Ideally, quantification of xenobiotic concentration at the site of toxic insult or injury would afford the most direct information on exposure–response relationship and dynamics of response over time. Serial sampling of relevant biological tissues following dosing can be cost-prohibitive during in vivo studies in animals and is nearly impossible to perform in human exposure studies. The most accessible and simplest means of gathering information on absorption, distribution, metabolism, and elimination of a compound is to examine the time course of blood or plasma toxicant concentration over time. If one assumes that the concentration of a chemical in blood or plasma is in some describable dynamic equilibrium with its concentrations in tissues, then changes in plasma toxicant concentration should reflect changes in tissue toxicant concentrations and relatively simple kinetic models can adequately describe the behavior of that toxicant in the body system.
Classic toxicokinetic models typically consist of a central compartment representing blood and tissues that the toxicant has ready access and equilibration is achieved almost immediately following its introduction, along with one or more peripheral compartments that represent tissues in slow equilibration with the toxicant in blood (Fig. 7-1). Once introduced into the central compartment, the toxicant distributes between central and peripheral compartments. Elimination of the toxicant, through biotransformation and/or excretion, is usually assumed to occur from the central compartment, which should comprise the rapidly perfused visceral organs capable of eliminating the toxicant (eg, kidneys, lungs, and liver). The obvious advantage of compartmental toxicokinetic models is that they do not require information on tissue physiology or anatomic structure. These models are useful in predicting the toxicant concentrations in blood at different doses or exposure levels, in establishing the time course of accumulation of the toxicant, either in its parent form or as biotransformed products during continuous or episodic exposures, in defining concentration–response (vs dose–response) relationships, and in guiding the choice of effective dose and design of dosing regimen in animal toxicity studies (Rowland and Tozer, 2011).
Compartmental toxicokinetic models. Symbols for 1-compartment model: ka is the first-order absorption rate constant, and kel is the first-order elimination rate constant. Symbols for 2-compartment model: ka is the first-order absorption rate constant into the central compartment (1), k10 is the first-order elimination rate constant from the central compartment (1), k12 and k21 are the first-order rate constants for distribution between central (1) and peripheral (2) compartment.
The most straightforward toxicokinetic assessment entails quantification of the blood or more commonly plasma concentrations of a toxicant at several time points after a bolus intravenous (iv) injection.
Often, the data obtained fall on a straight line when they are plotted as the logarithm of plasma concentration versus time; the kinetics of the toxicant is said to conform to a one-compartment model (Fig. 7-2). Mathematically, this means that the decline in plasma concentration over time profile follows a simple exponential pattern as represented by the following mathematical expressions:
or its logarithmic transform
where C is the plasma toxicant concentration at time t after injection, C0 is the plasma concentration achieved immediately after injection, and kel is the exponential constant or elimination rate constant with dimensions of reciprocal time (eg, minute–1 or hour–1). The constant 2.303 in Equation (7-2) is needed to convert natural logarithm into base-10 logarithm. It can be seen from Equation (7-2) that the elimination rate constant can be determined from the slope of the log C versus time plot (ie, kel = –2.303 · slope).
Plasma concentration versus time curves of toxicants exhibiting kinetic behavior conforming to a 1-compartment model (top row) and a 2-compartment model (bottom row) following iv bolus injection. Left and middle panels show the plots on a rectilinear and semilogarithmic scale, respectively. Right panels illustrate the relationship between tissue (dash lines) and plasma (solid line) concentrations over time. The right panel for the 1-compartment model shows the concentration–time profile for a typical tissue with a higher concentration than plasma. Note that tissue concentration can be higher, nearly the same, or lower than plasma concentration. Tissue concentration peaks almost immediately, and thereafter declines in parallel with plasma concentration. The right panel for the 2-compartment model shows concentration–time profiles for typical tissues associated with the central (1) and peripheral (2) compartments as represented by short and long dash lines, respectively. For tissues associated with the central compartment, their concentrations decline in parallel with plasma. For tissues associated with peripheral compartment, toxicant concentration rises, while plasma concentration declines rapidly during the initial phase; it then reaches a peak and eventually declines in parallel with plasma in the terminal phase. Elimination rate constant kel for 1-compartment model and the terminal exponential rate constant β are determined from the slope of the log–linear concentration versus time curve. Half-life (T1/2) is the time required for plasma toxicant concentration to decrease by one-half. C0 is the concentration of a toxicant for a 1-compartment model at t = 0 determined by extrapolating the log–linear concentration–time curve to the Y-axis.
The elimination rate constant kel represents the overall elimination of the toxicant, which includes biotransformation, exhalation, and/or excretion pathways. When elimination of a toxicant from the body occurs in an exponential fashion, it signifies a first-order process, that is, the rate of elimination at any time is proportional to the amount of toxicant remaining in the body (ie, body load) at that time. This means that following an iv bolus injection, the absolute rate of elimination (eg, milligrams of toxicant eliminated per minute) continually changes over time. Shortly after introduction of the dose, the rate of toxicant elimination will be at the highest. As elimination proceeds and body load of the toxicant is reduced, the elimination rate will decline in step. As a corollary, it also means that at multiple levels of the toxicant dose, the absolute rate of elimination at corresponding times will be proportionately more rapid at the higher doses. This mode of elimination offers an obvious advantage for the organism to deal with increasing exposure to a toxicant. First-order kinetics occur at toxicant concentrations that are not sufficiently high to saturate either metabolic or transport processes.
In view of the nature of first-order kinetics, kel is said to represent a constant fractional rate of elimination. Thus, if the fractional elimination rate is constant, for example, 0.3 hour–1, the percentage of dose or plasma concentration remaining in the body (C/C0 · 100) and the percentage of the dose or concentration eliminated from the body after 1 hour, that is, 1 – (C/C0 · 100), are 74% and 26%, respectively, regardless of the dose administered (Table 7-1). The reason why the amount remaining at 1 hour is slightly more than 70%, or the amount eliminated is less than 30%, is because the amount in the body declined continually over the 1-hour period and thus the actual amount eliminated is less than anticipated based on the starting amount in the body. Again, this illustrates the fact that under first-order kinetics, the actual elimination rate declines with the depletion in body load, but the percentage eliminated over a given period of time is the same regardless of dose, that is, the fractional rate of elimination of the toxicant remains constant over time after iv injection or any acute exposure. Because a constant percentage of toxicant present in the body is eliminated over a given time period regardless of dose or the starting concentration, it is more intuitive and convenient to refer to an elimination half-life (T1/2), that is, the time it takes for the original blood or plasma concentration to fall by 50% or to eliminate 50% of the original body load. By substituting C/C0 = 0.5 into Equation (7-1), we obtain the following relationship between T1/2 and kel:
where 0.693 is the natural logarithm of 2. Simple calculations reveals that it would take about 4 half-lives for >90% (exactly 93.8%) of the dose to be eliminated, and about 7 half-lives for >99% (exactly 99.2%) elimination. Thus, given the elimination T1/2 of a toxicant, the length of time it takes for near-complete washout of a toxicant after discontinuation of its exposure can easily be estimated. As will be seen in next section, the concept of T1/2 is also applicable to toxicants that exhibit multiexponential kinetics.
Table 7-1Elimination of a Toxicant That Follows First-Order Kinetics (kel = 0.3 h−1) by 1 Hour After iv Administration at Four Different Dose Levels ||Download (.pdf) Table 7-1 Elimination of a Toxicant That Follows First-Order Kinetics (kel = 0.3 h−1) by 1 Hour After iv Administration at Four Different Dose Levels
|DOSE (mg) ||TOXICANT REMAINING (mg) ||TOXICANT ELIMINATED (mg) ||TOXICANT ELIMINATED (% of dose) |
|10 ||7.4 ||2.6 ||26 |
|30 ||22 ||8 ||26 |
|90 ||67 ||23 ||26 |
|250 ||185 ||65 ||26 |
We can infer from the monoexponential decline of blood or plasma concentration that the toxicant equilibrates very rapidly between blood and the various tissues relative to the rate of elimination, such that extravascular equilibration is achieved nearly instantaneously and maintained thereafter. Depiction of the body system by a one-compartment model does not mean that the concentration of the toxicant is the same throughout the body, but it does assume that the changes that occur in the plasma concentration reflect proportional changes in tissue toxicant concentrations (Fig. 7-2 upper, right panel). In other words, toxicant concentrations in tissues are expected to decline with the same elimination rate constant or T1/2 as in plasma; tissue and plasma concentrations should decline in parallel.
After rapid iv administration of some toxicants, the semilogarithmic plot of plasma concentration versus time does not yield a straight line but a curve that implies more than one dispositional phase (Fig. 7-2). In these instances, it takes some time for the toxicant to be taken up into certain tissue groupings, and to then reach an equilibration with the concentration in plasma; hence, a multicompartmental model is needed for the description of its kinetics in the body (Fig. 7-1). The concept of tissue groupings with distinct uptake and equilibration rates of toxicant becomes apparent when we consider the factors that govern the uptake of a lipid-soluble, organic toxicant. Table 7-2 presents data on the volume and blood perfusion rate of various organs and tissues in a standard size human. From these data and assuming reasonable partitioning ratios of a typical lipid-soluble, organic compound in the various tissue types, we can estimate the uptake equilibration half times of the toxicant in each organ or tissue region during constant, continuous exposure. The results suggest that the tissues can be grouped into rapid-equilibrating visceral organs, moderately slow-equilibrating lean body tissues (mainly skeletal muscle), and very slow-equilibrating body fat; these groupings give rise to three distinct uptake phases, that is, half times of <2 minutes for the rapid-equilibrating tissues, several tens of minutes for the moderately slow-equilibrating lean body tissues, and hours for the very slow-equilibrating body fat. By inference, three distinct phases of washout should also be evident following a brief exposure to the toxicant, such as after a bolus iv injection. The relative prominence of these distributional phases will vary depending on the average lipid solubility of the toxicant in each tissue grouping and any other sequestration and export mechanisms of a toxicant in particular tissues (eg, tight binding to tissue proteins, active influx into or efflux out of the tissue cell types), as well as the competing influence of elimination from the visceral organs. For example, very rapid metabolism or excretion at the visceral organs would limit distribution into the slow or very slow tissue groupings. Also, there are times when equilibration rates of a toxicant into visceral organs overlaps with lean body tissues, such that the distribution kinetics of a toxicant into these two tissue groupings become indistinct with respect to the exponential decline of plasma concentration, in which case two instead of three tissue groupings may be observed. The concept of tissue groupings with respect to uptake or washout kinetics serves to justify the seemingly simplistic, yet pragmatic, mathematical description of extravascular distribution by the classic two- or three-compartment models.
Table 7-2Prediction of Equilibration Half-Times for Tissues in the Groupings of Highly Perfused Visceral Tissues, Poorly Perfused Lean Tissues, and Adipose Tissues for a Lipid-Soluble, Organic Toxicant With Assumed Typical Tissue-to-Blood Partitioning Ratios (P)* ||Download (.pdf) Table 7-2 Prediction of Equilibration Half-Times for Tissues in the Groupings of Highly Perfused Visceral Tissues, Poorly Perfused Lean Tissues, and Adipose Tissues for a Lipid-Soluble, Organic Toxicant With Assumed Typical Tissue-to-Blood Partitioning Ratios (P)*
|GROUPING ||TISSUE ||PERFUSION (L/min)† ||VOLUME (L)† ||EQUILIBRATION T1/2(min)‡ |
|Highly perfused viceral tissues, P = 1 ||Heart ||0.20 ||0.28 ||0.97 |
| ||Lungs ||5.0 ||1.1 ||0.15 |
| ||Liver, Gut ||1.4 ||1.6 ||0.79 |
| ||Kidneys ||1.1 ||0.35 ||0.22 |
| ||Brain ||0.70 ||1.4 ||1.4 |
|Poorly perfused tissues, P = 0.5 ||Muscle (resting) ||0.75 ||30 ||14 |
| ||Skin (cool weather) ||0.30 ||7.7 ||9.0 |
|Adipose tissue, P = 5 ||— ||0.20 ||14 ||243 |
Plasma concentration–time profile of a toxicant that exhibits multicompartmental kinetics can be characterized by multiexponential equations. For example, a two-compartment model can be represented by the following biexponential equation:
where A and B are coefficients in units of toxicant concentration, and α and β are the respective exponential constants for the initial and terminal phases in units of reciprocal time. The initial (α) phase is often referred to as the distribution phase, and terminal (β) phase as the postdistribution or elimination phase. The lower, middle panel of Fig. 7-2 shows a graphical resolution of the two exponential phases from the plasma concentration–time curve. It should be noted that the α constant is the slope of the residual log-linear plot and not the initial slope of the decline in the observed plasma toxicant concentration, that is, the initial rate of decline in plasma concentration approximates, but is not exactly equal, the α rate constant in Equation (7-4).
It should be noted that distribution into and out of the tissues and elimination of the toxicant are ongoing at all times, that is, elimination does occur during the “distribution” phase, and distribution between compartments is still ongoing during the “elimination” phase. As illustrated in Fig. 7-3, the dynamics of a multicompartmental system is such that the governing factor, that is, be it intercompartmental distribution or elimination, for each of the phases depends on the relative magnitude of the rate constants for intercompartmental exchange and the elimination process. In the usual case, that of rapid distribution and relatively slow elimination (left panel of Fig. 7-3), the initial rapid phase indeed reflects extravascular distribution, and the later, slow phase reflects elimination after a dynamic equilibration between tissue and blood is attained. There are cases where distribution into some tissue group is much slower than elimination. Then, the initial phase could largely reflect elimination, and the later phase is controlled by the slow redistribution of the toxicant from the tissues associated with the peripheral compartment to the central compartment, where it is eliminated (ie, washout in the terminal phase is rate-limited by redistribution from the peripheral compartment). The aminoglycoside antibiotic gentamicin is a case in point (Schentag and Jusko, 1977). Following an iv injection of gentamicin in patients with normal renal function, serum gentamicin concentration exhibits biphasic kinetics. Serum concentration of gentamicin initially falls very quickly with a half-life of around 2 hours reflecting rapid excretion by the kidneys; a slow terminal phase does not emerge until serum concentration has fallen to less than 10% of initial concentration. The terminal half-life of serum gentamicin is in the range of 4 to 7 days. This protracted terminal half-life reflects the slow turnover of gentamicin sequestered in the kidneys. In fact, repeated administration of gentamicin leads to accumulation of gentamicin in the kidneys, which is a risk factor for its nephrotoxicity. Because of the interplay of distribution and elimination kinetics, it has been recommended that multiphasic disposition should be simply described as consisting of early and late or rapid and slow phases; mechanistic labels of distribution and elimination should be applied with some caution. Lastly, the initial phase may last for only a few minutes or for hours. Whether multiphasic kinetics becomes apparent depends to some extent on how often and when the early blood samples are obtained, and on the relative difference in the exponential rate constants between the early and later phases. If the early phase of decline in toxicant concentration is considerably more rapid than the later phase or phases, the timing of blood sampling becomes critical in the ability to resolving two or more phases of washout.
Effects of interplay between kinetics of distribution and elimination processes on time course of exchange between body compartments and removal from the body for toxicants whose disposition conforms to a 2-compartment model. Left panel depicts the more common scenario of rapid distribution between compartments relative to elimination, in which case elimination from the body occurs largely during the terminal phase when a dynamic equilibration between the central and peripheral compartment has been reached. Accordingly, half-life of the terminal phase is an appropriate measure of elimination. Right panel depicts the scenario of very slow distribution relative to elimination, in which case a substantial (>90% of dose) loss of toxicant occurs during the initial phase. The terminal phase reflects the slow redistribution of the toxicant sequestered in the peripheral site to the central site where it can be eliminated (ie, washout is rate-limited by redistribution). Under this scenario, the initial phase reflects elimination kinetics, whereas the terminal phase reflects tissue distribution kinetics. (Adapted from Tozer and Rowland, 2006.)
Sometimes three or even 4 exponential terms are needed to fit a curve to the plot of log C versus time. Such compounds are viewed as displaying characteristics of three- or four-compartment open models. The principles underlying such models are the same as those applied to the two-compartment open model, but the mathematics is more complex and beyond the scope of this chapter.
Apparent Volume of Distribution
For a one-compartment model, a toxicant is assumed to equilibrate between plasma and tissues immediately following its entry into the systemic circulation. Thus, a consistent relationship should exist between plasma concentration and the amount of toxicant in each tissue and, by extension, to the entire amount in the body or body burden. The apparent volume of distribution (Vd) is defined as the proportionality constant that relates the total amount of the toxicant in the body to its concentration in plasma, and typically has units of liters or liters per kilogram of body weight (Rowland and Tozer, 2011). Vd is the apparent fluid space into which an amount of toxicant is distributed in the body to result in a given plasma concentration. As an illustration, envision the body as a tank containing an unknown volume (L) of well-mixed water. If a known amount (mg) of dye is placed into the water, the volume of that water can be calculated indirectly by determining the dye concentration (mg/L) that resulted after the dye has fully dispersed, that is, by dividing the amount of dye added to the tank by the resultant concentration of the dye in water. Analogously, the apparent volume of distribution of a toxicant in the body is determined after iv bolus administration, and is mathematically defined as the quotient of the amount of toxicant in the body and its plasma concentration. Vd is calculated as
where Doseiv is the iv dose or known amount of toxicant in the body at time zero, β is the elimination rate constant, and Display Formula is the area under the toxicant concentration versus time curve from time zero to infinity. Display Formula is estimated by numerical methods, the most common one being the trapezoidal rule (Gibaldi and Perrier, 1982). The product, β · Display Formula, in unit of concentration, is the theoretical concentration of toxicant in plasma if dynamic equilibration were achievable immediately after introduction of the toxicant into the systemic circulation. For a 1-compartment model, immediate equilibration of the toxicant between plasma and tissues after an acute exposure does hold true, in which case Vd can be calculated by a simpler and more intuitive equation
where C0 is the concentration of toxicant in plasma at time zero. C0 is determined by extrapolating the plasma disappearance curve after iv injection to the zero time point (Fig. 7-2, upper, middle panel).
For the more complex multicompartmental models, Vd is calculated according to Equation (7-5) that involves the computation of area under the toxicant concentration–time curve. Moreover, the concept of an overall apparent volume of distribution is strictly applicable to the terminal exponential phase when equilibration of the toxicant between plasma and all tissue sites are attained. This has led some investigators to refer to the apparent volumes of distribution calculated by Equation (7-5) as Vβ (for a two-compartment model) or Vz (for a general multicompartmental model); the subscript designation refers to the terminal exponential phase (Gibaldi and Perrier, 1982). It should also be noted that when Equation (7-6) is applied to the situation of a multicompartmental model, the resultant volume is the apparent volume of the central compartment, often times referred to as Vc. By definition, Vc is the proportionality constant that relates the total amount of the toxicant in the central compartment to its concentration in plasma. It has limited utility, for example, it can be used to calculate an iv dose of the toxicant to target an initial plasma concentration.
Vd is appropriately called the apparent volume of distribution because it does not correspond to any real anatomical volumes. The magnitude of the Vd term is toxicant-specific and represents the extent of distribution of toxicant out of plasma and into extravascular sites (Table 7-3). Equation (7-7) provides a more intuitive interpretation of an apparent volume of distribution:
where Vp is the plasma volume and the Σ represents the summation of the apparent volume of each tissue region (t,i) as represented by the product of Pt,i (ie, partition ratio or tissue-to-plasma concentration ratio at dynamic equilibrium) and Vt,i (ie, anatomical volume of tissue). At one extreme, a toxicant that predominantly remains in the vasculature will have a low Vd that approximates the volume of blood or plasma, that is, the minimum Vd for any toxicant is the plasma volume. For toxicants that distribute extensively into extravascular tissues, Vd exceeds physiological fluid spaces, such as plasma or blood volume, interstitial fluid, or extracellular fluid. A toxicant that is highly sequestered in tissues (ie, high Pt,i) can have a volume of distribution larger than average body size (>1 L/kg). The mechanisms of tissue sequestration include partitioning of a toxicant into tissue fat, high-affinity binding to tissue proteins, trapping in specialized organelles (eg, pH trapping of amine compounds in acidic lysozomes), and concentrative uptake by active transporters. In fact, the equation below is an alternate form of Equation (7-7), which features the interplay of binding to plasma and tissue proteins in determining the partitioning of a toxicant in that only free or unbound drug can freely diffuse across membrane and cellular barriers.
where fup is the unbound fraction of toxicant in plasma and fut,i is the effective unbound fraction in a tissue region. Here, Pt,i is governed by the ratio fup / fut,i. Thus, a toxicant that has high affinity for plasma proteins (eg, albumin and/or α1-acid glycoprotein) relative to tissue proteins has a restricted distribution volume; for example, the anticoagulant warfarin with a plasma-bound fraction of 0.995 or an fup of 0.005 (Table 7-3). On the contrary, a toxicant that has a high affinity for tissue proteins and lesser affinity for plasma proteins can have a very high Vd. For example, the tricyclic antidepressant nortriptyline has a good affinity for plasma proteins with a bound fraction of 0.92 or an fup of 0.08; however, binding of nortriptyline to tissues constituents is so much higher such that it has a Vd of 18 times the body weight in adult humans.
Table 7-3Volume of Distribution (Vd) and Unbound or Free Fraction in Plasma (fup) for Several Drugs That Are of Clinical Toxicological Interest ||Download (.pdf) Table 7-3 Volume of Distribution (Vd) and Unbound or Free Fraction in Plasma (fup) for Several Drugs That Are of Clinical Toxicological Interest
|CHEMICAL ||Vd (L/kg) ||fup |
|Chloroquine ||~200 ||~0.45 |
|Nortriptyline ||18 ||0.080 |
|Oxycodone ||2.0 ||0.55 |
| ||Body size = 1.0 || |
|Acetaminophen ||0.95 ||<0.20 |
| ||Total body water = 0.60 || |
|Phenytoin ||0.64 ||0.11 |
| ||Extracellular fluid = 0.27 || |
|Warfarin ||0.14 ||0.005 |
|Epoetin alfa ||0.05 ||- |
| ||Plasma volume = 0.045 || |
In addition to its value as a parameter to indicate the extent of extravascular distribution of a toxicant, Vd also has practical utility. Once the Vd for a toxicant is known, it can be used to estimate the amount of toxicant remaining in the body at any time when the plasma concentration at that time is known, that is,
where XB is the amount of toxicant in the body or body burden, and C is the plasma concentration at a time after equilibration between tissue and plasma had been achieved.
Toxicants are cleared from the body via various routes, for example, excretion by the kidneys into urine or via bile into the intestine ending in feces, biotransformation by the liver, or exhalation by the lungs. Clearance is an important toxicokinetic parameter that relates the rate of toxicant elimination from the whole body in relation to plasma concentration (Wilkinson, 1987). Although terminal T1/2 is reflective of the rate of removal of a toxicant from plasma or blood, as was explained earlier, it is also subject to the influence of distributional processes, especially for toxicants exhibiting multicompartmental kinetics. Clearance, on the contrary, is a parameter that solely represents the rate of toxicant elimination; it is not influenced by extravascular distribution.
A formal definition of total body clearance is the ratio of overall elimination rate of a toxicant divided by plasma concentration at any time after an acute exposure or during repetitive or continuous exposure (ie, elimination rate/C); in effect, clearance expresses the overall efficiency of the elimination mechanisms. Because this measure of elimination efficiency is in reference to blood or plasma toxicant concentration, it is also named blood or plasma clearance. High values of clearance indicate efficient and generally rapid removal, whereas low clearance values indicate slow and less efficient removal of a toxicant from the body. By definition, clearance has the units of flow (eg, mL/min or L/h). In the classic compartmental context, clearance is portrayed as the apparent volume containing the toxicant that is cleared per unit of time. After iv bolus administration, total body clearance (Cl) is calculated by
Clearance can also be calculated if the volume of distribution and elimination rate constants are known within the context of a compartmental model; that is, Cl = Vd · kel for a 1-compartment model and Cl = Vb · β for a 2-compartment model. It should be noted that the preceding relationship is a mathematical derivation and does not imply that clearance is dependent on the distribution volume (see later section for further commenting). A clearance of 100 mL/min can be visualized as having 100 mL of blood or plasma completely cleared of toxicant in each minute during circulation.
The biological significance of total body clearance is better appreciated when it is recognized that it is the sum of clearances by individual eliminating organs (ie, organ clearances):
where Clr depicts renal, Clh hepatic, and Clp pulmonary clearance. Each organ clearance is in turn determined by blood perfusion flow through the organ (i) and the fraction of toxicant in the arterial inflow that is irreversibly removed, that is, the extraction fraction (Ei). Various organ clearance models have been developed to provide quantitative description of clearance that is related to blood perfusion flow (Qi), free fraction in blood (fub), and intrinsic clearance (Clint,i) (Wilkinson, 1987). For example, for hepatic clearance (Clh), if the delivery of the toxicant to its intracellular site of removal is rate-limited by liver blood flow (Qh) and the toxicant is assumed to have equal, ready access to all the hepatocytes within the liver (ie, the so-called well-stirred model):
where Clint,h is the hepatic intrinsic clearance that embodies a combination of factors that determine the access of the toxicant to the enzymatic sites (eg, plasma protein binding and sinusoidal membrane transport) and the biochemical efficiency of the metabolizing enzymes (eg, Vmax/KM for an enzyme obeying Michaelis kinetics). Clint,h would also embody canalicular transport activity if the toxicant is subject to biliary excretion. Equation (7-12) dictates that hepatic clearance of a toxicant from the blood is bounded by either liver blood flow or intrinsic clearance (ie, fub · Clint,h). Note that when fub · Clint,h is very much higher than Qh, Eh approaches unity (ie, near-complete extraction during each passage of toxicant through the hepatic sinusoid or high extraction) and Clh approaches Qh. Put in another way, hepatic clearance cannot exceed the hepatic blood flow rate even if the intrinsic rate of metabolism in the liver is more rapid than the rate of hepatic blood flow because the rate of overall hepatic clearance is limited by the delivery of the toxicant to the metabolic enzymes in the liver via the blood. At the other extreme, when fub · Clint,h is very much lower than Qh, Eh becomes quite small (ie, low extraction) and Clh nearly equals fub · Clint,h. In this instance, the intrinsic clearance is relatively inefficient; hence, alteration in liver blood flow would have little, if any, influence on liver clearance of the toxicant. Thus, the concept of clearance is grounded in the physiological and biochemical mechanisms of an eliminating organ.
Relationship of Elimination Half-Life to Clearance and Volume
Elimination half-life (T1/2) is probably the most easily understood toxicokinetic concept and is an important parameter as it measures the persistence of a toxicant following discontinuation of exposure. As will be seen in a later section, elimination half-life also governs the rate of accumulation of a toxicant in the body during continuous or repetitive exposure. Elimination half-life is dependent on both volume of distribution and clearance. T1/2 can be calculated from Vd and Cl:
The above relationship among T1/2, Vd, and Cl is another illustration that care should be exercised in interpretation of data when relying upon T1/2 as the sole representation of elimination of a chemical in toxicokinetic studies, since T1/2 is influenced by both the volume of distribution for the toxicant and the rate by which the toxicant is cleared from the blood. For a fixed Vd, T1/2 decreases as Cl increases because the chemical is being removed from this fixed volume faster as clearance increases (Fig. 7-4). Conversely, as Vd increases, T1/2 increases for a fixed Cl since the volume of fluid that must be cleared of chemical increases but the efficiency of clearance does not.
The dependence of T1/2 on Vd and Cl. Consider the hypothetical case of a toxicant that is eliminated entirely by renal excretion. Renal Cl values of 60, 130, and 650 mL/min represent partial reabsorption, glomerular filtration, and tubular secretion, respectively. Values for Vd of 3, 18, and 40 L represent approximate volumes of plasma water, extracellular fluid and total body water, respectively, for an average-sized person. Note that T1/2 depends on both Cl and Vd.
Absorption and Bioavailability
For most chemicals in toxicology, exposure occurs mostly via extravascular routes (eg, inhalation, dermal, or oral), and absorption into the systemic circulation is often incomplete. The extent of absorption of a toxicant can be experimentally determined by comparing the plasma toxicant concentration after iv and extravascular dosing. Because iv dosing assures full (100%) delivery of the dose into the systemic circulation, the AUC ratio should equal the fraction of extravascular dose absorbed and reaches the systemic circulation in its intact form, and is called bioavailability (F). In acute toxicokinetic studies, bioavailability can be determined by using different iv and non-iv doses according to the following equation, provided that the toxicant does not display dose-dependent or saturable kinetics.
where AUCnon-iv, AUCiv, Dosenon-iv, and Doseiv are the respective area under the plasma concentration versus time curves and doses for non-iv and iv administration. Bioavailabilities for various chemicals range in value between 0 and 1. Complete availability of chemical to the systematic circulation is demonstrated by F = 1. When F < 1, less than 100% of the dose is able to reach the systemic circulation. Because the concept of bioavailability is judged by how much of the dose reaches the systemic circulation, it is often referred to as systemic availability. Systemic availability is determined by how well a toxicant is absorbed from its site of application and any intervening processes that could remove or inactivate the toxicant between its point of entry and the systemic circulation. Specifically, systemic availability of an orally administered toxicant is governed by its absorption at the gastrointestinal barrier, metabolism within the intestinal mucosa, and metabolism and biliary excretion during its first transit through the liver. Metabolic inactivation and excretion of the toxicant at the intestinal mucosa and the liver prior to its entry into the systemic circulation is called presystemic extraction or first-pass effect. The following equation accounts for the action of absorption and sequential first-pass extraction at the intestinal mucosa and the liver as determinants of the bioavailability of a toxicant taken orally:
where fg is the fraction of the applied dose that is released and absorbed across the mucosal barrier along the entire length of the gut, Em is the extent of loss due to metabolism within the gastrointestinal mucosa, and Eh is the loss due to metabolism or biliary excretion during first-pass through the liver. Note that Eh in this equation is same as the hepatic extraction Eh defined in Equation (7-12), which refers to hepatic extraction of a toxicant during recirculation. This means that low oral bioavailability of a chemical can be attributed to multiple factors. The chemical may be absorbed to a limited extent because of low aqueous solubility preventing its effective dissolution in the gastrointestinal fluid or low permeability across the brush-border membrane of the intestinal mucosa. Extensive degradation by metabolic enzymes residing at the intestinal mucosa and the liver may also minimize entry of the chemical in its intact form into the systemic circulation.
The rate of absorption of a toxicant via an extravascular route of entry is another critical determinant of outcome, particularly in acute exposure situations. As shown in Fig. 7-5, slowing the absorption rate of a toxicant, while maintaining the same extent of absorption or bioavailability, leads to a delay in time to peak plasma concentration (Tp) and a decrease in the maximum concentration (Cmax) (compare case 2 to case 1). The converse is true; accelerating absorption shortens Tp and increases Cmax. The dependence of Tp and Cmax on absorption rate has obvious implication in the speed of onset and maximum toxic effects following exposure to a chemical. Case 3 in Fig. 7-5 illustrates the peculiar situation when the absorption rate is so much slower than the elimination rate (eg, ka << kel for a one-compartment model). The terminal rate of decline in plasma concentration reflects the absorption rate constant, instead of the elimination rate constant; that is, the washout of toxicant is rate-limited by slow absorption until the applied dose is completely absorbed or removed, beyond which time the toxicant remaining in the body will be cleared according the elimination rate-constant. This means that continual absorption of a chemical can affect the persistence of toxic effect following an acute exposure, and that it is important to institute decontamination procedure quickly after overdose or accidental exposure to a toxicant. This is especially a consideration in occupational exposure via dermal absorption following skin contact with permeable industrial chemicals.
Influence of absorption rate on the time to peak (Tp) and maximum plasma concentration (Cmax) of a toxicant that exhibit 1-compartment kinetics. The left panel illustrates the change in plasma concentration–time curves as the first-order absorption rate constant (ka) decreases, while keeping the extent of absorption or bioavailability (F), hence the AUC, constant. The right panel displays the same curves in a semilogarithmic plot. Time to peak plasma concentration shows a progressive delay as ka decreases, along with a decrease in Cmax. In case 1 and 2, the terminal decline in plasma concentration is governed by elimination half-life; hence, the parallel decline in the semilogarithmic plot. In case 3 where ka << kel, the absorption becomes so slow that decline in plasma concentration in the terminal phase reflects the absorption half-life, that is, washout of toxicant is rate-limited by absorption. Accordingly, the terminal decline is slower than in case 1 and 2.
The toxicity of a chemical is in some cases attributed to its biotransformation product(s). Hence, the formation and subsequent disposition kinetics of a toxic metabolite is at times of interest. In biological monitoring, urinary excretion of a signature metabolite often serves as a surrogate measure of exposure to the parent compound (see later section). As expected, the plasma concentration of a metabolite rises as the parent drug is transformed into the metabolite. Once formed, the metabolite is subject to further metabolism to a nontoxic byproduct or undergoes excretion via the kidneys or bile; hence at some point in time, the plasma metabolite concentration peaks and falls thereafter. Fig. 7-6 illustrates the plasma concentration–time course of a primary metabolite in relation to its parent compound under contrasting scenarios. The left panel shows the case when the elimination rate constant of the metabolite is much greater than the overall elimination rate constant of the parent compound (ie, km >> kp). The terminal decline of the metabolite parallels that of the parent compound; the metabolite is cleared as quickly as it is formed or its washout is rate-limited by conversion from the parent compound. The right panel shows the opposite case when the elimination rate constant of the metabolite is much lower than the overall elimination rate constant of the parent compound (ie, km << kp). The slower terminal decline of the metabolite compared to the parent compound simply reflects a longer elimination half-life of the metabolite. It should also be noted that the Display Formula of the metabolite relative to the parent compound is dependent on the partial clearance of the parent drug to the metabolite and clearance of the derived metabolite. A biologically active metabolite assumes toxicological significance when it is the major metabolic product and is cleared much less efficiently than the parent compound.
Plasma concentration–time course of a primary metabolite and its parent compound under contrasting scenarios: when elimination of the metabolite is much more rapid than its formation (km >> kp lower left panel) and when elimination of the metabolite is much slower than its formation (km << kp, lower right panel). Semilogarithmic plots are shown to compare the slope of the terminal decline of parent compound and its metabolite. The top panel shows the model for conversion of the parent compound to a single metabolite. Note that the elimination rate constant for the parent compound (kp) includes both the rate constants for metabolism and extra-metabolic routes of elimination. km is the elimination rate constant for the derived metabolite. When km >> kp, the terminal decline of the metabolite parallels that of the parent compound, that is, metabolite washout is rate-limited by its formation. When km << kp, the terminal decline of metabolite concentration is much slower than that of parent compound, that is, metabolite washout is rate-limited by its elimination.
As already mentioned, the distribution and elimination of most chemicals occurs by first-order processes. Under first-order elimination kinetics, the elimination rate constant, apparent volume of distribution, clearance, and half-life are expected not to change with increasing or decreasing dose (ie, dose independent). As a result, a semilogarithmic display of plasma concentration versus time over a range of doses shows a set of parallel plots. Furthermore, plasma concentration at a given time or the Display Formula is strictly proportional to dose; for example, a 2-fold change in dose results in an exact 2-fold change in plasma concentration at a given time after dosing or Display Formula. However, for some toxicants, as the dose of a toxicant increases, its volume of distribution and/or clearance may change, as shown in Fig. 7-7. This is generally referred to as nonlinear or dose-dependent kinetics. Biotransformation, active transport processes, and protein binding have finite capacities and can be saturated. For instance, most metabolic enzymes operate in accordance to Michaelis–Menten kinetics (Gibaldi and Perrier, 1982). As the dose is escalated and concentration of a toxicant at the site of metabolism approaches or exceeds the KM (substrate concentration at one-half Vmax, the maximum metabolic capacity), the increase in rate of metabolism becomes less than proportional to the dose and eventually approaches a maximum at exceedingly high doses. The transition from first-order to saturation kinetics is important in toxicology because it can lead to prolonged persistence of a compound in the body after an acute exposure and excessive accumulation during repeated exposures. Some of the salient characteristics of nonlinear kinetics include the following: (1) the decline in the concentrations of the chemical in the body is not exponential; (2) Display Formula is not proportional to the dose; (3) Vd, Cl, kel (or β), or T1/2 change with increasing dose; (4) the composition of excretory products changes quantitatively or qualitatively with the dose; and (5) dose–response curves show an abruptly steeper change in response to an increasing dose, starting at the dose at which saturation effects become evident.
Changes in Vd, Cl, and T1/2 following first-order toxicokinetics (left panels) and following saturable toxicokinetics (right panels). Vertical dashed lines in the right panels represent point of departure from first-order to saturation toxicokinetics. Pharmacokinetic parameters for toxicants that follow first-order toxicokinetics are independent of dose. When plasma protein binding or elimination mechanisms are saturated with increasing dose, pharmacokinetic parameter estimates become dose-dependent. Vd may increase, for example, when protein binding is saturated, allowing more free toxicant to distribute into extravascular sites. Conversely, Vd may decrease with increasing dose if tissue protein binding saturates. Then, toxicant may redistribute more freely back into plasma. When toxicant concentrations exceed the capacity for biotransformation by metabolic enzymes, overall clearance of the toxicant decreases. These changes may or may not have effects on T1/2 depending upon the magnitude and direction of changes in both Vd and Cl.
Inhaled methanol provides an example of a chemical whose metabolic clearance changes from first-order kinetics at low level exposures to zero-order kinetics at near toxic levels (Burbacher et al., 2004). Fig. 7-8 shows predicted blood methanol concentration–time profiles in female monkeys followed a 2-hour controlled exposure in an inhalation chamber at 2 levels of methanol vapor, 1200 and 4800 ppm. Blood methanol kinetics at 1200 ppm exposure follows typical first-order kinetics. At 4800 ppm, methanol metabolism is fully saturated, such that the initial decline in blood methanol following the 2-hour exposure occurs at a constant rate (ie, a fixed decrease in concentration per unit time independent of blood concentration), rather than a constant fractional rate. As a result, a rectilinear plot of blood methanol concentration versus time yields an initial linear decline, whereas a convex curve is observed in the semilogarithmic plot (compare left and right panels of Fig. 7-8). In time, methanol metabolism converts to first-order kinetics when blood methanol concentration falls below KM (ie, the Michaelis constant for the dehydrogenase enzyme), at which time blood methanol shows an exponential decline in the rectilinear plot and a linear decline in the semilogarithmic plot. Moreover, Fig. 7-8 shows the greater than proportionate increase (ie, >4-fold) in Cmax and Display Formula as the methanol vapor concentration is raised from 1200 to 4800 ppm. It should be noted that a constant T1/2 or kel does not exist during the saturation regimen; it varies depending upon the saturating methanol dose.
Predicted time course of blood methanol concentration following a 120-minute exposure to 1200 and 4800 ppm of methanol vapor in the female monkey based on the toxicokinetic model reported by Burbacher et al. (2004) that features a saturable (Michaelis–Menten type) metabolic clearance. The left panel is a rectilinear plot of the simulated blood methanol concentration–time curves at the 2 exposure levels. The right panel shows the same simulations in a semilogarithmic plot. The washout of blood methanol following the 120-minute inhalation exposure at 1200 ppm follows a typical concave or exponential pattern in the rectilinear plot (left panel) and is linear in a semilogarithmic plot (right panel). The postexposure profile at 4800 ppm shows a linear segment during the first 120 minutes of washout and becomes exponential thereafter in the rectilinear plot (left panel). The linear segment reflects saturation of alcohol dehydrogenase, which is the principal enzyme responsible for the metabolism of methanol. The in vivo kM for this simulation was set at 32.7 μg/mL. At concentrations well above kM, the kinetics approach zero-order kinetics. At concentrations below kM, washout kinetics become first-order with a half-life of about 60 minutes. The right panel shows a characteristic convex semilogarithmic plot for the initial phase of zero-order kinetics and becomes linear as expected for first-order kinetics when the concentration falls below kM. It should also be noted that the maximum blood methanol following 4800 ppm exposure is predicted to be 5.9-times higher than that following 1200 ppm. Also, the area under the blood concentration–time curve (AUC) from time zero to 480 minutes at 4800 ppm exposure is 8-fold higher than at 1200 ppm. Under linear kinetics, the increase in maximum blood methanol and AUC ought to be proportionate to the dose increase, that is, a 4-fold increment. Here, we observe a more than proportionate increase in blood methanol concentration in relation to the dose, which is another hallmark of saturation kinetics.
In addition to the complication of dose-dependent kinetics, there are toxicants whose clearance kinetics changes over time (ie, time-dependent kinetics). A common cause of time-dependent kinetics is autoinduction of xenobiotic metabolizing enzymes; that is, the substrate is capable of inducing its own metabolism through activation of gene transcription. The classic example of autoinduction is with the antiepileptic drug, carbamazepine. Daily administration of carbamazepine leads to a continual increase in clearance and shortening in elimination half-life over the first few weeks of therapy (Bertilsson et al., 1986).
Accumulation During Continuous or Intermittent Exposure
It stands to reason that continual or chronic exposure to a chemical leads to its cumulative intake and accumulation in the body. For a chemical that follows first-order elimination kinetics, the elimination rate increases as the body burden increases. Therefore, at a fixed level of continuous exposure, accumulation of a toxicant in the body eventually reaches a point when the intake rate of the toxicant equals its elimination rate, from thereon the body burden stays constant. This is referred to as the steady state. Fig. 7-9 illustrates the rise of toxicant concentration in plasma over time during continuous exposure and the eventual attainment of a plateau or the steady state. Steady-state concentration of a toxicant in plasma (Css) is related to the intake rate (Rin) and clearance of the toxicant.
Accumulation of plasma toxicant concentration over time during constant, continuous exposure as a function of exposure level (left panel) and elimination half-life (right panel). These simulations are based on a 1-compartment model at a constant apparent volume of distribution. Case 1 serves as the reference with an elimination half-life set equal to 1 arbitrary time unit. In the left panel, which illustrates accumulation of toxicant as a function of exposure level, exposure level is raised by 2-fold in case 2 and lowered by 50% in case 3. The changes in eventual steady state concentration are proportional to the changes in exposure level, that is, increased by 2-fold in case 2 and decreased by 50% in case 3. During continuous exposure, 50% of steady state is achieved in 1 half-life. Near plateau or steady state (>90%) is reached after 4 half-lives. Since the elimination half-life is constant across cases 1–3 in the left panel, the time it takes to attain 50% of steady state concentration (see arrows) is the same, that is, 1 time unit. Right panel illustrates the influence of elimination half-life and clearance on accumulation at a fixed constant rate of exposure. Case 4 represents a 50% decrease in clearance and a corresponding 2-fold increase in elimination half-life compared to case 1. Case 5 represents a 2-fold increase in clearance and a corresponding 50% decrease in elimination half-life. Changes in both the time to attain steady state and the steady state concentration are evident. In case 4, the steady state concentration increased by 2-fold as a result of a 50% reduction in clearance, and the time to achieved 50% of steady state increased by 2-fold as a result of the prolonged elimination half-life. In case 5, the steady state concentration is reduced by 50%, while the time to reach 50% steady state is shortened by 50%.
For a one-compartment model, an exponential rise in plasma concentration is expected during continuous exposure and the time it takes for a toxicant to reach steady state is governed by its elimination half-life. It takes 1 half-life to reach 50%, 4 half-lives to reach 93.8%, and 7 half-lives to reach 99.2% of steady state. Time to attainment of steady state is not dependent on the intake rate of the toxicant. The left panel of Fig. 7-9 shows the same time to 50% steady state at 3 different rates of intake; on the contrary, the steady-state concentration is strictly proportional to the intake rate. The change in clearance of a toxicant often leads to a corresponding change in elimination half-life (see right panel of Fig. 7-9), in which case both the time to reach and magnitude of steady state concentration are altered. The same steady state principle applies to toxicants that exhibit multicompartmental kinetics; except that, accumulation occurs in a multiphasic fashion reflective of the multiple exponential half-lives for intercompartmental distribution and elimination. Typically, the rise in plasma concentration is relatively rapid at the beginning, being governed by the early (distribution) half-life, and becomes slower at later times when the terminal (elimination) half-life takes hold.
The concept of accumulation applies to intermittent exposure as well. Fig. 7-10 shows a typical occupational exposure scenario to volatile chemicals at the workplace over the course of a week. Whether accumulation occurs from day to day and further from week to week depends on the intervals between exposure and the elimination half-life of the chemicals involved. For a chemical with relatively short half-life compared to the interval between work shifts and the “exposure holiday” over the weekend, little accumulation is expected. In contrast, for a chemical with elimination half-life approaching or exceeding the between-shift intervals (>12–24 hours), progressive accumulation is expected over the successive workdays. Washout of the chemical may not be complete over the weekend and result in a significant carry forward of body burden into the next week. It should also be noted that the overall internal exposure as measured by the AUC over the cycle of a week is dependent on the toxicant clearance.
Simulated accumulation of plasma concentration from occupational exposure over the cycle of a work week for 2 industrial chemicals with short and long elimination half-lives. Shading represents the exposure period during the 8-hour workday, Monday through Friday. Intake of the chemical into the systemic circulation is assumed to occur at a constant rate during exposure. Exposure is negligible over the weekend. For the chemical with the short elimination half-life of 8 hours, minimal accumulation occurs from day to day over the workdays. Near-complete washout of the chemical is observed when work resumes on Monday (see arrow). For the chemical with the long elimination half-life of 24 hours, progressive accumulation is observed over the 5 workdays. Washout of the longer half-life chemical over the weekend is incomplete; a significant residual is carried into the next work week. Because of its lower clearance, the overall AUC of the long half-life chemical over the cycle of a week is higher by 3-fold.
For many chemicals, blood or plasma chemical concentration versus time data can be adequately described by a one- or two-compartment, classic pharmacokinetic model when basic assumptions are made (eg, instantaneous mixing within compartments and first-order kinetics). In some instances, more sophisticated models with increased numbers of compartments will be needed to describe blood or plasma toxicokinetic data; for example, if the chemical is preferentially sequestered and turns over slowly in select tissues. The parameters of the classic compartmental models are usually estimated by statistical fitting of data to the model equations using nonlinear regression methods. A number of software packages are available for both data fitting and simulations with classic compartmental models; examples include WinNonlin (Pharsight Corp, Palo Alto, CA), SAAM II (The Epsilon Group, Charlottesville, VA), ADAPT II (University of Southern California, Los Angeles, CA), and PK Solutions (Summit Research Services, Montrose, CO).
Knowledge of toxicokinetic data and compartmental modeling are useful in deciding what dose or dosing regimen of chemical to use in the planning of toxicology studies (eg, targeting a toxic level of exposure), in choosing appropriate sampling times for biological monitoring, and in seeking an understanding of the dynamics of a toxic event (eg, what blood or plasma concentrations are achieved to produce a specific response, how accumulation of a chemical controls the onset and degree of toxicity, and the persistence of toxic effects following termination of exposure).