De Bruijn sequence

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.

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.

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 statistical tools able to deal with complex models. That is true for both toxicity risk assessment and therapeutic drug development.

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 models, 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.

History

The first pharmacokinetic model described in the scientific literature was in fact a PBPK model. It led, however, to computations intractable at that time. The focus shifted then to simpler models , 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 physiological models in the early 1970s. For substances with complex kinetics, or when inter-species extrapolations were required, simple models were insufficient and research continued on physiological models . 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.

Building a PBPK model

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 . 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.)

Basic transport equations

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_i of substance, which defines the rate of change in this quantity, is:

${dQ_{i} \over dt}=F_{i}(C_{art}-{{Q_{i}} \over {P_{i}V_{i}}})$

where F_i is blood flow (noted Q in the Figure above), C_art incoming blood concentration, P_i the tissue over blood partition coefficient and V_i the volume of compartment i.

A complete set of differential equations for the 7-compartment model shown above could therefore be:

Gut:

${dQ_{g} \over dt}=F_{g}(C_{art}-{{Q_{g}} \over {P_{g}V_{g}}})$

Kidney:

${dQ_{k} \over dt}=F_{k}(C_{art}-{{Q_{k}} \over {P_{k}V_{k}}})$

Poorly-perfused tissues (muscle and skin):

${dQ_{p} \over dt}=F_{p}(C_{art}-{{Q_{p}} \over {P_{p}V_{p}}})$

Brain:

${dQ_{b} \over dt}=F_{b}(C_{art}-{{Q_{b}} \over {P_{b}V_{b}}})$

Heart and lung:

${dQ_{h} \over dt}=F_{h}(C_{art}-{{Q_{h}} \over {P_{h}V_{h}}})$

Pancreas:

${dQ_{pn} \over dt}=F_{pn}(C_{art}-{{Q_{pn}} \over {P_{pn}V_{pn}}})$

Liver:

${dQ_{l} \over dt}=F_{a}C_{art}+F_{g}({{Q_{g}} \over {P_{g}V_{g}}})+F_{pn}({{Q_{pn}} \over {P_{pn}V_{pn}}})-(F_{a}+F_{g}+F_{pn})({{Q_{l}} \over {P_{l}V_{l}}})$

The above equations include only transport terms and do not account for inputs or outputs. Those can be modelled with specific terms, as in the following.

Modelling inputs

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.

Ingestion

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_a:

${dQ_{g} \over dt}=F_{g}(C_{art}-{{Q_{g}} \over {P_{g}V_{g}}})+K_{a}Q_{ing}$

That requires defining an equation for the quantity ingested and present in the gut lumen:

${dQ_{ing} \over dt}=-K_{a}Q_{ing}$

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_ing in units of mass over time):

${dQ_{g} \over dt}=F_{g}(C_{art}-{{Q_{g}} \over {P_{g}V_{g}}})+R_{ing}$

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 .

Skin depot

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.

Skin absorption is quite complex in fact (to be added)...

Intra-venous injection

Intravenous injection is a common clinical route of administration. (to be completed)

Inhalation

Inhalation occurs through the lung and is hardly dissociable from exhalation (to be completed)

Modelling excretion

... (to be added)

Modelling metabolism

... (to be added)

Uses of PBPK modeling

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:

• Doses: e.g., from the high concentrations typically used in laboratory experiments to those found in the environment

<here show a Figure with time courses at different doses and explain>

• Exposure duration: e.g., from continuous to discontinuous, or single to multiple exposures
• Routes of administration: e.g., from inhalation exposures to ingestion
• 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
• Individuals: e.g., from males to females, from adults to children, from non-pregnant women to pregnant
• From in vitro to in vivo.

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).

Limits and extensions of PBPK modelling

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., ).

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.

References

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Further references:

• Balant L.P. Gex-Fabry M. (1990) Physiological pharmacokinetic modelling Xenobiotica 20: 1241-1257.

• 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.

• Menzel D.B. (1987) Physiological pharmacokinetic modeling Environmental Science and Technology 21:944-950.

• 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.

• Reddy M. et al. (2005) Physiologically Based Pharmacokinetic Modeling : Science and Applications, Wiley-Interscience.

• Peters S.A (2012) Physiologically-Based Pharmacokinetic (PBPK) Modeling and Simulations, Wiley.

Forums

• pbpk.org
• Ecotoxmodels is a website on mathematical models in ecotoxicology.

Software

Dedicated software:

• BioDMET
• Cloe Predict
• GastroPlus
• Maxsim2
• PK-Sim
• PKQuest
• Simcyp Simulator

General software:

• ADAPT 5
• ACSL X
• Berkeley Madonna
• Ecolego
• GNU MCSIM: - Free simulation software
• GNU Octave
• Matlab PottersWheel
• ModelMaker
• NONMEM
• PhysioLab
• R deSolve package
• SAAM II
• Phoenix WinNonlin/NLME/IVIVC/Trial Simulator