Physiologically Based Pharmacokinetic (PBPK) Modeling and Simulations. Sheila Annie Peters

the greater its extent of plasma protein binding. The binding equilibrium can be represented as:

      [P] is protein concentration; C_u and C_b are the unbound and bound concentrations of the drug at equilibrium. The equilibrium constant, K_A, also called the affinity constant is given by

      (1.19)

n is the number of binding sites per mole of the binding protein. Since the therapeutic concentrations of most drugs are low relative to the total protein concentration, [P] can be assumed to be the total protein concentration [P]_Total. The fraction unbound in plasma (f_up ) can be obtained from Equation 1.20 in terms of [P]_Total or in terms of the concentrations of α1‐acidic glycoprotein (AGP), [P]_AGP and albumin [P]_albumin:

(1.20)

The fraction unbound in plasma (f_up ) thus depends on the concentrations of plasma proteins and the affinity of the drug to the plasma proteins. Albumin is the principal protein to which many drugs bind, followed by AGP. Other plasma proteins include lipoproteins and globulins. The concentrations of various plasma proteins are shown in Table 1.1. Albumin is distributed in intravascular (plasma: 43 g/kg organ) and extravascular organs (muscle: 2.3 g/kg, skin: 7.7 g/kg, liver: 1.4 g/kg, gut: 5 g/kg, and other tissues: 3 g/kg). Albumin exists abundantly in the interstitial fluids.

TABLE 1.1. Plasma proteins.

Plasma proteins	Binding	Molecular weight (Da)	Concentration (μM)
Albumin	Binds mainly to anionic compounds	67 000	500–700
α1‐acidic glycoprotein (AGP)	Binds mainly to cationic drugs. E.g.: tricyclic antidepressants	42 000	9–23
Lipoproteins		200 000	Variable
α‐, β‐ and γ‐globulins		53 000	0.6–1.4

Albumin has six distinct binding sites, two of which specifically bind to long‐chain fatty acids, another selectively binds to bilirubin and two others bind to acidic and lipophilic drugs. One of these two drug binding sites binds drugs like warfarin, and phenylbutazone, while the other binds drugs such as diazepam and ibuprofen. Drugs binding to different binding sites do not compete with one another. When more than 20% of the sites are occupied, concentration dependence of binding begins to get appreciable, ultimately leading to saturation at higher concentrations. Saturation of albumin is rare and restricted to drugs (especially acids) with high therapeutic concentration. However, the binding sites of a few drugs such as tolbutamide and some sulfonamides are saturated even at therapeutic concentrations. AGP concentrations being much lower compared to albumin, saturation of AGP occurs at lower therapeutic concentrations. The concentrations of several plasma proteins can be altered by many factors including stress, surgery, liver dysfunction, and pregnancy. Most commonly, disease states increase AGP concentration while reducing albumin concentration. Higher levels of AGP have been reported in obese patients with nephrosis. Stress, cancer, and arthritis have been associated with lower AGP levels. Neonates have higher AGP levels. AGP is associated with a higher inter‐individual variability compared to albumin. Reduced levels of albumin have been reported in myalgia patients. Drugs that are highly bound to plasma proteins are confined to the vascular space and are not readily available for distribution to other tissues and organs. Many carboxylic acid drugs are not easily displaced from plasma proteins and have a low distribution volume. However, this is not true if the affinity of a drug to tissue proteins is higher than that to plasma proteins. Ultrafiltration and equilibrium dialysis are the two commonly employed methods for the determination of plasma protein binding (Wright et al., 1996). Albumin is the principal drug‐binding protein in tissues followed by ligandin. Measurement of tissue binding is not as straight forward as that in plasma, as the tissue must be disrupted, and it is not readily accessible for sampling. Tissue proteins cannot be easily separated into its constituents and cannot easily be quantified.

      The clearance (CL) of many low hepatic‐extraction drugs is limited by protein binding. Only the unbound drug is available for glomerular filtration and therefore for renal elimination. An increase in unbound drug concentration due to a reduced plasma protein binding will enable higher tissue distribution and higher CL. However, since the half‐life of a drug is directly proportional to the distribution volume and inversely proportional to CL, there is no net effect on the half‐life. Thus, changes in plasma protein binding of a drug are not likely to be clinically relevant (Benet and Hoener, 2002) except in the following cases:

      1 The drug is >98% bound to plasma proteins. In this case, even a small shift in plasma protein binding can have a substantial effect on the clearance but less so on the distribution volume, thus temporarily altering the unbound drug concentrations.

      2 The drug has high hepatic extraction. The clearance of such drugs will be dependent only on the hepatic blood flow rate and not on the product of fup × CLint . Thus, an increase in distribution volume is not sufficiently compensated for by an increase in CL, leading to a temporary increase in unbound drug concentrations.

      3 There is a rapid equilibrium between drug concentration and pharmacological response (e.g., lidocaine with a PK‐PD equilibration time of two minutes) compared to the time required for the body to regain equilibrium (about 30 minutes). Many anti‐arrhythmic drugs and anesthetics require only a short time for a change in concentration to cause a change in drug effect. In these cases, the response is sensitive to small transient changes in unbound drug concentrations.

      Differences in unbound drug concentrations discussed in (i) or (ii) will have a greater impact on a drug with a narrow therapeutic window/safety margin. Scaling of pharmacokinetic (PK) parameters like clearance or volume of distribution or translation of pharmacodynamic properties (Mager et al., 2009) from preclinical species to man should always be done with the unbound parameters. Any comparisons/correlations of PK parameters should also be done with unbound values.

      Some drugs also bind to and distribute into erythrocytes, the main drivers being lipophilicity, pKa and active uptake into the erythrocytes. Binding sites within erythrocytes are hemoglobin, proteins like carbonic anhydrase, and plasma membrane. The blood–plasma concentration ratio (R) of a drug is a measure of its binding and distribution to erythrocytes relative to plasma. A compound having a similar extent of binding to the constituents of erythrocytes and plasma has a blood–plasma ratio of 1. Acids tend to have R values of around 0.5 and rarely exceed 1, and bases tend to have a higher range of values, often exceeding 1 while neutrals and ampholytes have values of around 1 (Hinderling, 1997). Uchimura et al. (2010) describe several methods to determine R. Commonly, it is determined by measuring the concentrations of ¹⁴C‐labelled drug in erythrocytes (C_e ) and plasma (C_p ) in freshly collected blood. Then, knowing the hematocrit, H (the volume fraction of blood occupied by erythrocytes),