(CANCER RESEARCH 52. 5838-5844, October 15, 1992]

Advances in Brief

Pharmacokinetic Analysis of the Perivascular Distribution of BifunctionalAntibodies and Haptens: Comparison with Experimental Data1

Laurence T. Baxter, Fan Yuan, and Rakesh K. JainEdwin L. Steele Laboratory, Department of Radiation Oncology, Harvard Medical School, Massachusetts General Hospital, Boston, Massachusetts 02114

Abstract

A mathematical model is developed to describe the concentrationprofiles around individual tumor blood vessels for two-step approachesto cancer treatment. The model incorporates plasma pharmacokinetics,interstitial diffusion, reversible binding between antibody and haptenand between antibody and tumor-associated antigens, and physiologicalparameters to evaluate present experimental approaches and to suggestnew guidelines for the effective use of two-step approaches. Resultsshow considerable interaction between the binding kinetics, initial drugdoses, and antigen density, with optimal parameter ranges depending onthe desired goal: treatment or detection. The hapten concentration intumors was found to be nonuniform because of specific binding to antibodies. While binding of the hapten to the bifunctional antibody isnecessary for improved retention, too large a binding affinity may leadto very poor penetration of the hapten into regions far away from bloodvessels. The time delay between antibody and hapten injection wasfound to be an important parameter. Longer time delays were found tobe advantageous, subject to constraints such as internalization of theantibody and tumor growth during treatment. A proper combination ofinitial doses for the two species was also seen to be crucial for maximumeffectiveness. Comparison of the model with the experimental data of LeDoussal et al. (Cancer Res., 51: 6650-6655, 1991) and Stickney et al.(Cancer Res., 50: 3445-3452, 1990) suggests two novel, yet testable,hypotheses: (a) the early pharmacokinetics of low molecular weightagents can have an important effect on later concentrations using two-step approaches; and (b) metabolism may play an important role inreducing concentrations in the tumor and tumorplasma concentrationratios. These results should help in the effective design of two-stepstrategies.

Introduction

Two-step approaches to cancer detection and treatment havebeen developed which seek to combine the selectivity of certainhigh molecular weight agents (i.e., antibodies and their fragments) with the superior tissue penetration and rapid plasmaclearance of low molecular weight agents. These two-stepapproaches include BFA2 and haptens (1-3), enzyme-conjugated antibodies (4, 5), and other antibody-based localizationschemes (6, 7). The concept is to preinject a macromoleculewhich binds specifically within the tumor, using a high dose andwaiting long enough to achieve good localization. Later, a second molecule of low molecular weight is injected. It shouldselectively be taken up (or converted into an active agent) wherethe macromolecule concentration is high, presumably in the

Received 7/30/92; accepted 8/31/92.The costs of publication of this article were defrayed in part by the payment of

page charges. This article must therefore be hereby marked advertisement in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

1Supported by grants from the National Cancer Institute (CA-36902 andCA-49792).

2 The abbreviations used are: BFA, bifunctional antibody; AUC, area under thecurve (concentration versus time); <C>, average tissue concentration weighted byvolume; DTP A, diethylenetriaminepentaacetic acid, i.d./g, injected dose per g oftissue; SAT, fractional saturation; SR, specificity ratio; TAA, tumor-associatedantigen; TI, therapeutic index; T:P, tumorplasma concentration ratio; UNI, uniformity index.

tumor. In these approaches there are many physiological andkinetic parameters which influence uptake and distribution ofthe active agent.

Although these two-step approaches have shown significantprogress, current strategies may be suboptimal. There is a definite need for mathematical modeling of two-step approaches to

design experiments, to guide therapies, to understand the factors involved in biodistribution, and to identify the key parameters which limit or enhance delivery. One postulated advantageof using haptens in conjunction with antibodies is better penetration into tumor tissue, yet this has not been evaluated fortwo-step approaches. We have previously developed distributedparameter models which investigate different aspects of antibody penetration in tumors, on both macroscopic (8) and microscopic scales (9). We have also developed a lumped parameter compartmental model for the uptake of haptens andprodrugs in two-step therapies (10). The present investigationbuilds upon our previous models to study for the first time theconcentration profiles of both antibody and hapten around individual tumor blood vessels in two-step approaches. Yuan etal. (10) presented guidelines for improving two-step approaches

based on a compartmental model, but the lack of spatial concentration profiles was an acknowledged limitation which hasnow been overcome. The comparison of our model results withthe experimental data of Le Doussal et al.(\) and Stickney et al.(2) offers novel insights into the role of plasma pharmacokinetics and antigen internalization in two-step therapies.

Model Development

The tissue is modeled as a Krogh cylinder (11) in which a microvesselof radius p is surrounded by tissue of radius R. A three-compartmentmodel is used to describe plasma pharmacokinetics. Diffusion-reactionequations are written to describe the extravascular concentrations as afunction of both time and radial position (Appendix B). A permeabilityboundary condition governs the transvascular flux, and there is a no-flux condition due to symmetry between surrounding Krogh cylinders.These differential equations (in both tissue and plasma) were solvedsimultaneously using a numerical finite difference method.

Parameter Values. The baseline parameters chosen for the simulations are given in Table 1. The model BFA-hapten system is based onthe work of Le Doussal et al. (1) for " 'In-labeled hapten (In-DTPA)with a BFA made up of the G7A5 Fab' against human melanoma and

the 734 clone of an anti-indium-DTPA antibody. A time delay (r) of 24h between BFA injection and hapten injection was reported to workwell for detection with good T:P ratios for hapten achievable after 3 h.Initial doses used in Ref. l were 2 Mgof BFA and 1.5 pmol of hapten,typical for detection purposes. Our choice of 2.0 x 10~8 Mfor Cp°Aisbased on this antibody dose, while the value of 1.0 x 10~7 Mfor Cp°H

is much higher and more typical of treatment. For the BFA, we have fitthe human pharmacokinetic data of Stickney et al. (2) for F(ab')2. For

the hapten we have used the pharmacokinetic data of Houston et al.(12) for free ' "In-DTPA in human patients. Cellular uptake of the

antibody (with rapid internalization of surface antigen and metabolismof the antibody) may play an important role (13, 14). The relevance of

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Table 1 Baseline model parameters

SymbolD(cm2/s)P(cm/s)A/(M-'s-')Ms-')#«(M-')Ms-')Ag°(M)Ci°

(M)«i-^«2X,

(min-')lX2(min-')fX3(min-')JP

(>ím)A(íim)r

(h)DefinitionDifTusion

coefficientPermeabilitycoefficientAssociation

rateconstantDissociationrateconstantBinding

affinityMetabolism/internalizationrateconstantAntigen

densityInitialplasmaconcentration•

Pharmacokinetic parameters for freespecies-^

= a, exp(-X,/) + «2exp(-X¡/)(-p+

(1-a,-a¡)exp(-X3í)Blood

vesselradiusIntercapillaryhalf-distanceTime

delay between BFA and hapten injections(Fab'h

BFA2.0

x10-8(18)a9.0x10-7(18)04.33

x\0<(14)2.07x 10-* -ktfIKai2.1

x109(1)0(2.17x10-5)c (seetext)1.6

x10-7(1)'2.0xI0-8(l)0.76

(2)0.244.44

x IO-3(ZCE-025/CHA-255)4.10xJO'4n.a.1010024Hapten4.3

x10-6(I9)*l.Ox10-4(20)*8.6

x107=*2r/ia22.0x IO-2(O4.3x109(1)0.0

(ma.)1*n.a.l.Ox

10-7(10)0.69(12)0.240.588

("Mn-DTPA)3.34x10~25.20x10-'(9,

18)(9,18)(1)

" Scaled for molecular weight by correlation for tumor diffusivity, D ~ (A/,)"1-14, (8, 19).* Scaled for molecular weight from sucrose, 342, to a 600 M, hapten (2) by correlation for aqueous diffusivity, Da, ~ (M,)~c Baseline value is zero, but the simulation which includes metabolism uses a value of 2.17 x 10"! s"1 (14).'' n.a., not applicable.' Le Doussal et al. (\) reported 3.4 x 10* sites/cell, taking 2.83 x 10" cells/liter (assumed in Ref. 14), A¿>= l.6 x 10~7 M.

'(19).

this process as well as the corresponding rate constant would varygreatly from system to system. To study the effect of antigen internal-ization on hapten distribution in two-step therapies, we have included asimulation with ke = 1.9 day"1 (as used in Ref. 14).

Results and Discussion

Baseline Simulations

Fig. 1 shows the solution to the distributed parameter modelfor our baseline case. The extravascular concentration is plottedas a function of time and distance from the center of the bloodvessel. The total concentrations of BFA (Fig. IA) and hapten(Fig. IB) have been plotted, inasmuch as these would be theonly relevant values for treatment or detection using radiola-beled agents. In general the profiles are steep initially, becoming more uniform with time as material diffuses away from theblood vessel. When the plasma concentration of free speciesdrops below the free extravascular concentration, material begins to be reabsorbed into the bloodstream from the tissue, andconcentration levels thereafter decrease with time. Extravascular binding helps retard this reabsorption process and maintainhigh concentration levels for a longer time period. Free haptenis present throughout the Krogh cylinder, even in regions wherelittle antibody is present, but the distribution of bound haptenmimics the BFA distribution. These results indicate a trade-offbetween high specificity and uniformity of distribution.

A number of indices were calculated to help compare different simulations. The average T:P ratio is a measure of how thesolutes localize in the tissue and is an important parameter fortumor detection. <C> is the sum of the concentrations for eachspecies in various forms, averaged over the volume of thecylinder. A related quantity, the percentage of i.d./g, may becalculated as

% Injected dose/g tissue =<C>/Cp°

plasma volume x tissue densityx 100%

For cancer treatment, the total dose received is often measuredby the AUC, or its ratio with the plasma AUC, the TI. To helpunderstand the role of binding in the two-step approach we havealso calculated the SAT, which is the total BFA-hapten complex concentration divided by the total BFA (for hapten-BFAbinding), or the total BFA concentration divided by the tumor

antigen density (for BFA-TAA binding). SR is the average totalconcentration in the tumor for a binding species divided by theconcentration of an equivalent species which does not bind atall. UNI is a measure of the spatial uniformity of the distribution which is simply the volume around the blood vessel inwhich one-half of the BFA or hapten has distributed, divided bythe volume required for a uniform distribution (i.e., one-half thetotal volume). UNI is a function of time, equal to zero when allthe material is just outside blood vessel, and equal to 1 foruniform distribution. The importance of this parameter depends on the mode of detection or treatment. Lumped or com-partmental models (such as in Ref. 10) are not capable of calculating this index. For cancer detection, high (and early) T:Pand SR are important, while successful treatment also requiressufficiently high absolute concentrations throughout the tumor.These analytical indices for all of our simulations are given inTable 2.

Sensitivity Analysis

Since there is a wide range in the choice of doses, affinities,and other parameters, we have carried out simulations to showhow the drug distributions change with different sets of parameters. A close examination of Table 2 reveals some of the keypoints of these perturbation simulations. The reported valuesare obtained 24 h after hapten injection, except for the AUCand TI which are calculated 2 weeks postinjection (as an approximation of the total dose received). For the baseline parameter values used, the T:P ratio, <C>, UNI, AUC, and TIare not significantly improved for the hapten compared to theBFA, at 24 h after hapten injection. The SR is, however, muchhigher at that time, since most of the nonbinding hapten hasbeen cleared from the bloodstream. For a system without antigen internalization, T:P and SR continue to increase with time,while the average hapten concentration in the tumor reaches apeak approximately 30 min after injection (for the baselineparameters). Many of the results of these sensitivity simulationsare the same as in our previous compartmental model of two-step approaches (10). In particular, the earlier model showedthat antibody doses greater than those needed for saturationmay reduce SR and T:P and that clearing antibody from thebloodstream, longer time delays, and proper choice of binding

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I

B

48

Fig. 1. Model baseline simulation. Extravascular concentration pronies areshown for the baseline simulation for total antibody (in all forms, free and bound)(A) as a function of time (t = -24 to +48 h, with arrow at / = 0 denoting time ofhapten injection) and distance from the center of the blood vessel (r - 10 to 100Mm). B, total hapten (t = 0 to 48 h).

parameters can lead to greater uptake and SR values. Similar toresults in Ref. 10, multiple injections of hapten (4 injectionsspaced 1 day apart, each one-fourth of total dose) led to approximately 40% increases in the AUC and TI, and a similarincrease in T:P (results not shown).

Tumor Vasculature. An advantage of this distributed modelis that the effect of the PS/V product can be analyzed by separating it into the components of vascular permeability andvascular density (or S/V, reflected by a change in the inter-capillary distance or vessel radius). It showed that the two arenot equivalent on a microscopic scale. With the present model,increasing the permeability increased all analytical indices(Table 2). A Krogh cylinder of smaller radius (i.e., high densityof blood vessels indicating high S/V) also increased concentration levels, SR, etc., and resulted in a more uniform profile (dueto a smaller distance). A simulation with 10-fold lower effectivepermeability [as might be found in the center of solid tumors(8)] shows a large decrease in the uptake and in the analyticalindices. The difference between the simulations with increasedpermeability or decreased Krogh cylinder radius shows the extent of the error introduced by compartmental models in situations of high affinity binding. In the case of high affinity bind

ing, the average extravascular concentration is much less thanthe concentration at the wall, and the compartmental modeloverestimates the uptake. This is seen in the fact that the average BFA concentrations in the high vascular density simulationare essentially 10 times the baseline, while the more heterogeneous high permeability simulation shows less than a 6-foldincrease. The difference between an increase in permeability asopposed to vascular density is greatest in a high binding affinitysituation.

Clearing BFA. Clearing the antibody from the plasma beforehapten injection results in very high T:P ratio for the antibodyand improved T:P for the hapten by decreasing the amount ofhapten trapped in the bloodstream. At the same time, there islittle change in the TI or spatial distribution of hapten, whilethe SAT and the SR actually decrease (due also to quickerelimination from the bloodstream). The combination of alonger delay with BFA clearance is more effective with IgG(results not shown).

Initial Doses. If the antibody dose is increased by a factor of10 (which may be possible only for an unlabeled antibody), theconcentration and SAT in the tumor increase, the drug distributions are more uniform, and a TI of 15 is obtained for hapten.For a 100-fold increase in Cp°A(results not shown) the increase

in BFA concentration in the tumor was not proportional to theincrease in dose, and the T:P hapten ratios decreased as haptenwas trapped in the plasma. Similar results are seen when highconcentrations of BFA and hapten are used together. When thehapten dose alone is increased by a factor of 10, the increase inits average concentration in tumor is approximately 2-fold, butthe TI and SR are sharply reduced.

Binding Kinetics. The choice of binding parameters plays animportant role in determining uptake and distribution. Itshould be noted that AUC and TI for nonbinding hapten arewithin an order of magnitude of those in the baseline bindingcase. The difference is the distribution of BFA and haptenwithin the tissue cylinder; the profiles are very heterogeneous,with one-half of all material found within 7.5% of the volumesurrounding the blood vessel, while the distribution of free hapten is uniform. Table 2 shows that increasing the antigen density also makes the spatial distribution very nonuniform. Increasing the BFA-hapten binding affinity has very similarresults in this range of parameters. However, increasing thisbinding affinity by a factor of 100 results in trapping of haptenin the plasma, with the AUC and TI decreased below baseline,decreasing the effectiveness of the two-step approach (resultsnot shown).

Time Delay between Injections and Antigen Internalization.An important parameter which can be controlled is the timedelay between BFA and hapten injection. Twenty-four h is insufficient time for the antibody to penetrate throughout thetumor tissue, and the average concentration in the tumor is stillbelow the initial plasma concentration. Since the time scale forthe BFA clearance from the bloodstream is 2-3 days (especiallyfor an IgG molecule), a T of 1 or 2 weeks is needed to significantly reduce the plasma concentration (unless the BFA iscleared artificially). Increasing T from 1 day to 2 weeks increased T:P 40-fold for both BFA and hapten, with higherconcentration levels, AUC, TI, and a more uniform distributionthroughout the Krogh cylinder. Of course there are practicalconstraints in a clinical setting including the tumor growthduring treatment. In addition, internalization of the antibody bythe cells and, to a lesser extent, antigen shedding, can severely

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PERIVASCULAR DISTRIBUTION OF BFA/HAPTEN

Table 2 Analytical indices normalized with respect to baseline"

CaseBaselineClear

BFA frombloodAg°x10CpOj

x10Cp°„x10Cp°jand Cp°„x10K„,

x10Ka2x101

wk between BFA andhaptenIgG(instead ofFfab'h)'*P*

-10PAX10Vascular

density x10Metabolism(Ac= 1.87day-')Nonbinding

BFA and haptenT:BFA1.008.27001.070.970.990.961.040.9843.80.290.115.769.090.150.19PHap1.0010.820.01.070.860.810.811.050.8339.60.160.099.548.030.120.09<i

(10BFA1.001.180.751.079.751.009.781.041.001.251.490.115.759.320.140.19O'»M)Hap1.000.300.281.1023.21.7528.01.092.870.792.430.099.808.930.120.00SATBFA1.000.070.750.119.751.019.771.051.011.261.490.115.789.330.150.00Hap1.000.250.371.002.371.692.851.002.850.611.600.711.690.960.720.00UNIBFA1.000.261.150.131.871.001.870.281.001.920.700.851.631.920.541.92Hap1.000.271.150.141.831.001.830.280.991.870.701.011.601.870.641.87SRBFA1.005.300.901.070.971.000.971.041.0040.00.460.238.2613.90.140.19Hap1.007700.781.0722.70.1727.41.052.810.772.380.099.579.300.120.00AUC*(IO"5 Mmin)BFA1.0043.40.761.079.261.009.261.051.001.321.970.115.147.560.190.02Hap1.001.011.031.9170.44.9585.21.878.891.806.070.3822.014.60.500.27TI"BFA1.004.0011.751.080.931.000.931.061.005.330.740.115.157.550.190.22Hap1.003.77f0.971.073.850.422.681.042.791.670.840.2112.237.960.280.25

" Indices shown are spatial averages, for time = 24 h post-hapten (Hap) injection (which was 24 h after BFA injection), except where otherwise noted.* AUC and TI calculated at 2 weeks after hapten injection, to more closely approximate the total doses received.r These are the absolute values of the indices for the baseline simulation."*Parameter values are baseline plus: n, = 0.46. X, = 0.0117 min-'. X2= 0.000135 min"1 (for B72.3) (21): D, J)3 = 1.39 x 10-«cmVs (22); />,,P, = 5.73 x 10~7 cm/s

(8. 18).

limit the gains received by long time delays. The effect of antibody metabolism or antigen internalization was completely detrimental to diagnostic or therapeutic goals. All analytical indices were decreased compared to the baseline and in some caseseven worse than for a nonbinding system without antigeninternalization. The results were worse as r or the metabolicrate constant, A;c,were increased.

Effect of BFA Molecular Weight. Larger BFA would notpenetrate as far away from the vessel, and they remain in theblood and tumor tissue longer. Table 2 includes a simulation fora BFA (size of a whole IgG, A/r 150,000). This gave lower T:Pand UNI values, higher tumor concentrations after 1 day, and aslightly lower TI compared to a F(ab')2 BFA. Therefore the

disadvantages of using a larger BFA molecule would be lessuniform distribution and a longer time period to reach a peakT:P ratio. Alternately, the use of a pair of sFv antibody fragments or molecular recognition units (combined molecularweights approximately 54,000 and 10,000, respectively) wouldresult in more rapid penetration in the perivascular space andsomewhat more uniform hapten profiles. The T:P ratios wereincreased for smaller BFA, with lower concentrations in thetumor and only marginal increases in the TI (results notshown).

Comparison with Experimental Data

The baseline simulations and parameter perturbations haveconsidered the two-step approach in a human system. To compare the model with experimental data we have carried outsimulations for two murine systems. In the first we use parameters reported by Le Doussal et al. (1) (initial hapten plasmaconcentration = 1.0 x 10~9 M; BFA pharmacokinetic parameters obtained by fit to data: a, = 0.74; X, = 0.0184 min"1; X2=0.00127 min"1). No plasma data were given for free hapten;

therefore, total hapten data were used in conjunction with ourcompartmental model for pharmacokinetics (Appendix B) toobtain the parameters: a, = 0.98; X, =4.0 min"1; a2 = 0.0184;X2= 0.030 min"1; and X, = 0.0004 min"1. All other parameters

used were the same as in Table 1.Fig. 1A shows the results of the compartmental model for

plasma concentrations using the pharmacokinetic rate constants of Ref. 12. The hapten shows similar decay rates whether

injected alone or with BFA, and there is poor agreement withthe murine data of Ref. l. In order to obtain proper haptenconcentrations in the plasma (which is the driving force forthe distributed parameter model) it is necessary to assume atriexponential plasma decay with a large and rapid initial drop(a, = 0.98; X, = 4.0 min"1). In addition, a slower decay rate at

long time periods (days) is also needed to explain the plasmadata. Using the above parameters to match the plasma concentrations, our model results are in good agreement with the datafor the T:P and percentage of i.d./g from Le Doussal et al. (1).(Fig. 2, B and C). The effects of metabolism (ke) on the modelresults are also shown in Fig. 2.

We have also compared our model to the experimental dataof Stickney et al. (2). These investigators coinject 3 Mgof BFAas a carrier along with the hapten (8 pmol inIn-EOTUBE) 24

h after the original BFA injection, to minimize the very rapidinitial clearance phase of free hapten. No plasma concentrationdata were available for free hapten or BFA; therefore the pharmacokinetic rate constants given above for Le Doussal's data

were used again. The presence of the carrier leads to a strongdependence of the hapten plasma concentration on the BFApharmacokinetics, since the BFA and hapten are in equilibriumbefore the injection is made. The T:P ratios obtained with thissimulation are compared with experimental data in Fig. ID.Since these data were collected over a longer, 5-day period,metabolism was incorporated as an adjustable parameter whichserved to decrease concentration values at later times. Fig. 2Dshows the T:P data of Stickney et al. (2) with our model resultsfor no metabolism and for metabolism rate constants near0.001 min"'. The T:P at day 5 is over 5000 without metabolismbut is reduced to 5 with ke = 0.001 min"1.

Role of Plasma Pharmacokinetics. Using the baseline valuesfor pharmacokinetic parameters, the plasma concentrations approach equilibrium values very quickly, within 1 min after injection. Yet the plasma concentration data of Le Doussal et al.(1) could not be explained by equilibrium kinetics, which predicted much higher plasma concentrations than those seen experimentally. Within the first minute the free hapten took afinite time to mix within the plasma and to associate with theBFA. This allowed the free hapten concentration to drop beforereaching steady state. This effect is most pronounced for

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10°B

4.

3.

2-

I.

10-5

48

24

20.

16 H

12.

8.

Ko

24

20

16.

12.

8.

12 24

Time (hr)

36 48

12 24 36

Time (hr)

48

24 48 72 96

Time (hr)

120

Fig. 2. Comparison of model to murine data of Le Doussal et al. ( l ) and Stickney el al. (2). A, plasma pharmacokinetics ( I ); B, percentage of i.d./g ( I ); C, T:P ratio( I ); and D, T:P ratio using BFA as a carrier at the time of hapten injection (2). The plasma concentrations show the two-step hapten data of Le Doussal et al. (1), withmodel results for hapten injected alone and total hapten when injected with BFA, for two sets of parameters. The first set was chosen to give a good fit between thecompartmental model and the data for hapten concentration in the presence of BFA. The second set of parameters are the rate constants of free hapten in humanpatients (12). Note that the latter concentrations are much too high at early time points, while dropping far too rapidly at later times. The metabolism rate constantis varied in B. C, and D to compare the model to the data and to show the effect of antigen internalization. Note that metabolism is more important at later times.

smaller binding affinities (Ka2) and slow pharmacokinetic rateconstants. Using a biexponential function (as reported in Ref.1) instead of a triexponential led to a very poor fit between theplasma model and the data and yielded exaggerated values forthe percentage of i.d./g in tumor at all time periods. It is notsurprising that this initial clearance phase could be important.The time scale for BFA-hapten association is approximatelyl s (=ln 2/C, ,/c2f)-The first-phase time scale was 1.2 min (=ln2/X|) in humans (12), but the physiological time scale for clearances in mice is approximately one-eighth that in humans [bodyweight"0-25 (15)], which would be 9 s. The plasma mixing time

scale has been estimated to be 12 s (16) in rabbits and would besmaller in mice. Initial loss of hapten from the bloodstreambefore complete mixing or binding is consistent with observations that hapten injected without a carrier is very rapidly eliminated from the plasma.1 Such an early phase effect would make

the percentage of i.d./g dependent on processes occurring in thefirst min after injection. Fortunately, in a human system the

3 D. Mackensen, personal communication.

binding is just as rapid while clearance times are longer, reducing the magnitude of this effect. Fig. 2A also suggests the existence of a physiological compartment with a very slow decayrate, only evident at later times. The plasma concentrations arevery small after 24 h, and their lack of relevance in single-stepapproaches with haptens might explain why long-term, slow-decay rates have not been reported. However, the presenceof BFA for binding and the longer persistence times may lead toa significant contribution to the AUC, which should not beneglected.

Implications and Limitations

It should be emphasized that the difference between effectivedetection and treatment is not simply an increase in dose.Rather, the whole set of physiological, kinetic, and schedulingparameters should be optimized for the desired use of the two-step procedure. If the primary goal is detection the advantagesof the two-step approach are the greater T:P and SR obtained

compared to that for BFA or hapten alone. In addition these5842

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PERIVASCULAR DISTRIBUTION OF BFA/HAPTEN

peak ratios occur much more rapidly with the hapten than withantibody alone. Nonuniform distributions are not a problem inthis case, and the highest possible BFA-TAA binding affinities

and antigen densities should be sought. For effective treatment,where absolute concentration levels and the time-averageddoses are important and a more uniform distribution is desired,different parameter values must be chosen, with higher doses,somewhat lower BFA-TAA affinities, and increased Tand P. Inboth cases the BFA-hapten binding affinity must be chosen withregard to plasma trapping and the need for uniformity on alocal scale. Targeting of antigens which have a low internaliza-

tion rate and the development of antibodies which have a weakimmunogenic response and slow metabolism are important areas of research. Knowledge that a specific BFA will not beinternalized much over a long time period would allow for theenhanced hapten localization possible with a time delay greaterthan the typical 1-2 days.

The advantage of this two-step method versus that of antibody alone appears to be marginal in two respects: (a) theperivascular penetration is hindered because of rapid bindingwith the heterogeneously distributed BFA; (b) the TI for haptenin a two-step treatment is not significantly better than for BFAalone (if an equivalent dose of radiolabeled BFA could be given). Currently the dose of radiolabeled antibody alone is limitedby toxicity in normal tissues. A major advantage of the two-stepapproach is the higher dose of (unlabeled) BFA which is possible and which may allow greater tumor uptake and penetration. Other definitive advantages of the BFA-hapten approachare as follows: (a) the SR are much greater for hapten than forBFA; (b) peak T:P ratios occur very quickly; (c) the haptenclears from the bloodstream much more quickly, which may beuseful for many radioactive agents; (d) hapten is expected topenetrate more effectively into necrotic or avascular regions;and (<')repeated or continuous hapten injections may be given,

as an immune response is less likely for low molecular weightagents.

The following limitations remain in the present model: (a)there is a need for a whole body pharmacokinetic model fortwo-step approaches to allow for scaling between species, toincorporate sensitive organs of the body, and to provide neededinformation on early times; and (b) the present model does notaddress issues of heterogeneities within the tumor on a macroscopic scale (8); the effectiveness of the BFA/hapten systemmust be addressed in the light of necrosis, variable blood supply, and elevated interstitial pressure (as suggested by the sensitivity of the present results to the vascular permeability coefficient and vascular density).

Acknowledgments

We thank D. Mackensen and J. M. Le Doussal for their help.

Appendix A: Nomenclature

^. = Extravascular concentration, mols/liter, where i = 1, 2, 3for the mobile species free BFA, free hapten, BFA-haptencomplex, and /' = 4, 5, 6 for the immobile species, TAA-

BFA complex, TAA-BFA-hapten complex, and the available TAA bindings sites.

Ag° = Af, (t = 0) = Total (initial) concentration of binding sites

in the extravascular spaceCjj = Concentration of the free species above (i = I, 2, 3),

mols/liter, where./ = 1, 2, 3 represents the plasma and thewell perfused and poorly perfused peripheral compart

D,

k,

k\f, k2r

k\„k2f

Kai, Ka2

P¡

r

R

S/V

t

a,, X,

PT

ments, respectively. Cp°^,Cp°/,are synonymous for initial concentrations Cn°and C2i°

= Interstitial diffusion coefficient for the mobile species(i= 1,2, 3), cm2/s

= Elimination (metabolism) rate constant for species boundto TAA, (A4 and A5), s~'

= Forward binding rate constant for BFA-TAA and BFA-hapten, respectively, M~'S

= Reverse (dissociation) rate constant for BFA-TAA andBFA-hapten, s~'

= Binding affinity for BFA-TAA (=*,//* lr) and BFA-hapten (= ky/k2r), respectively.

= Effective vascular permeability coefficient for the mobilespecies (/ = 1, 2, 3), cm/s

= Radial position (distance from center of blood vessel), Mm

= one-half of mean intercapillary distance (also radius of

Krogh cylinder), urn= Surface area per unit volume for transcapillary exchange,

cm-'

= Time, min

= Pharmacokinetic constants for free species in plasma(i= 1,2, 3) (\, in min-')

= Blood vessel radius, ^m

= Time interval between the BFA and hapten injections, h.

Appendix B: Model Equations

Five extravascular species are considered in the model. We assumethat transport of BFA and its binding to the TAA is not changed whencomplexed with hapten (a simplifying but not fundamental assumption). The equations for unsteady-state diffusion and reaction for theextravascular species are:

- klfAtAt + k,,A4 - kvAtA2 + k2rA3

a = £>2V142- kvA2(A, +A4) + k2r(A,

kvA,A2-

"aT"

-

~~ *lr^4 ~~ KlfAjAt "*" *2r-^5 ~ * eA t

- klrA,, + k2fA,A4 - k2,As- kfAf

(A)

(B)

(C)

(D)

(E)

where V2 is the Laplacian operator in cylindrical coordinates. Metabolism (antigen internalization) is treated as a first-order process whicheliminates BFA and BFA-hapten bound to the TAA (antigen, however,is renewed immediately). In addition the number of free binding sites onthe tumor cells, A,,, equals A6°—¿�At —¿�A*,.

There is a no-flux boundary condition for mobile species at thesurface of the cylinder (r = R) because of symmetry with surroundingidentical Krogh cylinders:

U« = 0 0=1,2,3) (F)

The boundary condition at the blood vessel wall relates the solute fluxto the concentration gradient across the blood vessel wall and transportcoefficients:

-Drsr U. =nc,i-A, U) (/ = i,2,3) (G)

5843Unlike our previous study (10), we do not assume equilibrium binding

on July 29, 2020. © 1992 American Association for Cancer Research. cancerres.aacrjournals.org Downloaded from

PERIVASrULAR DISTRIBUTION OF BFA/HAPTEN

between hapten and BFA. In the present study a three-compartmentmodel is used for hapten (and two for BFA):

dC,

(i =1,2, 3)

' = 2, 3)

(H)

(I)

where <t>¡= 1 for i = l, 2, and <t>¡= -1 for /' = 3;y represents the three

compartments, with kjk.irepresenting the transfer coefficient of speciesi from compartment y to k, and klu the rate constant for eliminationfrom the plasma (e.g., urine). These compartmental transfer coefficientsmay be related to experimental data for the plasma concentration (C,/)of free species (17).

¿7—¿� : «,exp(—X,/)+ a2exp(—X2/)

+ (1 -a, - a2)exp(-X3f)(J)

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5844

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1992;52:5838-5844. Cancer Res   Laurence T. Baxter, Fan Yuan and Rakesh K. Jain  Experimental DataBifunctional Antibodies and Haptens: Comparison with Pharmacokinetic Analysis of the Perivascular Distribution of

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