Initial Conditions and Dose-Mass Bookkeeping

Why Initial Amounts Need a Dose-History Check

A PBPK simulation can have dimensionally correct equations and still begin with the wrong amount of drug. A dose may be omitted, entered twice, placed in the wrong compartment, or represented both as an initial amount and as a dose event. Such errors can distort every subsequent concentration profile. The key audit is not simply “does the model contain the dose?” but “does each modeled starting amount and input correspond to the intended administration history?”

This tutorial treats dose as an amount of active drug, such as milligrams, and compartment states as amounts in the same mass unit. It focuses on the mass assigned at the start of a simulation and the location and timing of dose inputs. It does not assume that every administered dose is immediately present in systemic compartments: an extravascular dose may initially reside in a depot, and an infusion enters over time.

Definition: Let \(A_i(t)\) denote the amount of drug in model compartment \(i\) at time \(t\). The modeled drug amount at time \(t\) is \(M(t)=\sum_i A_i(t)\), summed over the compartments included in the chosen inventory. A dose history specifies the amounts, routes, and administration times that the model is intended to represent. A bookkeeping audit checks that the initial state and subsequent dose inputs encode that history without omissions or duplication.

Separate the Predose State From the Dose Event

For a simulation beginning at time \(t_0\), distinguish the state just before the start, \(A_i(t_0^-)\), from the state just after any dose administered at that time, \(A_i(t_0^+)\). If an IV bolus of amount \(D\) is assigned to compartment \(j\), the event is represented by

A_j(t_0^+)=A_j(t_0^-)+D,\qquad A_i(t_0^+)=A_i(t_0^-)\quad (i\ne j).

Thus, the total amount increases by \(D\) at the event. If there is no predose drug, the initial state immediately before the dose is zero. The post-dose state contains the dose. A simulation may implement this by setting the post-dose initial amount directly or by applying a dose event at \(t_0\); it must not do both for the same dose.

For an IV bolus, the full dose enters the modeled injection or central compartment designated by the model. For an infusion, the dose is ordinarily represented by an input rate over an interval, not by placing the entire planned infusion amount into the body at its start. For an extravascular dose, the administered amount ordinarily starts in a depot or absorption compartment. These assignments encode different dose histories even when the nominal dose is identical.

Worked Example: IV Bolus With a Predose Amount

A hypothetical model begins at \(t_0\) with 3 mg in the central compartment and 1 mg in a tissue compartment, representing a known predose state. At \(t_0\), a 100 mg IV bolus is given into the central compartment. Find the post-dose amounts and check the total.

Before the dose, the modeled total is \(3+1=4\) mg. The bolus adds 100 mg to the central compartment only:

A_{\mathrm{central}}(t_0^+)=3+100=103\ \mathrm{mg},\qquad A_{\mathrm{tissue}}(t_0^+)=1\ \mathrm{mg}.

The post-dose total is \(103+1=104\) mg, which also equals the 4 mg predose amount plus the 100 mg bolus. Initializing the central compartment to 103 mg and then applying the same 100 mg bolus event would instead produce 204 mg total: the dose would be counted twice.

Bookkeeping for Boluses, Infusions, and Depots

A bolus is a discrete amount input; an infusion is a rate input. If an infusion begins at \(t_0\) at a constant rate \(R\) and runs for duration \(T\), its administered amount is \(RT\). This is an input to the model over the interval, not automatically the body amount at \(t_0\). Drug may also leave the modeled system during the infusion, so the amount remaining in compartments at a later time need not equal the cumulative administered amount.

For an extravascular dose \(D\), a simple depot model may have a depot amount \(A_{\mathrm{dep}}\) and first-order absorption rate constant \(k_a\). If systemic availability is represented by a fraction \(F\), with \(0\le F\le1\), one bookkeeping convention is to send fraction \(F\) of the depot’s absorption flux into systemic compartments and count the remainder as presystemic loss. In that convention, the depot begins with the full administered dose, not \(FD\), because the fraction not entering the systemic circulation is still represented as drug in the depot at the start.

A_{\mathrm{dep}}(0)=D,\qquad \text{systemic input}=F k_a A_{\mathrm{dep}},\qquad \text{presystemic loss}=(1-F)k_a A_{\mathrm{dep}}.

Other model structures can represent presystemic metabolism in explicit compartments. The accounting must match the chosen structure: if the model already represents the relevant loss explicitly, do not add another loss term for the same process. Conversely, if only the fraction entering systemic circulation is tracked, the unrepresented fraction should be identified as outside that tracked inventory rather than silently treated as systemic drug.

Worked Example: Oral Depot With Explicit Presystemic Loss

Consider a hypothetical 200 mg extravascular dose. The model has one depot, with \(k_a=0.50\ \mathrm{h}^{-1}\), and represents systemic availability as \(F=0.80\). At time zero, the depot contains the full dose and systemic compartments contain none. Calculate the initial inputs and reconcile the mass after one hour, assuming no other elimination or distribution losses for this bookkeeping calculation.

The initial depot amount is \(A_{\mathrm{dep}}(0)=200\) mg. The initial absorption flux is \(k_aA_{\mathrm{dep}}=0.50\times200=100\) mg/h. Of this, systemic input is \(0.80\times100=80\) mg/h and presystemic loss is \(0.20\times100=20\) mg/h. The two pathways sum to the 100 mg/h leaving the depot.

After one hour, the depot amount is

A_{\mathrm{dep}}(1)=200e^{-0.50(1)}=121.3\ \mathrm{mg}.

The amount that left the depot is \(200-121.3=78.69\) mg. The systemic amount delivered is \(0.80\times78.69=62.96\) mg, and presystemic loss is \(0.20\times78.69=15.74\) mg. The accounted amount is \(121.3+62.96+15.74=200.0\) mg, allowing for rounding. If the initial depot had instead been set to 160 mg while also applying \(F=0.80\), the model would apply the availability fraction twice in this convention.

Worked Example: An IV Infusion Is Not an Initial Bolus

A hypothetical patient receives a 60 mg IV infusion at a constant rate over five hours. The model begins immediately before the infusion, with no predose drug, and uses an input rate into the central compartment. Find the infusion rate and state what amount should be present initially.

The infusion rate is

R=\frac{60\ \mathrm{mg}}{5\ \mathrm{h}}=12\ \mathrm{mg/h}.

Immediately before the infusion starts, the initial amount attributable to this dose is 0 mg. The model applies 12 mg/h during the five-hour infusion; the cumulative administered amount by the end is \(12\times5=60\) mg. The amount in the body at five hours can be less than 60 mg if elimination or other losses have occurred. Setting the initial central amount to 60 mg and also applying the infusion would encode 120 mg of dose input, not the intended 60 mg.

A Mass-Increment Check and Its Limits

A useful first check is to compare the total modeled amount immediately after an event with the total immediately before it. For a bolus of \(D\) entering the modeled inventory, the expected increment is \(D\). For a depot dose, the expected increment in the inventory containing the depot is also \(D\), even though the systemic amount initially may be zero. For an infusion, the cumulative input over an interval is the integral of the rate over that interval. These checks are about amount, so all terms must be expressed in compatible mass units.

The check can be extended when the model tracks losses. Let \(L(t)\) be cumulative drug loss from the chosen inventory, measured in amount units. If no other dose is given between 0 and \(t\), the inventory relationship is

M(t)+L(t)=M(0)+D_{\mathrm{in}}(0,t),

where \(D_{\mathrm{in}}(0,t)\) is the cumulative dose input during the interval. This identity follows by accounting for each amount entering or leaving: internal transfers between compartments cancel when the compartment amounts are summed; external inputs increase the total; and external losses are recorded in \(L(t)\). If there are multiple dose events, their amounts are included in \(D_{\mathrm{in}}(0,t)\). This is a bookkeeping identity for the stated inventory, not a claim that the amount remaining in the body equals the administered dose at all times.

A correct total does not guarantee correct allocation. For example, a 120 mg IV bolus assigned as 90 mg central, 20 mg liver, and 10 mg tissue sums to 120 mg. Yet it does not represent an immediate bolus into the central compartment if the intended model places the whole bolus there. The mass check passes, but the initial concentrations and subsequent exchange fluxes can be wrong. Check both the sum and the compartment-by-compartment allocation against the route and timing.

Practical Audit and Failure Modes

Write the dose history.
Record each dose amount, route, and administration time, and identify any predose drug that the simulation is intended to include.

Map each input to a model state.
Identify the bolus compartment, infusion input, or extravascular depot. Confirm whether a dose is entered as an initial amount, an event, or a rate—not more than one representation for the same dose.

Reconcile amounts and units.
Sum the initial amounts and compare them with the intended predose state and any dose at the start. Convert concentration to amount with the compartment volume used by the model, \(A_i=V_iC_i\), and keep units consistent.

Check allocation and timing.
Verify that the dose enters the correct compartment at the correct time. For an infusion, integrate the input rate; for a depot, distinguish administered depot amount from systemic entry.

Inspect the simulation at the event boundary.
Compare compartment amounts immediately before and after a dose. Check for an unexpected jump, missing jump, negative amount, or solver/event convention that applies a time-zero dose twice or not at all.

Common warning signs include a nonzero initial systemic amount for a first-ever extravascular dose with no modeled absorption yet; a time-zero jump twice the intended bolus; a depot initialized to a bioavailability-adjusted amount while the same availability fraction is applied again; and an infusion whose rate has units of amount rather than amount per time. A concentration-to-amount conversion can also be wrong if the volume does not correspond to the model’s compartment definition.

Different implementations are valid when their accounting is explicit. A dose event that adds an amount and direct initialization to the post-dose state can represent the same bolus, but the state should not receive both. A depot with an explicit loss pathway can represent presystemic loss, while a simplified systemic input can track only the portion entering the modeled systemic inventory. These approaches are not interchangeable without adjusting the states and losses. The limitation of a mass-only audit is that it does not validate physiology, parameter values, or the correctness of the route-specific model structure; it is a necessary check, not a complete model evaluation.

Common Mistake

Common mistake: “The sum of the initial compartment amounts equals the dose, so the dose setup must be correct.” The total can be right while the dose is assigned to the wrong compartments or route; a depot dose can be mistaken for an immediately systemic dose; and an infusion can be entered as an initial bolus. The correction is to check total amount, compartment allocation, route, and timing separately, then verify the dose-event jump or cumulative input.

Check Your Understanding

Use the stated route and model structure as well as the total amount when answering.

A first-ever 80 mg IV bolus is given into the central compartment at time zero. The model has no predose drug. What are the intended central amount and total amount immediately after the dose?
A 150 mg oral dose is placed in a depot. The model sends 70% of depot absorption into systemic compartments and tracks the remainder as loss. What is the initial depot amount, and what fraction is initially systemic?
A 24 mg IV infusion runs at 6 mg/h for four hours. What is the initial amount attributable to the dose, and how much is cumulatively administered after three hours?
A 100 mg bolus is entered as 100 mg in the initial central state and also as a time-zero dose event. What is wrong with this setup if there is no predose drug?
After a dose, the sum across all compartments equals the administered amount, but some of the bolus was placed in a tissue compartment even though the intended model injects the entire IV bolus centrally. Has the audit passed?

Solutions. (1) The central amount and total amount are each 80 mg immediately after the bolus. (2) The depot starts at 150 mg; the initial systemic amount is zero because absorption has not yet occurred, and 70% of the absorption flux is directed systemically. (3) The initial amount attributable to the dose is zero; after three hours, \(6\times3=18\) mg has been administered. (4) The dose has been counted twice, giving 200 mg total instead of the intended 100 mg. (5) No. The total-mass check passes, but the compartment allocation conflicts with the intended IV bolus location and can produce incorrect initial concentrations and exchange fluxes.