Why binding inputs need a consistent concentration basis
PBPK calculations use drug concentrations defined for particular compartments. A plasma fraction unbound cannot be applied directly to a whole-blood concentration unless the conversion between those concentration bases is accounted for. Likewise, a fraction unbound measured in diluted tissue homogenate is not, by itself, the fraction unbound in intact tissue. These are arithmetic conversions, but each depends on a clear definition of the input measurements and the assumptions connecting them.
This tutorial works through conversions involving plasma fraction unbound, blood-to-plasma concentration ratio, hematocrit, and tissue fraction unbound corrected for homogenate dilution. The earlier material on fraction unbound in plasma, blood-to-plasma concentration ratios, and tissue homogenate dilution provides the underlying measurement concepts; here the emphasis is on carrying the values through calculations without changing their meaning.
Converting plasma binding to whole-blood binding
The conversion from \(f_{u,p}\) to \(f_{u,b}\) uses the blood-to-plasma ratio. It assumes that the unbound drug concentration is the same in plasma and red-cell water at equilibrium, and that the relevant concentrations refer to the same drug species and comparable conditions. Under this assumption, the unbound concentration in whole blood equals the unbound concentration in plasma. Since \(C_b=R_bC_p\), the whole-blood fraction unbound is:
The ratio in the denominator matters: a larger total blood concentration relative to plasma means a smaller fraction unbound on a whole-blood concentration basis, all else equal. This does not mean the unbound concentration itself changes across the two compartments under the stated equilibrium assumption. The numerator and denominator used to define the fraction have changed together.
Hematocrit connects \(R_b\) to the concentration ratio between red cells and plasma. Whole blood consists of a plasma volume fraction \(1-Hct\) and a red-cell volume fraction \(Hct\). If the plasma concentration is \(C_p\), the red-cell concentration is \(R_{bc}C_p\); the volume-weighted whole-blood concentration is therefore:
This relationship treats hematocrit as a volume fraction and assumes the reported whole-blood concentration is the volume-weighted concentration of plasma and red cells. Rearranging gives a red-cell-to-plasma ratio implied by \(R_b\) and \(Hct\):
Worked Example: Converting Plasma Binding and Recovering the Red-Cell Ratio
Suppose hypothetical measurements give \(f_{u,p}=0.080\), \(R_b=1.30\), and \(Hct=0.45\). Calculate whole-blood fraction unbound and the implied red-cell-to-plasma concentration ratio.
Whole-blood fraction unbound. Under the equal-unbound-concentration assumption, use \(f_{u,b}=f_{u,p}/R_b\):
Thus the whole-blood fraction unbound is approximately 0.0615, or 6.15%. It is dimensionless, as are the input fractions and ratios.
Red-cell-to-plasma ratio. First express hematocrit as a fraction, \(0.45\), rather than as 45:
The implied red-cell concentration is about 1.67 times the plasma concentration. Substitution checks the calculation: \((1-0.45)+(0.45)(1.667)\approx1.30\), matching the supplied \(R_b\).
Correcting tissue binding for homogenate dilution
Tissue homogenization dilutes tissue binding components in the assay mixture. Under a linear-binding approximation, dilution reduces the bound-to-free ratio in the homogenate. Define \(D\) as the fold dilution of the tissue binding material: \(D=4\), for example, means the homogenate contains one part tissue material per four parts of the corresponding undiluted volume. With the same free drug concentration basis, the bound-to-free ratio in intact tissue is \(1/f_{u,t}-1\). In the diluted homogenate, that ratio is reduced by a factor of \(D\).
This assumption yields a direct correction. Starting from the definition of fraction unbound in the diluted homogenate:
\(1/f_{u,hom}-1=(1/f_{u,t}-1)/D\), because dilution by \(D\) reduces binding material concentration by that factor under linear binding.
Multiplying by \(D\) gives \(D(1/f_{u,hom}-1)=1/f_{u,t}-1\).
Adding 1 and taking the reciprocal gives \(f_{u,t}=1/[1+D(1/f_{u,hom}-1)]\).
For \(D>1\) and \(0<f_{u,hom}<1\), this correction gives \(f_{u,t}\leq f_{u,hom}\): the diluted homogenate generally appears less bound than intact tissue because its binding components are less concentrated. The relationship is not a universal correction for every assay. It relies on the stated dilution definition and assumptions including linear, reversible binding over the relevant concentration range, adequate equilibration, and no important change in binding from the assay procedure.
Worked Example: Correcting a Four-Fold-Diluted Tissue Homogenate
A hypothetical assay reports \(f_{u,hom}=0.20\) for a tissue homogenate diluted four-fold, so \(D=4\). Calculate the corrected fraction unbound in intact tissue.
Insert the measured value and dilution factor into the correction:
The corrected tissue fraction unbound is about 0.0588, or 5.88%, rather than the measured homogenate value of 20%. The smaller corrected value is expected under the dilution model: intact tissue has more binding material per unit volume than the diluted homogenate.
Putting the calculated values into one PBPK input set
The conversions can be combined, but each quantity must retain its compartment label. Plasma fraction unbound, whole-blood fraction unbound, and corrected tissue fraction unbound are not interchangeable values. The following example calculates them from a single hypothetical set of measurements and then checks how the blood-related quantities relate to total and unbound concentrations.
Worked Example: Building a Consistent Set of Binding Inputs
Suppose hypothetical data for one compound are \(f_{u,p}=0.040\), \(R_b=0.80\), \(Hct=0.45\), \(f_{u,hom}=0.12\), and \(D=5\). Assume the blood and plasma measurements are at comparable conditions and the equal-unbound-concentration assumption applies. Calculate the derived binding inputs.
Whole-blood fraction unbound:
Implied red-cell-to-plasma ratio:
Corrected tissue fraction unbound:
These calculations give a plasma fraction unbound of 0.0400, a whole-blood fraction unbound of 0.0500, an implied red-cell-to-plasma ratio of 0.5556, and a corrected tissue fraction unbound of 0.02655. They refer to different compartments or ratios and should be stored with those labels.
As a concentration-basis check, let a hypothetical total plasma concentration be \(C_p=2.00\ \mathrm{mg/L}\). Then \(C_b=R_bC_p=(0.80)(2.00)=1.60\ \mathrm{mg/L}\). The unbound plasma concentration is \((0.040)(2.00)=0.0800\ \mathrm{mg/L}\), while the unbound whole-blood concentration calculated using \(f_{u,b}\) is \((0.0500)(1.60)=0.0800\ \mathrm{mg/L}\). Their equality is consistent with the assumption used in the conversion. If a hypothetical total tissue concentration were \(10.0\ \mathrm{mg/L}\), the corresponding unbound tissue concentration under the corrected fraction would be \((0.02655)(10.0)=0.2655\ \mathrm{mg/L}\). That tissue calculation uses a tissue total concentration; it does not convert the plasma or blood concentration into tissue concentration.
Assumptions, checks, and common mistakes
A calculation is only as reliable as the definitions and assumptions attached to its inputs. Before using these conversions, confirm that the blood-to-plasma ratio is \(C_b/C_p\), not its reciprocal; that \(Hct\) is entered as a fraction; and that \(D\) describes the fold dilution of the tissue binding material. The tissue correction also presumes that a single fraction unbound is meaningful over the relevant concentration range. Concentration-dependent binding, non-equilibrium measurements, or assay-related changes in binding can make the simple correction inappropriate.
Several elementary checks help catch arithmetic or definition errors without replacing careful evaluation of the measurements. Fractions unbound should lie between 0 and 1. For \(Hct>0\), the equation \(R_b=(1-Hct)+HctR_{bc}\) implies \(R_b\geq1-Hct\) when the red-cell concentration ratio is nonnegative. The corrected \(f_{u,t}\) should be no larger than \(f_{u,hom}\) when \(D\geq1\) and \(0<f_{u,hom}\leq1\), under the stated correction. Values outside these mathematical expectations call for checking the inputs, units, definitions, and assumptions before the values are used.
- Reversing the blood-to-plasma ratio. If \(R_b=C_b/C_p\), then \(f_{u,b}=f_{u,p}/R_b\), not \(f_{u,p}R_b\). Multiplication reverses the intended concentration-basis conversion.
- Using percent rather than fraction for hematocrit. Enter \(Hct=0.45\) for 45%. Entering 45 in the weighted-concentration equation does not represent a volume fraction.
- Using the uncorrected homogenate value as intact-tissue binding. A diluted homogenate and intact tissue have different binding-material concentrations. Apply the stated correction when its assumptions are reasonable.
- Swapping compartment-specific fractions. \(f_{u,p}\), \(f_{u,b}\), and \(f_{u,t}\) describe different concentration bases. A calculated value should stay attached to its compartment and should not be substituted for another merely because all are fractions.
There are alternatives to inferring these values from conversions. Whole-blood fraction unbound may be measured directly, rather than calculated from \(f_{u,p}\) and \(R_b\). Tissue fraction unbound may be measured using an approach that avoids homogenate dilution, or corrected using a different validated assay model. Direct measurements can reduce reliance on a conversion assumption, but they still depend on assay conditions, recovery, and the definition of the measured concentration. The calculation method is especially useful when only a subset of measurements is available or when a model requires inputs on a particular compartment basis.
These equations do not establish that inputs from different assays are biologically compatible, nor do they resolve whether measurements taken under different conditions can be combined. They provide the conversion conditional on stated definitions and assumptions. In particular, a mathematically valid \(f_{u,b}\) or corrected \(f_{u,t}\) is not by itself evidence that the underlying measurements are reliable.
Check Your Understanding
Try each calculation using the definitions and equations above. Brief solutions follow the questions.
- If \(f_{u,p}=0.060\) and \(R_b=1.20\), what is \(f_{u,b}\)? State the assumption needed for this conversion.
- For \(R_b=0.90\) and \(Hct=0.40\), calculate \(R_{bc}\).
- A tissue homogenate has \(f_{u,hom}=0.25\) and was diluted three-fold. Calculate corrected \(f_{u,t}\).
- Why should an \(f_{u,t}\) value not automatically be used as \(f_{u,p}\) in a PBPK input set?
Solutions. (1) \(f_{u,b}=0.060/1.20=0.0500\), assuming equal unbound concentrations in plasma and red-cell water at equilibrium. (2) \(R_{bc}=[0.90-(1-0.40)]/0.40=(0.90-0.60)/0.40=0.750\). (3) \(f_{u,t}=1/[1+3(1/0.25-1)]=1/[1+3(4-1)]=1/10=0.100\). (4) \(f_{u,t}\) describes unbound fraction relative to total tissue concentration, whereas \(f_{u,p}\) is defined relative to total plasma concentration; they are compartment-specific quantities.