[CANCER RESEARCH 56. 3771-3781. August 15. 19%]

Physiologically Based Kinetic Model of Effector Cell Biodistribution in Mammals:Implications for Adoptive Immunotherapy1

Hui Zhu, Robert J. Melder, Laurence T. Baxter, and Rakesh K. Jain2

Steele Laboratory, Department of Radiation Oncology, Massachusetts General Hospital and Harvard Medical School, Boston, Massachusetts 02114 ¡H.Z., R.J.M., L T. B.,R. K, J.]. and Radiological Sciences Program, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 ¡H.Z.]

ABSTRACT

The goal of the present investigation was to develop a physiologicallybased kinetic model to describe the biodistribution of immunologicallyactive effector cells in normal and neoplastic tissues of mammals based onthe current understanding of lymphocyte trafficking pathways and signals. The model was used to extrapolate biodistribution among differentanimal species and to identify differences among different effector populations and between intra-arterial and systemic injections. Most impor

tantly, the model was used to discern critical parameters for improvingthe delivery of effector cells. In the model, the mammalian body wasdivided into 12 organ compartments, interconnected in anatomic fashion.Each compartment was characterized by blood flow rate, organ volumeand lymphatic flow rate, and other physiological and immunologicalparameters. The resulting set of 45 differential equations was solvednumerically. The model was used to simulate the following biodistributiondata: (a) nonactivated T lymphocytes in rats; (ft) interleukin 2-activatedtumor-infiltrating lymphocytes in humans; (c) nonactivated natural killer(NK) cells in rats; and u/i interleukin 2-activated adherent NK cells in

mice. Comparisons between simulations and data demonstrated the feasibility of the model and the scaling scheme. The similarities as well asdifferences in biodistribution of different lymphocyte populations wererevealed as results of their trafficking properties. The importance oflymphocyte infiltration from surrounding normal tissues into tumor tissuewas found to depend on lymphocyte migration rate, tumor size, and hostorgan. The study confirmed that treatment with effector cells has not beenas impressive as originally promised, due, in part, to the biodistributionproblems. The model simulations demonstrated that low effector concentrations in the systemic circulation greatly limited their delivery to tumor.This was due to high retention in normal tissues, especially in the lung.Reducing normal tissue retention through decreasing attachment rate oradhesion site density in the lung by 50% could increase the tumor uptakeby —¿�40%for tumor-infiltrating lymphocytes and by —¿�60%for adherent

NK cells. Our analysis suggested the following strategies to improveeffector cell delivery to tumor: (a) bypassing the initial lung entrapmentwith administration to the arterial supply of tumor; (/>) reducing normaltissue retention using effector cells with high deformability or blockinglymphocyte adhesion to normal vessels; and (c) enhancing tumor-specific

capture and arrest by modifying the tumor microenvironment.

INTRODUCTION

Adoptive immunotherapy with lymphokine-activated and expandedeffector cells or gene-modified lymphocytes, such as IL-23 activated

TILs and natural killer cells (lymphokine-activated killer cells orA-NK cells), has had limited success in advanced cancer patients(1-3). Although a conclusive mechanism remains elusive, numerousin vitro findings and limited in vivo data suggest that the cancer-

Received 6/16/95: accepted 6/17/96.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 with18 U.S.C. Section 1734 solely to indicate this fact.

1This work was supported by a grant from the American Cancer Society. R. K. J. is

the recipient of Outstanding Investigator Grant R35-CA-56591 from the National CancerInstitute.

2 To whom requests for reprints should be addressed, at Steele Laboratory. Department

of Radiation Oncology, Massachusetts General Hospital. Boston. MA 02114. Phone:(617) 726-4083; Fax: (617) 726-4172.

' The abbreviations used are: IL-2. interleukin 2; TIL. tumor-infiltrating lymphocyte;

NK, natural killer: A-NK. adherent NK.

specific cytolytic activity of the effector cells is realized in multiplesteps. A prerequisite, however, is the optimal delivery of effector cellsto the target tissues while minimizing the toxicity in normal tissues(4-6).

Experimental evidence together with theoretical considerationsbased on effector cell functions indicate that the ability of adoptiveimmunotherapy to eradicate an established tumor is quantitativelydetermined by the initial tumor burden, growth pattern, and themagnitude of immunological response generated by effector and otheraccessory cells at the site of tumor (5-10). Thus, there is great interest

in studying the effector cell accumulation in tumors after their adoptive transfer. In addition, dose-limiting side effects also make biodis-

tribution study important for normal tissues. Vascular leak syndrome,the most common side effect associated with IL-2 administration, is

mediated by activated cells in normal tissues and may cause edema,hypotension, and renal dysfunction (11, 12). Thus, to achieve tumoreradication and avoid normal tissue toxicity, it will be highly beneficial to understand the global and microscopic aspects of lymphocytedistribution and quantitatively assess the factors governing the effector cell uptakes in tumor and critical normal tissues.

Although prior investigations of the effector cell biodistribution incancer patients have been carried out, practical considerations makethese studies difficult in humans (13). As a result, biodistribution datain patients are highly limited. Fortunately, preclinical models provideimportant information on effector cell biodistribution in both normaland tumor tissues (14-22). Aiming at integrating microscopic knowl

edge into a unified global picture, kinetic modeling can serve as auseful tool to the study of effector cell biodistribution. By systematically examining the effects of changing individual model parameters,it can help quantitatively formulate hypotheses for experimental evaluation, identify key parameters and their values, and suggest possiblestrategies for improvements in biodistribution. Therefore, throughmathematical modeling of the kinetics of effector cell biodistribution,we can quantitatively identify the barriers to effector cell delivery,suggest improvements in the treatment by overcoming these barriers,and explore the maximum therapeutic potential.

Compared with other kinetic modeling methods, such as multi-

exponential curve fitting, a physiologically based kinetic modelingapproach has unique advantages. It uses measurable quantities, suchas organ volumes and blood flow rates, as model parameters and,hence, may permit a priori prediction of biodistribution and extrapolation from one species to another by appropriately scaling the modelparameters. Although physiologically based kinetic modeling hasbeen successfully applied to the pharmacokinetics of chemotherapeu-tic agents (23, 24) and macromolecules (25-27), such as methotrexate

and antibodies, and used to design treatment protocols, there has beenno effort to extend the method to cell biodistribution because of thecomplexities involved in the latter. In this study, instead of themolecular (microscopic) processes involved in the cell interactions,we focused on the whole-body (global) processes and established a

simple physiologically based kinetic model for the biodistribution ofadoptively transferred cells. This model is not limited to effector cellsbut applicable to all circulating cells (e.g., lymphocytes, cancer cells,and peripheral blood stem cells) and provides a general framework for

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KINETIC MODEL OF IMMUNE CELL BIODISTRIBUTION

the study of cell biodistribution. It may provide input to futuremicroscopic studies, and can, in principle, incorporate microscopicdetails as they become available.

Being the first effort to model the biodistribution of adoptivelytransferred cells, the model inevitably invoked a number of hypotheses (assumptions) and parameter estimates due to our limited quantitative knowledge of cell biodistribution. The model can assess theimpact of these hypotheses and estimations through sensitivity analysis and may serve as an impetus for further investigations.

It is also important to note that the present study was based onlimited, sometimes fragmentary, experimental data. Our purpose inthis study was not to fit a particular set of experimental data but toprovide a conceptual framework and to guide future experiments. Inthe long term, the predictive power of such a model is likely toimprove with the incorporation of new parameter values as theybecome available.

The cells considered in this study were nonactivated and activatedlymphocytes with either T-cell or NK cell lineage, including nonactivated T lymphocytes, nonactivated NK cells, and IL-2-activated and-expanded TILs and A-NK cells. The majority of the TILs exhibit aT-lymphocyte phenotype (CD4+ or CD8+), and most of A-NK cellsare of natural killer cell lineage (CD3 CD56+; Refs. 4, 6, 28, and 29).

In this study, biodistribution of lymphocytes with T-cell lineage,including nonactivated T lymphocytes in rats and IL-2-activated TILs

in humans, was first simulated with a minimum number of adjustableparameters. Based on the simulation for humans, we projected abaseline biodistribution for activated TILs in mice by scaling downthe model parameters from humans.

The same simulation and scaling scheme was then applied tolymphocytes with NK cell lineage. The biodistributions of nonactivated NK cells in rats and IL-2-activated A-NK cells in mice were

simulated by adjusting the same set of parameters. With parametersscaled up from mice, the model predicted biodistribution for activatedA-NK cells in humans in the absence of clinical data. Through model

simulation and parameter estimation, the study revealed the similarities and differences between nonactivated and activated lymphocytesand between lymphocytes with T-cell and NK-cell lineages. Finally,

the model enabled us to identify the key factors that determineeffector cell delivery to human tumor through sensitivity analysis.

MATERIALS AND METHODS

Published Data. The biodistribution of cells of the immune system isusually studied using the tracer principle, in which a sample population ofmarked cells is followed over a period of time. Since this approach requires thecells to be labeled with radioactivity or fluorescence, it strongly depends on thestability of the label and may not faithfully track the traffic of the cells.

We selected the following literature data for nonactivated lymphocytes:'"In-oxine-labeled T lymphocytes and large granular lymphocytes (NK cells)in rats (30). For IL-2-activated effectors, we chose: 125I-PKH95-labeled TILsin mice (22). "'In-labeled TILs in humans (13), and 125IdUrd or positronemitter "C-labeled A-NK cells in mice (16-20).

Model Development. The model included the following key processesinvolved in the traffic of lymphocytes after adoptive transfer: (a) transport viathe systemic circulation: (b) initial reversible lymphocyte capture (temporaryadhesion or entrapment in the form of random contact and rolling) at theendothelial wall: (c) arrest (stable adhesion) at the endothelial wall followingcapture; (d) transmigration across the endothelial wall of arrested lymphocytes:(e) lymphocyte (except NK cell) recirculation via the lymphatic system aftertransmigration; (/) limited TIL recirculation from tumor; (g) residence ofrecirculating lymphocytes in the lumped (model) lymph node before theirreturn to systemic circulation; (/i) lymphocyte proliferation in the extravascularspace; (/') lymphocyte migration from surrounding normal tissues (e.g., thepulmonary tissue) into tumor: and (/') possible lymphocyte depletion (e.g.,

mechanical disruption or apoptosis) in tissues. The model assumed that free

labels were excreted through urine at a much faster rate than their spontaneousrelease from cells, and thus their influence on the observed biodistribution wasignored. The major lymphoid tissues included in the model were bone marrow,spleen, and the lumped whole-body peripheral lymph node. Peyer's patch, an

important mucosal lymphoid tissue for the recirculation and homing of Blymphocytes, plays a minor role for T lymphocytes and was thus omitted in themodel (31 ). In addition to lymphoid tissues, other peripheral organs and tissuessuch as blood, heart, lung, liver, kidney, gastrointestinal tract, skin, muscle,and tumor were considered in the model. These organs and tissues are arrangedaccording to their anatomic relationship, as illustrated in Fig. 1.

The biodistribution of adoptively transferred lymphocytes is the result ofrecirculation and homing regulated by various molecular signals (reviewed inRef. 32). Although little is known regarding the traffic of IL-2-activated TILs,

we assumed that they acted as activated T lymphocytes and that their accumulation in tumor was the result of antigen recognition. Thus, within thetumor, activated TILs were mainly retained in the extracellular matrix afterextravasation and were prevented from recirculation. NK cells, however,experience little recirculation by the lymphatic system (33). Thus, the same setof model equations was applied for all lymphocytes but with different values

i^ I Spleen

Q-L

OC, \

LC,S0

Fig. 1. Top, schematic of the whole-body compartments for the physiologically basedkinetic model. Botiom, the internal view showing the subcompartments and traffic eventsmodeled for lymphocyte localization in the study. Solid lines with arrows indicate bloodtlow directions, whereas dashed lines with arrows indicate lymphatic flow directions. Thelymphocyte traffic is illustrated with the following nomenclature: AR, adherent lymphocyte arrest rate; 7V.,, lymphocyte transmigration rate; A\ lymphocyte attachment rate; fC,lymphocyte detachment rate; L lymphatic flow rate; Q. blood flow rate; and S, fractionof recirculating lymphocytes; C(. free lymphocyte concentration in the vascular space;C*, captured lymphocyte concentration in the vascular space; C?, arrested (stable adhe

sion) concentration in the vascular space; C,. lymphocyte concentration in the extravascular space.

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Table 1 Physiological parameters"

Blood now rale (ß)(ml/min) Lymph flow rate (L) (ml/min) Total volume (VT) (ml) Vascular volume (Vv) (ml) Interstitial volume (V{) (ml)

OrganPlasmaBoneHeartKidneyLiverLungMuscleSkinSpleenG.I.LymphnodeTumorMouse4.380.170.280.801.104.380.801.210.050.900.050.10Rat84.63.31.612.84.784.622.423.30.9514.60.950.4Human300013812063080030004132201384681380.564Mouse06.0

X10~5l.OX10~51.7

XIO"42.0X10~41.0X10~46.0X10~41.0X10~52.0X10~67.0X10~"1.8X10~37.0

X 10~5Rat06.2

X10~4l.OX10~41.8

XIO"32.1XIO"3l.OX

10~36.4X10~3l.OX

10~42.1X10~57.3X10~31.9X10~27.3

X 10~4Human02.6

X10~24.3XIO"37.4XIO"28.7XIO"24.3X10~22.6X10~'4.3X10~38.7X10~43.0X10~'7.7X10""'3.0

X IO"2Mouse0.7741.5000.1330.2980.9510.1917.9242.9400.1003.4500.1000.472Rat19.651.11.23.719.62.1245.066.81.3020.11.302.0Human2700.01500.0300.0284.01809.0999.035000.06800.0173.62147.01734.020.0Mouse0.7740.0800.0070.0300.0950.0190.1500.2000.0100.1000.0100.033Rat19.62.30.0860.371.90.214.64.50.130.580.130.14Human2700.0150.015.028.4180.999.9700462.017.043.017.01.4Mouse0.0000.2800.0190.1010.1900.0571.0320.9990.0200.6000.0200.258Rat06.40.161.23.90.6331.922.70.263.50.261.1Human0279.042.996.6361.8299.7455822734.7373.234.710.9

" In addition, the parameter values for the lumped lymph node were assumed the same as for the spleen in the absence of literature date (24, 25).

for attachment rate, arrest rate, and recirculation fraction as model parametersto reflect the differences.

To model the lymphocyte traffic described above, each tissue was furtherdivided into two subcompartments: vascular and extravascular spaces. First-

order, reversible, saturable binding kinetics were used for initial lymphocytecapture or entrapment, under the assumptions of similar lymphocyte activationstate and independent lymphocyte interaction with endothelium (34, 35).Under the same assumptions, irreversible first-order kinetics were used for the

arrest of adherent lymphocytes to the vessel wall. We also assumed that thelymphocyte transcapillary migration obeyed first-order kinetics, which means

simply that the rate of lymphocyte extravasation is proportional to the numberof cells arrested to the vascular endothelium. This should be a reasonableassumption for independent lymphocyte transmigration at low concentration.

For simplicity, in the lumped peripheral lymph node extravascular compartment, first-order kinetics with efferent lymph flow were used to characterize

the temporary residence of lymphocytes. In addition, the model also accountedfor lymphocyte depletion with first-order kinetics by assuming homogeneity in

each lymphocyte population. This subcompartment schematic for each tissue isillustrated in Fig. 1. Lymphocyte proliferation was neglected during the lymphocyte distribution phase in this study because of the lack of experimentaldata. This process should not be neglected during the effector phase and can betaken into consideration should pertinent data become available. Similarly, theextent to which lymphocyte migration from surrounding normal tissues intotumor tissue contributes to localization is not known because of the lack ofexperimental data. The role of this pathway for effector cell infiltration intotumor tissue was, nevertheless, assessed quantitatively with the model andsensitivity analysis. The injection volume, which is proportionately larger insmall rodents than in humans, was assumed to be excreted quickly via thekidney, leaving little impact on the total vascular volume. The mathematicalequations for the model are given in Appendix B.

Model Parameters. We used in vivo literature values as model parameterswhenever possible or in vitro estimates when necessary to minimize thenumber of adjustable parameters.

The physiological parameters used to describe the circulation of lymphocytes in the body were: blood flow rate: lymph flow rate; and vascular, interstitial,and total volume for each organ. They were obtained from the literature, exceptfor tumor sizes, for which hypothetical values were assumed in the model (25,36). The values of these parameters are summarized in Table 1.

The initial lymphocyte capture by the endothelium, i.e., random contact androlling, is characterized by quick attachment but slow detachment (37). An invivo dynamic study with mixed populations of lymphocytes and high endo-

thelial venule in murine Peyer's patch by Bjerknes el al. (37) has shown that

lymphocytes readily attach to high endothelial venules. However, most of theattached cells detach within a few seconds, with the majority (about two of

three) of lymphocytes detaching in a characteristic time of 0.3 s. Although thisstudy was limited to the high endothelial venule and confounded by theheterogeneity of lymphocytes, it provided a baseline estimation of the Tlymphocyte detachment rate. Thus, in the model, the detachment rate for the

initial lymphocyte capture was estimated with a characteristic time constant of0.3 s. This estimation is consistent with the measured off rate of 3.5 s" ' for the

P-selectin-carbohydrate bond at physiological stress (35). A similar in vivo

study for NK cells has also been carried out in syngeneic C3H mice bearingMCalV carcinoma, and a characteristic detachment time of 10 s for NK cellswas estimated (Table 2A; Ref. 21).

Because of the difficulties with in vivo measurement, no data are availableto determine the adhesion site density for each tissue. We. therefore, estimated

an ad hoc adhesion site density by assuming that 15% of the vascular spacewas available for cell adhesion in all tissues except lung, liver, and spleen, inwhich 30% was assumed because of the sinusoidal structure of vessels in theseorgans. With an assumed average 10 /xnr for each adherent cell surface areaand 30-/im diameter for adhesive vessels, the estimated values of adhesion sitedensity were 1.0 x IO9 cm"3 of the vascular space for lung, liver, and spleen,and 5.0 X 10s cm"3 for other organs.

Studies have revealed that activated T lymphocytes have much higher

motility and transcapillary migration rate than nonactivated T lymphocytes(38). Quantitative studies using endothelial monolayers derived from blood-

retinal barrier have shown that about 50% of activated T lymphocytes migratedthrough the monolayer compared with 5% for nonactivated T lymphocytes in4 h (38). Accordingly, we estimated the baseline transcapillary migration rates tobe 2.1 X 10~3 min"' for activated TILs and 2.9 x IO"4 min"1 for nonactivated

T lymphocytes. In the absence of experimental data, the same transcapillarymigration rates were assumed for nonactivated and activated NK cells.

The efficiency of lymphocyte recirculation is described by the fraction ofrecirculating lymphocytes with lymph flow in the extravascular space. Forsimplicity, the recirculating fractions for nonactivated T lymphocytes andIL-2-activated TILs were assumed to be 100%, except 0% was assumed for all

lymphocytes in tumor, and 1.8% was estimated for nonactivated T lymphocytes in the lumped lymph node based on the 12-h recirculation time in rats

(39). Zero recirculation was assumed in all tissues for both nonactivated NKcells and activated A-NK cells. The values of all the estimated parameters are

summarized in Table 2. The impact of uncertainties associated with these

Table 2 Parameters estimated from literature

Transmigration rale" (min ')OrganLymph

nodeTumorOthers2.12.12.1NKX10~4X10~4X

IO"4A-NK2.9

X10~32.9X10~32.9X 10~3T2.1

X2.1X2.1

Xio-4IO"4IO'4TIL2.9

X2.9X2.9

XIO"3IO"3IO"3NK0.00.00.0Recirculating

free cellfraction*A-NK0.00.00.0T0.0180.01.0TIL0.0180.01.0

" From Greenwood and Calder (38).* Calculated from data from Butcher and Ford (39).

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KINETIC MODEL OF IMMUNE CELL BIODISTRIBUTION

estimated parameters on effector cell biodistribution was also considered andquantified using sensitivity analysis (see "Discussion").

For parameters without any independent experimental estimates, includinglymphocyte initial attachment rates, arrest rates, and depletion rates, a leastsquare nonlinear regression was carried out sequentially in lung, spleen, liver,and tumor. For other tissues, no lymphocyte capture or arrest was assumedbecause of their minor influences on cell biodistribution. The lymphocytedepletion rates in all tissues were set to zero, except the lung, to further reducethe number of adjustable parameters used in the model.

Lymphocyte Migration from Surrounding Normal Tissues. In additionto extravasation, effector cells may localize in the tumor through migrationfrom surrounding normal tissues. To assess the importance of this infiltrationpathway, the lymphocyte migration rate from surrounding normal tissues intotumor tissue was estimated. This rate was estimated in two different scenariosfor A-NK cells based on the numbers of infiltrating A-NK cells observed in

B16 melanoma murine métastasesand in the surrounding pulmonary tissue atdifferent time points (16).

In the first scenario, we assumed that migration from surrounding normaltissues was the sole pathway for effector cell infiltration. The lymphocyte migration rate can be estimated according to a mass balance equation:KumordCMmJdt = MR,ungSAtumorC, ,ung.The estimate gives an upper limit for

the lymphocyte migration rate. In the second scenario, we assumed that whileboth pathways contribute to the effector cell infiltration, the rate of lymphocyteextravasation per unit volume was constant for métastases of differentsizes, while the infiltration rate varied through migration from surroundingnormal tissues due to the changes in the surface area to volume ratio fortumor. The mass balance equation is then: dCtamarildt —¿�dCtamor2/dt =MRlaaf <£4lumur,/VIllmorl)CUuog - Ai/?,ung (SAtamaa/Vtamm2i CUang.

Based on the estimated migration rates, the role of lymphocyte migrationfrom surrounding normal tissues on effector cell infiltration into tumor wasassessed with model simulations for different tumor sizes and host organs.

RESULTS

Comparison of Model with Data. Lymphocyte biodistributiondata from the literature were used to estimate lymphocyte initialattachment rates, arrest rates, and lymphocyte depletion rates. Table 3gives the estimated values of these parameters for all lymphocytepopulations. Figs. 2 and 3 show the corresponding model simulationsfor nonactivated T lymphocytes in rats, IL-2-activated TILs in humans, nonactivated NK cells in rats, and IL-2-activated A-NK cells in

mice.For cells with T-lymphocyte lineage, comparison of model simu

lations with literature data in Fig. 2 showed that the model accuratelysimulated the biodistribution kinetics in most of the tissues. Examination of biodistribution profiles of nonactivated and activated Tlymphocytes revealed that they shared the same qualitative features,characterized by a decreasing cell concentration in lung following ahigh initial uptake, a stable concentration in liver, slow accumulationsin spleen and tumor, and a decreasing low concentration in blood.

As for cells with NK lineage. Fig. 3 demonstrates that the modelcan simulate the biodistribution kinetics. Nonactivated NK cells exhibited a similar biodistribution profile to lymphocytes with T-celllineage except for a lower lung concentration at late times. IL-2-activatedA-NK cells, however, showed significantly decreasing concentrations in

liver and spleen during the later course of biodistribution.Based on the estimated parameters for IL-2-activated effector cells,

we then scaled the model simulations between two different species(mice and humans) to assess the prediction capability of the model inthe presence of limited experimental data and to provide baselineestimations in the absence of any experimental data. To scale themodel among species, the values of physiological parameters in themodel had to be changed correspondingly for different animals. Formany parameters, experimental values were available, such as bloodflow rates to each tissue (25, 26). For parameters without experimental measurements, a more sophisticated scaling scheme was adopted

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KINETIC MODEL OF IMMUNE CELL BIODISTRIBUTION

Fig. 2. The comparison of model simulation (solid lines} with literature data tor lymphocytes with T cell lineage: A, nonaclivated Tlymphocytes in blood (D), lung ( 0 ). liver (•).and spleen ( * ) in rats(30); and B, IL-2-activated TILs in blood, lung (D). liver (•).spleen( * ). and tumor (•)in humans (13). The lymphocyte concentrations areexpressed in percentage of injected dose (cells) per gram of tissue. Theadjustable parameters in the simulations were T-lymphocyte attachmentand arrest rales for each organ as well as the lymphocyte depletion ratein lung. The model simulation showed that 43% of the injected nunac-tivated T lymphocytes remained in rats at 48 h, and 60% of 1L-2-activated TILs remained in humans at 168 h.

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according to the underlying physiology. For each effector population,we assumed constant attachment and detachment rates between animal species, together with arrest rates, transmigration rates, and re-circulation fractions. The lymphocyte depletion rates (in min" ') were

scaled according to (body weight) "4 in accordance to the scaling

routines for the clearance rates for drugs (40). The values of the scaledparameters are listed in Table 3 for activated TILs in mice fromhuman values and for A-NK cells in humans from mouse values. The

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lung

blood

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spleen

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10 20 30 40 50

Time (h)20 30

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Fig. 3. The comparison of model simulation (soliti lines} with literaturedata for lymphocytes with NK cell lineage: A. nonactivated NK cells inblood (D). lung ( 0 ), liver (•).and spleen ( * ) in rats (30): and B,IL-2-activated A-NK cells in lung (D). liver (•).spleen ( 4>). and tumor inmice (16-20). The lymphocyte concentrations are expressed in percentageof injected dose (cells) per gram of tissue. The adjustable parameters in thesimulations were NK cell attachment and arrest rates for each organ and thelymphocyte depletion rate in lung. The model simulation showed that 63%of the injected nonactivated NK cells remained in rats at 48 h. and 8% ofIL-2-activated A-NK cells remained in mice at 24 h.

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KINETIC MODEL OF IMMUNE CELL BIODISTRIBUTION

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Fig. 4. Scaled biodistribution simulations (solid linex). The parameters were scaled for the animal size, but there was no adjustableparameter in these simulations. A, down-scaled IL-2-activated TILbiodistribution in blood, lung (Q), liver (•),and spleen ( t ) in micefrom humans in comparison with literature data (22). B, up-scaledIL-2-activated A-NK cell biodistribution in blood. lung, liver, spleen,

and tumor in humans. The lymphocyte concentrations are expressed inpercentage of injected dose (cells) per gram of tissue. For IL-2-activated

TILs in mice, a better fil could also be achieved by further adjustinglymphatic flow rate (doited lines and items denoted with *). The modelsimulation showed that 22% of the injected IL-2-activated TILs remained in mice al 120 h. and 50% of IL-2-activated A-NK cells

remained in humans at 168 h.

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human data is shown and compared with available experimental datain Fig. 4A, and the scaled up biodistribution of A-NK cells in humans

from mouse data is shown in Fig. 4B.The comparison of the scaled down TIL biodistribution in mice

with limited experimental data in Fig. 4/4 demonstrated that the modelcould provide a baseline biodistribution by simply scaling modelparameters between different species. However, there were considerable discrepancies between model predictions and available data insome tissues, especially during late times due to the limitations inmodel assumptions, the scaling scheme, and the variability associatedwith the experimental data. Nevertheless, the model could simulatethe experimental data by adjusting a single additional parameter, suchas the lymph flow rate, as shown in Fig. 4/4.

Sensitivity Analysis. For any kinetic model, it is important notonly to estimate the values of parameters but also determine the effecteach parameter may have on the model simulation due to parametervariability and uncertainty. The relative sensitivity coefficients for theparameters were calculated [(dC/Q/(d/VP)], i.e., the percentage ofchange in the number of effector cells per gram of tumor divided bythe percentage of change in parameter value, P). The sensitivitycoefficients for IL-2-activated TILs and A-NK cells in humans at the

late stage of biodistribution after cell injection are given in Fig. 5. Weselected 150 h, when cell biodistribution in most of the organs reachesthe steady state. Negative values for the sensitivity coefficients indicate a decrease in concentration for an increase in the parameter; verysmall absolute values indicate that the effector cell concentration isinsensitive to that parameter.

The sensitivity analysis showed that lymphocyte attachment andarrest rate, together with adhesion site density in the tumor (proportional to the vascular space available for binding and inversely proportional to average adherent cell surface area), were key parameters

determining effector cell accumulation. Most importantly, the accumulation of effector cells in normal tissues, especially in lung, adversely affected the delivery of effector cells to tumor, evident by thesensitivity of tumor accumulation to lymphocyte attachment rate andadhesion site density in the lung. The sensitivity analysis furtherrevealed a difference between TILs and A-NK cells in their different

recirculation and homing patterns. Increasing tumor blood flow rate,although leading to a higher tumor accumulation for A-NK cells, was

found to be ineffective to improve TIL accumulation in tumor.Because of the low efficiency of systemic delivery, an alternative

route bypassing lung entrapment, administration into the arterial supply of the tumor tissue, was examined. The accumulations in humantumor are shown in Fig. 6 for TILs and A-NK cells. The simulationsindicated that intra-arterial administration had dramatic advantage

over systemic delivery, with more than 1000 times higher effector cellaccumulation in tumor than with systemic administration under theassumption that all of the injected cells go through tumor vasculature.This advantage is most pronounced under conditions of high retentionin normal tissues and rapid arrest at the tumor endothelium.

Lymphocyte Migration from Surrounding Normal Tissues.Under the assumption in the first scenario that migration from surrounding normal tissues is the sole pathway for effector cell infiltration, the migration rate for A-NK cells was estimated as 1.56-4.12

/Ltm/h.Whereas in the second scenario, which assumed that the rate ofextravasation per unit volume was constant for métastasesof differentsizes, the A-NK cell migration rate was estimated to be nearly zero.

This indicated that a minimal contribution of lymphocyte migrationfrom surrounding normal tissues into tumor in such a scenario. Theestimated A-NK cell migration rate was significantly lower than the

speed of locomotion of a vigorously moving lymphocyte (about 1.20mm/h) (31, 41) or A-NK cells (about 60 jum/h: Ref. 42). The A-NK

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KINETIC MODEL OF IMMUNE CELL BIODISTRIBUTION

1.5

1.0

0.5

o.o

-0.5

-i.o

HLB

let

i.o

0.5

0.0

-0.5

-1.0

-1.5

Fig. 5. The sensitivity analysis for IL-2-aciivated TILs in humans (A) and IL-2-activaled A-NK cells in humans at 150 h (B) following cell transfer. Small absolute values

indicate that the effector cell concentration is insensitive to that parameter; negative valuesfor the sensitivity coefficients mean that the concentration decreases when the parameterincreases. The same nomenclature is used here as in Fig. 1: v4/?lumor,adherent lymphocytearrest rate in tumor; ß,ul)1()r,tumor adhesion site density (proportional to the availablevascular space for binding and inversely proportional to average adherent cell surface areain tumor); Blmf. adhesion site density in lung (proportional to the available vascular spacefor binding and inversely proportional to average adherent cell surface area in the lung);J. lymphocyte transmigration rate; /^lumor. lymphocyte attachment rate in tumor; ^„„g,lymphocyte attachment rate in lung; Q. Iunior blood flow rate; L. tumor fluid leakage rateto the peripheral normal tissues; 5, fraction of recirculating lymphocytes; and E. lymphocyte depletion rate in tumor.

cell migration rate was expected to be between the estimates of thesetwo scenarios.

The kinetics of A-NK cell infiltration through lymphocyte migra

tion from surrounding normal tissues were simulated for differentlymphocyte migration rates (41.2 jam/h, 4.12 /xm/h, and 0.412 /xm/h).different tumor sizes (an established cancer with tumor mass 20 g anda micrometastasis with tumor mass 0.0042 g), and at two of the mostfrequent sites for cancer metastasis, the liver and lung (Fig. 7).

DISCUSSION

The general framework of cell biodistribution developed in thisstudy is useful not only for immune effector cells but also for othercell types, such as neoplastic and peripheral blood stem cells. It canserve as a quantitative tool to study adoptive immunotherapy, cellulartherapy, cancer metastasis, immune cell trafficking, bone marrowtransplantation, and cell population kinetics. The physiologicallybased kinetic model enabled us to quantitatively assess the lymphocyte biodistribution under widely different conditions and in different

species. Through simulation and sensitivity analysis, the modelshowed that: (a) effector cell adhesion to tumor vasculature has animportant impact on tumor accumulation; and (b) the low effectorconcentrations in systemic circulation due to high retention in lungand liver greatly limit their delivery to tumor.

Comparison of Model with Data. For lymphocytes of T-celllineage, nonactivated and activated lymphocytes shared many common features in biodistribution regardless of their states of activation.The most important was the high initial uptake in lung, followed byliver and spleen, and the low effector cell concentrations in blood. Asimilar pattern of biodistribution was observed for nonactivated lymphocytes of NK-cell lineage. This implied that common mechanisms

of distribution, such as the common anatomic route of lymphocytecirculation and mechanical entrapment in tissues with sinusoidalvessels, were dominant in the initial phase of biodistribution. At latertimes, the observed differences in liver and spleen between nonactivated NKs and activated A-NKs may be the result of different

cell labels (18). Further experiments are needed to discern thesemechanisms.

The model also revealed the differences in biodistributions betweendifferent lymphocyte populations. Such differences can be explainedprimarily with different lymphocyte attachment and arrest rates. Forexample, model simulation indicated that activated A-NK cells exhibited a higher lymphocyte attachment rate in tumor (3.0 X 10~6)than in other normal tissues (1.5 X 10~6) except in the lung(9.0 X 10~5; Table 3). On the other hand, activated TILs showed alower estimated lymphocyte attachment rate in tumor (6.9 X 10~9)than other tissues (from 3.1 X 10~6 to 7.9 x 10~9 min"1; Table 3).

The estimated higher lymphocyte attachment rates in tumor for acti-

10

sa

HA

0.001 -

0.0001

intra-arterialinjection

50 100 150

Time(h)

B

-if•¿�d

0.11

o.oi-

0.001 -

0.001)1

intra-arterialinjection

systemicinjection

50 100 150

Time (h)200

Fig. 6. Plot of simulated effector cell concentrations in tumor under intra-arterial versus

systemic administration under the assumption that all of the injected cells go throughtumor vasculature for IL-2-activated TILs in humans (A) and IL-2-activated A-NK cellsin humans (ß).The intra-arterial administration of effector cells was simulated in the study

by supplying all of the effector cells to tumor vascular space at the time of injection.

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KINETIC MODEL OF IMMUNE CELL BIODISTRIBUTION

1E-01

'S« lE-02i^5

U 1E-03-o

§H 1E-04-J

IE-OS100 150 200

Time (h)

n: in

•¿�S»1E-02-

H 1E-04-

1E-05

Lung 168.mm

Liver 168 mm

100 150

Time (h)

2IMI

Fig. 7. The model simulated kinetics of IL-2-activated A-NK cell infiltration intotumor through migration from surrounding normal tissues for: A. different migration ratesfor 20 g of tumor in the lung; B. different tumor sizes (l g with 1-mm radius and 20 g with168-mm radius) and different host organs (the liver and the lung) with a constantmigration rate of 4.12 jim/h.

vated A-NK cells cannot be uniquely determined by fitting the cell

biodistribution data. These estimates were, nevertheless, consistentwith experimental observations that activated A-NK cells exhibited a

higher adhesion to tumor vasculature (15).It is also important to note that besides effector cell preferential

adhesion to tumor vascular endothelium, other mechanisms might alsoaccount for the preferential tumor accumulation observed experimentally (16, 17, 43, 44). These mechanisms may include reduced lymphocyte recirculation in tumor as a result of impaired lymphocytefunctions (45) and lymphocyte migration from the surrounding normal tissues, such as the lung, into tumor (17).

Differences between activated effector cells and nonactivated lymphocytes were also revealed by the model. Activated A-NK cells, due

to their larger diameter and higher rigidity compared to other lymphocytes, may be more easily trapped in vessels with sinusoidalstructures, such as the lung, immediately after the transfer (46-48).

For TILs, a higher accumulation in spleen was expected during thelate period as a result of their higher adhesion in spleen throughlymphocyte homing receptors (31).

Implications of Sensitivity Analysis. The sensitivity analysisshowed that adhesion site density, lymphocyte attachment, and arrestrates were important parameters, as would be expected. However, themost significant limiting factor on effector cell accumulations intumor was found to be the low effector cell availability. This limitation was imposed by lymphocyte accumulations in normal tissues andcharacterized with low effector concentrations in the systemic circulation, as also observed in clinical trials (13). Therefore, the systemic

treatment with effector cells has not been as impressive as originallypromised, due, in part, to the biodistribution problems.

To circumvent this limitation and increase the availability of effector cells for tumor, it is essential to reduce the uptake of effector cellsin normal tissues by decreasing mechanical entrapment and adhesion.For example, since the effector cell rigidity as well as effector celladhesion to endothelium may promote entrapment and adhesion innormal tissues (46, 48-50), reducing the effector cell rigidity or

blocking adhesion molecules may decrease the cell retention in normal tissues and increase their delivery to tumor. Sensitivity analysisshowed that reducing either lymphocyte attachment rate or adhesionsite density in lung by one-half elevated the effector cell accumulations in tumor by 40% for TILs and 60% for A-NK cells (Fig. 5). This

limitation also suggested that administration of effector cells to thearterial supply of tumor bypassing the initial lung entrapment represented the best case scenario and would lead to a higher delivery to thetumor, as shown in Fig. 6. Unfortunately, such a local treatment isneither practical nor desirable for a systemic disease.

Another interesting feature revealed by sensitivity analysis was thedifferent implications for TILs and A-NK cells due to their different

attachment and arrest rates. Simulation showed that increasing tumorblood flow rate may significantly increase the A-NK cell accumulation in tumor because of the higher attachment rate of A-NK cells to

tumor microvessels. This, however, may have little or negative impacton TIL accumulation in tumor due to loss of adherent TILs fromtumor with increased tumor blood flow.

Sensitivity analysis also enabled us to assess the impact of theuncertainties associated with our parameter estimation on effector cellbiodistribution. For example, the low sensitivity for activated TILrecirculation fraction in tumor (Fig. 5A) suggests that uncertaintyassociated with the estimated baseline value would only have amoderate influence on lymphocyte distribution.

Lymphocyte Migration from Surrounding Normal Tissues.Previous studies have shown that effector cells may localize intocancer métastasesof different origins and establish cell-to-cell contact

with both cancer cells and tumor endothelial cells (16, 17). Experimental evidence has correlated the number of infiltrating A-NK cells

into pulmonary métastaseswith the density of microvessels and themetastatic subtypes (compact and loose; Ref. 51). However, the mainpathway of lymphocyte infiltration into tumor, either through extravasation across tumor vessel or migration from surrounding normaltissues, remain to be elucidated.

Although a mathematical model alone cannot discern the twopossible pathways of lymphocyte localization in tumor, model simulations indicated that the lymphocyte migration rate is an importantparameter for effector cell infiltration into tumor. In addition, tumorsize is also an important factor in effector cell infiltration, becausesmaller tumors have higher surface area-to-volume ratio than larger

tumors. For example, a micrometastasis with a radius of 1 mm isexpected to have a 14-fold higher effector cell infiltration at 160 hthan a 20-g tumor with a radius of 1.68 cm under the same conditions.

This is consistent with the observed higher effectiveness of adoptiveimmunotherapy against micrometastases than established cancerswith high tumor mass (1). Similarly, the host organ in which tumorsgrow was also found to play an important role in effector cell infiltration. Tumors growing in organs with higher effector cell retention,such as the lung, may have increased infiltration compared to tumorsin organs with lower effector cell retention, such as the liver.

Model Limitations. Although the model was able to simulate thebiodistribution of adoptively transferred cells, there were some discrepancies between simulations and literature data, especially forscaled simulations. Thus, the results must be interpreted with caution,since the model simplified the processes of lymphocyte recirculation

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KINETIC MODEL OF IMMUNE CELL BIODISTRIBUTION

Table 4 Comparison of cell labels1

LabelsFealureType

oflabelHighlabelingefficiencyEasy

detectionNontoxicityto the studiedcellsNo

known interference with celltrafficNospontaneous release from viablecellsRapid

excretion from the body after celldeathHighresolutionProvide

information on pattern of microscopic migration within the tissueIdealN/AYyvYYYYYNa25lCrO4y

emitterYYYYM*NNN"'in-oxineyemitterNYNYNYNNl25ldUrdyemitterNYY'YNYNN"C

methylalionpositron

emitterYYY'YY'YYN''

" Y, yes; N. no.* Possible label uptake by macrophages.' Covalent linkage.d Resolution limited by the positron emission tomography camera.

and homing. In addition to the errors associated with literature data,these disparities may be caused by cell labels, model assumptions,lymphocyte heterogeneity, assumed parameter values, and thescale-up schemes.

Due to the limited data in the literature, the biodistribution data weconsidered were obtained using different cell labels for different cellpopulations. Although most of these labels are well established forcell trafficking, their reliability as tracers may vary among differentcell populations and during different distribution periods. For everystudied cell population except IL-2-activated A-NK cells in mice (twocell labels: '2<iIdUrd and positron emitter "C), only one type of cell

label was used. Even when different labels were used, the data wereconsistent (see Figs. 2-4, except 4A). On the other hand, for IL-2-

activated TILs in mice (Fig. 4/4). the data showed high variability,even when the same label was used. As a result, more emphasisshould be put on the scaled-down predictions, while simulations

fitting the data should be interpreted with caution.Ideally, to obtain reliable data on cell biodistribution, a cell label

must have such properties as nontoxicity to the studied cells, nointerference with cell traffic, no spontaneous release from viable cells,and rapid excretion from the body after cell death (39). However, suchan ideal label may not exist, and various cell labels could be used toprovide complementary data on cell biodistribution. To evaluate theexperimental data used in this study and to facilitate further experiments, the major features of the cell labels are listed in Table 4.

Certain limitations lie in the basic assumptions of all lumpedmodels. Such models can only provide an averaged biodistributionwithout considering the spatial heterogeneity. This may not only leadto disparities between model simulations and experimental data butalso imply that high effector cell concentration in tumor may notresult in high cell-cell contact between effector cells and their targets.

Furthermore, the values of many parameters are not available in theliterature due to the experimental limitations. Consequently, ad hocparameter estimations and scaling scheme had to be used, which maylead to some disparities between model simulations and literaturedata. Nevertheless, the model identified the parameters that need to bemeasured and modified to improve effector cell delivery to tumor.

Summary. This study demonstrated the feasibility of our modelthrough the comparisons between simulations and experimental data.The study confirmed that treatment with effector cells has not been asimpressive as originally promised, due, in part, to the biodistributionproblems. The model simulation and data are consistent with thehypothesis that activated A-NK cells exhibit a preferential adhesion to

tumor vasculature, whereas activated TILs do not. This suggested thatmechanisms other than preferential adhesion to tumor vasculaturemay be responsible for tumor preferential uptake of TILs. As a result,increasing tumor blood flow, while leading to a higher A-NK accu

mulation, may not increase TIL concentration in tumor. The impor

tance of lymphocyte migration from surrounding normal tissues oneffector cell infiltration into tumor tissue was found to depend onlymphocyte migration rate, tumor size, and host organ. The modelquantitatively predicted that reducing normal tissue retention in thelung by 50% could increase the tumor uptake by —¿�40%for TILs andby -60% for A-NK cells.

Based on the analysis, strategies to improve the delivery of effectorcells to tumor include: (u) bypassing the initial lung entrapment withadministration to the arterial supply of tumor: (/?) reducing entrapmentin normal tissues using effectors with high deformability or modifyinglocal lymphocyte adhesion; and (r) enhancing tumor-specific capture

and arrest by modulating tumor microenvironment (44).

APPENDIX A: Nomenclature

/\A,,r(..,n Lymphocyte arrest rate (min"')Morgan Lymphocyte adhesion site density (ml"1)

C[ an Free lymphocyte concentration in the vascular space (ml ' )

Morgan Captured lymphocyte concentration in the vascular space(ml'1)

d.orgun Arrested (stable adhesion) lymphocyte concentration in thevascular space (ml ')

C¡organ Lymphocyte concentration in the extravascular space(ml"1)

Surg,,,, Fraction of lymphocyte in the extravascular space that canrecirculate

•¿�WC,organ)

AMNiran

Gorgan

"T.org,,,,

Lymphocyte depletion rate (min"')

Lymphocyte proliferation rate in the extravascular spaceLymphocyte transmigration rate (min ')Attachment (ml min ') for lymphocyte captureDetachment rates (min ') for lymphocyte captureLymph flow rate (ml min ')

Lymphocyte migration rate from the pulmonary tissue totumor (cm min" ')Blood How rate (ml min" ')

Tumor surface area (cm2)

Interstitital space (ml)Vascular space (ml)Tolal organ volume (ml)

APPENDIX B: Mathematical Model and GoverningEquations

The mass balance equations for the kinetic model of lymphocytes describethe transpon of transferred cells throughout the body via systemic circulation(Fig. I ). Each organ is further divided into two subcompartments. the vascularspace, and extravascular space (Fig. 1). Inside these organs or tissues, theinitial attachment of lymphocytes is quantified using saturable, reversible,first-order kinetics. The net rates of lymphocyte arrest and transcapillarymigration are also assumed to follow first-order kinetics. The equations were

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KINETIC MODEL OF IMMUNE CELL BIODISTRIBUTION

solved using LSODE. a software package using Gear's method for stiff B.4 Mass Balance Equations for Lung

equations (52).Vascular Space:

B.I Mass Balance Equations for Major Organs

For most of the organs, including the gastrointestinal tract, spleen, skin.muscle, bone, kidney, and heart, the mass balance equations are listed belowaccording to the model scheme illustrated in Fig. 1.

= (ß„v„- ¿„.jeu, + (ßk,dney- t^

+ (ß,™ - AurnJCt,,,™, + (ßskin -

Vascular Space:

i = 0 C* —¿�(O —¿�L IC*»¿organ̂ v.blood vȣorgan '-'organ'*-v.t

—¿�i-f tfí —¿�r* —¿�r3 n''organ V"organ ^ v.organ ** v.organ/

organ v.organ v.organ

.lung

'.organ' v.organ v.urgai

—¿�AJ f* V —¿�AR r* Vorgan *" v.organ v.organ ri"organ ^v,organ ' v.orgai

—¿�F r* v^ organ *-v.organ 'v.organ

—¿�F r* vorgan ^v.organ v.organ

Extravascular Space:

v,„,m«ic,„fjd»=y„fganQors„vv„rsan- ¿„ganc,^ sorg„+<wc,.organ)

In each organ, the average concentration is the weighted average of theconcentrations within each subcompartment.

B.2 Mass Balance Equations for Tumor

Vascular Space:

Vv,un,,,r«,um,M) = ö,„™ Ct,blral - (ß,um„, - ¿,u„,o,>C,.u„„„r

—¿�k1 (R —¿�C* —¿�C* ir4 V"•lumorv*'^uinoT ^v.mmor ^v.iumor'^v.tumor v.tumor

+ L-r r*1 vAmmor ^v.mmor vv.wmor

jdt) =*L„(ß,un,„- ct.mmo,- c-„am,)C,Mamvv.,un„

—¿�i1 r*1 v —¿�AR r* v"•luinor*"v.tumor v v.tumor n" tumor ^ v.iumor ' v.iumor

—¿�F r41 v^lumor ^v.iumor v.iumor

,,^) =AÄ,um„C,umwv,,umut- y,um,,rq,um„vv,,umor

—¿�F r° v^lumiir v.iumor v.iumor

4- (O —¿�I \C* + (O —¿�Õ i(Msihcan Mican'*-v.heart ' Vtilnodc Mnodc'1

' Mnodc ^i.lnode "inodc ~~ ÕCílung~ Mung^v.lung

+ if C* Vlung *"v.lung v.lung

—¿�xjr* v —¿�ARc* vRtung ^v.lung *v.tung ^"lung ^v.lung 'v.lung

—¿�r r* vMung ""v.lung vv,lung

AR r** v —¿�i râ„¢ v"'lung ^v.lung 'v.lung •¿�'lung*"v.lung v.lung

—¿�F r*j v^lung *-v.lung v v.lung

Extravascular Space:

W^Jung«') = -/lung Qlung ^v.,u„g ~ Aung C|.l»ng 8,ung + <i>(C,,ung)

-««lungQlungSA^

B.5 Mass Balance Equation for Lymph Nodes

Vascular Space:

—¿�r* —¿�r* \f~* vdc "-v.lnode "-v.lnode'^v.lnodc vv.ln

if f* V*lnodc ^-v.lnode 'v.lnodc

—¿�f f* V —¿�AR"imxle Lv.lnudc Vv,ln.»lc '^''in. v.lnodc vv.lniKic

Extravascular Space:

V,Mma,(dC,M^d,) = 7,um,,r V,,

B.3 Mass Balance Equation for Blood

Vv.b,<»dWCfv.b1ood/rf') = (ß,u„g - ¿,ung)C:,ung - (ß„v„+ /-g, + /-spleen + ßkKtay

+ ö,„,„,„+ ß>kl„+ ßmilsc,e + ßb», + ßhean + ß,^)^,.,,

There is an additional constraint on the volumetric flow rates:

ö,ung = ßl,v„ - ¿l,v„ + ßlud,«, - 4,d„c> + öum™ ~ ¿,um„,+ ßsk,„~ ¿sk.n + ßm

- ¿n,„.,c + ßhonc - ¿bone + öh.an ~ ¿hean + ö,„,*

and

¿inode = ¿,ung + ¿l,v„+ ¿g, + ¿Iple.„+ ¿hean + ¿k.dno, + ¿,um« + ¿>k,n + ¿mu»c.

= AR C*1 V —¿�/ C* V^"inode "-v.lnodc Kv.lnode Jlnode Lv.lmxk Kv.lnodc

—¿�F r*d v^Inodc ^v.lnodc Kv,lnodc

Extravascular Space:

v idr ifit\ = / r* v +1 c1 Ävi.lnodcVUl-i.[node/"1' Jlnode '-'v.lnode ^v.lnode Mung ^i.lung ulung

+ ¿,iv„Ci.liVBS„vcr+ ¿g,C,gl Sgl + LsplecnC,lpl„„osplra,

* Mtidney ^i.kidney "kidney ^tumor ^-i.iumor "tumor

+ ¿skin Ci,Skm ßskin + ¿músete Q.muscte ^muscle + ¿bone Q.lx,« '

Mieart ^i.hcart *-Tiean Mnodc ^i.lnode "inode

+ <«c,,nude)

B.6 Mass Balance Equations for Liver

Vascular Space:

V,,ivc,WCf,.l,veA'') = (ßgi- í-gi)^ + (ß,p,«„- ¿5p,een)Cv.5plce„

+ (O, - O - O + L + L IC?v»liver *ígi »¿spleen gì spleen' v.blood

—¿�if) —¿�/ \C*UAn Cliver'1'v.liver

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KINETIC MODEL OF IMMUNE CELL BIODISTRIBUTION

_ /r r* vHiver '-v.liver "v.liver

K,i,,CTWQ.n,e/¿')= AR„„,C*hv„Vv|1VCT- y,iverq llverVvli

- Cliver Q,„e, K.,™

Extravascular Space:

=y„vctq,„vcrvvllver- t,iv„cMlv„s„vcr

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1996;56:3771-3781. Cancer Res   Hui Zhu, Robert J. Melder, Laurence T. Baxter, et al.   ImmunotherapyBiodistribution in Mammals: Implications for Adoptive Physiologically Based Kinetic Model of Effector Cell

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