Kp Definitions, Reference Phases, and Concentration Basis

Why the Definition of \(K_p\) Matters

A tissue-to-plasma partition coefficient is often written as a single number, but that number is meaningful only when its reference phases and concentration bases are clear. “Tissue concentration” might mean amount per wet tissue mass, amount per tissue volume, or amount per tissue-water volume. “Plasma concentration” might mean total drug or unbound drug. These are not interchangeable quantities, even when they describe the same sample.

In PBPK models, a tissue-to-plasma coefficient commonly relates total drug concentration in a tissue to total drug concentration in plasma at distribution equilibrium. It can be used to relate tissue drug amount to plasma concentration, but only after matching the coefficient’s concentration definition to the model’s tissue-volume convention. This tutorial focuses on defining and converting those quantities, not on predicting their values from tissue composition.

The subscripts in \(K_{p,t:p}\) identify the two phases in the ratio: tissue \(t\) over plasma \(p\). The notation is deliberately explicit because an unlabelled \(K_p\) may conceal whether the reference was plasma or blood and how tissue concentration was calculated. The ratio does not, by itself, say how quickly equilibrium is reached, whether it holds at every dose, or whether free drug concentrations are equal.

Amounts, Volumes, and Concentration Bases

A concentration is an amount divided by a specified denominator. Let \(A_t\) be the drug amount assigned to a tissue sample, \(m_t\) its wet mass, \(V_t\) its tissue volume, and \(V_{t,w}\) its water volume. The same amount can therefore be represented by different concentrations:

The first quantity is a wet-weight concentration, often reported as amount per gram of wet tissue. The second is a tissue-volume concentration, such as amount per litre of tissue. The third is amount per litre of tissue water. The underlying amount is unchanged; only the denominator changes. Therefore, these concentrations are not numerically interchangeable.

If \(V_{t,w}\) is a fraction \(f_w\) of tissue volume, so that \(f_w=V_{t,w}/V_t\), then \(C_{t,\mathrm{vol}}=f_w C_{t,w}\). If tissue density is \(\rho_t=m_t/V_t\), then \(C_{t,\mathrm{vol}}=\rho_t C_{t,\mathrm{wet}}\). These identities convert concentration bases without making a claim about how drug partitions within the tissue.

The units provide an immediate check. For example, if \(C_{t,\mathrm{wet}}\) is in mg/g and \(\rho_t\) is in g/mL, their product is mg/mL. A wet-weight concentration divided directly by a plasma concentration in mg/mL has units mL/g, not a dimensionless ratio. That quotient can be reported as a mass-basis ratio, but it is not the same numerical quantity as a volume-basis \(K_p\) unless the density conversion is applied.

Converting a Tissue Concentration to the Model Basis

A PBPK compartment typically tracks drug amount in a tissue and represents tissue concentration as amount per compartment volume. To use a reported partition coefficient in that setting, first determine whether its tissue concentration already has this volume basis. If it is a wet-weight value, convert using tissue density; if it is a tissue-water value, convert using the water-volume fraction. In either case, the denominator used in the conversion must describe the same tissue sample and condition as the reported concentration.

The conversion follows directly from the amount definition. For wet weight, \(C_{t,\mathrm{wet}}=A_t/m_t\), while \(V_t=m_t/\rho_t\). Thus:

For the water basis, \(V_{t,w}=f_w V_t\), so:

These conversions are valid when each concentration represents the same drug amount \(A_t\). If, for example, a value is specifically the amount in tissue water while another includes drug in all tissue phases, they are not two bases for the same amount and this conversion cannot make them equivalent.

Reference Phases: Plasma, Blood, and Unbound Drug

A plasma-referenced coefficient uses plasma concentration in its denominator. A blood-referenced coefficient instead uses whole-blood concentration. Define the whole-blood-to-plasma concentration ratio as \(R_{b:p}=C_b/C_p\), with both concentrations measured on compatible total-drug bases. For the same tissue concentration, \(K_{p,t:b}=C_t/C_b\) and \(K_{p,t:p}=C_t/C_p\). Substitution gives:

Thus a tissue-to-blood coefficient is not generally numerically equal to a tissue-to-plasma coefficient. The conversion requires a blood-to-plasma ratio measured or specified for the same drug and relevant conditions. A serum-referenced quantity also has its own reference phase; it should not be silently relabelled as plasma-referenced.

A second important distinction is total versus unbound concentration. The \(K_p\) defined here uses total concentrations unless otherwise stated. A ratio formed from unbound tissue and unbound plasma concentrations is a different quantity, commonly denoted \(K_{p,uu}\). A total-concentration \(K_p\) cannot be interpreted as \(K_{p,uu}\) merely because both are tissue-to-plasma ratios. Their numerators and denominators differ.

Worked Example: Converting a Tissue-Water Basis

Diagnostics, Alternatives, and Common Pitfalls

Before using a reported coefficient, trace the units from the original amount and denominators. A useful audit records the tissue numerator basis, the plasma or blood reference phase, whether concentrations are total or unbound, and the measurement condition. Confirm that all quantities refer to the same drug and that mass units have not been mixed with molar units. For the same compound, a consistent mass-to-molar conversion applies to both sides of a ratio, but converting only one side changes the result and its units.

• Check dimensionality. A ratio of mg/L to mg/L is dimensionless. A ratio of mg/g to mg/L has units L/g and requires a density conversion before it is treated as a volume-basis coefficient.
• Check the stated denominator. “Tissue concentration” is incomplete if it is unclear whether the denominator is wet mass, total tissue volume, or tissue-water volume.
• Check the reference phase. A plasma-based number cannot be used as blood-based without \(C_b/C_p\); the conversion direction matters.
• Check the amount represented. A concentration in tissue water and a total tissue concentration may represent different drug amounts, not just different denominators.
• Check the sampling condition. The equilibrium interpretation requires tissue and reference-phase concentrations from a condition in which distribution equilibrium is appropriate. A ratio at a transient sampling time need not be an equilibrium partition coefficient.
• Check sample processing. If tissue was homogenized with added buffer, the added volume should not inadvertently replace the original tissue mass or volume as the concentration denominator when the intended quantity is an in-vivo tissue concentration.

A wet-weight ratio can be useful when assay results are naturally reported per gram and the intended application is also mass based. A volume-basis ratio is usually the direct choice when a PBPK model represents tissue amount per tissue volume. A tissue-water basis can be useful when the relevant data or calculation is explicitly water normalized. These are alternative representations, not competing physical amounts; selecting among them is a matter of matching the coefficient to its intended use and retaining the conversion information.

A common mistake is to divide a value in mg/g by a plasma value in mg/L, call the numerical answer \(K_p\), and treat it as dimensionless. The arithmetic may be correct, but the quotient has units L/g. The correction is to report that mass-basis quotient with its units or convert the tissue value to amount per tissue volume using density before calculating a dimensionless \(K_p\). Another mistake is to assume that total \(K_p\) equals an unbound ratio. The correction is to identify the numerator and denominator concentrations explicitly; total and unbound ratios answer different questions.