Chapter 22. Physiologic Pharmacokinetic Models, Mean Residence Time, and Statistical Moment Theory

The study of pharmacokinetics describes the absorption, distribution, and elimination of the active drug and metabolites in quantitative terms (see Chapter 1). Ideally, a pharmacokinetic model uses the observed time course for drug concentration in the body and, from this data, obtains various pharmacokinetic parameters to predict drug dosing outcomes, pharmacodynamics, and toxicity.

In developing a model, certain underlying assumptions are made by the pharmacokineticist as to the type of pharmacokinetic model, the order of the rate process, the blood flow to a tissue, the method for the estimation of the plasma or tissue volume, and other factors. Even with a more general approach such as model-independent analysis, first-order drug elimination is often assumed in the calculation of In selecting a model for data analysis, the pharmacokineticist may choose more than one method of modeling, depending on many factors, including experimental condition, study design, and completeness of data. The goodness-of-fit to the model and the desired pharmacokinetic parameters are other considerations. Each estimated pharmacokinetic parameter has an inherent variability because of the variability of the biologic system and the observed data. Moreover, because pharmacokinetic studies are performed on a limited number of subjects, the estimated pharmacokinetic parameters may not be representative of the entire population.

In spite of difficulties in the construction of these pharmacokinetic models, such models have been extremely useful in describing the time course of drug action, improving drug therapy by enhancing drug efficacy, and minimizing adverse reactions through more accurate dose regimens. Pharmacokinetic models are also applied to the development of new drug delivery systems. Three main types of pharmacokinetic models—physiologic, compartment, and statistical moment approach models—are discussed in this chapter.

The human body is composed of organ systems containing living cells bathed in an extracellular aqueous fluid (see Chapter 10). Both drugs and endogenous substances, such as hormones, nutrients, and oxygen, are transported to the organs by the same network of blood vessels (arteries). The drug concentration within a target organ depends on plasma drug concentration, the rate of blood flow to an organ, and the rate of drug uptake into the tissue. Physiologically, uptake (accumulation) of drug by organ tissues occurs from the extracellular fluid, which equilibrates rapidly with the capillary blood in the organ. Some drugs cross the plasma membrane into the interior fluid (intracellular water) of the cell (Fig. 22-1).

In addition to drug accumulation, some organs of the body are involved in drug elimination, either by excretion (eg, kidney) or by metabolism (eg, liver). The elimination of drug by an organ may be described by drug clearance in the organ (see Chapters 6 and 11). The liver is an example of an organ with drug metabolism and drug uptake (accumulation). Physiologic pharmacokinetic models consider all processes of drug uptake and elimination.

Drugs are carried by blood flow from the administration (input) site to various body organs, where the drug rapidly equilibrates with the interstitial water in the organ. Physiologic pharmacokinetic models are mathematical models describing drug movement and disposition in the body based on organ blood flow and the organ spaces penetrated by the drug. In its simplest form, a physiologic pharmacokinetic model considers the drug to be blood flow limited. Drugs are carried to organs by arterial blood and leave organs by venous blood (Fig. 22-2). In such a model, transmembrane movement of drug is rapid, and the capillary membrane does not offer any resistance to drug permeation. Uptake of drug into the tissues is rapid, and a constant ratio of drug concentrations between the organ and the venous blood is quickly established. This ratio is the tissue/blood partition coefficient:

where P is the partition coefficient.

The magnitude of the partition coefficient can vary depending on the drug and on the type of tissue. Adipose tissue, for example, has a high partition for lipophilic drugs. The rate of drug carried to a tissue organ and tissue drug uptake depend on the rate of blood flow to the organ and the tissue/blood partition coefficient, respectively.

The rate of blood flow to the tissue is expressed as Qt (mL/min), and the rate of change in the drug concentration with respect to time within a given tissue organ is expressed as

where Cart is the arterial blood drug concentration and Cven is the venous blood drug concentration. Qt is blood flow and represents the volume of blood flowing through a typical tissue organ per unit of time.

If drug uptake occurs in the tissue, the incoming concentration, Cart, is higher than the outgoing venous concentration, Cven. The rate of change in the tissue drug concentration is equal to the rate of blood flow multiplied by the difference between the blood drug concentrations entering and leaving the tissue organ. In the blood flow–limited model, drug concentration in the blood leaving the tissue and the drug concentration within the tissue are in equilibrium, and Cven may be estimated from the tissue/blood partition coefficient in Equation 22.1. Substituting in Equation 22.3 with Cven = Ctissue/Ptissue yields

Equation 22.4 describes drug distribution in a noneliminating organ or tissue group. For example, drug distribution to muscle, adipose tissue, and skin is represented in a similar manner by Equations 22.5, 22.6, and 22.7, respectively, as shown below. For tissue organs in which drug is eliminated (Fig. 22-3), parameters representing drug elimination from the liver (kLIV) and kidney (kKID) are added to account for drug removal through metabolism or excretion. Equations 22.8 and 22.9 are derived similarly to those for the noneliminating organs above.

Removal of drug from any organ is described by drug clearance (Cl) from the organ. The rate of drug elimination is the product of the drug concentration in the organ and the organ clearance.

Rate of drug elimination

The rate of drug elimination may be described for each organ or tissue (Fig. 22-4).

where LIV = liver, SP = spleen, GI = gastrointestinal tract, KID = kidney, LU = lung, FAT = adipose, SKIN = skin, and MUS = muscle.

The mass balance for the rate of change in drug concentration in the blood pool is

Lung perfusion is unique because the pulmonary artery returns venous blood flow to the lung, where carbon dioxide is exchanged for oxygen and the blood becomes oxygenated. The blood from the lungs flows back to the heart (into the left atrium) through the pulmonary vein, and the quantity of blood that perfuses the pulmonary system ultimately passes through the remainder of the body. In describing drug clearance through the lung, perfusion from the heart (right ventricle) to the lung is considered venous blood (Fig. 22-4). Therefore, the terms in Equation 22.11 describing lung perfusion are reversed compared to those for the perfusion of other tissues. With some drugs, the lung is a clearing organ besides serving as a merging pool for venous blood. In those cases, a lung clearance term could be included in the general model.

After intravenous drug administration, drug uptake in the lungs may be very significant if the drug has high affinity for lung tissue. If actual drug clearance is at a much higher rate than the drug clearance accounted for by renal and hepatic clearance, then lung clearance of the drug should be suspected, and a lung clearance term should be included in the equation in addition to lung tissue distribution.

The system of differential equations used to describe the blood flow–limited model is usually solved through computer programs. The Runge–Kutta method is often used in computer methods for using a series of differential equations. Because of the large number of parameters involved in the mass balance, more than one set of parameters may fit the experimental data. This is especially true with human data, in which many of the organ tissue data items are not available. The lack of sufficient tissue data sometimes leads to unconstrained models. As additional data become available, new or refined models are adopted. For example, methotrexate was initially described by a flow-limited model, but later work described the model as a diffusion-limited model.

Because invasive methods are available for animals, tissue/blood ratios or partition coefficients can be determined accurately by direct measurement. Using experimental pharmacokinetic data from animals, physiologic pharmacokinetic models may yield more reliable predictions.

Physiologic Pharmacokinetic Model with Binding

The physiologic pharmacokinetic model assumes flow-limited drug distribution without drug binding to either plasma or tissues. In reality, many drugs are bound to a variable extent in either plasma or tissues. With most physiologic models, drug binding is assumed to be linear (not saturable or concentration dependent). Moreover, bound and free drug in both tissue and plasma are in equilibrium. Further, the free drug in the plasma and in the tissue equilibrates rapidly. Therefore, the free drug concentration in the tissue and the free drug concentration in the emerging blood are equal:

where fb is the blood free drug fraction, ft is the tissue free drug fraction, Ct is the total drug concentration in tissue, and Cb is the total drug concentration in blood.

Therefore, the partition ratio, Pt, of the tissue drug concentration to that of the plasma drug concentration is

By assuming linear drug binding and rapid drug equilibration, the free drug fraction in tissue and blood may be incorporated into the partition ratio and the differential equations. These equations are similar to those above except that free drug concentrations are substituted for Cb. Drug clearance in the liver is assumed to occur only with the free drug. The inherent capacity for drug metabolism (and elimination) is described by the term Clint (see Chapter 11). General mass balance of various tissues is described by Equation 22.16:

or

For liver metabolism,

The mass balance for the drug in the blood pool is

The influence of binding on drug distribution is an important factor in interspecies differences in pharmacokinetics. In some instances, animal data may predict drug distribution in humans by taking into account the differences in drug binding. For the most part, extrapolations from animals to humans or between species are rough estimates only, and there are many instances in which species differences are not entirely attributable to drug binding and metabolism.

Blood Flow-Limited versus Diffusion-Limited Model

Most physiologic pharmacokinetic models assume rapid drug distribution between tissue and venous blood. Rapid drug equilibrium assumes that drug diffusion is extremely fast and that the cell membrane offers no barrier to drug permeation. If no drug binding is involved, the tissue drug concentration is the same as that of the venous blood leaving the tissue. This assumption greatly simplifies the mathematics involved. Table 22-1 lists some of the drugs that have been described by a flow-limited model. This model is also referred to as the perfusion model. A more complex type of physiologic pharmacokinetic model is called the diffusion-limited model or the membrane-limited model. In the diffusion-limited model, the cell membrane acts as a barrier for the drug, which gradually permeates by diffusion. Because blood flow is very rapid and drug permeation is slow, a drug concentration gradient is established between the tissue and the venous blood (Lutz and Dedrick, 1985). The rate-limiting step of drug diffusion into the tissue depends on the permeation across the cell membrane rather than blood flow. Because of the time lag in equilibration between blood and tissue, the pharmacokinetic equation for the diffusion-limited model is very complicated.

Application and Limitations of Physiologic Pharmacokinetic Models

The physiologic pharmacokinetic model is related to drug concentration and tissue distribution using physiologic and anatomic information. For example, the effect of a change in blood flow on the drug concentration in a given tissue may be estimated once the model is characterized. Similarly, the effect of a change in mass size of different tissue organs on the redistribution of drug may also be evaluated using the system of physiologic model differential equations generated. When several species are involved, the physiologic model may predict the pharmacokinetics of a drug in humans when only animal data are available. Changes in drug–protein binding, tissue organ drug partition ratios, and intrinsic hepatic clearance may be inserted into the physiologic pharmacokinetic model.

Most pharmacokinetic studies are modeled based on blood samples drawn from various venous sites after either IV or oral dosing. Physiologists have long recognized the unique difference between arterial and venous blood. For example, arterial tension (pressure) of oxygen drives the distribution of oxygen to vital organs. Chiou (1989) and Mather (2001) have discussed the pharmacokinetic issues when differences in drug concentrations in arterial and venous are considered (see Chapter 10). The implication of venous versus arterial sampling is hard to estimate and may be more drug dependent. Most pharmacokinetic models are based on sampling of venous data. In theory, mixing occurs quickly when venous blood returns to the heart and becomes reoxygenated again in the lung. Chiou (1989) has estimated that for drugs that are highly extracted, the discrepancies may be substantial between actual concentration and concentration estimated from well-stirred pharmacokinetic models.

Interspecies Scaling

Various approaches have been used to compare the toxicity and pharmacokinetics of a drug among different species. Interspecies scaling is a method used in toxicokinetics and the extrapolation of therapeutic drug doses in humans from nonclinical animal drug studies. Toxicokinetics is the application of pharmacokinetics to toxicology and pharmacokinetics for interpolation and extrapolation based on anatomic, physiologic, and biochemical similarities (Mordenti and Chappell, 1989; Bonate and Howard 2000; Mahmood, 2000, 2007; Hu and Hayton, 2001; Evans et al, 2006).

The basic assumption in interspecies scaling is that physiologic variables, such as clearance, heart rate, organ weight, and biochemical processes, are related to the weight or body surface area of the animal species (including humans). It is commonly assumed that all mammals use the same energy source (oxygen) and energy transport systems across animal species (Hu and Hayton, 2001). Interspecies scaling uses a physiologic variable, y, that is graphed against the body weight of the species on log–log axes to transform the data into a linear relationship (Fig. 22-5). The general allometric equation obtained by this method is

where y is the pharmacokinetic or physiologic property of interest, b is an allometric coefficient, W is the weight or surface area of the animal species, and a is the allometric exponent. Allometry is the study of size.

Both a and b vary with the drug. Examples of various pharmacokinetic or physiologic properties that demonstrate allometric relationships are listed in Table 22-2. In the example shown in Fig. 22-5, methotrexatevolume of distribution is related to body weight B of five animal species by the equation Vβ = 0.859B0.918.

The allometric method gives an empirical relationship that allows for approximate interspecies scaling based on the size of the species. Not considered in the method are certain specific interspecies differences such as gender, nutrition, pathophysiology, route of drug administration, and polymorphisms. Some of these more specific cases, such as the pathophysiologic condition of the animal or human, may preclude pharmacokinetic or allometric predictions.

Interspecies scaling has been refined by considering the aging rate and life span of the species. In terms of physiologic time, each species has a characteristic life span, its maximum life-span potential (MLP), which is controlled genetically (Boxenbaum, 1982). Because many energy-consuming biochemical processes, including drug metabolism, vary inversely with the aging rate or life span of the animal, the allometric approach has been used for drugs that are eliminated mainly by hepatic intrinsic clearance.

Through the study of various species in handling several drugs that are metabolized predominantly by the liver, some empirical relationships regarding drug clearance of several drugs have been related mathematically in a single equation. For example, drug hepatic intrinsic clearance of biperiden in rat, rabbit, and dog was extrapolated to humans (Nakashima et al, 1987). Equation 22.20 describes the relationship between biperiden intrinsic clearance with body weight and MLP:

where MLP is the maximum life-span potential of the species, B is the body weight of the species, and Clint is the hepatic intrinsic clearance of the free drug.

Although further model improvements are needed before accurate prediction of pharmacokinetic parameters can be made from animal data, some interesting results were obtained by Sawada et al (1985) on nine acid and six basic drugs. When interspecies differences in protein–drug binding are properly considered, the volume of distribution of many drugs may be predicted with 50% deviation from experimental values (Table 22-3).

The application of MLP to pharmacokinetics has been described by Boxenbaum (1982). Initially, hepatic intrinsic clearance was considered to be related to volume or body weight. Indeed, a plot of the log drug clearance versus body weight for various animal species resulted in an approximately linear correlation (ie, a straight line). However, after correcting intrinsic clearance by MLP, an improved log–linear relationship was achieved between free drug Clint and body weight for many drugs. A possible explanation for this relationship is that the biochemical processes, including Clint, in each animal species are related to the animal's normal life expectancy (estimated by MLP) through the evolutionary process. Animals with a shorter MLP have higher basal metabolic rates and tend to have higher intrinsic hepatic clearance and thus metabolize drugs faster. Boxenbaum (1982, 1983) postulated a constant “life stuff” in each species, such that the faster the life stuff is consumed, the more quickly the life stuff is used up. In the fourth-dimension scale (after correcting for MLP), all species share the same intrinsic clearance for the free drug.

Extensive work with caffeine in five species (mouse, rat, rabbit, monkey, and humans) by Bonati and associates (1985) verified this approach. Caffeine is a drug that is metabolized predominantly by the liver. For caffeine,

where B is body weight, L is liver weight, and Q is blood flow.

Hepatic clearance for the unbound drug did not show a direct correlation among the five species. After intrinsic clearance was corrected for MLP (calculation based on brain weight), an excellent relationship was obtained among the five species (Fig. 22-6).

More recently, the subject of interspecies scaling was investigated using Cl values for 91 substances for several species by Hu and Hayton (2001). These investigators used Y = a (BW)b in their analysis, similar to Equation 22.19 above but with different symbols: Y = biological variable dependent on the body weight of the species, a = allometric coefficient, b = allometric exponent, and BW = body weight of the species. One issue discussed by Hu and Hayton is the uncertainty in the allometric exponent (b) of xenobiotic clearance (CL). Published literature has focused on whether the basal metabolic rate scale is a 2/3 or 3/4 power of the body mass (BW). When the uncertainty in the determination of a b value is relatively large, a fixed-exponent approach might be feasible according to Hu and Hayton. In this regard, 0.75 might be used for substances that are eliminated mainly by metabolism or by metabolism and excretion combined, whereas 0.67 might apply for drugs that are eliminated mainly by renal excretion. The researchers pointed out that genetic (intersubject) difference may be a limitation for using a single universal constant.

Brightman et al (2006) demonstrated the application of a PK-PD model, based on human parameters to estimate plasma pharmacokinetics of xenobiotics in humans. The model was parameterized through an optimization process, using a training set of in vivo data taken from the literature. On average, the vertical divergence of the predicted plasma concentrations from the observed data was 0.47 log units, on a semilog concentration-time plot. They also evaluated the method against other predictive methods that involve scaling from in vivo animal data. In terms of predicting human clearance for the test set, the model was found to match or exceed the performance of three published interspecies scaling methods, which tend to give overprediction. The article concludes that the generic physiologically based pharmacokinetic model is a means of integrating readily determined in vitro and/or in silico data, and useful for predicting human xenobiotic kinetics in drug discovery.

Physiologic versus Compartment Approach

Compartmental models represent a simplified kinetic approach to describe drug absorption, distribution, and elimination (see Chapters 3 and 4). The major advantage of compartment models is that the time course of drug in the body may be monitored quantitatively with a limited amount of data. Generally, only plasma drug concentrations and limited urinary drug excretion data are available. Compartmental models have been applied successfully to prediction of the pharmacokinetics of the drug and the development of dosage regimens. Moreover, compartmental models are very useful in relating plasma drug levels to pharmacodynamic and toxic effects in the body.

The simplicity and flexibility of the compartment model is the principal reason for its wide application. For many applications, the compartmental model may be used to extract some information about the underlying physiologic mechanism through model testing of the data. Thus, compartment analysis may lead to a more accurate description of the underlying physiologic processes and the kinetics involved. In this regard, compartmental models are sometimes misunderstood, overstretched, and even abused. For example, the tissue drug levels predicted by a compartment model represent only a composite pool for drug equilibration between all tissue and the circulatory system (plasma compartment). However, extrapolation to a specific tissue drug concentration is inaccurate and analogous to making predictions without experimental data. Although specific tissue drug concentration data are missing, many investigators may make general predictions about average tissue drug levels.

Compartment models account accurately for the mass balance of the drug in the body and the amount of drug eliminated. Mass balance includes the drug in the plasma, the drug in the tissue pool, and the amount of drug eliminated after dosage administration. The compartment model is particularly useful for comparing the pharmacokinetics of related therapeutic agents. In the clinical pharmacokinetic literature, drug data comparisons are based on compartment models. Though alternative pharmacokinetic models have been available for approximately 20 years, the simplicity of the compartment model allows easy tabulation of parameters such as VDSS, alpha t1/2, and beta t1/2. The alternative pharmacokinetic models, including the physiologic and statistical moment (mean residence time) approaches, are used much less frequently, even though a substantial body of data has been generated using both of these models.

In spite of these advantages, the compartmental model is generally regarded as somewhat empirical and lacking physiologic relevance. Many disease-related changes in pharmacokinetics are the result of physiologic changes, such as impairment of blood flow or a change in organ mass. These pathophysiologic changes are better evaluated using a physiologic-based pharmacokinetic model.

Because of its simplicity, the compartment model often serves as a “first model” that requires further refinement in order to describe the physiologic and drug distribution processes in the body accurately. The physiologic pharmacokinetic model—which accounts for processes of drug distribution, drug binding, metabolism, and drug flow to the body organs—is much more realistic. Disease-related changes in physiologic processes are more readily related to changes in the pharmacokinetics of the drug. Furthermore, organ mass, volumes, and blood perfusion rates are often scalable, based on size, among different individuals and even among different species. This allows a perturbation in one parameter and the prediction of changing physiology on drug distribution and elimination.

The physiologic pharmacokinetic model may also be modified to include a specific feature of a drug. For example, for an antitumor agent that penetrates into the cell, both the drug level in the interstitial water and the intracellular water may be considered in the model. Blood flow and tumor size may even be included in the model to study any change in the drug uptake at that site.

The physiologic pharmacokinetic model can calculate the amount of drug in the blood and in any tissues for any time period if the initial amount of drug in the blood is known and the dose is given by IV bolus. In contrast, the tissue compartment in the compartmental model is not related to any actual anatomic tissue groups. The tissue compartment is needed when the plasma drug concentration data are fitted to a multicompartment model. In theory, when tissue drug concentration data are available, the multiple-compartment models may be used to fit both tissue and plasma drug data together, including the drug concentration in a specific tissue. In such a case, the compartment model would mimic the system of equations used in the physiologic model, except that in place of blood flows, transfer constants would be used to describe the mass transfer in the model. The latter approach would probably, at best, yield less useful information than that obtained from the physiologic model.

Physiologic Pharmacokinetic Model Incorporating Hepatic Transporter-Mediated Clearance

It is now well recognized that drug transporters play important roles in the processes of absorption, distribution, and excretion and should be accounted for in models. Predicting human drug disposition, especially when involving hepatic transport, is difficult during drug development. However, drug transport may be a critical process in overall drug disposition in the body such that without a realistic description of transport processes in the body, model accuracy may be deficient. Watanabe et al (2009) describe a model with hepatobiliary excretion mediated by transporters, organic anion-transporting polypeptide (OATP) 1B1 and multidrug resistance–associated protein (MRP) 2, for the HMG-CoA reductase inhibitor drug, pravastatin. While the classical blood flow–based physiologic pharmacokinetic models developed 40 years ago using systems of differential equations that are still useful in describing the mass balance and transfer of drug within major organs, the models are inadequate in light of new discoveries in molecular biology and pharmacogenomics. Drug disposition and drug targeting are better understood based upon using influx/efflux and binding mechanisms in micro structures such as interior cellular structures, membrane transporters, surface receptors, genomes, and enzymes. The liver is a complex organ intimately connected to drug transport and bile movement. Compartment concepts are needed to track the mass of drug transfer in and out of those fine structures as shown by the example in Fig. 22-7. Human liver microsomes are used to help predict the metabolic clearance of drugs in the body.

The PBPK model with pravastatin (Watanabe et al, 2009) is used to evaluate the concentration-time profiles for drugs in the plasma and peripheral organs in humans using physiological parameters, subcellular fractions (cells lysed and contents fractionated based on density), and drug-related parameters (unbound fraction and metabolic and membrane transport clearances extrapolated from in vitro experiments). The principle of the prediction was as follows. First, subcellular fractions were obtained by comparing in vitro and invivo parameters in rats. Then, the in vitro human parameters were extrapolated in vivo using the subcellular fractions obtained in rats. Pravastatin was selected as the model compound because many studies have investigated the mechanisms involved in the drug disposition in rodents, and clinical data after intravenous and oral administration are available.

When multiple drug metabolites are involved, the physiologic model of the cascade events can be quite complicated and an abbreviated approach may be used. St-Pierre et al (1988) developed a simple one-compartment open model, based on the liver as the only organ of drug disappearance and metabolite formation. The model was used to illustrate the metabolism of a drug to its primary, secondary, and tertiary metabolites. The model encompassed the cascading effects of sequential metabolism (Fig. 22-8). The concentration-time profiles of the drug and metabolites were examined for both oral and intravenous drug administration. Formation of the primary metabolite from drug in the gut lumen, with or without further absorption, and metabolite formation arising from first-pass metabolism of the drug and the primary metabolite during oral absorption were considered. Mass balance equations, incorporating modifications of the various absorption and conversion rate constants, were integrated to provide the explicit solutions.

After an intravenous bolus drug dose (D0), the drug molecules distribute throughout the body. These molecules stay (reside) in the body for various time periods. Some drug molecules leave the body almost immediately after entering, whereas other drug molecules leave the body at later time periods. The term mean residence time (MRT) describes the average time for all the drug molecules to reside in the body. MRT may be considered also as the mean transit time or mean sojourn time. The residence time for the drug molecules in the dose may be sorted into groups i (i = 1, 2, 3, …, m) according to their residing time. The total residence time is the summation of the number of molecules in each group i multiplied by the residence time, ti, for each group. The summation of ni (number of molecules in each group) is the total number of molecules, N. Thus, MRT is the total residence time for all molecules in the body divided by the total number of molecules in the body, as shown in Equation 22.23:

where ni is the number of molecules and ti is the residence time of the ith group of molecules.

The drug dose (mg) may be converted to the number of molecules by dividing the dose (mg) by 1000 and the molecular weight of the drug to obtain the number of moles of drug, and then multiplying the number of moles of drug by 6.023 × 1023 (Avogadro's number) to obtain the number of drug molecules. For convenience, Equation 22.23 may be written in terms of milligrams (instead of molecules) by substitution of ni with Dei × f, where Dei is the number of drug molecules (as mg) leaving the body with residence time ti (i = 1, 2, 3, …, m). The f is a conversion factor. The number of molecules or milligrams of drug cancels out in Equation 22.24, showing that MRT is independent of mass:

where Dei(i = 1, 2, 3, …, m) is the amount of drug (mg) in the ith group with residence time ti.

Drug molecules may have a residence time ranging from values near zero (eg, 0.1, 0.2) to very large values (100, 1000, 10,000). The number of i groups may be large and the summation approach to calculate MRT will be only an approximation. Also, for the summation process to be accurate, data must be collected continuously in order not to miss any groups. Integration is an accurate method that replaces summation when the data or function needs to be continuously summed over time.

Mean Residence Time—IV Bolus Dose

The drug concentration in the body after an IV bolus injection for a drug that follows the pharmacokinetics of a one-compartment model is given by

where VD is the apparent volume of distribution, k is the first-order elimination rate constant, and t is the time after the injection of the drug. The drug exit rate was generated in Table 22-4 with a numerical example until most drug was eliminated.

The rate of change in the amount of drug in the body with respect to time (dDp/dt) reflects the rate at which the drug molecules leave the body at any time t. Although all drug molecules enter the body at the same time, the exit time, or the residing time, for each molecule is different. Equation 22.27 is obtained by taking the derivative of Equation 22.26, with all the drug molecules exiting the body from t = 0 to ∞ (Table 22-4):

Alternatively, the rate of drug molecules exiting at any time t is given by

Rearranging yields

At any time t, dDe molecules exit. Therefore, multiplying Equation 22.29 by t on both sides yields the residence for each molecule exiting with a residence time t. Summation of the residence time for each drug molecule, and division by the total number of molecules, estimates the mean residence time (Equation 22.30):

As shown in Equation 22.30, the MRT is related to the product of the elimination rate constant k and the function describing drug elimination in the body. MRT is the integrated normalized form of the differential function representing drug amount (or concentration) in the body. The term differential probability is used to reflect that the function is a probability density function (PDF), which represents the residence time probability of a molecule in the population. The mean residence time is the normalized (divided by D0) differential of the function governing drug elimination in the body. When a function is normalized, it becomes dimensionless, without units.

Equation 22.30 was derived in terms of amount of drug. Because D0 = C0pVD, substituting for D0 with C0pVD into the right side of Equation 22.30 yields

Equation 22.32 may be used to determine MRT directly or may be rearranged to Equation 22.33 by dividing the numerator and denominator by k to yield a moment equation.

The plasma concentration equation [function f(t)] multiplied by time and integrated from 0 = ∞ gives a term called the first moment of the plasma drug curve. The denominator is the area under the curve, The is equal to or D0/V0k. Because C0p = D0/VD, the denominator of Equation 22.33 is the of the time–concentration curve (= D0/kVD); see Chapter 6. Equation 22.33 is used in pharmacokinetics to determine MRT; the equation is abbreviated in the literature as shown in Equation 22.34:

where AUMC is the area under the (first) moment-versus-time curve from t = 0 to infinity. AUC is the area under plasma time-versus-concentration curve from t = 0 to infinity. AUC is also known as the zero moment curve.Table 22-5 shows how summation may be used to calculate MRT from generated data when the function is known.

The example is basically a verification of Equation 22.30. The approximation shows that when the function is not known, MRT may be estimated from the rate of drug excreted or from the plasma drug concentration–time curve. In practice, unless the entire dose is known, there is no assurance that all drug molecules are excreted through the plasma compartment if multiple compartments are involved.

MRT may also be calculated using the compartmental approach by considering the MRT for a drug after IV bolus injection as the reciprocal of the elimination rate constant, k. In this case, MRT is inversely related to the elimination constant, and MRT = 1/k. Therefore, MRT = 1/0.231 h–1 = 4.329 h. MRT was estimated earlier as 4.309 hours using the moment method. The slightly smaller value for the MRT here is due to approximation in the summation of the moment area and AUC.

Using the method of MRT = AUMC/AUC, MRT may be estimated using the AUC calculated from the Cp versus time curve (Table 22-4). Alternatively, the MRT can be estimated accurately using integration when the function that describes plasma drug concentrations is known (Table 22-5). A plot of De and Cp versus time, and the moment curve De × t versus t, are shown in Fig. 22-9. For drugs that follow the kinetics of a one-compartment model with drug elimination only from the plasma, MRT may be calculated by either the area method or from the first moment curve. In data analysis, the function that governs drug disposition is generally not known, and the MRT is generally calculated from the plasma drug concentration–time curve.

Statistical moment theory provides a unique way to study time-related changes in macroscopic events. A macroscopic event is considered the overall event brought about by the constitutive elements involved. For example, in chemical processing, a dose of tracer molecules may be injected into a reactor tank to track the transit time (residence time) of materials that stay in the tank. The constitutive elements in this example are the tracer molecules, and the macroscopic events are the residence times shared by groups of tracer molecules. Each tracer molecule is well mixed and distributes noninteractively and randomly in the tank. In the case of all the molecules that exit from the tank, the rate of exit of tracer molecules (–dDe/dt) divided by D0 yields the probability of a molecule having a given residence time t (Fig. 22-10C). A mathematical formula describing the probability of a tracer molecule exited at any time is a probability density function. MRT is merely the expected value or mean of the distribution.

MRT provides a fundamentally different approach than classical pharmacokinetic models, which involve the concept of dose, half-life, volume, and concentration. The classical approach is macroscopic and does not account for the observation that molecules in a cluster move individually through space and are more appropriately tracked as statistical distribution based on residence-time considerations. Consistent with the concept of mass and the dynamic movement of molecules within a region or “space,” MRT is an alternative concept to describe how drug molecules move in and out of a system. The concept is well established in chemical kinetics, where the relationships between MRT and rate constants for different systems are known.

In the last two decades, MRT has been well characterized for various pharmacokinetic models, although MRT application is less developed in pharmacodynamics. In the future, the residence-time concept will probably be applied further in pharmacodynamics if a useful model can be developed relating MRT to the intensity and duration of drug action that yields more insight than a conventional pharmacokinetic model. Empirically, it has been observed that changing from a slow rate of IV injection of a fast-acting drug may result in a quite different response in a subject compared to rapid injection of the same drug solution. This observation has been attributed to a transient change in concentration at the site. Alternatively, the response may be explained by modeling the MRT change of the drug at the site.

In the one-compartment model, AUC divided by C0 also yields MRT (432.896/100 = 4.33 hours). This example is another way to calculate MRT, as will be demonstrated later. MRT may also be computed by integrating f(t)/C0 (Fig. 22-10B). However, if f(t) represents a two- or multicompartment function, the computed MRT using this method is only for the central compartment, whereas using the AUMC/AUC approach leads to an MRT for the body (a larger value). The latter method treats the molecules as a single population within the body, tracking them only as they exit and without supposing any knowledge of molecular exchanges between the plasma and tissue compartments.

In the previous discussion, Equation 22.34 was derived to estimate MRT without applying the concept of probability density functions. However, the equation may be rearranged in the form of a PDF, as in Equation 22.31. This result is plotted in Fig. 22-10C for comparison. The moment theory facilitates calculation of MRT and related parameters from the kinetic function, as discussed below.

A probability density function f(t) multiplied by tm and integrated over time yields the moment curve (Equation 22.35). The moment curve shows the characteristics of the distribution.

where f(t) is the probability density function, t is time, and m is the mth moment.

For example, when m = 0, substituting for m = 0 yields Equation 22.36, called the zero moment,μ0:

If the distribution is a true probability function, the area under the zero moment curve is 1.

Substituting into Equation 22.35 with m = 1, Equation 22.37 gives the first moment μ1

The area under the curve f(t) times t is called the AUMC, or the area under the first moment curve. The first moment, μ1, defines the mean of the distribution.

Similarly, when m = 2, Equation 22.35 becomes the second moment,μ2:

where μ2 defines the variance of the distribution. Higher moments, such as μ3 or μ4, represent skewness and kurtosis of the distribution. Equation 22.35 is therefore useful in characterizing families of moment curves of a distribution.

The principal use of the moment curve is the calculation of the MRT of a drug in the body. The elements of the distribution curve describe the distribution of drug molecules after administration and the residence time of the drug molecules in the body.

In Equation 22.31, the plasma equation, f(t) = , was converted to a PDF [ie, f(t) = ke−kt]. It can be shown that μ0 for this function = 1 (total probability adds up to 1 by summing the zero moment), and the mean of the function is the area under the first moment curve (the mean is the MRT).

By comparison, Equation 22.25 is not a true PDF, because its mean is not given by the AUMC, and its AUC is not 1. Nonetheless, Equation 22.34 may be used to calculate MRT independent of the PDF concept, although some confusion over its application appears in the literature.

MRT for Multicompartment Model with Elimination from the Central Compartment

The moment theory provides a means for calculating MRT for the body from plasma drug concentration data obtained for drugs that follow one-compartment models. In this section, the MRT is determined for a drug that has a plasma (central) compartment and one or more tissue or peripheral compartments. The assumptions are (1) that the drug is eliminated only from the central compartment and (2) that all drug is eliminated by a linear process (constant clearance). The MRT for a multicompartment model drug is the summation of the residence time of the drug in each compartment,

where MRTpi represents MRT in the ith peripheral or tissue compartments and MRTc = MRT for the central compartment.

MRT for the body may be calculated from the plasma concentration curve using AUMC/AUC as discussed earlier, because the body (including all compartments) may be treated as a single compartment.

MRTc is known as the mean residence time or mean transit time for the central compartment. This is calculated by AUC/C0p, where AUC is the area under the plasma drug concentration curve and C0p is the initial plasma drug concentration at time = 0.

Derivation of MRTC

Let

• De = amount of drug eliminated at time t
• D0 = dose or drug at time zero
• Dp = amount of drug in the plasma compartment from which drug is eliminated at time t
• Cp = drug concentration in the plasma compartment (volume, VD)

At time t, dDe units of drug are eliminated, the residence time is tdDe, and integrating from 0 to yields the total residence time (see the numerator of Equation 22.41). Integrating De will yield the total units of drug eliminated (see the denominator of Equation 22.41), and MRT is obtained by dividing the total residence time by the total units of drugs. If elimination occurs only from the central compartment, then at any instant dt, dDe = –dDp.

Since

When a drug is distributed between a central and one or more peripheral compartments, the peripheral compartment is not available for sampling. Therefore, both the drug and tissue MRT have to be determined from the plasma data. If a two-compartment model is involved, the peripheral MRT is given by

The Model-Independent and Model-Dependent Nature of MRT

MRT evaluated from AUMC/AUC assumes that most drugs are excreted through the central compartment or that they are metabolized in highly vascular tissues that kinetically are considered part of the central compartment. For drugs eliminated through tissues that are not part of the central compartment, the MRT calculated by AUMC/AUC is smaller than that calculated by considering both peripheral and central compartment elimination.

Compare models A and B generated by Nakashima and Benet (1988) in Fig. 22-13 and Table 22-8. Both models A and B have identical rate constants except that model B has an elimination rate constant, k20, from the peripheral compartment. MRT calculated using the new approach, referred to as MRT (new), is 1.687 hours for model A and 2.212 hours for model B. The new equation for MRT (new) of model B is

where k20 = the elimination rate for drug eliminated in compartment 2; V2 = the distribution volume of compartment 2; E2 = the sum of rate constants exiting from compartment 2 (in model B, E2 = k20 + k21); Cl = total body clearance; MRTc = MRT from the central compartment; MRTp = MRT from the peripheral or tissue compartment, also referred to as MRTt; MRT = AUMC/AUC; and MRT (new) = MRT as calculated with correction for peripheral elimination.

The equation also applies for other multicompartment models. For a three-compartment model, Fig. 22-13 (model E), a third term is added to Equation 22.47 to reflect drug exiting in compartment 3, as in Equation 22.48.

From Table 22-8, MRT (new) calculated using Equations 22.47 and 22.48 is model dependent. The original MRT calculated using AUMC/AUC yields the same MRT value of 1.687 hours for models A, B, and C. The original method, frequently quoted as a model-independent method for calculating MRT from plasma data, in effect, treats the body as a single unit from which drugs are eliminated from the plasma pool regardless of the true nature of the model. Unfortunately, it is not possible to know, from plasma data alone, whether drug elimination occurs in the peripheral compartment. Therefore, it is not possible to interpret unambiguously the calculated MRT parameters without making some assumptions. Benet and Galeazzi (1979) and Nakashima and Benet (1988) showed that MRT is related to VSS and clearance (Equation 22.49), as illustrated in Table 22-8. The clearances among models A, B, and C are identical, showing that clearance is site independent and may be calculated through MRT. However, the three-compartment model clearances were clearly different.

MRT is useful in calculating the steady-state volume of distribution and additional parameters in compartmental and other models. In the compartmental models, MRT may give some idea of how long the drug molecules stay in the peripheral compartment. For example, for the drug digoxin, MRT for the body is 49.5 hours; for the central and peripheral compartments, MRT is 3.68 hours and 45.8 hours as calculated (Veng-Pedersen, 1989). The peripheral MRT is the mean total time the drug molecules spend in the peripheral tissue, considering the first entry as well as possibly subsequent entries into the peripheral tissue from the central, general systemic circulation. An overview of MRT values for various pharmacokinetic models is given in Table 22-9.

Mean Absorption Time (MAT) and Mean Dissolution Time (MDT)

After IV bolus injection, the rate of systemic drug absorption is zero, because the drug is placed directly into the bloodstream. The MRT calculated for a drug after IV bolus injection basically reflects the elimination rate processes in the body. After oral drug administration, the MRT is the result of both drug absorption and elimination. The relationship between the mean absorption time, MAT, and MRT is given by

For a one-compartment model, MRTIV = 1/k:

In some cases, IV data are not available and an MRT for a solution may be calculated. The mean dissolution time (MDT), or in vivo mean dissolution time, for a solid drug product is

MDT reflects the time for the drug to dissolve in vivo. Equation 22.52 calculates the in vivo dissolution time for a solid drug product (tablet, capsule) given orally. MDT has been evaluated for a number of drug products. MDT is most readily estimated for the drugs that follow the kinetics of a one-compartment model. MDT is considered model independent because MRT is model independent. MDT has been used to compare the in vitro dissolution versus in vivo bioavailability for immediate-release and extended-release drug products (see Chapters 14 and 17). Even with complete experimental data, the parameters obtained are quite dependent on the method of computation method employed.

Several objectives should be considered when using mathematical models to study rate processes (eg, pharmacokinetics of a drug). The primary objective in developing a model is to conceptualize the kinetic process in a quantitative manner that can be tested experimentally (see Chapter 1). A model that cannot be tested is weak and will not improve or yield new knowledge about the process. In contrast, a wrong model that can be tested may be useful if the proposed hypotheses and its subsequent rejection leads to the correct model. To be statistically vigorous, a hypothesis or model is tested with the null hypothesis (H0) (Appendix A). Only after rejection of the null hypothesis (tested beyond chance probability) is the hypothesis accepted. When the null hypothesis is rejected, the probability (eg, p < 0.05) means that the chance of error is less than 5% and the hypothesis or model is accepted. In fitting data using linear regression, the correlation of coefficient, r2, is calculated, where r2 is an indication of how well the data are predicted by the model. For example, r = 0.9 or r2 = 0.81 indicates that 81% of the data agree with the model. The r2 is not always a very good criterion. The sum of the squared differences (between the observed and predicted data) is a better criterion.

Adequate experimental design and the availability of valid data are important considerations in model selection and testing. For example, the experimental design should determine whether a drug is being eliminated by saturable (dose-dependent) or simple linear kinetics. A plot of metabolic rate versus drug concentration can be used to determine dose dependence, as in Figure 22-14. Metabolic rate can be measured at various drug concentrations using an in vitro system (see Chapter 11). In Fig. 22-14, curve B, saturation occurs at higher drug concentration.

For illustration, consider the drug concentration–time profile for a drug given by IV bolus. The combined metabolic and distribution processes may result in profiles like those in Fig. 22-15. Curve A represents a slow initial decline due to saturation and a faster terminal decline as drug concentration decreases. Curve C represents a dominating distributive phase masking the effect of nonlinear metabolism. Finally, a combination of A and C may approximate a rough overall linear decline (curve B). Notice that the drug concentration–time profile is shared by many different processes and that the goodness-of-fit is not an adequate criterion for adopting a model. For example, concluding linear metabolism based only on curve B would be incorrect. Contrary to common belief, complex models tend to mask opposing variables that must be isolated and tested through better experimental designs. In this case, a constant infusion until steady-state experiment would yield information on saturation without the influence of initial drug distribution.

The use of pharmacokinetic models has been critically reviewed by Rescigno and Beck (1987) and by Riggs (1963). These authors emphasize the difference between model building and simulation. A model is a secondary system designed to test the primary system (real and unknown). The assumptions in a model must be realistic and consistent with physical observations. On the other hand, a simulation may emulate the phenomenon without resembling the true physical process. A simulation without identifiable support of the physical system does little to aid understanding of the basic mechanism. The computation has only hypothetical meaning.

For example, the data in Fig. 22-16 may be described by three different equations: a straight line, a hyperbola, and an equation with trigonometric functions. Without physical support, the model gives little understanding of the mechanisms involved. Regression or data fitting (computation of parameters of a given equation so that divergence of the data from the equation is minimized) is considered to be modulation. All forms of modulation represent data manipulation and, as such, decrease information. Information comes from observations and observations only. If more points were available in this example, we could distinguish different processes if the data represent a straight line. Simulations within the realm of data allow for the determination of a quantitative relationship that is useful but does not actually increase understanding of the underlying physiologic processes governing the sojourn of the drug in the body.

Data dredging, such as manipulating the data to influence a stated study objective or model, is considered statistically undesirable because it may lead to a biased or a nonobjective analysis of the study. Ideally, statisticians prefer that details such as inclusion and exclusion criteria, the number of subjects per group, and the method of analysis or modeling be designed with sufficient power written into the protocols in advance. Pharmacokineticists are interested in understanding underlying kinetic processes that yield the results (data); statisticians are more interested in knowing that the conclusions drawn are real and not the result of random distribution.

It is perhaps refreshing to reexamine the basic paradigm of pharmacokinetics, or kinetics, which may be described as “observing how a given process changes over time.” For example, by monitoring a change in the drug concentration in a patient over time, drug absorption and elimination from the patient may be modeled. If a statistician wishes to extrapolate the conclusion to a larger population, he or she may choose to enroll a sufficient number of subjects, and consider inclusion or exclusion criteria such as age, sex, other diseases, and genetics.

Basic pharmacokinetic concepts should be simple with few assumptions. The clinician may choose to weigh the risk of extrapolating the data and conclusions from a study with a few patients to a larger patient population. With recent advances in genetics, there are thousands of known genomic and environmental factors that may potentially influence disease and pharmacotherapy. The combination of genetic and environmental factors is very large and may make it very difficult to apply average information to an individual subject (see Chapter 12). However, the concept of adjusting dosing based on individual pharmacokinetic/pharmacodynamic information is still the most reliable approach to optimize dosing and drug efficacy and avoid toxicity. Ultimately, individual pharmacokinetic information will be combined with pharmacogenetic information obtained from the patient to allow the development of even better models to improve drug therapy.

Physiologically based PK-PD models use a system of differential equations to describe drug transfer and accumulation in various tissues or organs in the body. Published data in the physiologic literature regarding size (mass) of organs and blood flow to each organ and body mass are used. Drug elimination in key organs is based on clearance concepts. The generic PK-PD model is useful as a crude model for rough estimates of plasma concentration versus time profiles after IV administration assuming no protein binding, specific transporters, or any unusual drug specific disposition characteristics. Despite this limitation, the PK-PD model is used when no kinetic or efficacy information is available for a drug when given the first time in humans. Typically, animal data about protein binding, drug efflux, and CYPs are used to fill in missing information due to the lack of human tissue data. Drug development methods such as Caco-2 cells, liver microsome metabolism, and binding to human serum proteins are also used to gather in vitro data about the drug in order to improve the accuracy of in vivo projection when using PK models. Toxicology data from animals are also collected to evaluate whether the first dose in humans is safe or not. Subsequently, single-dose and dose escalation studies, done under close medical supervision, will generate the first set of human PK data, which may then be used to refine the PK model in man. MRT (mean residence time) is a statistical approach that treats drug molecules as individual units that move through organ and body spaces according to kinetic principles, and allows independent development of many equations that are familiar to classical kineticists. MRT allows the determination of the time for mean residence of the molecules (eg, dose administered) in the body according to its route of administration. The variance of the residence time can also be determined based on statistical moment theory based on probability density function. The MRT approach allows another way of computing the clearance and volume of distribution of a drug through the derived equations. These equations are preferred by some scientists and are applied to AUC and clearance computation of new drugs. The approach is often referred to as the noncompartmental approach although all computation methods applied some form of interpolation or extrapolation at some stage of data analysis and parameter determination. An important consideration that should be kept in mind is the impact of the assumptions of the method rather than the semantics.

1. After an intravenous dose (500 mg) of an antibiotic, plasma–time concentration data were collected and the area under the curve was computed to be 20 mg/L h. The area was found to be 100 mg/L h2.

   1. What is the mean residence time of this drug?

   2. What is the clearance of this drug?

   3. What is the steady-state volume of distribution of this drug?

2. Why is MRT calculated from moment and AUC curves (ie, ) rather than directly, as the term was defined in Equation 22.43?

3. If the data in Question 1 above are fit to a one-compartment model with an elimination k that is found to be 0.25 hour, MRT may be calculated simply as 1/k. What different assumptions are used in here versus Question 1?

4. What are the principal considerations in interspecies scaling?

5. What are the key considerations in fitting plasma drug data to a pharmacokinetic model?