Development of a PBPK Model for Monoclonal Antibodies and Simulation of Human and Mice PBPK of a...

988 Current Pharmaceutical Design, 2009, 15, 988-1007

1381-6128/09 $55.00+.00 © 2009 Bentham Science Publishers Ltd.

Development of a PBPK Model for Monoclonal Antibodies and Simulation of Human and Mice PBPK of a Radiolabelled Monoclonal Antibody

Tomi Heiskanen1,*, Tomas Heiskanen

1 and Kalevi Kairemo

2

1Xemet Co. Ltd., Tikasniitynkuja 7, 02200, Espoo, Finland and

2Department of Oncology, Helsinki University Central

Hospital, 00029 HUS, Finland

Abstract: Physiology based pharmacokinetic (PBPK) modeling and simulation is a useful method for prediction of bio-

distribution of both macromolecules and small molecules. It can enhance our understanding of the underlying mechanisms

of biodistribution and hence may help in rational design of macromolecules used as diagnostic and therapeutic agents.

In this review we discuss PBPK modeling and simulation of a radiolabelled Monoclonal Antibody (111

In-DOTA-hAFP31

IgG) (“MAB”) in mice without tumor and in a human with tumor. This study is part of Xemet Co.’s effort to develop a

more accurate and reliable PBPK model and simulation platform, which is applicable both for small molecules and mac-

romolecules.

The simulated results were fitted to experimental time series data by varying parameters which were not fixed a priori. It

was demonstrated that the PBPK model describes the main features of the pharmacokinetics of the studied systems. It was

also shown that simulation can be used for evaluating the parameters of the system and scaling up the pharmacokinetics of

MAB from mice to man.

We identified several areas of improvement and further development needed to improve the accuracy of PBPK simulation

for MAB and other macromolecules.

It was concluded that the transvascular permeabilities are the most important parameters and more research is needed to

enable prediction of permeabilities from molecular characteristics of macromolecules. It would also be necessary to un-

derstand better and describe with a more detailed model the microstructure of the tumor and to measure or predict the an-

tigen concentration in tumor. Non-specific, non-saturable binding in other organs/tissues should be understood better and

the kinetic constants of the binding should be measured experimentally.

Although the metabolism and clearance were neglected in this study they need to be included in more detailed studies.

Also the intracellular trafficking of macromolecules, which was not included in this study, shall be included in the more

accurate models.

INTRODUCTION

Physiology based pharmacokinetic (PBPK) models de-scribe mathematically the mass transfer and reaction phe-nomena, also called ADME (absorption, distribution, me-tabolism and excretion), of xenobiotics in the body. Organs and tissues are described with idealized well mixed com-partments and the mass balances are formulated for each compartment. This leads to a system of simultaneous ordi-nary differential equations, which can be solved numerically with a computer. The solutions are typically concentrations in each compartment as a function of time. The biodistribu-tion for different systems can be simulated by solving the model equations at different values of parameters. The val-ues for the parameters can either be predicted with empirical correlations, or by fitting the simulated curves to experimen-tal time series data. Typically a combination of both ap-proaches is needed.

PBPK models have been applied mainly to small mole-cule drugs and chemicals, and to lesser extent to macromole-cules. Reddy et al. [1] have reviewed the applications of

*Address correspondence to this author at the Xemet Co. Ltd., Tikasni-

itynkuja 7, 02200, Espoo, Finland; E-mail: [email protected]

PBPK related to volatile organic compounds, environmental pollutants and pharmaceuticals. Baxter et al. [2,3] developed PBPK models for monoclonal antibodies and simulated the biodistribution in mice and man and compared the simulated results with experimental data.

We have used a more detailed PBPK model developed by Xemet Co.. Also this model is based on several simplifying assumptions which may need to be revised in further devel-opment. One of the purposes of this study was to identify the areas for improvement regarding the prediction of the pa-rameters and the refining of the model structure.

We applied PBPK simulation to the prediction of the pharmacokinetics of a radiolabelled Monoclonal Antibody (

111In-DOTA-hAFP31 IgG) (“MAB”) in mice without tumor

and a human with tumor.

An increased clinical utility of radiolabelled monoclonal antibodies (MoAb), recognizing a variety of different anti-gens expressed preferentially in malignant tissue, for localiz-ing primary, metastatic and recurrent cancer has been docu-mented in many investigations. At this time clinically vali-dated radioimmunodetection methods are available for colo-rectal, ovarian, breast, lung, hepatocellular, thyroid medu- llary, and head and neck carcinoma, and melanoma. Recent

Development of a PBPK Model for Monoclonal Antibodies Current Pharmaceutical Design, 2009, Vol. 15, No. 9 989

advances in humanization of MoAb significantly improved the prospects of effective antibody-guided radiotherapy in hematologic malignancies and lymphoma whereas limita-tions in the treatment of solid tumours occur [4,5].

Our purpose was also to study also the applicability of the simulation for interspecies extrapolation of pharmacoki-netics of MAB.

Additionally we simulated the biodistribution of MAB in a tumor. The results for the tumor are purely hypothetical because we did not have experimental data for MAB concen-tration in the tumor.

Accurate PBPK simulation would be a useful tool for studying the effects of different bio- and physicochemical parameters of MAB and other biological macromolecules on the biodistribution thus increasing the understanding of the mechanisms of biodistribution and enhancing rational design of macromolecular diagnostic and therapeutic agents.

This study is the first step in our development of a more accurate full scale PBPK model for macromolecules and some guidelines and recommendations for further develop-ment will be presented.

MATERIALS AND METHODS

Antibody Conjugate

The humanized 19G form was obtained by genetic engi-neering techniques, replacing more than 95% of the original murine antibody with anequivalent human sequence of amino acids. hAFP31 IgG was conjugated with the 1,4,7,10-tetraazacyclododecane-1,4,7,10-tetraacetic acid (DOTA) chelate (supplied by Macrocyclics, Dallas, TX), for radiola-beling with either In-111 radioisotope. The antibody DOTA conjugate was prepared using methods described in the lit-erature. The conjugation was carried out to yield a substitu-tion of approximately three to six DOTA chelates per anti-body molecule. Subsequent ELISA analyses demonstrated that hAFP3l IgG-DOTA showed no loss in immunoreactivity at those levels of substitution.

hAFP31 IgG is directed specifically against the AFP an-tigen [4,5], and was used to target AFP- producing tumors, with a high binding affinity Kd of 10 nM and an isoelectric point of ~9.5.

Preparation of the Radiolabelled DOTA-hAFP31 IgG

The radiolabelled DOTA-hAFP3l IgG study drug was prepared at the Radiopharmacy laboratory at the Hospital (HUCH) using DOTA-conjugated hAFP3l IgG supplied by Immunomedics, Inc. (Princeton, NJ) and radiopharmaceuti-cal/radiochemical (sterile, pyrogen-free) grade In-111 iso-tope purchased from CIS Bio international (Gif-sur-Yvette, France).

The radionuclide (mCi): antibody (mg) ratio was 2: 1 for In-111 during the radiolabeling procedure. The radionuclide was buffered with 2.5M ammonium acetate that equals 10% of the volume of radionuclide before addition of the DOTA- IgG conjugate. The radionuclide and DOTA-IgG conjugate in 0.25 M ammonium acetate buffer, pH 5.5 was preheated in separate vials for a10-minute incubation at 45° C. The appropriate calculated volume of the pre-heated DOTA-IgG

conjugate was removed aseptically from the vials and added to the preheated, prebuffered, sterile, pyrogen-free isotope vial for a15-minutes incubation at 45° C. To scavenge non--antibody bound radiometal, 0.1 M DTPA was then added to the radiolabelled antibody conjugate to equal 10% of the cumulative vial volume for an additional 5 minute incubation at room temperature.

The radiolabelled drug was then transferred aseptically to a dispensing container, reading a dose calibrator.). Quality control of the radiolabelled In-111 -DOTA-IgG conjugate was performed by instant thin layer chromatography (ITLC). Once prepared, the radiolabelled DOTA-hAFP3l IgG study drug was used within 2 hours.

In Vivo Distribution of the Radiolabelled DOTA-hAFP31 IgG in Animals

The in vivo distribution of the 90

Y-DOTA-hAFP31 IgG was studied in BALB/C mice. Thirty-five mice were injected with a mixture of

111In- and

88Y-DOTA-hAFP31 IgG, with

groups of five animals sacrificed at each of the following time points: 4, 24, 48, 72, 96, 168 and 240 hours post-injection.

The purpose of the experiment was two-fold: 1) to estab-lish the tissue biodistribution of both

111In- and

88Y-labelled

DOTA-hAFP31 IgG in normal animals, and 2) to determine whether

111In-labelled DOTA-hAFP31 IgG accurately pre-

dicted the biodistribution of 88

Y-DOTA-hAFP31 IgG.

Both compounds were injected in each animal, as 111

In and

88Y have substantially different decay energies in the

gamma spectrometer and, therefore, each could be assessed concurrently using appropriately set energy windows.

In this experiment, a 4.0 mL of injectable solution was prepared by adding 40 μCi of

88Yconjugate and 600 μCi of

111In-conjugate to 50 mM sodium acetate-150 mM sodium

chloride, pH 6.5, containing 1 % HSA. This resulted in an injectate volume of 100 microliters per animal, containing approximately 1 μCi of

88Y-DOTA-hAFP31 IgG, 15 μCi of

111In-DOTA-hAFP31 IgG and a protein dose of approxi-

mately 15-20 μg per animal. All labelled materials were tested for immunoreactivity retention by mixing an aliquot with a 20-fold molar excess of AFP antigen, heating for 30 minutes and analyzing by size-exclusion HPLC with radio-matic detection.

For each animal sacrificed, the tissues harvested for counting included liver, spleen, lungs, kidney, stomach, small intestine, large intestine and bone, in addition to blood. Entire femurs were taken for the quantisation of activity in whole bone while the contralateral femurs from each animal were also taken and washed of marrow with a saline solu-tion. Such ‘washed bone’ samples were then also counted alongside the other tissues to quantitate the nuclide uptake in cortical bone. During tissue activity counting, corrections to raw counts were made due to limited numbers of nuclide physical decays (‘cross-over’) of one nuclide within the other nuclide’s energy window.

Imaging

The early clearance (0 - 168 hours) of the activity was monitored by whole body planar images in the anterior pos-

990 Current Pharmaceutical Design, 2009, Vol. 15, No. 9 Heiskanen et al.

terior (AP) and posterior anterior directions (PA) direction acquired with a Toshiba GCA 7200A/UI gamma camera (Tochigi-ken, Japan). Whole body scans of the patients were also acquired from AP and PA directions with 7.5 -15 cm/min scan speed. The gamma energies of 177 keV and 245 keV of In-111 were registered using 20% gamma energy windows. Additionally a whole body transmission scan was performed before any injections using a Cobalt-57 flood source.

Cumulative Activity

The cumulative activity for the abdominal area was de-termined from the whole body scans for the In-111 therapy. The counts in these images are normalized by counts from a standard In-111 source. The counts from whole body AP and PA scans are combined using a geometric mean. We used the activity curves for liver, whole body and the abdominal area to determine the cumulative activity in the liver (tumor) and the abdominal area of the SPECT study.

We assigned the infused activity as the total activity for the whole body at 1hr. The physical decay constant of the radioisotope was used to calculate the cumulative activity in the patient abdominal area, liver and liver tumor region with the biological clearance data. The method includes the vary-ing biological clearance between normal liver tissue, liver tumor and rest of the abdominal area.

Patient History

A three and a half year old girl was diagnosed with hepa-toblastoma and treated with chemotherapy according to In-ternational Society of Pediatric Oncology protocol (SIOP PLADO; adriamycin – cisplatinum) for four cycles. The ini-tial serum AFP was 1,452,000 ng/l. Given that radical sur-gery was not considered to be possible, the patient was scheduled for liver Tx, which however could not be per-formed due to a single metastasis in the radix mesentery found at laparotomy. Therefore systemic chemotherapy was continued with a good response. At re-evaluation at age of four years, MRI and Tc-99m-label based imaging utilizing monoclonal anti-AFP antibody Fab’- fragments revealed active tumor tissue in the right liver lobe and near the lobe intersection, but no extrahepatic metastases. In two consecu-tive operations partial resection of the liver was performed and chemotherapy was continued. However, serum AFP started to rise after the initial postoperative decline and sub-sequently, MRI and the second anti-AFP imaging study re-vealed active tumor in the remaining liver, but no extrahe-patic metastases. Therefore, the patient was considered eligi-ble for LTx. An orthotopic liver Tx was performed at age 4 yr 6 mo and several biopsies from extrahepatic nodes re-vealed no tumor involvement. No postoperative chemother-apy was given. After LTx, serum AFP declined with half life of 4 to 5 days, reaching normal levels five weeks after LTx. It remained normal for 46 months when it slowly started to increase. Recurrent hepatoblastoma tumor in mesenterium was detected with FDG-positron emission tomography (PET) confirmed by excision of the tumor 52 months after the transplantation; at this time serum AFP was 7250 ng/l. DNA studies (AmpF/STR SGM Plus, AmpF/STR Profiler, Applied Biosystems, Foster City, CA) revealed that the tu-

mor originated from the patient and not from the donor liver. Chemotherapy was reinitiated. Despite of the treatments two tumor recurrences were subsequently found in the liver when FDG-PET was repeated. These were successfully treated with thermoablation. However, patient developed ileocuta-neous fistula. Peroral chemotherapy with etoposide and pa-clitaxel was continued. The decision of RIT treatment was made four months after latest surgery when serum AFP lev-els started to elevate again reaching 20,000 -40,000 ng/l at the institution of RIT. Despite RIT tumor mass continued to grow and 16 weeks after the RIT the patient died.

EXPERIMENTAL DATA

The animal in vivo biodistribution data is presented in Table D-1. The results of ROI analysis of the patient are pre-sented in Table D-2. “Fantom” refers to a cylindrical stan-dard radioactivity source filled with In-111 at the time of infusion and its volume was 300 ml.

MATHEMATICAL MODEL

In this study we used Xemet Co.’s PBPK simulation plat-form, which is based on a mathematical model of the sys-temic circulation and lymph flow of the system depicted in Fig. (A-1). Each organ and tissue is divided into three well mixed compartments of the arterial side and venous side of the vascular space and the interstitial space and shown in Fig. (A-2). A tumor in the liver is presented as a separate tissue divided into two compartments: vascular space and interstitial space, where vascular circulation is integrated with arterial side of liver’s blood circulation. This is a gross idealization, but will serve as the first approximation.

In each organ/tissue there is an additional phase within the interstitial space for binding sites: specific, saturable binding for tumor and non-specific, non-saturable binding for all the other tissues/organs.

Intracellular space is not included in this particular appli-cation since it is assumed that MAB will not enter the intra-cellular space.

So far other PBPK models for MAB have used two com-partments for each tissues/organ, one for venous side and one interstitial space [2,3]. Even though such models have a good predictive power, Xemet’s PBPK model can take into account the effect of re-absorption from interstitial space to venous blood in case the compound of interest has been de-pleted from the blood due to metabolism and/or excretion.

The previous models developed by Baxter et al. [3] de-scribe the transvascular flow and diffusion with the two pore model originally developed by Rippe and Haraldson [6].

Since such model has parameters, which are difficult or impossible to measure experimentally and there are some problems regarding the theory, we have replaced it with the following simple model which takes into account both transvascular diffusion and convection:

Transvascular transport from arterial side to interstitial space =

( ' )bxj oj xjP C C , when 'oj xjC C (1)


( ' ) 0bxj oj xjP C C = , when 'oj xjC C

Transvascular transport from interstitial space to venous

side =

( )xbj xj ojP C C , when xj ojC C (2)

( ) 0xbj xj ojP C C = , when xj ojC C

where bxjP and xbjP represent permeabilities times surface

area for transfer between vascular and interstitial space of the

tissue/organ j; 'ojC and ojC are concentrations of MAB in the

arterial side and venous side of vascular space and xjC is the

concentration in the interstitial space of the tissue/organ j.

Equation (2) would be taken into account only if there would

be re-absorption of MAB from interstitial space to vascular

space after the concentration of MAB in blood would have

been decreased below the concentration of MAB in intersti-

tial space of the organs/tissues due to metabolism and/or

excretion in other organs/tissues.

The general method of mathematical modeling of PBPK is to formulate dynamic mass balances for the individual component(s) in question, in this case for MAB, for each well mixed compartment of each organ/tissue. The mathe-matical description of organs/tissues with well mixed com-partments is a sufficiently good approximation for PBPK modeling although it is still far from a realistic description of the actual phenomena. More realistic description of the mass transfer phenomena would include modeling of distributed systems with partial differential equations but the numerical solutions would be more complicated. However, any distrib-uted system can be approximated with sufficient number of well mixed sub compartments, which leads to greatly simpli-fied mathematical treatment.

The dynamic mass balances for each sub compartment of each tissue are shown in general form in Appendix C. In the present case we have included the following organs/tissues: lung, heart muscle, muscles, brain, fat, bone, bone marrow, skin, kidney, spleen, pancreas and liver.

Additionally dynamic mass balances for arterial blood and venous blood are included. We have assumed that there is no accumulation of MAB in the lymph, and can approxi-mate the dynamic mass balance for lymph with stationary state mass balance as show in equation C-11 of Appendix C.

The non-specific, non-saturable binding of MAB in all organs/tissues except in tumor is described by first order kinetics in Equations C-3 and C-4, and specific/saturable binding to antigens in tumor (in case of the human) with second order kinetics in Equations C-6 and C-7.

The whole model consists of 65 simultaneous ordinary differential equations (ODE).

This system of ODEs is linear, if the metabolism in liver is approximated by a linear model, where rate of metabolism is first order with regard to concentration of MAB in liver and there is no tumor and hence no specific binding (in case of the mouse).

The specific binding in tumor is modeled with the non-linear equations C-6 and C-7 of Appendix C, thus making two of the ODEs non-linear. Both linear and non-linear sys-tems of ODEs can be easily solved with numerical methods. In this study we have used MATLAB’s “ode23” method.

MODEL PARAMETERS

The model includes both physiological parameters and kinetic parameters and their combinations. The physiological parameters are the volumes of each organ/tissue and their sub compartments, the volume flow rates of blood into and out of each organ/tissue and their sub compartments and volume flow rates of lymph from interstitial space of the organs/tissues (except brain and heart muscle). The kinetic parameters are the rate constants for binding and detachment of MAB for non-specific binding sites in all organs/tissues and for specific binding sites in tumor. Other parameters are the total concentration of antigen in tumor, the equilibrium ratios of MAB concentrations between the lymph and inter-stitial space and the permeabilities times surface area for transfer of MAB from arteries to the interstitial space and from the interstitial space to venous blood.

The physiological parameters for the mouse were ob-tained from Baxter et al. [2], and for the 15 kg human by scaling from the standard values for a 70 kg human by using the following scaling factors (exponents) as suggested in Truskey et al. [7]: 1.0 for blood, lung, heart, muscle and skin; 0.86 for kidney, liver, GI and spleen.

The simulations results were fitted to experimental re-

sults by varying the kinetic constants (N

fjk ,N

rjk ,S

ftk ,S

rtk ),

permeabilities times surface areas ( bxjP ) and the equilibrium

ratios for MAB concentration between lymph and the inter-

stitial space ( jZ ) for each organ/tissue.

The parameters used in simulations for the mouse are presented in Table E-1 (Appendix E) and the parameters for the human are presented in Table E-2.

SIMULATION RESULTS

Mouse

The simulated distribution of MAB versus experimental results for the mouse are shown in Fig. (1). The fitting of the curves to experimental results were done by varying the fol-lowing parameters: kinetic constants for non-specific bind-ing, permeability times surface area for transfer between vascular/arterial side and interstitial space, bxjP .

Logically, increasing the kinetic constants of non-specific

binding, N

fjk , or increasing the permeability, bxjP , increases

the level of %i.d./g in the respective organ/tissue, whereas

increasing the constant for detachment from non-specific

binding sites, N

rjk , or decreasing the permeability, bxjP , de-

creases the level of %i.d./g.

If these would be the only effects, it would be difficult to separate the effects of kinetic constants and permeabilities.


However, increasing the permeability also increases the ini-tial peak and shortens the response time, thus making it pos-sible to determine separately the kinetic constants and the permeabilities for each organ/tissue.

HUMAN

The simulated distribution of MAB versus experimental results for a 15 kg human (female) with a 100 g tumor in liver are shown in Fig. (2). It was assumed that 20% of the liver’s blood flow is circulated through the tumor.

The fitting of the curves to experimental results were

done by varying the following parameters: kinetic constants

for non-specific binding, permeability times surface area for

transfer between vascular/arterial side and interstitial space,

bxjP .

The effects of changing the kinetic constants and perme-

ability, bxjP , were naturally the same as for mice: increasing

the forward binding constants for non-specific binding, N

fjk ,

raises the level of %i.d./g and increasing the reverse binding

constants, N

rjk , decrease %i.d./g. The response time is sensi-

tive to the permeability times surface area for transfer be-

tween vascular/arterial side and interstitial space, bxjP . In-

creasing the permeability, bxjP , shortens the response time

meaning that MAB enters more rapidly the interstitial space.

We can conclude that the short term responses to i.v. injec-

tion could be used to estimate the values for transvascular

permeabilities, which is very difficult by other means.

The steady states are reached within 3-4 days, after which there is little change in the biodistribution.

In case of the human the experimental results for activity measurements were transformed into %i.d./kg by two meth-ods: an approximate method (A) where the average activity of each tissue/organ at each point of time was divided by the average activity of the whole body at the same point of time, and a more accurate method (B) where the activities in each organ was corrected by decay of

111In. We can conclude that

the approximate method A can be used if the excretion of MAB is neglible.

The simulated values for a 100 g tumor associated with liver are shown in Fig. (3). Since we do not have any ex-perimental values for the concentration of MAB in tumor, these are purely hypothetical results. However, they suggest that the concentration of MAB in tumor is very sensitive to the concentration of the antigen in tumor. Thus the optimal dosing would be highly dependent on the antigen concentra-tion in tumor, and hence the antigen concentration should be estimated before deciding the amount of dosing.

In order to make a realistic model for tumor detailed in-formation the internal structure and vasculature of tumor is needed as discussed by Jain [8]. However, each tumor has different structure and vasculature and both change over time, so modeling of the tumor shall be done for each case separately.

CONCLUSIONS

We have reviewed here the application of Xemet Co.’s PBPK model and simulation platform for the prediction of the biodistribution of a radiolabelled Monoclonal Antibody (

111In-DOTA-hAFP31 IgG) (“MAB”) in mice without tumor

and a human with tumor.

One purpose of this study was identification of the areas for further development of a more accurate and realistic PBPK model, which can be used for interspecies simulation of biodistribution of MAB and other macromolecules.

Fig. (1). Simulated biodistribution (continuous curves) of

111In-DOTA-hAFP31 IgG in a mouse compared with experimental results (circles);

%i.d./g as a function of time for liver, kidney, blood, bone, spleen lungs and stomach.


Such model and simulation can enhance our understand-ing of the underlying mechanisms and enable more rational design of the macromolecules for diagnostic and therapeutic purposes.

It was demonstrated that the simulated results can be fit-ted to the experimental results by varying the kinetic parame-ters for non-specific binding and the permeabilities for transvascular transport in each organ/tissue.

Furthermore it was shown that it is possible to separate the effects of the kinetic parameters from the effects of the permeabilities. Thus the simulation would enable the acqui-sition of the experimental values of these parameters from time series data for biodistribution.

This study clearly demonstrates that more information would be needed about the non-specific binding in all or-gans/tissues and about specific binding in tumor. The infor-

Fig. (2). Simulated and experimental biodistribution of 111In-DOTA-hAFP31 IgG in a human. Simulated results for liver, tumor and spleen

(continuous curves), and experimental results (A: squares, B: circles) for liver and spleen; experimental results were obtained with two meth-ods A and B as explained in the text.

Fig. (3). Simulated accumulation of MAB in tumor for different values of concentration of antigen in tumor AGM (mg/l).


mation about the concentration of antigens in tumor was shown to be critical for determining the dosing of MAB.

Although not proven by this study, we believe that a sim-ple presentation for transvascular transport, where the only parameter is permeability times surface area, would be suffi-cient and more justified than a complex formulation based on a two pore system, which has been used in some other PBPK models for macromolecules.

The two pore theory has variables and parameters which are difficult if not impossible to measure experimentally and the theory is based on some questionable assumptions.

In any case the parameters of transvascular transport are critically important and better understanding of the transport phenomena is still needed.

We have made several other simplifying assumptions in the model and simulation: we have assumed that metabolism and excretion are neglible and that there is no intracellular trafficking of the macromolecules. Our assumption of non-specific binding in all organs/tissues is probably unrealistic and should be checked against physiological facts and stud-ied experimentally in the further studies.

Xemet’s PBPK model is based on a three compartment approximation for each organ/tissue, except for tumor which is approximated with two compartments.

In the further developments a more detailed structure and model should be established for the tumor. However, the structure would depend on the type and location of the tumor and stage of the development. Therefore the model should be made for each case separately.

Also intracellular trafficking of macromolecules, metabo-lism and excretion shall be included in more accurate mod-els.

ACKNOWLEDGMENTS

The PBPK model has been developed by Xemet and the numerical solutions done with MATLAB. Xemet receives financing from TEKES for its technology development, which is highly acknowledged. The experimental nonclinical data was obtained from a larger study by Immunomedics Inc., which is greatly acknowledged. For the patient data the help of all co-authors in publications 4 and 5 has been valu-able and without them this study would have been impossi-ble.

APPENDIX A

SYSTEM FLOW DIAGRAMS

Fig. (A-1). System.


Fig. (A-2). Three compartment model of tissue/organ j.

Fig. (A-3). Two compartment model for tumor associated with liver’s blood circulation.

APPENDIX B

List of Symbols

C (mg/l) Concentration of MAB in arterial blood

,AG tC (mg/l) Concentration of MAB-antigen complex in tumor

,maxAGC (mg/l) Total concentration of antigen in tumor

bviC (mg/l) Concentration of MAB in venous blood (before lymph)

bvoC (mg/l) Concentration of MAB in venous blood (after lymph)

lC (mg/l) Concentration of MAB in lymph (after all lymph flows)

ljC (mg/l) Concentration of MAB in lymph from the organ/tissue j

'ojC (mg/l) Concentration of MAB in blood at arterial side of vascular space of the organ/tissue j

ojC (mg/l) Concentration of MAB in blood at venous side of vascular space of the organ/tissue j

xjC (mg/l) Concentration of free MAB in interstitial space of the organ/tissue j

b

xjC (mg/l) Concentration of non-specifically bound MAB in interstitial space of the organ/tissue j

N

fjk (1/min) Rate constant of non-specific binding (MAB to a non-specific binding site) in the organ/tissue j

N

rjk (1/min) Rate constant of detachment (MAB from a non-specific binding site) in the organ/tissue j

S

ftk (1/min) Rate constant of specific binding (MAB to tumor antigens)

S

rtk (1/min) Rate constant of detachment (MAB from tumor antigens)


. .i vm (mg/min) Mass flow rate of MAB into venous blood during the i.v. injection

bxjP (l/min) Permeability times surface area for transvascular transport (vascular/arterial side to interstitial space) in

the organ/tissue j

xbjP (l/min) Permeability times surface area for reverse transvascular transport (interstitial to vascular/venous side) in

the organ/tissue j

baQ (l/min) Volume flow rate of arterial blood

bviQ (l/min) Volume flow rate of venous blood (before lymph)

bvoQ (l/min) Volume flow rate of venous blood (after lymph)

ijQ (l/min) Volume flow rate of blood into the organ/tissue j

ljQ (l/min) Volume flow rate of lymph from organ/tissue j

ltotQ (l/min) Volume flow rate of lymph (total)

ojQ (l/min) Volume flow rate of blood out of the organ/tissue j

baV (l) Volume of arterial blood

bvV (l) Volume of venous blood

ijV (l) Volume of arterial side of vascular space of the organ/tissue j

lV (l) Volume of lymph

ojV (l) Volume of venous side of vascular space of the organ/tissue j

xjV (l) Volume of the interstitial space of the organ/tissue j

jZ ( ) Equilibrium ratio of MAB concentration between lymph and tissue in the organ/tissue j ( jZ = ljC / xjC )

Where j is Index for Organ/Tissue:

j = fa fat

j = br brain

j = bm bone marrow

j = bo bone

j = gi GI track

j = hm heart muscle

j = ki kidney

j = li liver

j = lu lung

j = mu muscle

j = ot other tissues

j = pa pancreas

j = sk skin

j = sp spleen

j = th thymus


APPENDIX C

Mathematical Model Equations

Mass balances for organs/tissues (j)

Vascular space – arterial side:

dC 'oj

dt=

Qij

Vij

CQ

oj

Vij

C 'oj

Pbxj

Vij

C 'oj

Cxj( )a C( ) C-1

where

( ) 1a C = , when 'oj xjC C

( ) 0a C = , when 'oj xjC C

Vascular space – venous side:

dCoj

dt=

Qoj

Voj

C 'oj

Qoj

Voj

Coj+

Pxbj

Voj

Cxj

Coj( )b C( ) C-2

where

( ) 1b C = , when xj ojC C

( ) 0b C = , when xj ojC C

Interstitial space – non-specific binding:

dCxj

dt=

Pbxj

Vxj

C 'oj

Cxj( )a C( )

Pxbj

Vxj

Cxj

Coj( )b C( )

Qlj

Vxj

ZjC

xjk

fj

NCxj

krj

NCxj

b( ) C-3

dCxj

b

dt= k

fj

NCxj

krj

NCxj

b C-4

Tumor

(j=t)

Vascular space

dCot

dt=

Qlt

Vit

C 'oli

Qlt

Vit

Cot

Pbxt

Vit

Cot

Cxt( )a C( ) +

Pxbt

Vit

Cxt

Cot( )b C( ) C-5

where

( ) 1a C = , when ot xtC C

( ) 0a C = , when ot xtC C

( ) 1b C = , when xt otC C

( ) 0b C = , when xt otC C


Interstitial space of tumor – specific binding:

( ) ( ) ( ) ( )

( ),max , ,

xt bxt xbtot xt xt ot

xt xt

S S

ft xt AG AG t rt AG t

dC P PC C a C C C b C

dt V V

k C C C k C

=

+

C-6

( ),

,max , ,

AG t S S

ft xt AG AG t rt AG t

dCk C C C k C

dt= C-7

Blood – arterial:

ba baolu

ba ba

dC Q QC C

dt V V= C-8

Blood – venous:

bvo bvi bvo ltotbvi bvo l

bv bv bv

dC Q Q QC C C

dt V V V= + C-9

where

. .

bvi bvi ohm ohm omu omu obr obr ofa ofa

obo obo obm obm osk osk oki oki oth oth

oot oot oli oli i v

Q C Q C Q C Q C Q C

Q C Q C Q C Q C Q C

Q C Q C m

= + + +

+ + + + +

+ + +

Lymph:

It is assumed that there is neglible accumulation in the lymph:

( )0l l

li li ltot l

d V CQ C Q C

dt= = C-10

where

lg lg

li li lmu lmu lfa lfa lbo lbo lbm lbm lsk lsk lki lki

lth lth lot lot lsp lsp lpa lpa i i lli lli llu llu

Q C Q C Q C Q C Q C Q C Q C

Q C Q C Q C Q C Q C Q C Q C

= + + + + +

+ + + + + + +

lg g

1(

)

l lmu mu xmu lfa fa xfa lbo bo xbo

ltot

lbm bm xbm lsk sk xsk lki ki xki lth th xth

lot ot xot lsp sp xsp lpa pa xpa i gi x i

lli li xli llu lu xlu

C Q Z C Q Z C Q Z CQ

Q Z C Q Z C Q Z C Q Z C

Q Z C Q Z C Q Z C Q Z C

Q Z C Q Z C

= + +

+ + + +

+ + + +

+ +

C-11

where

lgltot lmu lfa lbo lbm lsk lki lth lot lsp lpa i

lli llu

Q Q Q Q Q Q Q Q Q Q Q Q

Q Q

= + + + + + + + + + +

+ +


APPENDIX D

Experimental Data

Table D-1. Mouse – Concentration Data

Time 4 h 24 h 48 h 72 h 120 h 168 h 240 h

%i.d./g

Liver 10,2 7,2 6,9 6,5 6,1 5 5,5

Spleen 11,5 8,7 8,6 8,7 9,1 8,6 8,3

Kidney 9,2 6,6 6 5,9 5,7 5,5 5

Lungs 8,5 7,9 8,2 6,6 6,1 5,9 5,9

Blood 36,7 27,2 24,3 22,7 20,7 20,5 17,7

Stomach 2,1 1,4 1,7 1,9 1,4 1 1,6

S.I. 3,5 2,7 2,6 2,6 2,3 2 2,4

L.I. 1,8 1,8 1,7 1,9 1,8 1,2 1,5

Bone 4 3,7 3,3 3,1 3 2,9 3,4

W.B. 2,1 1,8 1,6 1,5 1,5 1,4 1,2

Table D-2. Human (15 kg) – Activity Data

Cells Total Av. Min Max Stdev Label

ROI (1h)

1 1829 384578 210 100 313 34 liver ap

3 1829 330973 181 67 256 29 liver pa

2 1157 393177 340 157 541 90 heart ap

4 1157 305863 264 159 410 50 heart pa

5 346 49968 144 34 208 30 spleen ap

6 355 61134 172 88 238 27 spleen pa

7 1504 209416 139 9 239 47 fantom ap

8 1504 184588 123 5 211 42 fantom pa

9 46315 2705023 58 0 541 70 WB, ap

10 46313 2409916 52 0 410 57 WB, pa

ROI (4h)

1 1829 336860 184 68 275 37 liver ap

3 1829 293700 161 43 244 32 liver pa

2 1157 347673 300 111 469 76 heart ap

4 1157 261142 226 119 363 45 heart pa


(Table D-2) contd….


ROI (1h)

5 346 27922 81 24 118 15 spleen ap

6 355 44710 126 78 164 15 spleen pa

7 1504 192429 128 16 223 35 fantom ap

8 1504 176459 117 20 202 31 fantom pa

9 46315 2477428 53 0 469 60 WB, ap

10 46313 2204838 48 0 363 49 WB, pa

ROI (24h)

1 1829 304826 167 57 241 30 liver ap

3 1829 232244 127 33 199 24 liver pa

2 1157 232188 201 86 323 52 heart ap

4 1157 176288 152 71 235 30 heart pa

5 346 27587 80 41 145 16 spleen ap

6 355 30847 87 38 131 18 spleen pa

7 1504 144143 96 3 217 33 fantom ap

8 1504 131929 88 3 192 29 fantom pa

9 46743 1930614 41 0 323 46 WB, ap

10 46743 1624569 35 0 235 35 WB, pa

ROI (48h)

1 1829 348528 191 78 277 35 liver ap

3 1829 269044 147 28 246 40 liver pa

2 1157 233046 201 98 316 48 heart ap

4 1157 190617 165 87 260 28 heart pa

5 346 22918 66 17 97 14 spleen ap

6 355 31991 90 48 129 13 spleen pa

7 1504 170457 113 20 221 34 fantom ap

8 1504 160369 107 11 205 34 fantom pa

9 46743 2115728 45 0 316 48 WB, ap

10 46743 1872336 40 0 260 40 WB, pa

ROI (96h)

1 1829 337176 184 69 258 33 liver ap

3 1829 258330 141 47 219 29 liver pa

2 1157 165196 143 62 234 35 heart ap


(Table D-2) contd….


ROI (96h)

4 1157 136220 118 59 195 22 heart pa

5 346 17560 51 10 80 15 spleen ap

6 355 24639 69 30 99 13 spleen pa

7 1504 150281 100 8 201 29 fantom ap

8 1504 135142 90 10 178 26 fantom pa

9 46315 1677256 36 0 258 41 WB, ap

10 46313 1471891 32 0 219 33 WB, pa

ROI (168h)

1 1829 181896 99 33 157 20 liver ap

3 1829 134990 74 8 137 21 liver pa

2 1157 66888 58 23 105 16 heart ap

4 1157 54749 47 20 74 10 heart pa

5 346 9784 28 7 52 9 spleen ap

6 355 12363 35 17 55 7 spleen pa

7 1504 69940 47 10 88 13 fantom ap

8 1504 67544 45 6 84 12 fantom pa

9 46315 750321 16 0 157 21 WB, ap

10 46313 645129 14 0 137 16 WB, pa

ROI (264h)

1 1829 46842 26 5 49 6 liver ap

3 1829 34230 19 4 37 5 liver pa

2 1157 15820 14 2 31 4 heart ap

4 1157 12958 11 2 25 3 heart pa

5 346 3078 9 0 23 4 spleen ap

6 355 3448 10 2 22 3 spleen pa

7 1504 13636 9 0 22 3 fantom ap

8 1504 11839 8 0 19 3 fantom pa

9 46315 211227 5 0 49 5 WB, ap

10 46313 173333 4 0 37 4 WB, pa


APPENDIX E

Parameters

Table E-1. Model Parameters for Mouse Physiological Parameters

Flow Rates(l/min)

Tissue/Organ

Qi (Blood in) Qo (Blood Out) QL (Lympha Out)

heart muscle (hm) Qihm = 5,10E-04 Qohm = 5,10E-04 QLhm = 0,00E+00

muscles (mu) Qimu = 1,45E-03 Qomu = 1,45E-03 QLmu = 6,00E-07

brain (br) Qibr = 1,60E-03 Qobr = 1,60E-03 QLbr = 0

fat (fa) Qifa = 1,20E-04 Qofa = 1,20E-04 QLfa = 0

bone (bo) Qibo = 3,10E-04 Qobo = 3,10E-04 QLbo = 6,00E-08

bone marrow (bm) Qibm = 3,10E-04 Qobm = 3,10E-04 QLbm = 6,00E-08

skin (sk) Qisk = 2,20E-03 Qosk = 2,20E-03 QLsk = 1,00E-08

kidney (ki) Qiki = 1,45E-03 Qoki = 1,45E-03 QLki = 1,70E-07

thymus (th) Qith = 6,00E-05 Qoth = 6,00E-05 QLth = 9,60E-09

other tissues (ot) Qiot = 1,20E-04 Qoot = 1,20E-04 QLot = 0,00E+00

spleen (sp) Qisp = 9,00E-05 Qosp = 9,00E-05 QLsp = 2,00E-09

pancreas (pa) Qipa = 9,00E-05 Qopa = 9,00E-05 QLpa = 2,00E-09

gastrointestinal (gi) Qigi = 1,64E-03 Qogi = 1,64E-03 QLgi = 7,00E-07

liver (li) Qili = 2,00E-03 Qoli = 3,82E-03 QLli = 2,00E-07

lung (lu) Qilu = 1,20E-02 Qolu = 1,20E-02 QLlu = 1,00E-07

blood-arterial (ba) Qba = 1,20E-02

Tumor (t) Qlt = 0,00E+00

blood-venous (bv) Qbv = 1,19E-02

lymph (l) QLtot = 1,91E-06

Qotot = 1,19E-02

Volumes(l)

V Vi Vo Vx

Vhm = 1,33E-04 Vihm = 3,50E-06 Vohm = 3,50E-06 Vxhm = 1,90E-05

Vmu = 7,92E-03 Vimu = 7,50E-05 Vomu = 7,50E-05 Vxmu = 1,03E-03

Vbr = 5,00E-04 Vibr = 2,50E-05 Vobr = 2,50E-05 Vxbr = 1,00E-04

Vfa = 2,00E-03 Vifa = 1,00E-04 Vofa = 1,00E-04 Vxfa = 4,00E-04

Vbo = 1,50E-03 Vibo = 2,00E-05 Vobo = 2,00E-05 Vxbo = 2,80E-04

Vbm = 1,50E-03 Vibm = 2,00E-05 Vobm = 2,00E-05 Vxbm = 2,80E-04

Vsk = 2,94E-03 Visk = 1,00E-04 Vosk = 1,00E-04 Vxsk = 9,99E-04

Vki = 2,98E-04 Viki = 1,50E-05 Voki = 1,50E-05 Vxki = 1,01E-04


(Table E-1) contd….

Volumes(l)

V Vi Vo Vx

Vth = 2,00E-05 Vith = 1,00E-06 Voth = 1,00E-06 Vxth = 4,00E-06

Vot = 2,00E-04 Viot = 1,00E-05 Voot = 1,00E-05 Vxot = 4,00E-05

Vsp = 1,00E-04 Visp = 5,00E-06 Vosp = 5,00E-06 Vxsp = 2,00E-05

Vpa = 1,00E-04 Vipa = 5,00E-06 Vopa = 5,00E-06 Vxpa = 2,00E-05

Vgi = 3,45E-03 Vigi = 5,00E-05 Vogi = 5,00E-05 Vxgi = 6,00E-04

Vli = 9,51E-04 Vili = 4,75E-05 Voli = 4,75E-05 Vxli = 1,90E-04

Vlu = 1,91E-04 Vilu = 9,50E-06 Volu = 9,50E-06 Vxlu = 5,70E-05

Vba = 7,00E-04 VIt = 1,00E-05 Vxt = 1,00E-05

Kinetic and Mixed Parameters

Kinetic Constants (1/min)

kNf kNr

kNfhm = 1,00E-05 kNrhm = 1,00E-01

kNfmu = 1,00E-03 kNrmu = 1,00E-01

kNfbr = 0,00E+00 kNrbr = 0,00E+00

kNffa = 1,00E-04 kNrfa = 1,00E-02

kNfbo = 1,00E-03 kNrbo = 1,00E-01

kNfbm = 1,00E-03 kNrbm = 1,00E-01

kNfsk = 1,00E-03 kNrsk = 1,00E-01

kNfki = 1,00E-04 kNrki = 1,00E-01

kNfth = 1,00E-04 kNrth = 1,00E-02

kNfot = 1,00E-03 kNrot = 1,00E-01

kNfsp = 4,00E-02 kNrsp = 1,00E-01

kNfpa = 1,00E-05 kNrpa = 1,00E-01

kNfgi = 1,00E-03 kNrgi = 1,00E-01

kNfli = 1,00E-03 kNrli = 1,00E-01

kNflu = 1,00E-04 kNrlu = 1,00E-01

Permeabilities (l/min) and Z ( )

Pbx Z

Pbxhm = 1,75E-07 Zhm = 0

Pbxmu = 4,99E-07 Zmu = 1



Pbx Z

Pbxbr = 0,00E+00 Zbr = 0

Pbxfa = 8,26E-08 Zfa = 1

Pbxbo = 3,41E-07 Zbo = 2

Pbxbm = 2,48E-07 Zbm = 2

Pbxsk = 1,26E-07 Zsk = 1

Pbxki = 2,32E-07 Zki = 1

Pbxth = 2,06E-09 Zth = 1

Pbxot = 4,13E-09 Zot = 1

Pbxsp = 1,08E-06 Zsp = 1

Pbxpa = 3,10E-08 Zpa = 1

Pbxgi = 4,10E-07 Zgi = 2

Pbxli = 8,00E-06 Zli = 3

Pbxlu = 3,19E-07 Zlu = 1

Table E-2. Model Parameters for the Human Body (15 kg)

Flow Rates(l/min)

Tissue/Organ


heart muscle (hm) Qihm = 4,30E-02 Qohm = 4,30E-02 QLhm = 0,00E+00

muscles (mu) Qimu = 1,48E-01 Qomu = 1,47E-01 QLmu = 5,57E-05

brain (br) Qibr = 1,80E-01 Qobr = 1,80E-01 QLbr = 0

fat (fa) Qifa = 7,00E-02 Qofa = 7,00E-02 QLfa = 0

bone (bo) Qibo = 4,90E-02 Qobo = 4,90E-02 QLbo = 5,57E-06

bone marrow (bm) Qibm = 4,90E-02 Qobm = 4,90E-02 QLbm = 5,57E-06

skin (sk) Qisk = 7,86E-02 Qosk = 7,86E-02 QLsk = 9,20E-07

kidney (ki) Qiki = 2,25E-01 Qoki = 2,25E-01 QLki = 1,59E-05

thymus (th) Qith = 0,00E+00 Qoth = 0,00E+00 QLth = 0,00E+00

other tissues (ot) Qiot = 2,80E-02 Qoot = 2,80E-02 QLot = 0,00E+00

spleen (sp) Qisp = 4,93E-02 Qosp = 4,93E-02 QLsp = 1,86E-07

pancreas (pa) Qipa = 4,93E-02 Qopa = 4,93E-02 QLpa = 1,86E-07

gastrointestinal (gi) Qigi = 1,67E-01 Qogi = 1,67E-01 QLgi = 6,43E-05

liver (li) Qili = 3,90E-01 Qoli = 6,56E-01 QLli = 1,86E-05

lung (lu) Qilu = 1,53E+00 Qolu = 1,53E+00 QLlu = 9,21E-06

blood-arterial (ba) Qba = 1,53E+00

Tumor (t) Qlt = 5,70E-02



Flow Rates(l/min)

Tissue/Organ


blood-venous (bv) Qbv = 1,53E+00

lymph (l) QLtot = 1,76E-04

Volumes (l)

V Vi (5%) Vo (5%) Vx (20%)

Vhm = 6,40E-02 Vihm = 3,20E-03 Vohm = 3,20E-03 Vxhm = 1,28E-02

Vmu = 7,50E+00 Vimu = 3,75E-01 Vomu = 3,75E-01 Vxmu = 1,50E+00

Vbr = 4,00E-01 Vibr = 2,00E-02 Vobr = 2,00E-02 Vxbr = 8,00E-02

Vfa = 3,00E+00 Vifa = 1,50E-01 Vofa = 1,50E-01 Vxfa = 6,00E-01

Vbo = 1,50E+00 Vibo = 7,50E-02 Vobo = 7,50E-02 Vxbo = 3,00E-01

Vbm = 1,50E+00 Vibm = 7,50E-02 Vobm = 7,50E-02 Vxbm = 3,00E-01

Vsk = 1,46E+00 Visk = 7,30E-02 Vosk = 7,30E-02 Vxsk = 2,92E-01

Vki = 7,55E-02 Viki = 3,78E-03 Voki = 3,78E-03 Vxki = 1,51E-02

Vth = 1,00E-02 Vith = 5,00E-04 Voth = 5,00E-04 Vxth = 2,00E-03

Vot = 3,00E-01 Viot = 1,50E-02 Voot = 1,50E-02 Vxot = 6,00E-02

Vsp = 4,60E-02 Visp = 2,30E-03 Vosp = 2,30E-03 Vxsp = 9,20E-03

Vpa = 4,60E-02 Vipa = 2,30E-03 Vopa = 2,30E-03 Vxpa = 9,20E-03

Vgi = 5,70E-01 Vigi = 2,85E-02 Vogi = 2,85E-02 Vxgi = 1,14E-01

Vli = 4,80E-01 Vili = 2,40E-02 Voli = 2,40E-02 Vxli = 9,60E-02

Vlu = 2,14E-01 Vilu = 1,07E-02 Volu = 1,07E-02 Vxlu = 4,28E-02

Vba = 4,82E-01 Vit = 2,00E-02 Vxt = 3,00E-02

Vbv = 4,82E-01

Vl = 1,92E-01

Kinetic and Mixed Parameters

Kinetic Constants (1/min) (l/minxmg) (1/min)

kNf kNr kSf kSr

kNfhm = 2,00E-01 kNrhm = 1,00E-03 kSfhm = 0 kSrhm = 0

kNfmu = 2,00E-01 kNrmu = 1,00E-03 kSfmu = 0 kSrmu = 0

kNfbr = 2,00E-01 kNrbr = 1,00E-03 kSfbr = 0 kSrbr = 0

kNffa = 2,00E-01 kNrfa = 1,00E-03 kSffa = 0 kSrfa = 0

kNfbo = 2,00E-01 kNrbo = 1,00E-03 kSfbo = 0 kSrbo = 0



Kinetic Constants (1/min) (l/minxmg) (1/min)

kNf kNr kSf kSr

kNfbm = 2,00E-01 kNrbm = 1,00E-03 kSfbm = 0 kSrbm = 0

kNfsk = 2,00E-01 kNrsk = 1,00E-03 kSfsk = 0 kSrsk = 0

kNfki = 2,00E-01 kNrki = 1,00E-03 kSfki = 0 kSrki = 0

kNfth = 2,00E-01 kNrth = 1,00E-03 kSfth = 0 kSrth = 0

kNfot = 2,00E-01 kNrot = 1,00E-03 kSfot = 0 kSrot = 0

kNfsp = 2,00E-01 kNrsp = 2,50E-04 kSfsp = 0 kSrsp = 0

kNfpa = 2,00E-01 kNrpa = 1,00E-03 kSfpa = 0 kSrpa = 0

kNfgi = 2,00E-01 kNrgi = 1,00E-03 kSfgi = 0 kSrgi = 0

kNfli = 4,00E-01 kNrli = 1,67E-04 kSfli = 0 kSrli = 0

kNflu = 2,00E-01 kNrlu = 1,00E-03 kSflu = 0 kSrlu = 0

kSft = 1,00E-02 kSrt = 1,00E-05

Permeabilities (l/min) and Z ( )

Pbx Z

Pbxhm = 1,72E-04 Zhm = 0

Pbxmu = 5,90E-04 Zmu = 1

Pbxbr = 7,20E-04 Zbr = 1

Pbxfa = 2,80E-04 Zfa = 1

Pbxbo = 1,96E-04 Zbo = 1

Pbxbm = 1,96E-04 Zbm = 1

Pbxsk = 3,14E-04 Zsk = 1

Pbxki = 9,00E-04 Zki = 1

Pbxth = 0,00E+00 Zth = 1

Pbxot = 1,12E-04 Zot = 1

Pbxsp = 3,35E-04 Zsp = 1

Pbxpa = 1,97E-04 Zpa = 1

Pbxgi = 6,68E-04 Zgi = 2

Pbxli = 7,80E-03 Zli = 3

Pbxlu = 6,10E-03 Zlu = 1

Pbxt = 4,56E-04


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https://www.researchgate.net/publication/23159826_A_Combined_Approach_with_Rituximab_Plus_Anti-TRAIL-R_Agonistic_Antibodies_for_the_Treatment_of_Haematological_Malignancies?el=1_x_8&enrichId=rgreq-4f2ef58baf41a4e34d5cf2fa0f87a757-XXX&enrichSource=Y292ZXJQYWdlOzI0MTkwODU3O0FTOjk4NzEzMzg5MjQwMzMwQDE0MDA1NDY1MTQ2ODA=




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