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| [[Image:wholeBody wiki.svg| thumb |Graphic representation of a physiologically based whole body model. Here, it is dissected into seven tissue/organ compartments: brain, lungs and heart, pancreas, liver, gut, kidney and adipose/muscle tissue. Blood flows, Q, and concentration, [X], of a substance of interest are depicted.]]
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| '''Physiologically based pharmacokinetic (PBPK) modeling''' is a [[mathematical modeling]] technique for predicting the absorption, distribution, metabolism and excretion (ADME) of synthetic or natural chemical substances in humans and other animal species. PBPK modeling is used in pharmaceutical research and drug development, and in health [[risk assessment]] for cosmetics or general chemicals.
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| PBPK models strive to be mechanistic by mathematically transcribing anatomical, physiological, physical, and chemical descriptions of the phenomena involved in the complex ADME processes. A large degree of residual simplification and empiricism is still present in those models, but they have an extended domain of applicability compared to that of classical, empirical function based, [[pharmacokinetic]] models. PBPK models may have purely predictive uses, but other uses, such as statistical inference, have been made possible by the development of [[Bayesian statistics|Bayesian]] statistical tools able to deal with complex models.<ref>{{cite journal|last=Gelman|first=A.|coauthors=Bois, F.Y., Jiang, J.|title=Physiological pharmacokinetic analysis using population modeling and informative prior distributions|journal=Journal of the American Statistical Association|year=1996|volume=91|pages=1400–1412}}</ref> That is true for both toxicity risk assessment and therapeutic drug development.
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| PBPK models try to rely ''a priori'' on the anatomical and physiological structure of the body, and to a certain extent, on biochemistry. They are usually [[multi-compartment model]]s, with compartments corresponding to predefined organs or tissues, with interconnections corresponding to blood or lymph flows (more rarely to diffusions). A system of [[differential equations]] for concentration or quantity of substance on each compartment can be written, and its parameters represent blood flows, pulmonary ventilation rate, organ volumes etc., for which information is available in scientific publications. Indeed the description they make of the body is simplified and a balance needs to be struck between complexity and simplicity. Besides the advantage of allowing the recruitment of ''a priori'' information about parameter values, these models also facilitate inter-species transpositions or extrapolation from one mode of administration to another (''e.g.'', inhalation to oral). An example of a 7-compartment PBTK model, suitable to describe the fate of many solvents in the mammalian body, is given in the Figure on the right.
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| ==History==
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| The first pharmacokinetic model described in the scientific literature
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| <ref>{{cite journal|last=Teorell|first=T.|title=Kinetics of distribution of substances administered to the body|journal=Archives Internationales de Pharmacodynamie et de Thérapie|year=1937|volume=57|pages=205–240}}</ref>
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| was in fact a PBPK model. It led, however, to computations intractable at that time. The focus shifted then to simpler models
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| ,<ref>{{cite book|last=Gibaldi M., and Perrier D.|title=Pharmacokinetics, , Second Edition, revised and expanded ed.|year=1982|publisher=Marcel Dekker|location=New-York}}</ref>
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| for which analytical solutions could be obtained (such solutions were sums of exponential terms, which led to further simplifications.) The availability of computers and numerical integration algorithms marked a renewed interest in [[mathematical physiology|physiological models]] in the early 1970s.<ref>{{cite journal|last=Bischoff|first=K.B.|coauthors=Dedrick, R.L., Zaharko, D.S., Longstreth, J.A.|title=Pharmacokinetics of 1-beta-D-arabinofuranosylcytosine (ARA-C) deamination in several species|journal=[[Biochemical Pharmacology]]|year=1971|volume=22|pages=1128–1133}}</ref><ref>{{cite journal|last=Dedrick|first=R.L.|coauthors=Forrester, D.D., Cannon, J.N., El Dareer, S.M., Mellett, L.B.|title=Methotrexate pharmacokinetics|journal=Journal of Pharmaceutical Sciences|year=1973|volume=60|pages=2405–2417}}</ref>
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| For substances with complex kinetics, or when inter-species extrapolations were required, simple models were insufficient and research continued on physiological models
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| <ref>{{cite journal|last=Gerlowski|first=L.E.|coauthors=Jain, R.K.|title=Physiologically based pharmacokinetic modeling: principles and applications|journal=Journal of Pharmaceutical Sciences|year=1983|volume=72|pages=1103–1127}}</ref>
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| <ref>{{cite journal|last=Droz|first=P.O.|coauthors=Guillemin, M. P.|title=Human styrene exposure - V. Development of a model for biological monitoring|journal=International Archives of Occupational and Environmental Health|year=1983|volume=53|pages=19–36}}</ref>
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| .<ref>{{cite journal|last=Lutz|first=R.J.|coauthors=Dedrick, R.L., Tuey, D., Sipes, G., Anderson, M.W., Matthews, H.B.|title=Comparison of the pharmacokinetics of several polychlorinated biphenyls in mouse, rat, dog, and monkey by means of a physiological pharmacokinetic model|journal=Drug Metabolism and Disposition |year=1984|volume=12|pages=527–535}}</ref>
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| By 2010, hundreds of scientific publications have described and used PBPK models, and at least two private companies are basing their business on their expertise in this area.
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| ==Building a PBPK model==
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| The model equations follow the principles of mass transport, fluid dynamics, and biochemistry in order to simulate the fate of a substance in the body
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| .<ref>{{cite journal|last=Campbell|first=JL, Jr.|coauthors=Clewell RA, Gentry PR, Andersen ME, Clewell HJ, III|title=Physiologically based pharmacokinetic/toxicokinetic modeling|journal=Computational Toxicology, Methods in Molecular Biology Series|year=2012|pages=439–499}}</ref>
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| Compartments are usually defined by grouping organs or tissues with similar blood perfusion rate and lipid content (''i.e.'' organs for which chemicals' concentration ''v.s.'' time profiles will be similar). Ports of entry (lung, skin, intestinal tract...), ports of exit (kidney, liver...) and target organs for therapeutic effect or toxicity are often left separate. Bone can be excluded from the model if the substance of interest does not distribute to it. Connections between compartment follow physiology (''e.g.'', blood flow in exit of the gut goes to liver, ''etc.'')
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| === Basic transport equations ===
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| Drug distribution into a tissue can be rate-limited by either perfusion or permeability. Perfusion-rate-limited kinetics apply when the tissue membranes present no barrier to diffusion. Blood flow, assuming that the drug is transported mainly by blood, as is often the case, is then the limiting factor to distribution in the various cells of the body. That is usually true for small lipophilic drugs. Under perfusion limitation, the instantaneous rate of entry for the quantity of drug in a compartment is simply equal to (blood) volumetric flow rate through the organ times the incoming blood concentration. In that case; for a generic compartment ''i'', the differential equation for the quantity ''Q<sub>i</sub>'' of substance, which defines the rate of change in this quantity, is:
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| <center>
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| <math>{dQ_i \over dt} = F_i (C_{art} - {{Q_i} \over {P_i V_i}})</math>
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| </center>
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| where ''F<sub>i</sub>'' is blood flow (noted ''Q'' in the Figure above), ''C<sub>art</sub>'' incoming blood concentration, ''P<sub>i</sub>'' the tissue over blood partition coefficient and ''V<sub>i</sub>'' the volume of compartment ''i''.
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| A complete set of differential equations for the 7-compartment model shown above could therefore be:
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| Gut:
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| <center>
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| <math>{dQ_g \over dt} = F_g (C_{art} - {{Q_g} \over {P_g V_g}})</math>
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| </center>
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| Kidney:
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| <center>
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| <math>{dQ_k \over dt} = F_k (C_{art} - {{Q_k} \over {P_k V_k}})</math>
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| </center>
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| Poorly-perfused tissues (muscle and skin):
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| <center>
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| <math>{dQ_p \over dt} = F_p (C_{art} - {{Q_p} \over {P_p V_p}})</math>
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| </center>
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| Brain:
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| <center>
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| <math>{dQ_b \over dt} = F_b (C_{art} - {{Q_b} \over {P_b V_b}})</math>
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| </center>
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| Heart and lung:
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| <center>
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| <math>{dQ_h \over dt} = F_h (C_{art} - {{Q_h} \over {P_h V_h}})</math>
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| </center>
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| Pancreas:
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| <center>
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| <math>{dQ_{pn} \over dt} = F_{pn} (C_{art} - {{Q_{pn}} \over {P_{pn} V_{pn}}})</math>
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| </center>
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| Liver:
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| <center>
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| <math>{dQ_l \over dt} = F_a C_{art}
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| + F_g ({{Q_g} \over {P_g V_g}})
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| + F_{pn} ({{Q_{pn}} \over {P_{pn} V_{pn}}})
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| - (F_a + F_g + F_{pn}) ({{Q_l} \over {P_l V_l}}) </math>
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| </center>
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| The above equations include only transport terms and do not account for inputs or outputs.
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| Those can be modelled with specific terms, as in the following.
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| === Modelling inputs ===
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| Modelling inputs is necessary to come up with a meaningful description of a chemical's pharmacokinetics. The following examples show how to write the corresponding equations.
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| ==== Ingestion ====
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| When dealing with a bolus dose (''e.g.'' ingestion of a tablet), first order absorption is a very common assumption. In that case the gut equation is augmented with an input term, with an absorption rate constant ''K<sub>a</sub>'':
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| <center>
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| <math>{dQ_g \over dt} = F_g (C_{art} - {{Q_g} \over {P_g V_g}}) + K_a Q_{ing}</math>
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| </center>
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| That requires defining an equation for the quantity ingested and present in the gut lumen:
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| <center>
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| <math>{dQ_{ing} \over dt} = - K_a Q_{ing}</math>
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| </center>
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| In the absence of a gut compartment, input can be made directly in the liver. However, in that case local metabolism in the gut may not be correctly described. The case of approximately continuous absorption (''e.g. via'' drinking water) can be modelled by a zero-order absorption rate (here ''R<sub>ing</sub>'' in units of mass over time):
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| <center>
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| <math>{dQ_g \over dt} = F_g (C_{art} - {{Q_g} \over {P_g V_g}}) + R_{ing}</math>
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| </center>
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| More sophisticated gut absorption model can be used. In those models, additional compartments describe the various sections of the gut lumen and tissue. Intestinal pH, transit times and presence of active transporters can be taken into account
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| .<ref>{{cite journal|last=Darwich|first=AS|coauthors=Pade D, Ammori BJ, Jamei M, Ashcroft DM, Rostami-Hodjegan A|title=A mechanistic pharmacokinetic model to assess modified oral drug bioavailability post bariatric surgery in morbidly obese patients: interplay between CYP3A gut wall metabolism, permeability and dissolution|journal=Journal of Pharmacy and Pharmacology|year=2012}}</ref>
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| ==== Skin depot ====
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| The absorption of a chemical deposited on skin can also be modelled using first order terms. It is best in that case to separate the skin from the other tissues.
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| Skin absorption is quite complex in fact (to be added)...
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| ==== Intra-venous injection ====
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| Intravenous injection is a common clinical route of administration. (to be completed)
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| ==== Inhalation ====
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| Inhalation occurs through the lung and is hardly dissociable from exhalation (to be completed)
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| === Modelling excretion ===
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| ... (to be added)
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| === Modelling metabolism ===
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| ... (to be added)
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| ==Uses of PBPK modeling==
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| PBPK models are compartmental models like many others, but they have a few advantages over so-called "classical" pharmacokinetic models, which are less grounded in physiology. PBPK models can first be used to abstract and eventually reconcile disparate data (from physico-chemical or biochemical experiments, ''[[in vitro]]'' or ''[[in vivo]]'' pharmacological or toxicological experiments, ''etc.'') They give also access to internal body concentrations of chemicals or their metabolites, and in particular at the site of their effects, be it therapeutic or toxic. Finally they also help interpolation and extrapolation of knowledge between:
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| *Doses: e.g., from the high concentrations typically used in laboratory experiments to those found in the environment
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| <here show a Figure with time courses at different doses and explain>
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| *Exposure duration: ''e.g.'', from continuous to discontinuous, or single to multiple exposures
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| *Routes of administration: ''e.g.'', from inhalation exposures to ingestion
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| *Species: ''e.g.'', transpositions from rodents to human, prior to giving a drug for the first time to subjects of a clinical trial, or when experiments on humans are deemed unethical, such as when the compound is toxic without therapeutic benefit
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| *Individuals: ''e.g.'', from males to females, from adults to children, from non-pregnant women to pregnant
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| * From ''in vitro'' to ''in vivo''.<ref>{{cite journal|last=Bouvier d'Yvoire|first=M.|coauthors=Prieto P., Blaauboer B.J., Bois F., Boobis A., Brochot C., Coecke S., Freidig A., Gundert-Remy U., Hartung T., Jacobs M. N., Lavé T., Leahy D.E., Lennernäs H., Loizou G.D., Meek B., Pease C., Rowland M., Spendiff M., Yang J., Zeilmaker M.|title=Physiologically-based kinetic modelling (PBK modelling): meeting the 3Rs agenda - The report and recommendations of ECVAM Workshop 63a|journal=Alternatives to Laboratory Animals|year=2007|volume=35|pages=661–671}}</ref>
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| Some of these extrapolations are "parametric" : only changes in input or parameter values are needed to achieve the extrapolation (this is usually the case for dose and time extrapolations). Others are "nonparametric" in the sense that a change in the model structure itself is needed (''e.g.'', when extrapolating to a pregnant female, equations for the foetus should be added).
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| ==Limits and extensions of PBPK modelling==
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| After numerical values are assigned to each PBPK model parameter, specialized or general computer software is typically used to numerically integrate a set of ordinary differential equations like those described above, in order to calculate the numerical value of each compartment at specified values of time (see Software). However, if such equations involve only linear functions of each compartmental value, or under limiting conditions (e.g., when input values remain very small) that guarantee such linearity is closely approximated, such equations may be solved analytically to yield explicit equations (or, under those limiting conditions, very accurate approximations) for the time-weighted average (TWA) value of each compartment as a function of the TWA value of each specified input (see, e.g., <ref>{{cite journal|last=Bogen|first=K.T.|title=Pharmacokinetics for regulatory risk analysis: The case of trichloroethylene|journal=[[Regulatory Toxicology and Pharmacology]]|year=1988|volume=8|pages=447–466}}</ref><ref>{{cite journal|last=Bogen|first=K.T.|last=Hall|first=L.C.|title=Pharmacokinetics for regulatory risk analysis: The case of 1,1,1-trichloroethane (methyl chloroform)|journal=Regulatory Toxicology and Pharmacology|year=1989|volume=10|pages=26–50}}</ref>).
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| PBPK models can rely on chemical property prediction models ([[QSAR]] models or predictive chemistry models) on one hand. They also extend into, but are not destined to supplant, [[systems biology]] models of metabolic pathways. They are also parallel to [[physiome]] models, but do not aim at modelling physiological functions beyond fluid circulation in detail. In fact the above four types of models can reinforce each other when integrated.<ref>{{cite journal|last=Bois|first=F.Y.|title=Physiologically-based modelling and prediction of drug interactions|journal=Basic and Clinical Pharmacology and Toxicology|year=2009|volume=106|pages=154–161}}</ref>
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| ==References==
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| {{Reflist}}
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| Further references:
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| *Balant L.P. Gex-Fabry M. (1990) ''Physiological pharmacokinetic modelling'' Xenobiotica 20: 1241-1257.
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| *Bois F.Y., Jamei M., Clewell H.J. (2010) ''PBPK modelling of inter-individual variability in the pharmacokinetics of environmental chemicals'', Toxicology, 278:256–267.
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| *Menzel D.B. (1987) ''Physiological pharmacokinetic modeling'' Environmental Science and Technology 21:944-950.
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| *Ramsey J.C., Andersen M. (1984) ''A physiologically based description of the inhalation pharmacokinetics of styrene in rats and humans'' [[Toxicology and Applied Pharmacology]] 73: 159-175.
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| *Reddy M. ''et al.'' (2005) ''Physiologically Based Pharmacokinetic Modeling : Science and Applications'', Wiley-Interscience.
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| *Peters S.A (2012) ''Physiologically-Based Pharmacokinetic (PBPK) Modeling and Simulations'', Wiley.
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| ==Forums==
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| *[http://www.pbpk.org pbpk.org]
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| *[http://www.ecotoxmodels.org Ecotoxmodels] is a website on mathematical models in ecotoxicology.
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| ==Software==
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| Dedicated software:
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| *[http://pdsl.research.ge.com/BioDMET BioDMET]
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| *[https://www.cloegateway.com/services/cloe_pk/pages/about_service.php Cloe Predict]
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| *[http://www.simulations-plus.com/Products.aspx?grpID=3&cID=16&pID=11 [[Simulations Plus|GastroPlus]]]
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| *[http://www.maxsim2.com Maxsim2]
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| *[http://www.pk-sim.com PK-Sim]
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| *[http://www.pkquest.com PKQuest]
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| *[http://www.simcyp.com [[Simcyp Simulator]]]
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| General software:
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| *[http://bmsr.usc.edu ADAPT 5]
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| *[http://www.acslx.com/solutions/pharmaceutical.shtml ACSL X]
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| *[http://www.berkeleymadonna.com Berkeley Madonna]
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| *[http://www.ecolego.facilia.se [[Ecolego]]]
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| *[http://www.gnu.org/software/mcsim ''[[MCSim|GNU MCSIM]]: - Free simulation software'']
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| *[http://www.octave.org GNU Octave]
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| *[http://www.mathworks.com Matlab [[PottersWheel]]]
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| *[http://www.modelkinetix.com ModelMaker]
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| *[http://www.icondevsolutions.com NONMEM]
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| *[http://www.entelos.com PhysioLab]
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| *[http://www.r-project.org R deSolve package]
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| *[http://depts.washington.edu/saam2 SAAM II]
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| *[http://www.certara.com/products Phoenix WinNonlin/NLME/IVIVC/Trial Simulator]
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| [[Category:Toxicology]]
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| [[Category:Toxicokinetics]]
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| [[Category:Pharmacokinetics]]
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| [[Category:Pharmacology]]
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| [[Category:Pharmacy]]
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