Whole-Body Mass-Balance Residuals

Why Check Whole-Body Mass Balance?

A PBPK model can move drug among blood, organs, and tissue subcompartments while also adding drug through a dose and removing it through elimination. Internal movement changes where drug is located, but it does not by itself change the total amount represented by the model. A whole-body mass-balance residual summarizes whether the simulated total amount is consistent with the external input and elimination over a specified time interval.

This check is useful because it compares quantities with a direct physical accounting relationship. It does not establish that tissue concentrations, flows, or physiological assumptions are correct. It asks a narrower question: given the model's stated boundaries, do the changes in total modeled drug amount agree with the amount added and removed?

Definition: Let $A_i(t)$ be the amount of drug in modeled compartment $i$ at time $t$. The whole-body amount is $A_{\mathrm{body}}(t)=\sum_i A_i(t)$, where the sum includes every compartment in the declared accounting system. Let $D_{\mathrm{in}}(t)$ and $D_{\mathrm{el}}(t)$ be the cumulative administered input and cumulative elimination, respectively, from time zero through time $t$. The cumulative whole-body mass-balance residual is $$ R(t)=A_{\mathrm{body}}(t)-A_{\mathrm{body}}(0)-D_{\mathrm{in}}(t)+D_{\mathrm{el}}(t). $$ For a mass-balanced simulation, $R(t)$ is zero in the exact model and should be close to zero in a numerical simulation, subject to numerical error and the precision of recorded outputs.

All terms in this equation are amounts and must use compatible units, such as mg or µmol. If input and elimination are given as rates, their time integrals are amounts. A dose already present at time zero belongs in $A_{\mathrm{body}}(0)$; it must not also be counted as input after time zero. A bolus administered later is included as an increment in cumulative input at its administration time.

The accounting boundary must also be explicit. For example, if the modeled quantity is parent drug, conversion of parent drug to an untracked metabolite is an elimination from the parent-drug system, even if drug-derived material remains in the body. If that metabolite is included as a modeled compartment, the transfer is internal to the expanded system and should not be counted as an external loss. The chosen species and boundary determine what “elimination” means.

Deriving the Residual

Suppose each compartment has an amount balance. External input and elimination act on the whole modeled system; intercompartmental exchange moves drug between compartments. When the compartment equations are added, each internal transfer appears once as an amount leaving one compartment and once as an equal amount entering another. The two terms cancel. This cancellation is why internal redistribution does not appear in the whole-body balance.

\frac{dA_{\mathrm{body}}(t)}{dt}=I(t)-E(t)

Here $I(t)$ is the total external input rate and $E(t)$ is the total rate of elimination from the defined system, both in amount per unit time. Integrating this equation from time zero to time $t$ gives the whole-body amount change:

A_{\mathrm{body}}(t)-A_{\mathrm{body}}(0) =\int_0^t I(u)\,du-\int_0^t E(u)\,du =D_{\mathrm{in}}(t)-D_{\mathrm{el}}(t).

Rearranging yields $R(t)=0$ for the exact mass-balanced model. For an interval from $t_a$ to $t_b$, the corresponding residual is

R_{[t_a,t_b]}= A_{\mathrm{body}}(t_b)-A_{\mathrm{body}}(t_a) -\left[D_{\mathrm{in}}(t_b)-D_{\mathrm{in}}(t_a)\right] +\left[D_{\mathrm{el}}(t_b)-D_{\mathrm{el}}(t_a)\right].

A positive residual means that the simulated increase in stored drug exceeds the net amount expected from input minus elimination. A negative residual means that the simulated amount has fallen more, or risen less, than the accounted input and elimination imply. The sign describes the direction of the discrepancy; it does not identify its cause.

Interpretation: A residual has units of amount. $R=0$ means the amount change is consistent with the recorded cumulative input and elimination at the precision considered. A residual that is not zero is a mass discrepancy for the chosen boundary and interval; it is not itself a statistical test or a measure of model fit to concentration data.

Calculating an Interval Residual

Worked Example: A Single Interval With Known Amounts

Consider a hypothetical simulation in which the whole-body amount is 12.0 mg at the start of an interval and 15.6 mg at the end. During that interval, the model administers 5.0 mg and eliminates 1.4 mg. Calculate the interval residual.

The amount change is $15.6-12.0=3.6$ mg. Substituting the interval quantities into the residual equation gives

R_{[t_a,t_b]}=(15.6-12.0)-5.0+1.4 =3.6-5.0+1.4=0.0\ \mathrm{mg}.

The accounting prediction is also $12.0+5.0-1.4=15.6$ mg at the interval end, matching the simulated amount. The zero residual indicates agreement for this interval, to the reported precision. It does not show whether the drug was distributed among organs correctly.

A key practical step is to form $A_{\mathrm{body}}$ by summing amounts, not concentrations. Concentrations in different tissues cannot generally be added to obtain a whole-body amount. If a compartment state is concentration $C_i$ and its applicable volume is $V_i$, its amount is obtained using the model's amount–concentration relationship (commonly $A_i=C_iV_i$); the compartment amounts can then be summed. The volume and concentration units must yield the same amount units used for input and elimination.

Worked Example: Summing Amounts Across Compartments

In a hypothetical PBPK simulation, the amounts at the start of an interval are 8.0 mg in central blood, 2.0 mg in liver, 1.0 mg in kidney, and 4.0 mg in the remaining modeled tissues. At the end, the corresponding amounts are 6.5, 4.2, 1.8, and 4.0 mg. The interval input is 3.0 mg and elimination is 1.5 mg. Calculate the whole-body residual.

First sum the compartment amounts at each endpoint:

A_{\mathrm{body}}(t_a)=8.0+2.0+1.0+4.0=15.0\ \mathrm{mg}, \qquad A_{\mathrm{body}}(t_b)=6.5+4.2+1.8+4.0=16.5\ \mathrm{mg}.

The whole-body amount increased by $16.5-15.0=1.5$ mg. The net external amount expected is $3.0-1.5=1.5$ mg, so

R_{[t_a,t_b]}=(16.5-15.0)-3.0+1.5 =1.5-3.0+1.5=0.0\ \mathrm{mg}.

The central amount fell while liver and kidney amounts rose, but those changes are part of redistribution within the system. Only the summed whole-body amount is compared with external input and elimination.

Using a Simulated Trajectory

A simulation often reports input and elimination rates at discrete time points rather than cumulative amounts. To calculate a residual, integrate the rates over each interval, using the same time unit as the rates. For example, a rate in mg/h must be integrated over hours to obtain mg. Numerical integration introduces approximation error; the residual can therefore depend on the solver and on how rates are integrated from the saved outputs.

For a rate $q(t)$ recorded at two interval endpoints separated by $\Delta t$, the trapezoidal approximation to the interval amount is $\Delta t[q(t_a)+q(t_b)]/2$. This approximation is appropriate when it represents the chosen numerical integration scheme or is a reasonable approximation to the rate trajectory. It is not interchangeable with an exact cumulative input or elimination amount already supplied by the simulation.

Worked Example: Residuals Along a Hypothetical Trajectory

A hypothetical PBPK simulation reports whole-body amounts at $t=0,1,2$ h of 10.00, 12.10, and 12.03 mg. The input rates at those times are 4.0, 2.0, and 0 mg/h; the elimination rates are 0.8, 1.0, and 1.2 mg/h. Approximate interval input and elimination by the trapezoidal rule, then calculate the cumulative residuals at 1 h and 2 h. Assume no bolus doses occur between the reported times.

For the first hour, the input is approximately

D_{\mathrm{in}}(1)=1\ \mathrm{h}\times\frac{4.0+2.0}{2}\ \mathrm{mg/h}=3.0\ \mathrm{mg}.

The first-hour elimination is $1\ \mathrm{h}\times(0.8+1.0)/2\ \mathrm{mg/h}=0.9$ mg. Thus,

R(1)=12.10-10.00-3.0+0.9=0.00\ \mathrm{mg}.

For the second hour, the input is $1\ \mathrm{h}\times(2.0+0)/2\ \mathrm{mg/h}=1.0$ mg, and elimination is $1\ \mathrm{h}\times(1.0+1.2)/2\ \mathrm{mg/h}=1.1$ mg. Cumulative amounts through 2 h are therefore 4.0 mg input and 2.0 mg eliminated. The cumulative residual is

R(2)=12.03-10.00-4.0+2.0=0.03\ \mathrm{mg}.

The trajectory has a residual of 0.03 mg at 2 h using the stated rate integration. This small nonzero value could reflect rounding or numerical integration, but its acceptability cannot be decided from this number alone; it depends on the simulation's numerical precision and the scale of the amounts. The result is an accounting check, not evidence that the trajectory is physiologically accurate.

Choosing and Interpreting the Check

The cumulative residual $R(t)$ summarizes the net discrepancy from the initial time to $t$. The interval residual $R_{[t_a,t_b]}$ instead summarizes the discrepancy within a selected interval. These are complementary views: a cumulative residual can reveal a discrepancy that has accumulated over a long run, while interval residuals show whether the discrepancy is concentrated in particular time ranges. A cumulative residual near zero at a final time does not guarantee that every interval was balanced; discrepancies of opposite signs can offset.

Another useful view is the rate balance. In a continuous-time model, the whole-body derivative should satisfy $dA_{\mathrm{body}}/dt=I-E$. Comparing this relation at time points is a local check, whereas the integrated residual compares accumulated amounts. Rate-based checks can be sensitive to noisy numerical derivatives; integrated checks avoid directly differentiating saved amounts but require reliable rate integration. Neither replaces the other in every setting.

When reporting a residual, state the time interval, the amount units, whether input and elimination were integrated or taken from cumulative outputs, and the numerical precision of the result. A relative or scaled residual can help compare runs with very different dose scales, but there is no single universal scaling convention or acceptance threshold. Preserve the absolute residual as well, so the discrepancy remains interpretable in amount units.

Assumptions, Diagnostics, and Limits

The mass-balance calculation relies on several conditions. The compartment sum must include all states inside the declared system; amounts must use consistent units; and cumulative input and elimination must refer to the same drug species and time interval as the amounts. Internal exchange must represent transfer rather than creation or destruction of drug. Any additional external source or sink must be included in the input or elimination accounting.

Warning signs include a residual much larger than the reporting precision, a residual that grows steadily over time, abrupt jumps not aligned with a recorded dose, or residuals that change substantially when the output time grid or integration method changes. These signs indicate that the accounting result deserves closer examination; they do not, by themselves, identify a specific implementation error. A zero residual is also not a full validation: compensating errors can preserve the total while producing incorrect compartment amounts.

A common mistake is to add concentrations from several organs and treat the sum as total drug amount. Concentration is amount per volume, so this operation generally has the wrong units and ignores differences in compartment volume. The correction is to convert each concentration state to its compartment amount using the model's defined volume relationship, sum those amounts, and only then calculate the residual. A second frequent mistake is to count internal tissue exchange as elimination. Internal fluxes cancel in the whole-body sum; only transfers across the chosen system boundary belong in external input or elimination.

Key takeaway: The whole-body residual is the change in summed compartment amounts minus cumulative input plus cumulative elimination. It should be near zero for a correctly accounted simulation, but it checks conservation of the chosen modeled quantity—not the biological validity of the model.

Check Your Understanding

Try each question, then compare your reasoning with the solutions below.

A model starts with 2.0 mg in the body, receives 6.0 mg, eliminates 1.5 mg, and ends with 6.8 mg. What is the residual?
Why do intercompartmental fluxes not appear in the whole-body residual?
A residual is reported in mg/h. What is wrong with that report if it is a cumulative residual?
Over one interval, total body amount changes by $-2.0$ mg, input is 1.0 mg, and elimination is 3.2 mg. Calculate and interpret the residual.

Solutions. (1) $R=6.8-2.0-6.0+1.5=0.3$ mg. The modeled amount is 0.3 mg greater than the amount implied by the recorded initial amount, input, and elimination. (2) Each internal transfer is a loss from one compartment and an equal gain to another, so the terms cancel when compartment balances are summed. (3) A cumulative residual is an amount, not a rate; mg/h would indicate that a rate quantity was reported or that units were not integrated consistently. (4) $R=-2.0-1.0+3.2=0.2$ mg. The residual is positive: the simulated amount fell 0.2 mg less than expected from the stated input and elimination.