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Core Curriculum

Toxicokinetics

Definitions

Xenobiotic (X.): A chemical that is not native in the body, or is present in much higher concentration than normal.

Toxic effect: A change in physiological conditions caused by an effect of xenobiotics on the cellular level creating a decrease in health or behavior.

Toxicodynamics: Mechanism of the toxic effect, reactivity, receptors and organ types.

Toxicokinetics: Uptake, transport and lingering time/concentration of X.

Absorption: Transport from the place of disposition to blood with a rate constant <math>k_a</math>.

Bolus: A dosage of X. administered directly into the plasma.

Elimination: Biotransformation, exhalation or excretion of X. X. does not need to be removed from the body, only made unavailable in its original form.

Introduction

There are two main ways to model toxicokinetics: Compartmental models and physiological models. The compartmental models are described more in detail below, and involve modeling organ systems by simple relations without involving physiology, i.e. the rate constants used are acquired from measurements alone. The physiological model looks at theoretical, or physiological, models to predict rate constants of the organs in the body. This involves factors such as:

• Blood flow through organs
• Absorption of the small intestine
  • Villi and microvilli in the intestine: These greatly increase the intestinal area, so absorption for selected X. is greatly enhanced here.
  • Active and passive diffusion: Some substances can diffuse directly across tissues, but most require some form of transport proteins. The mechanisms of these proteins determine how effectively and selectively xenobiotics are absorbed.
  • There is also metabolism in the intestine, by e.g. the cytochrome P450 3A4 (CYP3A4) enzyme which can activate many prodrugs.
  • Drug export from cells via P-glycoprotein is a very important mechanism which greatly reduces the amount of many xenobiotics that are absorbed.
• The portal vein collects blood from the intestine and goes directly to the liver, where many substances are metabolized and their bioavailability is reduced. This is called first-pass metabolism, where the drugs are metabolized before reaching general systemic circulation.
• After being metabolized in the liver many xenobiotics are conjugated and marked for excretion into the bile. The bile is excreted in the small intestine, where the drugs can be un-conjugated and reabsorbed, passing into the liver again. This is called the entero-hepatic circulation, and keeps plasma concentration of xenobiotics low in general.
• Other special barriers, such as the blood-brain barrier and the placenta also greatly effect the distribution of xenobiotics.

Compartmental models

A model often used to model toxicokinetics is the compartmental model. In the compartmental model there is a central compartment representing the blood plasma and rapidly equilibrating tissues (e.g. liver and kidney), and side-compartments of more slowly equilibrating tissues. The simplest such model is the one-compartment model. Here there is only one compartment, which means all the modeled tissues are rapidly equilibrating. In this model a bolus will decay exponentially, i.e. measuring the logarithm of the plasma concentration over time gives a linear plot. Conversely, if experimental data holds with this description, it can be modeled by the one-compartment model. The decay is elimination, and elimination happens from the central compartment.

Rate constants and elimination

There are several rate constants involved in toxicokinetics. There are elimination and absorption rate constant, <math>k_e</math> and <math>k_a</math>, which describes elimination from and absorption into the central compartment (see below) if the dose is administered e.g. orally. In multi-compartment models there are also distribution and redistribution constants, e.g. <math>k_{12}</math> and <math>k_{21}</math>, which describes rates between the compartments.

An example of a rate constant is the excretion rate constant through the kidney, <math>k_r</math>. In the kidney, glomerular filtration has a certain rate, tubular excretion another, and and reabsorption into the tubules a third. Thus, the excretion from the kidneys is given by <math>k_r=k_{gf}+k_{te}-k_{tr}</math>. Similar models can be made for other organs, both absorbative and eliminative.

The elimination rates can follow different rate laws. Generally, in a one-compartment model, there is a first-order rate law, e.g. <math>-\frac{d C(t)}{dt}=k_e * C(t)</math>. Other rate laws hold if e.g. the elimination system is saturated, then <math>-\frac{d C(t)}{dt}=const.</math>.

Integrating the formula above gives

<math>C(t)=C_0 e^{-k_{el} t}</math>,

and further manipulation gives e.g. the half-life of X. in the blood to be <math>t_{1/2}=\frac{ln 2}{k_e}</math>.

Often the concentration is plotted on a semilogarithmic plot versus time. If this yields a straight line, we have a one-compartment model. <math>k_e</math> can be predicted from the slope, and <math>C_0</math> by extrapolation.

If the semilogarithmic plot of plasma concentration of X. versus time does not yield a straight line, higher compartmental models must be used. In the higher-compartment model the tissues connected to the plasma equilibrate more slowly with the plasma, so the plasma concentration falls off more rapidly in the beginning, in what is called the distribution phases, before the concentration profile again is as for the one-compartment model above. If there are two phases, one distribution phase and one linear phase (the eliminiation phase), we have a two-compartment model, which usually can be modeled by:

<math>C(t)=A e^{-\alpha t}+B e^{-\beta t}</math>,

where <math>\beta</math> corresponds to <math>k_{e}</math> above, and can be treated the same way.

If C is measured for e.g. an orally distributed drug there is also an absorption phase where the concentration increases over a certain time.

Toxicokinetic Parameters

There are several parameters that can be used to describe the models in more experiment-friendly terms. At the heart is C(t), the plasma concentration of X. at a given time. X is the total amount of X. in the body. The parameter V, called the volume of distribution, which relates X and C. V tells how large a volume is needed to distribute the total amount of the xenobiotic (X), so the concentration of X. in V is the same as in the blood (C). Mathematically, this gives <math>V=\frac{X}{C}</math>.

D is the administered dosage. AUC is the area under the concentration/time curve from 0 to infinity. The bioavailability of X. is given as <math>F=\frac{AUC_{a}}{AUC_{i.v.}}</math>, which gives the fraction in plasma when administered e.g. orally compared to intra venously. This gives another relation: <math>V=\frac{D \times F}{k_{el} \times AUC}</math> for a non-i.v. delivered drug. The denominator term is the plasma concentration. For a one-compartment model this can often be approxomated as <math>V=\frac{D \times F}{C_0}</math>, or <math>V=\frac{X}{C}</math> as above for an i.v. delivered dosage (D=X).

Clearance (Cl) is a term the that describes the volume of plasma that is cleared of X. per unit time, and can be given as the sum of clearances from each of the eliminating organs (<math>Cl_{total}=Cl_{renal}+Cl_{hepatic}+...</math>). The total body clearance is given by <math>Cl=\frac{D_{i.v.}}{AUC}</math>, which gives units of volume/time. Using the relations from above this can be seen to be equivalent to <math>Cl_t=V \times k_{el}</math> for a one-compartment model.

If more than one dose is given, the dosage interval is given by <math>\tau</math>. Giving a dose either continuously, or with a certain interval, allows one to reach a steady state concentration, where there is a balance between absorption and elimination. By definition, this is equal to <math>5 \times C(t_{1/2})</math>. Equivalent equations for this is:

• <math>C_{ss}=\frac{F\times D}{Cl_t \times \tau}=\frac{F\times D}{k_e\times V \times \tau}</math>

If the steady state is reached by a dosage D every <math>\tau</math>, there is naturally an oscillation of steady state values, given by <math>\frac{C_{ss}^{max}}{C_{ss}^{min}}=e^{k_e \tau}</math>. By replacing the bioavailable dosage per time (<math>\frac{F \times D}{\tau}</math>) with an constant infusion rate <math>k_0</math> on obtains <math> C_{ss}=\frac{k_0}{Cl_t}</math>. Often it is desirable to reach steady state concentration as quickly as possible. In this case a bolus dose that immediately gives <math>C_{ss}</math> in the plasma. This dose is then given by <math>D_{bolus}=C_{ss}\times V</math>.

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