$$ \newcommand{\esp}[1]{\mathbb{E}\left(#1\right)} \newcommand{\var}[1]{\mbox{Var}\left(#1\right)} \newcommand{\deriv}[1]{\dot{#1}(t)} \newcommand{\prob}[1]{ \mathbb{P}\!(#1)} \newcommand{\eqdef}{\mathop{=}\limits^{\mathrm{def}}} \newcommand{\by}{\boldsymbol{y}} \newcommand{\bc}{\boldsymbol{c}} \newcommand{\bpsi}{\boldsymbol{\psi}} \def\pmacro{\texttt{p}} \def\like{{\cal L}} \def\llike{{\cal LL}} \def\logit{{\rm logit}} \def\probit{{\rm probit}} \def\one{{\rm 1\!I}} \def\iid{\mathop{\sim}_{\rm i.i.d.}} \def\simh0{\mathop{\sim}_{H_0}} \def\df{\texttt{df}} \def\res{e} \def\xomega{x} \newcommand{\argmin}[1]{{\rm arg}\min_{#1}} \newcommand{\argmax}[1]{{\rm arg}\max_{#1}} \newcommand{\Rset}{\mbox{$\mathbb{R}$}} \def\param{\theta} \def\setparam{\Theta} \def\xnew{x_{\rm new}} \def\fnew{f_{\rm new}} \def\ynew{y_{\rm new}} \def\nnew{n_{\rm new}} \def\enew{e_{\rm new}} \def\Xnew{X_{\rm new}} \def\hfnew{\widehat{\fnew}} \def\degree{m} \def\nbeta{d} \newcommand{\limite}[1]{\mathop{\longrightarrow}\limits_{#1}} \def\ka{k{\scriptstyle a}} \def\ska{k{\scriptscriptstyle a}} \def\kel{k{\scriptstyle e}} \def\skel{k{\scriptscriptstyle e}} \def\cl{C{\small l}} \def\Tlag{T\hspace{-0.1em}{\scriptstyle lag}} \def\sTlag{T\hspace{-0.07em}{\scriptscriptstyle lag}} \def\Tk{T\hspace{-0.1em}{\scriptstyle k0}} \def\sTk{T\hspace{-0.07em}{\scriptscriptstyle k0}} \def\thalf{t{\scriptstyle 1/2}} \newcommand{\Dphi}[1]{\partial_\pphi #1} \def\asigma{a} \def\pphi{\psi} \newcommand{\stheta}{{\theta^\star}} \newcommand{\htheta}{{\widehat{\theta}}} $$

\[ \newcommand{\deriv}[1]{\dot{#1}(t)} \def\ka{k{\scriptstyle a}} \def\ska{k{\scriptscriptstyle a}} \def\kel{k{\scriptstyle e}} \def\skel{k{\scriptscriptstyle e}} \def\cl{C{\small l}} \def\Tlag{T\hspace{-0.1em}{\scriptstyle lag}} \def\sTlag{T\hspace{-0.07em}{\scriptscriptstyle lag}} \def\Tk{T\hspace{-0.1em}{\scriptstyle k0}} \def\sTk{T\hspace{-0.07em}{\scriptscriptstyle k0}} \def\thalf{t{\scriptstyle 1/2}} \def\Imax{I\hspace{-0.15em}{\scriptstyle max}} \def\S0{S\hspace{-0.1em}{\scriptstyle 0}} \def\IC50{IC\hspace{-0.1em}{\scriptstyle 50}} \def\ke0{k{\scriptstyle e0}} \def\kin{k{\scriptstyle in}} \def\kout{k{\scriptstyle out}} \def\cc{C} \]

1 Introduction

A drug can be given to a patient in several ways: intravenously, orally, subcutaneously, or in less common ways such as intramuscularly, by skin patch or inhalation. Once a drug is administered, we usually describe subsequent processes within the organism by the pharmacokinetics (PK) process known as ADME: absorption, distribution, metabolism, excretion.

Absorption is the movement of the drug into the bloodstream from an extravascular site. The extent of absorption of a drug into the systemic circulation is known as the bioavailability.

After absorption, most drugs are distributed via the blood to body tissue. Distribution describes the reversible transfer of a drug between blood, tissue and organs.

Metabolism refers to the biotransformation within the body of a drug into other molecules, called metabolites. Metabolites are often pharmacologically inactive, but can also be active or toxic. Some drugs, called pro-drugs, are inactive until they are metabolized.

Excretion is the removal of a drug from the body. The kidneys, which excrete water soluble substances with urine, are the principal organ for this process. Bile flow from the liver is also an important route for elimination in feces. A drug may also leave the body by other natural routes: breath, tears, sweat, saliva, etc. Metabolism and excretion processes are often merged under the name elimination process.

Drug PK are very complex biological processes that are difficult to describe quantitatively. Pharmacometricians therefore simplify physiological processes by creating a compartmental model: the human body is described by a series of compartments in which the drug is distributed.

The depot compartment is the site at which a drug is deposited: the gut for oral administration, skin for the application of dermal patches, the bloodstream for intravenous administration, etc. The central compartment consists of blood and highly perfused organs (liver, kidney, lungs) and peripheral compartments of less perfused tissue (fat, skin, muscle).

This simplified framework can be turned into a mathematical model that is intended to give a good descriptive approximation of PK processes and is able to compute the concentration of a drug at any time in all parts of the body.

PK compartment models assume that drug concentration is perfectly homogeneous in each compartment of the body at all times. This strong assumption has the advantage of allowing a quantitative description of a PK model. Indeed, describing quantitatively a PK model turns into describing how the drug amount varies in each compartment. This means we typically need to describe three components:

The rate in describes how the drug moves from the depot compartment to the central compartment. The rate of distribution describes exchanges of the drug between the central and peripheral compartments. The rate out describes how the drug is eliminated from the central compartment.

The final model will then bring together these three components into one model.

2 Dynamical systems for characterizing the ADME process

A PK model is a dynamical system mathematically represented by a system of ordinary differential equations (ODEs) which describe transfers between compartments and elimination from the central compartment. Let us look as several examples of dynamical systems for different types of administration.

Intravenous administration. Consider first a system for iv administration with only one central compartment. The only process that needs to be described is elimination.

There exist several several different mathematical models to describe the elimination process. For instance, *linear elimination} (or first-order elimination) means that the rate of elimination is directly proportional to the amount:

\[\deriv{A_c} = - \kel \times A_c(t) , \] where \(A_c(t)\) and \(\deriv{A_c}\) are, respectively, the drug amount in the central compartment and its derivative at time \(t\). The proportionality constant \(\kel\) is the elimination rate constant. It is directly related to a corresponding half-life, i.e., the time required for half of the drug amount to leave the compartment: \[t_{1/2} = \log(2)/\kel .\]

Let \(V\) be the volume of distribution of the central compartment. Then, the clearance, defined as \(Cl=\kel/V\), relates the rate of elimination to the concentration \(\cc=A_c/V\):

\[\deriv{A_c} = - \cl \times \cc(t) .\] Clearance is expressed in volume per unit of time.

For drugs which are metabolized, the rate of elimination does not increase proportionately to drug concentration. Saturable elimination means that above a certain drug concentration, the elimination rate tends to reach a maximal value. Such capacity-limited elimination is best explained by the Michaelis-Menten equations:

\[\begin{aligned} \deriv{A_c} &= - \, \frac{V_m}{V\, K_m + A_c(t)} \, A_c(t) \\ &= - \, \frac{V_m}{K_m + \cc(t)} \, \cc(t). \end{aligned}\]

Here, \(V_m\) is the maximum elimination capacity, i.e., the total drug amount that can be eliminated per unit time at saturation, and \(K_m\) the drug concentration eliminated at half maximum capacity. When the drug concentration is quite small with respect to \(K_m\), the mixed-order process becomes similar to a first-order one and elimination is directly proportional to concentration. When the drug concentration is large in relation to \(K_m\), the elimination rate approaches the constant value \(V_m\).

Let us next introduce a peripheral compartment to the model and assume linear transfer between the central compartment and it. The mathematical representation of this model now consists of a system of two ODEs:

\[\begin{aligned} \deriv{A_c} &= k_{21}A_p(t) - k_{12}A_c(t) - \kel A_c(t) \nonumber \\ \deriv{A_p} &= -k_{21}A_p(t) + k_{12}A_c(t) , \end{aligned}\] where \(A_p\) is the amount in the peripheral compartment and \(k_{12}\) and \(k_{21}\) the distribution rate constants.

It is straightforward to extend this model to more than one peripheral compartment. For example, a three-compartment model is characterized by three ODEs: \[\begin{aligned} \deriv{A_c} &= k_{21}A_p(t) + k_{31}A_q(t)- (k_{12} +k_{13} + \kel) A_c(t) \\ \deriv{A_p} &= -k_{21}A_p(t) + k_{12}A_c(t) \\ \deriv{A_q} &= -k_{31}A_q(t) + k_{13}A_c(t) . \end{aligned}\]

Oral administration. Once swallowed, a drug reaches the gastrointestinal tract and is absorbed into the bloodstream. Only a fraction \(F\) of an orally administered dose may reach systemic circulation due to factors such as breakdown in the intestine, poor absorption and presystemic extraction. \(F\) is known as the bioavailability.

A zero-order absorption process assumes that a drug is transferred from the depot compartment with constant rate \(R_0\): \[\deriv{A_d} = - R_0 \times \one_{A_d(t)>0} .\]

A first-order absorption process assumes that the absorption rate is proportional to the drug amount in the depot compartment: \[\deriv{A_d} = - \ka \times A_d(t) .\] The absorption rate constant \(\ka\) is directly related to a corresponding absorption half-life: \[t_{\rm abs, 1/2} = \log(2)/\ka .\]

Remark: A zero-order absorption process can therefore be defined as the limit of the \(\alpha\)-absorption process: \[\deriv{A_d} = - R_\alpha \times A_d^\alpha(t) ,\] as \(\alpha\) tends to 0.

PK models for oral administration consist of a model each for absorption, distribution and elimination. For example, a one-compartment model with a first-order absorption process and linear elimination combines writes: \[\begin{aligned} \deriv{A_d} &= - \ka \, A_d(t) \\ \deriv{A_c} &= \ka \, A_d(t) - \kel \, A_c(t) . \end{aligned}\]

A two-compartment model with a zero-order absorption process and nonlinear elimination writes: \[\begin{aligned} \deriv{A_d} &= - R_0 \one_{A_d(t)>0} \\ \deriv{A_c} &= R_0 \one_{A_d(t)>0} + k_{21}A_p(t) - k_{12}A_c(t) - \frac{V_m}{V\, K_m + A_c(t)}A_c(t) \\ \deriv{A_p} &= -k_{21}A_p(t) + k_{12}A_c(t) . \end{aligned}\]

3 Putting doses into a system

PK models do not describe how a drug is administered. Administered doses are source terms, i.e., inputs that dynamically modify the state of a system. This is important to note because it means that the same PK model can be used for different dose regimens.

The plasmatic concentration predicted by a model is the solution of a system of ODEs for a given series of inputs and initial conditions (we suppose in general that all compartments are empty before the first dose). When the system of ODEs is linear, an analytical solution can be calculated. Otherwise, numerical methods for ODEs are required for approximating the solution.

In the case of iv bolus, a drug is administered intravenously over a negligible period of time and achieves instantaneous distribution throughout the central compartment.

Let \(D\) be the drug amount administered at time \(\tau\). Then, \[ A_c(\tau^+) = A_c(\tau^-) + D ,\] where \(\tau^-\) and \(\tau^+\) are the times just before and after drug administration. If the compartment is empty before \(\tau\), the solution is \[\cc(t) = \frac{D}{V}\, e^{- \skel \, (t-\tau)} \, \one_{t>\tau}.\]

If \(K\) doses \(D_1\), \(D_2\), ??? , \(D_K\) are administered at times \(\tau_1\), \(\tau_2\), ??? , \(\tau_K\), a superposition principle applies since the system is linear: \[\cc(t) = \frac{1}{V} \sum_{k=1}^{K} D_k e^{- \skel \, (t-\tau_k)} \, \one_{t>\tau_k}.\]

In the case of iv infusion, a dose is administered with a constant rate \(R_{\rm inf}\) during a certain infusion time period of length \(T_{\rm inf}=D/R_{\rm inf}\). Assuming a unique dose is administered at time \(\tau\), the solution is

\[\cc(t) = \left\{ \begin{array}{ll} 0 & \text{if } t<\tau \\ \frac{R}{V\,\kel}(1- e^{- \skel \, (t-\tau)}) & \text{if } \tau \leq t<\tau+T_{\rm inf} \\ \frac{R}{V\,\kel}(1- e^{- \skel \, T_{\rm inf}})e^{- \skel \, (t-\tau -T_{\rm inf})} & \text{if } t \geq \tau+T_{\rm inf} \end{array} \right. \] Note again that the same model was used for all these different examples; it is the changing inputs ??? here doses ??? which generate different solutions.

Consider now oral administration and let \(A_d\) be the drug amount in the depot compartment. If an amount \(D\) is instantaneously deposited at time \(\tau\) (imagine, for instance, that the dose is swallowed all at once), then, denoting \(F\) the bioavailability, state variable \(A_d\) is modified at time \(\tau\): \[ A_d(\tau^+) = A_d(\tau^-) + F \, D .\] If the central and depot compartments are empty before \(\tau\), the solution for a first order absorption process is \[\cc(t) = \frac{F \, D \, \ka}{V(\ka-\kel)}\, \left( e^{- \skel \, (t-\tau)} - e^{- \ska \, (t-\tau)} \right) \, \one_{t>\tau}.\] For a zero order absorption process, the solution is \[\cc(t) = \left\{ \begin{array}{ll} 0 & \text{if } t<\tau \\ \frac{D}{\Tk \,V\,\kel}(1- e^{- \skel \, (t-\tau)}) & \text{if } \tau \leq t<\tau+\Tk \\ \frac{D}{\Tk \,V\,\kel}(1- e^{- \skel \, \Tk })e^{- \skel \, (t-\tau -\Tk )} & \text{if } t \geq \tau+\Tk \end{array} \right. \] where \(\Tk = D/R_0\) is the duration of the absorption.

Similar types of solution can be obtained when there are two or more compartments, or when multiple doses are administered; the solutions are linear combinations of decaying exponentials.