa;liver CV;liver VV;liver

 total EB bB CV;lung ¼ ðQlung Llung ÞCV;lung CV;lung

Qhepatic þ Qlnode þ Qtumor þ Qbone þ QGI þQheart þ Qkidney þ Qmuscle þ Qskin þ Qspleen

fB þKfCD ;plasma CV;plasma fCDV;plasma VV;plasma

cEB cEB Eliver CV;liver VV;liver

! bB CV;plasma

2.4 Arrested EC – BFA complex in vascular space VV;liver

aEB dCV;liver dt

cEB ¼ Kfa;liver CV;liver VV;liver aEB aEB Eliver CV;liver VV;liver

bB KrCD ;plasma CV;plasma VV;plasma

aEB aEB Jliver CV;liver VV;liver

2. Liver 2.5 BFA bound to effector cells in vascular space 2.1 Free BFA in vascular space VV;liver fB dCV;liver VV;liver dt

¼

bB dCV;liver dt

bB ¼ Qhepatic CV;plasma

fB Qhepatic CV;plasma

bB EB total þðQspleen Lspleen ÞCV;spleen CV;spleen =CV;spleen

fB þðQspleen Lspleen ÞCV;spleen

bB EB total þðQGI LGI ÞCV;liver CV;GI =CV;GI

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bB EB total ðQliver Lliver ÞCV;liver CV;liver =CV;liver

nEB KrCD ;liver CEV;liver VEV;liver

fB þKfCD ;liver CV;liver fCD V;liver VV;liver

aEB aEB bB total þJliver CV;liver CV;liver VV;liver =CV;liver

bB KrCD ;liver CV;liver VV;liver

EB bB total Lliver CEV;liver FV;liver CEV;liver =CEV;liver

457

aEB aEB bB total Jliver CV;liver CV;liver VV;liver =CV;liver cEB cEB bB total Eliver CV;liver CV;liver VV;liver =CV;liver aEB aEB bB total Eliver CV;liver CV;liver VV;liver =CV;liver

2.6 Free BFA in the extravascular space dC fB VEV;liver EV;liver dt

3. Lung 3.1 Free BFA in vascular space VV;lung

fB dCV;lung dt

fB þðQkidney Lkidney ÞCV;kidney

fB ¼ Jliver

fB þðQtumor Ltumor ÞCV;tumor

fB Lliver CEV;liver

fB þðQskin Lskin ÞCV;skin

fB KfCD ;liver CEV;liver fCDEV;liver VEV;liver

fB þðQmuscle Lmuscle ÞCV;muscle

bB nEB þKrCD ;liver ðCEV;liver CEV;liver ÞVEV;liver

fB þðQbone Lbone ÞCV;bone

fB Kfn;liver CEV;liver VEV;liver

fB þðQheart Lheart ÞCV;heart

nB þKrn;liver CEV;liver VEV;liver

fB þQlnode CV;lnode fB þLlnode CEV;lnode

2.7 Nonspecifically bound BFA in the extravascular space VEV;liver

nB dCEV;liver dt

fB ðQlung Llung ÞCV;lung

fB ¼ Kfn;liver CEV;liver VEV;liver

fB Jlnode

nB Krn;liver CEV;liver VEV;liver

fB KfCD ;lung CV;lung fCDV;lung VV;lung

nB EB total KfCD ;liver CEV;liver CEV;liver fCDEV;liver VEV;liver =CEV;liver

bB þKrCD ;lung CV;lung VV;lung

nEB þKrCD ;liver CEV;liver VEV;liver

2.8 Free EC – BFA complex in the extravascular space dC EB VEV;liver EV;liver dt

¼

aEB aEB Jliver CV;liver VV;liver

3.2 Free EC – BFA complex in vascular space VV;lung

EB dCV;liver dt

EB þðQtumor Ltumor ÞCV;tumor

EB total Kfn;liver CEV;liver oCDEV;liver VEV;liver =CEV;liver

EB þðQskin Lskin ÞCV;skin

EB Lliver CEV;liver Fliver

EB þðQmuscle Lmuscle ÞCV;muscle

nB EB total KfCD ;liver CEV;liver CEV;liver fCDEV;liver VEV;liver =CEV;liver

EB þðQbone Lbone ÞCV;bone

nEB þKrCD ;liver CEV;liver VEV;liver

EB þðQheart Lheart ÞCV;heart EB þQlnode CV;lnode

2.9 Nonspecifically bound EC – BFA complex in the extravascular space nEB dCEV;liver dt

EB ðQlung Llung ÞCV;lung

nEB Krn;liver CEV;liver VEV;liver

EB Kfc;liver CV;liver fCV;liver VV;lung

nB EB total þKfCD ;liver CEV;liver CEV;liver fCDEV;liver VEV;liver =CEV;liver

cEB þKrc;liver CV;liver VV;lung

2.10 BFA bound to effector cells in the extravascular space bB dCEV;liver

dt

EB þLlnode CEV;lnode Flnode

EB total ¼ Kfn;liver CEV;liver oCDEV;liver VEV;liver =CEV;liver

nEB KrCD ;liver CEV;liver VEV;liver

VEV;liver

EB ¼ ðQliver Lliver ÞCV;liver EB þðQkidney Lkidney ÞCV;kidney

nEB þKrn;liver CEV;liver VEV;liver

VEV;liver

fB ¼ ðQliver Lliver ÞCV;liver

fB ¼ KfCD ;liver CEV;liver fCDEV;liver VEV;liver

3.3 Captured EC – BFA complex in vascular space VV;lung

cEB dCV;liver dt

EB ¼ Kfc;lung CV;lung fCV;lung VV;lung cEB Krc;lung CV;lung VV;lung

bB nEB KrCD ;liver ðCEV;liver CEV;liver ÞVEV;liver

cEB Kfa;lung CV;lung VV;lung

nB EB total þKfCD ;liver CEV;liver CEV;liver fCDEV;liver VEV;liver =CEV;liver

cEB cEB Elung CV;lung VV;lung

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3.4 Arrested EC – BFA complex in vascular space VV;lung

aEB dCV;liver dt

EB Llung CEV;lung Flung

cEB ¼ Kfa;lung CV;lung VV;lung

nB EB total KfCD ;lung CEV;lung CEV;lung fCDEV;lung VEV;lung =CEV;lung

aEB aEB Elung CV;lung VV;lung

nEB þKrCD ;lung CEV;lung VEV;lung

aEB aEB Jlung CV;lung VV;lung

3.5 BFA bound to effector cells in vascular space VV;lung

bB dCV;liver dt

3.9 Nonspecifically bound EC – BFA complex in the extravascular space

bB EB total ¼ ðQliver Lliver ÞCV;liver CV;liver =CV;liver bB EB total þðQkidney Lkidney ÞCV;kidney CV;kidney =CV;kidney

VEV;lung

nB EB total þKfCD ;lung CEV;lung CEV;lung fCDEV;lung VEV;lung =CEV;lung

bB EB total þðQskin Lskin ÞCV;skin CV;skin =CV;skin

nEB KrCD ;lung CEV;lung VEV;lung

bB EB total þðQmuscle Lmuscle ÞCV;muscle CV;muscle =CV;muscle

bB EB total þðQheart Lheart ÞCV;heart CV;heart =CV;heart bB EB total þQlnode CV;lnode CV;lnode =CV;lnode

3.10 BFA bound to effector cells in the extravascular space VEV;lung

nEB KrCD ;lung CEV;lung VEV;lung

fB þKfCD ;lung CV;lung fCDV;lung VV;lung

aEB aEB bB total þJlung CV;lung CV;lung VV;lung =CV;lung

bB KrCD ;lung CV;lung VV;lung

EB bB total Llung CEV;lung Flung CEV;lung =CEV;lung

aEB aEB bB total Jlung CV;lung CV;lung VV;lung =CV;lung

4. Lymph Node 4.1 Free BFA in vascular space

3.6 Free BFA in the extravascular space VV;lnode VEV;lung

fB dCEV;lung

dt

fB ¼ KfCD ;lung CEV;lung fCDEV;lung VEV;lung

nB EB total þKfCD ;lung CEV;lung CEV;lung fCDEV;lung VEV;lung =CEV;lung

bB EB total ðQlung Llung ÞCV;lung CV;lung =CV;lung

aEB aEB bB total Elung CV;lung CV;lung VV;lung =CV;lung

bB dCEV;lung dt

bB nEB KrCD ;lung ðCEV;lung CEV;lung ÞVEV;lung

bB EB total þLlnode CEV;lnode CEV;lnode Flnode =CEV;lnode

cEB cEB bB total Elung CV;lung CV;lung VV;lung =CV;lung

EB total ¼ Kfn;lung CEV;lung oCDEV;lung VEV;lung =CEV;lung

nEB Krn;lung CEV;lung VEV;lung

bB EB total þðQtumor Ltumor ÞCV;tumor CV;tumor =CV;tumor

bB EB total þðQbone Lbone ÞCV;bone CV;bone =CV;bone

nEB dCEV;lung dt

fB dCV;lnode dt

fB ¼ Jlung

fB ¼ Qlnode CV;plasma fB Qlnode CV;lnode

fB Llung CEV;lung

fB Jlnode

fB KfCD ;lung CEV;lung fCDEV;lung VEV;lung

fB KfCD ;lnode CV;lnode fCDV;lnode VV;lnode

bB nEB þKrCD ;lung ðCEV;lung CEV;lung ÞVEV;lung

bB þKrCD ;lnode CV;lnode VV;lnode

fB Kfn;lung CEV;lung VEV;lung nB þKrn;lung CEV;lung VEV;lung

4.2 Free EC – BFA complex in vascular space VV;lnode

3.7 Nonspecifically bound BFA in the extravascular space VEV;lung

nB dCEV;lung

dt

EB dCV;lnode dt

EB Qlnode CV;lnode EB Kfc;lnode CV;lnode fCV;lnode VV;lnode

fB ¼ Kfn;lung CEV;lung VEV;lung

cEB þKrc;lnode CV;lnode VV;lnode

nB Krn;lung CEV;lung VEV;lung nB EB total KfCD ;lung CEV;lung CEV;lung fCDEV;lung VEV;lung =CEV;lung nEB þKrCD ;lung CEV;lung VEV;lung

4.3 Captured EC – BFA complex in vascular space VV;lnode

3.8 Free EC – BFA complex in the extravascular space VEV;lung

EB dCEV;lung

dt

aEB aEB ¼ Jlung CV;lung VV;lung

nEB þKrn;lung CEV;lung VEV;lung

EB ¼ Qlnode CV;plasma

cEB dCV;lnode dt

EB ¼ Kfc;lnode CV;lnode fCV;lnode VV;lnode cEB Krc;lnode CV;lnode VV;lnode cEB Kfa;lnode CV;lnode VV;lnode cEB cEB Elnode CV;lnode VV;lnode

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4.4 Arrested EC – BFA complex in vascular space VV;lnode

aEB dCV;lnode dt

cEB ¼ Kfa;lnode CV;lnode VV;lnode

VV;lnode

bB dCV;lnode dt

EB dCEV;lnode dt

EB ¼ Llung CEV;lung Flung

aEB aEB Jlnode CV;lnode VV;lnode

EB þLliver CEV;liver Fliver

bB EB total Qlnode CV;lnode CV;lnode =CV;lnode fB þKfCD ;lnode CV;lnode fCDV;lnode VV;lnode bB KrCD ;lnode CV;lnode VV;lnode aEB aEB bB total Jlnode CV;lnode CV;lnode VV;lnode =CV;lnode cEB cEB bB total Elnode CV;lnode CV;lnode VV;lnode =CV;lnode aEB aEB bB total Elnode CV;lnode CV;lnode VV;lnode =CV;lnode

4.6 Free BFA in the extravascular space dC fB VEV;lnode EV;lnode dt

VEV;lnode

EB þLGI CEV;GI FGI

bB ¼ Qlnode CV;plasma

459

4.8 Free EC – BFA complex in the extravascular space

aEB aEB Elnode CV;lnode VV;lnode

4.5 BFA bound to effector cells in vascular space

Friedrich et al.

EB þLspleen CEV;spleen Fspleen EB þLheart CEV;heart Fheart EB þLkidney CEV;kidney Fkidney EB þLskin CEV;skin Fskin EB þLmuscle CEV;muscle Fmuscle EB þLbone CEV;bone Fbone EB þLtumor CEV;tumor Ftumor EB Llnode CEV;lnode Flnode aEB aEB þJlnode CV;lnode VV;lnode nEB þKrn;lnode CEV;lnode VEV;lnode EB total Kfn;lnode CEV;lnode oCDEV;lnode VEV;lnode =CEV;lnode nB EB total KfCD ;lnode CEV;lnode CEV;lnode fCDEV;lnode VEV;lnode =CEV;lnode

fB ¼ Jlnode

nEB þKrCD ;lnode CEV;lnode VEV;lnode

fB þLlung CEV;lung fB þLGI CEV;GI fB þLliver CEV;liver fB þLspleen CEV;spleen fB þLheart CEV;heart

4.9 Nonspecifically bound EC – BFA complex in the extravascular space

VEV;lnode

nB EB total þKfCD ;lnode CEV;lnode CEV;lnode fCDEV;lnode VEV;lnode =CEV;lnode

fB þLskin CEV;skin

nEB KrCD ;lnode CEV;lnode VEV;lnode

fB þLmuscle CEV;muscle

4.10 BFA bound to effector cells in the extravascular space

fB þLtumor CEV;tumor fB Llnode CEV;lnode fB KfCD ;lnode CEV;lnode fCDEV;lnode VEV;lnode bB nEB þKrCD ;lnode ðCEV;lnode CEV;lnode ÞVEV;lnode fB Kfn;lnode CEV;lnode VEV;lnode nB þKrn;lnode CEV;lnode VEV;lnode

4.7 Nonspecifically bound BFA in the extravascular space dC nB VEV;lnode EV;lnode dt

EB total ¼ Kfn;lnode CEV;lnode oCDEV;lnode VEV;lnode =CEV;lnode

nEB Krn;lnode CEV;lnode VEV;lnode

fB þLkidney CEV;kidney

fB þLbone CEV;bone

nEB dCEV;lnode dt

VEV;lnode

bB dCEV;lnode dt

EB bB total ¼ Llung CEV;lung Flung CEV;lung =CEV;lung

EB bB total þLGI CEV;GI FGI CEV;GI =CEV;GI EB bB total þLliver CEV;liver Fliver CEV;liver =CEV;liver EB bB total þLspleen CEV;spleen Fspleen CEV;spleen =CEV;spleen EB bB total þLheart CEV;heart Fheart CEV;heart =CEV;heart EB bB total þLkidney CEV;kidney Fkidney CEV;kidney =CEV;kidney EB bB total þLskin CEV;skin Fskin CEV;skin =CEV;skin EB bB total þLmuscle CEV;muscle Fmuscle CEV;muscle =CEV;muscle

¼

fB Kfn;lnode CEV;lnode VEV;lnode

EB bB total þLbone CEV;bone Fbone CEV;bone =CEV;bone

nB Krn;lnode CEV;lnode VEV;lnode

EB bB total þLtumor CEV;tumor Ftumor CEV;tumor =CEV;tumor

nB EB total KfCD ;lnode CEV;lnode CEV;lnode fCDEV;lnode VEV;lnode =CEV;lnode

fB þKfCD ;lnode CEV;lnode fCDEV;lnode VEV;lnode

nEB þKrCD ;lnode CEV;lnode VEV;lnode

bB nEB KrCD ;lnode ðCEV;lnode CEV;lnode ÞVEV;lnode

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nB EB total þKfCD ;lnode CEV;lnode CEV;lnode fCDEV;lnode VEV;lnode =CEV;lnode

5.5 Enhanced captured EC – BFA complex in vascular space

nEB KrCD ;lnode CEV;lnode VEV;lnode aEB aEB bB total þJlnode CV;lnode CV;lnode VV;lnode =CV;lnode

VV;tumor

agEB dCV;tumor dt

EB total ¼ Kfag ;tumor CV;tumor fAV;tumor oCDV;tumor VV;tumor =CV;tumor agEB Krag ;tumor CV;tumor VV;tumor

EB bB total Llnode CEV;lnode Flnode CEV;lnode =CEV;lnode

agEB Kfa;tumor CV;tumor VV;tumor

5. Tumor

agEB cEB Etumor CV;tumor VV;tumor agB EB total þKfCD ;tumor CV;tumor CV;tumor fCDV;tumor VV;tumor =CV;tumor

5.1 Free BFA in vascular space

agEB KrCD ;tumor CV;tumor VV;tumor

VV;tumor

fB dCV;tumor dt

fB ¼ Qtumor CV;plasma fB ðQtumor Ltumor ÞCV;tumor fB Jtumor fB KfCD ;tumor CV;tumor fCDV;tumor VV;tumor

5.6 Arrested EC – BFA complex in vascular space

VV;tumor

aEB dCV;tumor dt

cEB ¼ Kfa;tumor CV;tumor VV;tumor agEB þKfa;tumor CV;tumor VV;tumor

agEB bB þKrCD ;tumor ðCV;tumor CV;tumor ÞVV;tumor

aEB aEB Etumor CV;tumor VV;tumor

fB Kfag ;tumor CV;tumor fAV;tumor VV;tumor

aEB aEB Jtumor CV;tumor VV;tumor

agB þKrag ;tumor CV;tumor VV;tumor

5.2 BFA bound to vascular antigens in vascular space agB

VV;tumor

dCV;tumor dt

fB ¼ Kfag ;tumor CV;tumor fAV;tumor VV;tumor

5.7 BFA bound to effector cells in vascular space

VV;tumor

bB dCV;tumor dt

bB ¼ Qtumor CV;tumor

agB Krag ;tumor CV;tumor VV;tumor

bB EB total ðQtumor Ltumor ÞCV;tumor CV;tumor =CV;tumor

agB EB total KfCD ;tumor CV;tumor CV;tumor fCDV;tumor VV;tumor =CV;tumor

fB þKfCD ;tumor CV;tumor fCDV;tumor VV;tumor agEB bB KrCD ;tumor ðCV;tumor CV;tumor ÞVV;tumor

agEB þKrCD ;tumor CV;tumor VV;tumor

agB EB total þKfCD ;tumor CV;tumor CV;tumor fCDV;tumor VV;tumor =CV;tumor

5.3 Free EC – BFA complex in vascular space VV;tumor

EB dCV;tumor dt

agEB KrCD ;tumor CV;tumor VV;tumor aEB aEB bB total Jtumor CV;tumor CV;tumor VV;tumor =CV;tumor

EB ¼ Qtumor CV;plasma

cEB cEB bB total Etumor CV;tumor CV;tumor VV;tumor =CV;tumor

EB ðQtumor Ltumor ÞCV;tumor

agEB cEB bB total Etumor CV;tumor CV;tumor VV;tumor =CV;tumor

EB Kfc;tumor CV;tumor fCV;tumor VV;tumor

aEB aEB bB total Etumor CV;tumor CV;tumor VV;tumor =CV;tumor

cEB þKrc;tumor CV;tumor VV;tumor EB total Kfag ;tumor CV;tumor fAV;tumor oCDV;tumor VV;tumor =CV;tumor

5.8 Free BFA in the extravascular space

agEB þKrag ;tumor CV;tumor VV;tumor agB EB total KfCD ;tumor CV;tumor CV;tumor fCDV;tumor VV;tumor =CV;tumor agEB þKrCD ;tumor CV;tumor VV;tumor

5.4 Normal captured EC – BFA complex in vascular space VV;tumor

EB dCV;tumor dt

EB ¼ Kfc;tumor CV;tumor fCV;tumor VV;tumor

VEV;tumor

fB dCEV;tumor dt

fB ¼ Jtumor

fB Ltumor CEV;tumor fB KfCD ;tumor CEV;tumor fCDEV;tumor VEV;tumor agEB bB nEB þKrCD ;tumor ðCEV;tumor CEV;tumor CEV;tumor ÞVEV;tumor fB Kfag ;tumor CEV;tumor fAEV;tumor VEV;tumor

cEB Krc;tumor CV;tumor VV;tumor

agB þKrag ;tumor CEV;tumor VEV;tumor

cEB Kfa;tumor CV;tumor VV;tumor

fB Krn;tumor CEV;tumor VEV;tumor

cEB cEB Etumor CV;tumor VV;tumor

nB þKrn;tumor CEV;tumor VEV;tumor

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5.9 BFA bound to tumor antigens in the extravascular space agB

dC VEV;tumor EV;tumor dt

¼

fB Kfag ;tumor CEV;tumor fAEV;tumor VEV;tumor

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5.14 BFA bound to effector cells in the extravascular space bB dCEV;tumor dt

VEV;tumor

fB ¼ KfCD ;tumor CEV;tumor fCDEV;tumor VEV;tumor

agEb bB nEB KrCD ;tumor ðCEV;tumor CEV;tumor CEV;tumor ÞVEV;tumor

agB Krag ;tumor CEV;tumor VEV;tumor

agB EB total þKfCD ;tumor CEV;tumor CEV;tumor fCDEV;tumor VEV;tumor =CEV;tumor

agB EB total KfCD ;tumor CEV;tumor CEV;tumor fCDEV;tumor VEV;tumor =CEV;tumor

agEB KrCD ;tumor CEV;tumor VEV;tumor

agEB þKrCD ;tumor CEV;tumor VEV;tumor

nB EB total þKfCD ;tumor CEV;tumor CEV;tumor fCDEV;tumor VEV;tumor =CEV;tumor nEB KrCD ;tumor CEV;tumor VEV;tumor

5.10 Nonspecifically bound BFA in the extravascular space

aEB aEB bB total þJtumor CV;tumor CV;tumor VV;tumor =CV;tumor

VEV;tumor

nB dCEV;tumor

dt

EB bB total Ltumor CEV;tumor Ftumor CEV;tumor =CEV;tumor

fB ¼ Kfn;tumor CEV;tumor VEV;tumor

nB Krn;tumor CEV;tumor VEV;tumor nB EB total KfCD ;tumor CEV;tumor CEV;tumor fCDEV;tumor VEV;tumor =CEV;tumor nEB þKrCD ;tumor CEV;tumor VEV;tumor

6. Other Organs ( Bone, GI, Heart, Kidney, Muscle, Skin, and Spleen ) 6.1.a Free BFA in vascular space ( bone, GI, heart, muscle, skin, and spleen )

5.11 Free EC – BFA complex in the extravascular space VV;i dC EB VEV;tumor EV;tumor dt

fB dCV;i dt

fB ¼ Qi CV;plasma

aEB aEB ¼ Jtumor CV;tumor VV;tumor

fB ðQi Li ÞCV;i

EB total Kfag ;tumor CEV;tumor fAEV;tumor oCDEV;tumor VEV;tumor =CEV;tumor

JifB

agEB þKrag ;tumor CEV;tumor VEV;tumor

fB KfCD ;i CV;i fCDV;i VV;i

EB total Kfn;tumor CEV;tumor oCDEV;tumor VEV;tumor =CEV;tumor

bB þKrCD ;i CV;i VV;i

nEB þKrn;tumor CEV;tumor VEV;tumor

6.1.b Free BFA in vascular space ( kidney )

EB Ltumor CEV;tumor Ftumor agB EB total KfCD ;tumor CEV;tumor CEV;tumor fCDEV;tumor VEV;tumor =CEV;tumor

VV;kidney

fB dCV;kidney dt

fB ¼ Qkidney CV;plasma fB ðQkidney Lkidney ÞCV;kidney

nB EB total KfCD ;tumor CEV;tumor CEV;tumor fCDEV;tumor VEV;tumor =CEV;tumor

fB Jkidney

agEB þKrCD ;tumor CEV;tumor VEV;tumor

fB KfCD ;kidney CV;kidney fCDV;kidney VV;kidney

nEB þKrCD ;tumor CEV;tumor VEV;tumor

bB þKrCD ;kidney CV;kidney VV;kidney

5.12 Nonspecifically bound EC – BFA complex in the extravascular space

fB bB Ekidney CV;kidney

6.2 Free EC – BFA complex in vascular space dC nEB VEV;tumor EV;tumor dt

EB total ¼ Kfn;tumor CEV;tumor oCDEV;tumor VEV;tumor =CEV;tumor

nEB Krn;tumor CEV;tumor VEV;tumor

VV;i

EB dCV;i dt

EB ðQi Li ÞCV;i

nB EB total þKfCD ;tumor CEV;tumor CEV;tumor fCDEV;tumor VEV;tumor =CEV;tumor

EB Kfc;i CV;i fCV;i VV;i

nEB KrCD ;tumor CEV;tumor VEV;tumor

5.13 EC – BFA bound to tumor antigens in the extravascular space agEB

dC VEV;tumor EV;tumor dt

¼

EB total Kfag ;tumor CEV;tumor fAEV;tumor oCDEV;tumor VEV;tumor =CEV;tumor

EB ¼ Qi CV;plasma

cEB þKrc;i CV;i VV;i

6.3 Captured EC – BFA complex in vascular space VV;i

cEB dCV;i dt

EB ¼ Kfc;i CV;i fCV;i VV;i

agEB

cEB Krc;i CV;i VV;i

EB total þKfCD ;tumor CEV;tumor CEV;tumor fCDEV;tumor VEV;tumor =CEV;tumor

agB

cEB Kfa;i CV;i VV;i

agEB KrCD ;tumor CEV;tumor VEV;tumor

cEB EicEB CV;i VV;i

ag

Kr ;tumor CEV;tumor VEV;tumor

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Friedrich et al.

6.4 Arrested EC – BFA complex in vascular space

VV;i

aEB dCV;i dt

6.9 Nonspecifically bound EC – BFA complex in the extravascular space

cEB ¼ Kfa;i CV;i VV;i aEB EiaEB CV;i VV;i

VEV;i

nEB dCEV;i dt

EB total ¼ Kfn;i CEV;i oCDEV;i VEV;i =CEV;i nEB Krn;i CEV;i VEV;i

aEB JiaEB CV;i VV;i

nB EB total þKfCD ;i CEV;i CEV;i fCDEV;i VEV;i =CEV;i

6.5 BFA bound to effector cells in vascular space

VV;i

bB dCV;i dt

bB ¼ Qi CV;plasma

nEB KrCD ;i CEV;i VEV;i

6.10 BFA bound to effector cells in the extravascular space

bB EB total ðQi Li ÞCV;i CV;i =CV;i fB þKfCD ;i CV;i fCDV;i VV;i

VEV;i

bB dCEV;i dt

fB ¼ KfCD ;i CEV;i fCDEV;i VEV;i bB nEB KrCD ;i ðCEV;i CEV;i ÞVEV;i

bB KrCD ;i CV;i VV;i

nB EB total þKfCD ;i CEV;i CEV;i fCDEV;i VEV;i =CEV;i

aEB bB total JiaEB CV;i CV;i VV;i =CV;i

nEB KrCD ;i CEV;i VEV;i

cEB bB total EicEB CV;i CV;i VV;i =CV;i

aEB bB total þJiaEB CV;i CV;i VV;i =CV;i

aEB bB total EiaEB CV;i CV;i VV;i =CV;i

EB bB total Li CEV;i Fi CEV;i =CEV;i

6.6 Free BFA in the extravascular space

Appendix C dC fB VEV;i dtEV;i

¼

JifB fB Li CEV;i fB KfCD ;i CEV;i fCDEV;i VEV;i bB nEB þKrCD ;i ðCEV;i CEV;i ÞVEV;i fB Kfn;i CEV;i VEV;i nB þKrn;i CEV;i VEV;i

Other Model Equations 1. Plasma 1.1 Adhesion site density EB bB fCD V;plasma ¼ CV;plasma CDCV;plasma

2. Tumor 6.7 Nonspecifically bound BFA in the extravascular space 2.1 Transcapillary exchange rate of free BFA dC nB VEV;i dtEV;i

fB ¼ Kfn;i CEV;i VEV;i nB Krn;i CEV;i VEV;i nB EB total KfCD ;i CEV;i CEV;i fCDEV;i VEV;i =CEV;i nEB þKrCD ;i CEV;i VEV;i

fB Jtumor ¼ JL;tumor ð1L;tumor ÞCfB V;tumor fB fB þPSL;tumor ½CV;tumor ðCEV;tumor =ktumor ÞPeL;tumor =ðe PeL;tumor 1Þ fB þJS;tumor ð1S;tumor ÞCV;tumor fB fB þPSS;tumor ½CV;tumor ðCEV;tumor =ktumor ÞPeS;tumor =ðe PeS;tumor 1Þ

6.8 Free EC – BFA complex in the extravascular space 2.2 Total concentration of cells VEV;i

EB dCEV;i dt

aEB ¼ JiaEB CV;i VV;i nEB þKrn;i CEV;i VEV;i EB total Kfn;i CEV;i oCDEV;i VEV;i =CEV;i

agEB total EB cEB aEB CV;tumor ¼ CV;tumor þ CV;tumor þ CV;tumor þ CV;tumor

agEB total EB nEB CEV;tumor ¼ CEV;tumor þ CEV;tumor þ CEV;tumor

EB Li CEV;i Fi nB EB total KfCD ;i CEV;i CEV;i fCDEV;i VEV;i =CEV;i nEB þKrCD ;i CEV;i VEV;i

2.3 Adhesion site densities total bB fCDV;tumor ¼ CV;tumor CDCV;tumor

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Modeling Bifunctional Antibody Therapy total bB fCDEV;tumor ¼ CEV;tumor CDCEV;tumor

fB þJS;i ð1S;i ÞCV;i fB fB þPSS;i ½CV;i ðCEV;i =ki ÞPeS;i =ðe PeS;i 1Þ

bB oCDV;tumor ¼ CV;tumor

3.2 Total concentration of cells

bB oCDEV;tumor ¼ CEV;tumor

fAV;tumor ¼

Friedrich et al.

agEB agB AV;tumor CV;tumor CV;tumor

agEB

agB

fAEV ;tumor ¼ AEV ;tumor CEV ;tumor CEV ;tumor aEB fCV ;tumor ¼ Ctumor CVcEB ;tumor CV ;tumor

total EB cEB aEB CV;i ¼ CV;i þ CV;i þ CV;i

total EB nEB CEV;i ¼ CEV;i þ CEV;i

3.3 Adhesion site densities total bB fCDV;i ¼ CV;i CDCV;i

3. Other Organs ( Lung, Liver, Lymph Node, Bone, GI, Heart, Kidney, Muscle, Skin, and Spleen )

total bB fCDEV;i ¼ CEV;i CDCEV;i

3.1 Transcapillary exchange rate of free BFA

bB oCDV;i ¼ CV;i

fB JifB ¼ JL;i ð1L;i ÞCV;i fB fB þPSL;i ½CV;i ðCEV;i =ki ÞPeL;i =ðe PeL;i 1Þ

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bB oCDEV;i ¼ CEV;i cEB aEB fCV;i ¼ Ci CV;i CV;i

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