After reading this article you will learn about the concept and types of pharmacokinetic models.
Concept of Pharmacokinetic Models:
Drugs remain in dynamic state within the body and drug events often happen simultaneously. In order to describe a complex biologic system, assumptions are made concerning the movement of drugs. The mathematical models are used to describe the absorption, distribution and elimination of drugs.
Mathematical equations are used to describe the drug concentration in the body as a function of time. For example, if the drug is administered (i.v.) it is distributed rapidly in the body fluid. The pharmacokinetic model that will describe this situation would be a tank containing a volume of fluid which is rapidly equilibrated with the drug. In the animals body a fraction of drug is continuously eliminated as function of time.
The concentration of a drug after a given dose in governed by two important parameters:
(i) The fluid volume of the body.
(ii) The elimination of drug in unit time.
In pharmacokinetics the above parameters are assumed to be constants. If a known set of drug concentrations in the body is determined at various time intervals then the volume of the body fluid and the rate of drug elimination is established.
Types of Pharmacokinetic Models:
1. Compartment Models:
Similar to humans, the animal body is considered as a series of compartments. Each compartment communicate each other reversibly. A compartment is not a real anatomic region but a group of tissues which have similar blood flow. Within each compartment drug is distributed uniformly. Drugs move in and out of the compartment.
Rate constants. K12 and K21 are used to represent the constants for transfer of drug from central to peripheral and from peripheral to central compartment respectively. The model is called open model because the drug can be eliminated (Fig. 2.8).
The well perfused tissues like liver, kidney and heart make the central compartment.
The peripheral compartment consists of the less perfused tissues like skin, bone, cartilages etc. The brain and bones as well as other parts of the central nervous system is excluded, since most drugs have little penetration into these organs. The sketch diagram of the pharmacokinetic model is given in Fig. 2.9.
2. One Compartment Open Model:
When a drug is given as rapid i.v. bolus, the entire dose of the drug enters the body immediately. In this case, the rate of absorption is neglected in calculations. In most cases the drug distributes via the circulation to all the tissues in the body.
The most simple pharmacokinetic model for describing the dissolution of the drug is an apparent volume within the body. The one compartment model assumes that any change in the plasma levels of drug reflects proportional changes in tissues drug concentrations.
3. Two Compartment Open Model:
In this model it is assumed that the drug is distributed in two compartments. If drug is administered in animals body through i.v. injection, it is first distributed into the highly perfused tissues (central compartment) and thereafter to less perfused tissues (peripheral compartment). If plasma level-time profile is plotted on semi-logarithmic graph it gives a bi-exponential appearance (fig 2.8).
Why a Steep decline is Obtained:
After an i.v injection of a drug, concentrations in plasma decline rapidly because of drug distribution into peripheral compartment and the bi-exponential curve shows a steep line initially.
Distribution (α) and elimination (β) Phases:
The initial steep decline of a drug concentration in central compartment is known as distribution phase (α) of the curve. In time, the drug attains a state of equilibrium between the central compartment and the peripheral compartment.
After this equilibrium is established, the loss of the drug from the central compartment appears to be a single first-order process due to overall process of elimination of the drug from the body. This second, lower rate process is known as the elimination phase (β).
The theoretical tissue compartment of a drug can be calculated once the parameters for the model are determined. The drug concentration in the tissue compartment represents the average drug concentration in a group of tissues rather than any real anatomic tissue drug concentration.
Real tissue drug concentration can sometimes be calculated by the addition of compartments to the model unit. In-spite of the hypothetical nature of the tissue compartment the theoretical tissue level is still a valuable piece of information for clinicians.
Why Initial Experimental Samples are Essential?
In practice, samples of blood are removed from the central compartment and analyzed for the presence of drug. The plasma concentration time-curve represents distribution phase (α) followed by an elimination phase (β) after the tissue compartment has also been diffused with the drug.
The distribution phase may take minutes and may be missed entirely if the blood is sampled too late after administration of the drug. This is why the blood samples are collected initially starting from 2 min. or even earlier than this.
The following equations describe the change in drug concentration in plasma and in the tissue with respect to time:
Cp = Drug concentration in plasma,
Ct = Drug concentration in tissue;
D = i.v. dose,
t = time after dose administration; α and β are distribution and elimination rate constants respectively.
The mathematical relationship of α and β to the rate constants are as below:
where A and B are intercepts on Y axis for distribution and elimination phase respectively. The values of A and B may be obtained graphically by the method of residuals.
Method of Residuals:
This method is also known as feathering technique (Fig. 2.10) or peeling technique is an useful procedure for fitting a curve to the experimental data of a drug, which shows the necessity of a multi-compartment model.
For example, 1gm of cefgazolin was administered by rapid i.v. injection to a 12 kg healthy she goat. Blood samples were taken at different predetermined time intervals post drug administration and the each plasma sample was assayed for the drug. The following data were obtained.
When this data will be plotted on semi-logarithmic graph paper, a curved line should be observed (Fig. 2.9). The curved line relationship between the logarithm of the plasma concentration and time indicates that the drug is distributed in more than one compartment. From these data, a bi-exponential equation. (Equation 2.15) may be derived by method of residuals.
From the bi-exponential curve (Fig. 2.8) it can be seen that the initial distribution rate is more rapid than the elimination rate. This explains that the rate constant α will be larger than rate constant β. Therefore, at some later time the term Ae-α.t will approach zero while B will still have a value. At this time Equation 2.15 will reduce to:
Cp = Be-β.t
The rate constant β can be obtained from the slope (-β/2.3) of a straight line representing the terminal exponential phase. The t1/2β for the elimination phase can be derived from the following relationship.
The new line obtained by graphing the logarithm of the residual plasma concentration (Cp – C’p) against time, represents the a phase, (Fig. 2.11).
Different pharmacokinetic parameters may be derived by proper substitution of rate constants α, β and intercepts A and B into the following equations: