Compartment models concentration versus time

As with classic compartment pharmacokinetic models, PBPK models can be used to simulate drug plasma concentration versus time profiles. However, PBPK models differ from classic PK models in that they include separate compartments for tissues involved in absorption, distribution, metabolism and elimination connected by physiologically based descriptions of blood flow (Figure 10.1). 

In this chapter, only the one- and two-compartment models following IV dosing were described. Other models with extravascular dosing have an additional compartment with an absorption rate constant describing input into the central compartment. Models with three or greater compartments may be used if the drug concentration versus time may be described better with additional exponential terms. However, these models present greater complexity. 

As stated above, the Vss calculation using Eqs. (5) or (10) is valid only when elimination exclusively occurs from the sampling (plasma/blood) compartment. When some or all elimination occurs from the tissue compartment (Fig. 7.1), the concentration versus time profile will still be characterized by a bi-exponential equation however, the ability of modeling systems to quantify the micro rate constants is lost. That is to say, essentially identical bi-exponential concentration time profiles are possible with and without elimination from the tissue compartment. Therefore, when modeling from a plasma profile only, there is no way of determining if the exit of drug from the body is exclusive to the central compartment. 

The first step in performing PK modeling is to graph the plasma concentration versus time profile to examine the shape of the curve and to get some preliminary ideas whether the data would fit a one-, two- or a three-compartment PK model. From the semi-logarithmic plot (Figure 1), it was obvious that the compound exhibited either two- or three-compartment kinetics. 

Simultaneous integration of these two equations gives the explicit solution as a multiexponential equation, the exponents being expressed as a function of the distribution (a) and elimination rate constants (jS), and factoring in the volumes of the compartment (Vc). The following equation (eqn(3)) represents the concentration versus time for a drug which follows a two-compartment model ... 

Figure 33-9 Decline of plasma concentration versus time after intravenous administration of a drug best characterized by a two-compartment model.

Model-independent relationships are firequently used in evaluation of clinical pharmacokinetics, because there are fewer relationships to remember, fewer restrictive assumptions, a more general insight into elimination mechanisms, and easier computations. However, model-independent relationships are not without their disadvantages conceptualization of compartments or physiological spaces maybe lost, specific information that may be clinically relevant or pertinent to mechanisms of distribution or efimination can be lost, and the difficulty can be increased in constructing profiles of concentration versus time. 

Vancomycin requires multicompartment models to completely describe its serum-concentration-versus-time curves. However, if peak serum concentrations are obtained after the distribution phase is completed (usually V2 to 1 hour after a 1-hour intravenous infusion), a one-compartment model can be used for patient dosage calculations. Also, since vancomycin has a relatively long half-life compared with the infusion time, only a small amount of drug is eliminated during infusion, and it is usually not necessary to use more complex intravenous infusion equations. Thus simple intravenous bolus equations can be used to calculate vancomycin doses for most patients. Although a recent review paper questioned the clinical usefulness of measuring vancomycin concentrations on a routine basis, research articles" " have shown potential benefits in obtaining vancomycin concentrations... 

FIGURE 3.6 Compartmental analysis for different terms of volume of distribution. (Adapted from Kwon, Y., Handbook of Essential Pharmacokinetics, Pharmacodynamics and Drug Metabolism for Industrial Scientists, Kluwer Academic/Plenum Publishers, New York, 2001. With permission.) (a) Schematic diagram of two-compartment model for compound disposition. Compound is administrated and eliminated from central compartment (compartment 1) and distributes between central compartment and peripheral compartment (compartment 2). Vj and V2 are the apparent volumes of the central and peripheral compartments, respectively. kI0 is the elimination rate constant, and k12 and k21 are the intercompartmental distribution rate constants, (b) Concentration versus time profiles of plasma (—) and peripheral tissue (—) for two-compartmental disposition after IV bolus injection. C0 is the extrapolated concentration at time zero, used for estimation of V, The time of distributional equilibrium is fss. Ydss is a volume distribution value at fss only. Vj, is the volume of distribution value at and after postdistribution equilibrium, which is influenced by relative rates of distribution and elimination, (c) Time-dependent volume of distribution for the corresponding two-compart-mental disposition. Vt is the starting distribution space and has the smallest value. Volume of distribution increases to Vdss at t,s. Volume of distribution further increases with time to Vp at and after postdistribution equilibrium. Vp is influenced by relative rates of distribution and elimination and is not a pure term for volume of distribution. 

As discussed previously, the area under the plasma concentration versus time curve AUC) can be very useful in determining the bioavailability of a drug and other PK parameters. The AUC for the one-compartment IV infusion model is represented graphically as the shaded area in Figure 10.37. The value of AUC for this model can be calculated directly by the equation... 

The effects of different model parameters on the plasma concentration versus time relationship can be demonstrated by mathematical analysis of the previous equations, or by graphical representation of a change in one or more of the variables. The model equations indicate the plasma concentration (C ) is proportional to the intravenous dosing rate ( o.iv) and inversely proportional to the compartment 1 distribution volume (Vi). Thus an increase in or a decrease in Vi both yield an equivalent increase in Cp, as illustrated in... 

The plasma concentration versus time equation for the two-compartment first-order absorption model contains three exponential decay terms, and three corresponding phases in Figure 10.80. The first exponential decay term contains the larger hybrid rate constant (/li), which dominates just after t ax during the distribution phase. Hence li is called the hybrid distribution rate constant. The second exponential decay term contains the smaller hybrid rate constant I2), which dominates at later times during the elimination phase. The third exponential decay curve contains and dominates the early rising absorption phase. The two-compartment elimination half-lrfe is then written in terms of as... 

As with previous two-compartment models, the plasma concentration of drug (Cp) is given by dividing the amount of drug in compartment 1 (Ai) by distribution volume of compartment 1 (Fi). The generalized plasma concentration versus time equations are summarized in Table 10.2 for each type of drug absorption. Application of the generalized equations to one-, two-, and three-compartment models is provided to demonstrate the universality of these equations. For... 

Each of the exponential decay terms in the generalized multicompartment models represent a distinct phase or change in shape of the plasma concentration versus time curve. The extra (n+l) exponential term for first-order absorption always has the absorption rate constant (ka) in the exponent, and always represents an absorption phase. The exponential term with the smallest rate constant (A ) always represents the elimination phase, and this rate constant always represents the elimination rate constant and always equals the terminal line slope m= — A J. All other exponential terms represent distinct distribution phases caused by the different rates of distribution to different tissue compartments. 

Plasma Concentration (versus Time Relationships for Different Numbers of Model Compartments and Types of Absorption... 

In our case study, the experimental observations (i.e. concentration versus time data) were used as input to the conceptualization phase of a mathematical model with two sink compartments (the so-called two-sink model). Following the above discussion, this model (schematically represented in Fig. 2.3-1) can be eonsidered as a hybrid-empirical model. Conceptually, this model describes the test chamber kinetics of a VOC for the three types of experiments which have been carried out. The adsorption-desorption kinetics is described by the rate constants k, k, k. Given that the conceptualization... 

Figure 7.3 Predicted concentration profiles from models with slow absorption. Results from pharmacokinetic models containing a single compartment with elimination ti/2 = 60 min and absorption (j/2 = 15 min. (a) The concentration versus time curve for a single dose given at t = 0 the inset shows the curve calculation on a semi-log plot, (b) The sohd line indicates the total concentration versus time for three consecutive doses given at t = 0, 120, and 240 min. The dashed lines indicate the concentration that would be produced by each individual dose.

It is important to recognize that the selection of the compartment model is contingent upon the availability of plasma concentration versus time data. Therefore, the model selection process is highly dependent upon the following factors. 

The sensitivity of the procedure employed to analyze drug concentration in plasma samples. (Since inflections of the plasma concentration versus time curve in the low-concentration regions may not he detected when using assays with poor sensitivity, the use of a more sensitive analytical procedure will increase the probability of choosing the correct compartment model.)... 

Please note that there is a single phase in the concentration versus time plot and one exponential term in the equation required to describe the data. This indicates that a one-compartment model is appropriate in this case. 

The plasma concentration versus time profile presented in Fig. 1.10 represents a one-compartment model for a drug administered extravascularly. There are two phases in the profile absorption and elimination. However, the profile clearly indicates the presence of only one phase in the post-absorption period. Since distribution is the sole property that determines the chosen compartment model and, since the profile contains only one phase in the post-absorption period, these data can be described accurately and adequately by employing a one-compartment model. However, a biexponential equation would be needed to characterize the concentration versus time data accurately. The following equation can be employed to characterize the data ... 

Please note that a one-compartment model will provide an accurate description since there is only one post-absorption phase however, since there are two phases for the plasma concentration versus time data, a biexponential equation is required to describe the data accurately. 

Figure 4.6 Application of the trapezoidal rule to determine the area under the plasma concentration (Cp) versus time curve (AUC). (Rectilinear plot of plasma or serum concentration versus time following the administration of an intravenous bolus of a drug fitting a one-compartment model.)...

A lower creatinine clearance value will affect other so-called "constant" parameters such as the elimination and/or excretion rate constants (K or jy, the elimination half life (ti/2) and, possibly, the apparent volume of distribution. These, in turn, will influence the value of any other pharmacokinetic parameter mathematically related to them. (This example is for a one-compartment model). These parameters include plasma concentration (Cp) at any time t, the area under the concentration versus time curve from t— 0 to t= °°, and clearance. 

This problem set will provide you with the plasma concentration versus time data (questions 1, 2 and 4) as well as urinary data (questions 3 and 5), following the intravenous bolus administration of a drug that follows the first-order process and exhibits the characteristics of a one-compartment model. The following are our answers to these five questions. Please note that your answers may differ from these owing to the techniques employed in obtaining the best fitting straight line for the data provided. These differences will, therefore, be reflected in the subsequent answers. 

Figure 13.2 A typical concentration versus time profile for a drug in the peripheral compartment (also called the tissue compartment or compartment 2) and that obeys a twcKompartment model following intravenous bolus administration.

In many cases pharmacokinetic data (i.e. plasma drug concentration versus time data) cannot be fitted to an explicit equation equivalent to a system containing a discrete number of compartments into which dmg distributes. This data analysis requires some form of non-compartmental analysis (also referred to as model-independent analysis.) This is achieved by the use of statistical moment theory. 

A coefficient in the equation for plasma drug concentration over time for a two-compartment intravenous bolus model AUC area under the plasma drug concentration versus time curve... 

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