Calculating Unbound Concentration In Whole Blood

Why Convert a Whole-Blood Concentration?

A whole-blood measurement can be the concentration available from a clinical sample, while a PBPK calculation may require an unbound concentration to describe freely available drug. Converting one to the other requires two pieces of information: how total drug distributes between whole blood and plasma, and what fraction of plasma drug is unbound. The conversion is simple algebra, but its interpretation depends on matching the measurements and respecting their concentration bases.

The blood-to-plasma ratio accounts for the difference between total concentration measured per volume of whole blood and total concentration measured per volume of plasma. Plasma fraction unbound then maps total plasma concentration to unbound plasma concentration. Under an equilibrium assumption that the relevant unbound concentration is shared, this gives an inferred unbound concentration from whole-blood data.

Definition: Let \(C_b\) denote total drug concentration in whole blood, \(C_p\) total drug concentration in plasma, \(R_b\) the blood-to-plasma concentration ratio, and \(f_{u,p}\) the fraction of plasma drug unbound. The ratio is \(R_b=C_b/C_p\), for \(C_p>0\). The unbound plasma concentration is \(C_{u,p}=f_{u,p}C_p\). If the relevant unbound concentration is assumed to be the same at equilibrium in blood and plasma, the inferred unbound concentration from whole-blood data is denoted here by \(C_{u,b}\), with \(C_{u,b}=C_{u,p}\).

Deriving the Conversion

The derivation uses the definition of \(R_b\), followed by the definition of plasma fraction unbound. It does not require a separate model of red-cell binding once \(R_b\) and \(f_{u,p}\) are available. The ratio must be positive, and the measurements must describe the same drug under compatible conditions.

Recover total plasma concentration.
From \(R_b=C_b/C_p\), rearrange to obtain \(C_p=C_b/R_b\), provided \(R_b>0\).

Calculate unbound plasma concentration.
By definition, \(C_{u,p}=f_{u,p}C_p\).

Apply the equilibrium assumption.
If the relevant unbound concentration is shared between blood and plasma, set \(C_{u,b}=C_{u,p}\) and substitute the expression for \(C_p\).

C_{u,b}=f_{u,p}C_p=f_{u,p}\frac{C_b}{R_b}

This expression uses \(C_b\), \(R_b\), and \(f_{u,p}\) from compatible conditions. For example, if binding depends on concentration, \(f_{u,p}\) should correspond to the plasma concentration represented by the sample, rather than to an unrelated concentration from a different assay condition. If the ratio also changes with concentration or hematocrit, use a ratio appropriate to the sample being converted.

The assumptions are substantive. The whole-blood and plasma concentrations must refer to the same analyte and sampling condition; the blood-to-plasma ratio must be defined as total concentration in whole blood divided by total concentration in plasma; and the plasma fraction unbound must apply to that plasma concentration and matrix. In addition, the unbound species of interest must be at equilibrium across the relevant blood and plasma compartments so that the plasma unbound concentration is an appropriate estimate of the blood unbound concentration. If these assumptions fail, the formula may still produce a number, but that number may not represent the desired quantity.

Worked Example: Converting a Whole-Blood Concentration

Suppose a hypothetical matched sample has \(C_b=12.0\) mg/L, \(R_b=1.50\), and \(f_{u,p}=0.0800\). Assume the ratio and fraction unbound apply to this sample, and the relevant unbound concentration is shared at equilibrium. Find \(C_{u,b}\).

First recover the total plasma concentration:

C_p=\frac{C_b}{R_b} =\frac{12.0\ \mathrm{mg/L}}{1.50} =8.00\ \mathrm{mg/L}.

Then apply the plasma fraction unbound:

C_{u,b}=f_{u,p}C_p =0.0800(8.00\ \mathrm{mg/L}) =0.640\ \mathrm{mg/L}.

The inferred unbound concentration is \(0.640\) mg/L. The calculation is internally consistent: \(R_b=1.50\) implies plasma total concentration is lower than whole-blood total concentration, and 8.00 mg/L multiplied by an unbound fraction of 0.0800 gives 0.640 mg/L.

Concentration Bases and an Important Bound

All concentrations in the formula must be expressed in compatible units, such as mg/L, but the denominators refer to different sample volumes. \(C_b\) is drug amount per volume of whole blood; \(C_p\) is drug amount per volume of plasma. Consequently, the inferred unbound concentration \(C_{u,b}\) is not automatically bounded above by \(C_b\). Nonnegative \(C_b\), nonnegative \(f_{u,p}\), and positive \(R_b\) ensure \(C_{u,b}\) is nonnegative, but they do not guarantee \(C_{u,b}\le C_b\).

For instance, plasma may occupy half the whole-blood volume, with no drug partitioning into blood cells. If \(C_p=1.00\) mg/L, \(R_b=0.500\), and \(f_{u,p}=0.800\), then \(C_b=R_bC_p=0.500\) mg/L but \(C_{u,b}=f_{u,p}C_p=0.800\) mg/L. This does not violate the definitions: total whole-blood concentration is normalized by whole-blood volume, whereas the inferred unbound concentration is the free concentration on the plasma basis. Do not use \(C_{u,b}\le C_b\) as a general diagnostic unless an additional volume-basis convention or condition justifies that comparison.

Worked Example: A Ratio Below One

Consider hypothetical values \(C_b=0.500\) mg/L, \(R_b=0.500\), and \(f_{u,p}=0.800\). Assume matched measurements and the same equilibrium condition as above. Calculate \(C_p\) and \(C_{u,b}\), then interpret the fact that \(C_{u,b}>C_b\).

The implied total plasma concentration is:

C_p=\frac{0.500\ \mathrm{mg/L}}{0.500} =1.00\ \mathrm{mg/L}.

Thus:

C_{u,b}=0.800(1.00\ \mathrm{mg/L}) =0.800\ \mathrm{mg/L}.

Here \(0.800\) mg/L exceeds the whole-blood total concentration of \(0.500\) mg/L. That comparison is not, by itself, evidence of an arithmetic or biological inconsistency: the values use different volume bases. The calculation does require that \(R_b\), \(f_{u,p}\), and the equilibrium assumption be valid for the hypothetical sample.

Input Matching and Practical Diagnostics

A calculated value can be numerically precise while being poorly supported by its inputs. Before using the conversion in a PBPK model, verify the definitions, conditions, and units rather than relying on the arithmetic alone.

Check the ratio direction. The convention here is \(R_b=C_b/C_p\). If a data source reports plasma-to-blood ratio, invert it before applying this formula.
Match conditions. Use a plasma fraction unbound and blood-to-plasma ratio relevant to the same drug, species, concentration range, and sample conditions. If binding is concentration-dependent, an \(f_{u,p}\) from a different concentration may not be suitable.
Check units and sample basis. Convert, for example, µg/L to mg/L before combining quantities. Confirm that the reported ratio is based on total concentrations and not on a different definition.
Check mathematical conditions. \(R_b\) must be positive, and \(f_{u,p}\) should be between zero and one under its definition. For nonnegative \(C_b\), the formula then gives a nonnegative inferred concentration.
Investigate unusual inputs. A zero or negative ratio, a fraction unbound outside its allowable range, or an extreme result may indicate a transcription error, assay limitation, incompatible matrices, or a mismatched condition. These checks do not establish that the biological assumptions hold.

A useful independent check is to reconstruct the quantities in reverse. Given the inferred value and the inputs, \(C_{u,b}R_b/f_{u,p}\) should equal \(C_b\), when \(f_{u,p}>0\). Agreement verifies the algebra, not the biological validity of the inputs.

Worked Example: Consequence of Using a Mismatched Fraction Unbound

Suppose hypothetical whole-blood data give \(C_b=18.0\) mg/L and \(R_b=1.20\). A matched plasma binding measurement gives \(f_{u,p}=0.0500\). A separate assay at a different concentration reports \(f_{u,p}=0.100\). Calculate the matched estimate and show what happens if the separate value is used instead.

The implied plasma total concentration is:

C_p=\frac{18.0\ \mathrm{mg/L}}{1.20} =15.0\ \mathrm{mg/L}.

Using the matched fraction unbound:

C_{u,b}=0.0500(15.0\ \mathrm{mg/L}) =0.750\ \mathrm{mg/L}.

Using the separate value instead would give:

C_{u,b}^{\mathrm{mismatched}}=0.100(15.0\ \mathrm{mg/L}) =1.50\ \mathrm{mg/L}.

The second calculation is arithmetically correct but uses an input not established as applicable to this sample. If plasma binding changes with concentration, the two values need not estimate the same condition. The matched estimate is \(0.750\) mg/L under the stated assumptions; the alternative should not be substituted without justification.

What the Conversion Does and Does Not Provide

This conversion is an inference from total blood concentration, a total blood-to-plasma ratio, and plasma binding. It is not a direct measurement of unbound concentration in whole blood, nor does it by itself estimate a whole-blood fraction unbound. In particular, the quotient \(C_{u,b}/C_b\) should not automatically be interpreted as a fraction of total whole-blood drug: the two concentrations can have different volume bases, as the example above demonstrates.

An alternative is to measure the relevant unbound concentration directly in an appropriate blood matrix. A direct measurement may avoid some inference steps, but it has its own assay requirements and must still represent the intended in vivo condition. Another practical route is to measure plasma concentration directly and multiply by \(f_{u,p}\), when a matched plasma sample is available. When only whole-blood concentration is available, the ratio-based formula provides a transparent conversion, conditional on reliable and compatible inputs.

In PBPK applications, the distinction matters because model parameters and observations may be represented on different concentration bases. Keep track of whether an input is total whole-blood concentration, total plasma concentration, or an inferred unbound concentration. The conversion changes the concentration being represented; it does not establish that the unbound concentration is constant across time, tissues, or physiological conditions.

Common Mistakes

Reversing the ratio. Multiplying \(C_b\) by \(R_b\) instead of dividing gives the wrong implied plasma concentration under the stated convention. Since \(R_b=C_b/C_p\), the correct rearrangement is \(C_p=C_b/R_b\).
Using total blood concentration with plasma fraction unbound directly. The expression \(f_{u,p}C_b\) omits the conversion from whole-blood to plasma concentration. The appropriate expression is \(f_{u,p}C_b/R_b\).
Treating the result as a blood unbound fraction. The result is an inferred concentration, not a fraction. Its comparison with whole-blood concentration must account for their volume bases.
Rejecting a result because it exceeds \(C_b\). There is no general upper-bound rule \(C_{u,b}\le C_b\) for these operational definitions. Check assumptions and bases rather than imposing that inequality.
Combining unmatched inputs. A ratio and fraction unbound measured under different conditions may not jointly describe the sample. Mathematical consistency cannot repair biological mismatch.

Key takeaway: With \(R_b=C_b/C_p>0\), matched \(f_{u,p}\), and an equilibrium assumption that the relevant unbound concentration is shared, calculate \(C_{u,b}=f_{u,p}C_b/R_b\). Interpret the result as an inferred unbound concentration, and do not assume it must be less than total whole-blood concentration.

Check Your Understanding

Use the stated ratio convention and show the key calculation in each answer.

A hypothetical sample has \(C_b=9.00\) mg/L, \(R_b=0.750\), and \(f_{u,p}=0.0400\). Find \(C_p\) and \(C_{u,b}\).
Explain why \(f_{u,p}C_b\) is not generally the correct conversion from whole-blood concentration to inferred unbound concentration.
For \(C_b=0.600\) mg/L, \(R_b=0.600\), and \(f_{u,p}=0.750\), calculate \(C_{u,b}\). Does its value exceeding \(C_b\) alone demonstrate an error?
Name two conditions that should be checked before combining a measured \(R_b\) with a measured \(f_{u,p}\).

Solutions. (1) \(C_p=9.00/0.750=12.0\) mg/L, so \(C_{u,b}=0.0400(12.0)=0.480\) mg/L. (2) \(f_{u,p}\) multiplies total plasma concentration, not total whole-blood concentration; first calculate \(C_p=C_b/R_b\). (3) \(C_p=0.600/0.600=1.00\) mg/L and \(C_{u,b}=0.750(1.00)=0.750\) mg/L. It exceeds \(C_b=0.600\) mg/L, but that alone is not an error because the concentration volume bases differ. (4) For example, check that both measurements concern the same drug, species, and relevant concentration conditions, and confirm that the ratio is defined as whole-blood concentration divided by plasma concentration.