Physiologically based pharmacokinetic modelling

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.

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 PBPK model, suitable to describe the fate of many solvents in the mammalian body, is given in the Figure on the right.

The availability of computers and numerical integration algorithms marked a renewed interest in physiological models in the early 1970s.

[6][7][8] By 2010, hundreds of scientific publications had described and used PBPK models, and at least two private companies have based their business on their expertise in this area.

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.

[9] Compartments are usually defined by grouping organs or tissues with similar blood perfusion rate and lipid content (i.e. organs for which chemicals' concentration vs. time profiles will be similar).

Connections between compartment follow physiology (e.g., blood flow in exit of the gut goes to liver, etc.)

[10][11] Perfusion-rate-limited kinetics apply when the tissue membranes present no barrier to diffusion.

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.

When dealing with an oral 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 Ka:

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

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 .

Gajewska M., Kovarich S., Mauch K., Paini A., Péry A., Sala Benito J.V., Teng S., Worth A., in press, Multiscale modelling approaches for assessing cosmetic ingredients safety, Toxicology.

doi: 10.1016/j.tox.2016.05.026] Unexposed stratum corneum simply exchanges with the underlying viable skin by diffusion:

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.

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).

For example, if a drug compound showed lower-than-expected oral bioavailability, various model structures (i.e., hypotheses) and parameter values can be evaluated to determine which models and/or parameters provide the best fit to the observed data.

As such, PBPK modeling can be used, inter alia, to evaluate the involvement of carrier-mediated transport, clearance saturation, enterohepatic recirculation of the parent compound, extra-hepatic/extra-gut elimination; higher in vivo solubility than predicted in vitro; drug-induced gastric emptying delays; gut loss and regional variation in gut absorption.

However, it is possible (and commonly done) to model explicitly the correlations between parameters (for example, the non-linear relationships between age, body-mass, organ volumes and blood flows).

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.,[16][17]).

They also extend into, but are not destined to supplant, systems biology models of metabolic pathways.

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.
Simulated drug plasma concentration over time curves following IV infusion and multiple oral doses. The drug has an elimination half-life of 4 hours, and an apparent volume of distribution of 10 liters.
Pharmacokinetic model of drugs entering a tumor. (A) Schematic illustration of a tumor vessel illustrating loss of smooth muscle cells, local degradation of the extracellular matrix, and increased permeability of the endothelium. (B) Illustration of the pharmacokinetic model taking into account the EPR effect. The rate constants kp and kd describe exchange with the peripheral volume. The rate constants kepr and kb describe extravasation from circulation into the tumor, and intravasation back into the circulation, respectively. The rate constant kel represents clearance by the kidneys, MPS, and any other non-tumor elimination processes, such that when kb = 0, k10 = kepr + kel where kel is the elimination rate constant. (C) Standard two compartment model with central and peripheral compartments. c1 and c2 represent the drug concentration in blood (central compartment) and normal tissue (peripheral compartment), respectively. The first order rate constant k10 describes all elimination pathways, including clearance by the kidneys, uptake by the MPS, and tumor accumulation. The first order rate constants k12 and k21 describe exchange between the two compartments. Note that kp = k12, kd = k21. (D) Two compartment model defined in terms of the drug amount, where Nbl is the amount of drug in blood (mg), and Np is the amount in peripheral tissue (mg). (E) Three compartment model with the addition of a tumor “compartment” where Nt is the amount of drug in the tumor. Exchange with the tumor is described by the rate constants kepr and kb, respectively. The rate constant kel describes elimination pathways including clearance by the kidneys and uptake by the MPS, but does not include tumor accumulation. [ 13 ]