Читать книгу Physiologically Based Pharmacokinetic (PBPK) Modeling and Simulations - Sheila Annie Peters - Страница 20

The first‐order rate equation depicting the rate of change of drug concentrations in the blood (C) is given by

(1.1)

Figure 1.1. Temporal changes in drug concentrations for (a) zero‐order and (b) first‐order kinetics.

Figure 1.2. Linear (a) and semilogarithmic (b) plots of drug concentrations vs. time. Area under the curve (AUC) and slope are the two parameters that can be obtained from the plot. C₂ and C₁ are drug concentrations at times t₂ and t₁ respectively. k_el, first‐order elimination rate constant; CL, is the total drug clearance; and V, volume of distribution.

(1.2)

where A is the amount of drug in the body at any time, t, k_el is the first‐order elimination rate constant, and V is the volume of distribution of the drug. The product of k_el and V is defined as the total clearance, CL, of the drug from blood.

Integrating Equation 1.1 (−dC/dt = k_el × C),

(1.3)

Taking natural logarithms on both sides,

(1.4)

Thus, k_el may be obtained by measuring the slope of a semilogarithmic plot of drug concentration vs time (Figure 1.2).

Similarly, integrating Equation 1.2 (−dA/dt = k_el × A) yields

(1.5)

where A₀ is the initial amount of drug in the body, the dose administered as IV bolus. Bringing A₀ to the left‐hand side, Equation 1.5 becomes,

(1.6)

Taking the natural logarithms on both sides of the resulting equation leads to the following:

(1.7)

The half‐life (t_1/2) of a drug, defined as the time taken for half of the administered drug to get eliminated from the body (time taken for drug amount in body to go from A₀, to A₀/2, or time taken for the drug concentration to be halved), is given by:

(1.8)

Using Equation 1.8, the half‐life of a drug can be calculated from the elimination rate constant k_el which is obtained from the semilogarithmic plot of concentration vs. time (Figure 1.2).

Integrating the Equation 1.2 (−dA = CL × C dt) yields

(1.9)

where AUC is the area under the drug concentration‐time profile (Figure 1.2), which may be estimated from the plot by applying the trapezoidal rule. Recognizing that the integral dA over time 0 to t is the dose, Equation 1.9 becomes,

(1.10)

Knowing the dose administered and the AUC, clearance can be calculated using Equation 1.10. The volume of distribution, V, of the drug can be determined using Equation 1.11, knowing that clearance is the product of k_el and V.

(1.11)

Most small molecule drugs bind reversibly to plasma proteins such as albumin and alpha‐glycoprotein. Drug binding to plasma proteins is of major interest in pharmacokinetics as it impacts both clearance and volume of distribution. Thus far, the term clearance refers to blood clearance. However, measurements of drug concentrations are often done in plasma, as whole blood contains cellular elements (red and white blood cells, platelets etc.) and proteins (albumin, glycoproteins, globulin, lipoproteins etc.). The clearance of a drug determined using the AUC estimated from plasma drug concentration‐time profile is referred to plasma clearance. To convert plasma clearance to blood clearance, the distribution of a drug between blood and plasma should be measured. The ratio of drug concentrations in blood to plasma is known as blood–plasma ratio (R).

Mean residence time is a parameter closely related to half‐life and is defined as the average time drug molecules spend in the body before being eliminated. It is expressed as the sum of the residence times of all drug molecules, divided by the total number of molecules. If dA_e is the number of drug molecules exiting the body at time interval t, MRT is given by:

(1.12)

Differentiating Equation 1.5 (),

(1.13)

Recognizing that the rate of decline in A = − rate of amount exiting the body, A_e :

(1.14)

(1.15)

Substituting for dA_e using Equation 1.15 in Equation 1.12 and dividing both numerator and denominator by k_el, we get

(1.16)

The numerator of equation is the first moment of the concentration–time integral, or the area under the curve formed by time and the product of concentration and time, also called the area under the first moment curve (AUMC). The denominator of equation is the same as AUC as shown below:

(1.17)

Thus, MRT for an IV bolus is given by the ratio of AUMC and AUC

(1.18)