Initial Conditions for Tissue and Blood Amounts

Why Initial Conditions Need Careful Accounting

A mass-balance model describes how amounts change from a specified starting time. Even if every differential equation is correct, the simulated trajectory can be wrong if the initial amount is assigned to the wrong state, converted using an incompatible volume, or counted in two states at once. Initializing a model therefore requires more than entering concentration values: each value must be matched to the physical pool and state it describes.

For a perfused tissue, drug may be represented in tissue, arterial blood, and venous blood. Whether these are separate states depends on the model structure. The essential bookkeeping rule is that each modeled amount must refer to a distinct physical pool. If the blood contained within a tissue region is already included in its reported tissue amount, it cannot also be added as a separate vascular amount.

Definition: An initial condition is the value assigned to a model state at the start of simulation, conventionally written at \(t=0\). An initial amount is the amount of drug in that state at that time. For a well-mixed compartment with volume \(V\) and representative concentration \(C\), the amount is \(A(0)=C(0)V\), provided the concentration and volume refer to the same physical pool and use compatible units.

From Concentration to Amount

The conversion follows from the definition of concentration as amount per volume. In a homogeneous compartment, an amount \(A\) distributed through volume \(V\) at concentration \(C\) satisfies \(C=A/V\). Rearranging gives the initialization formula. If concentration varies across the compartment, the exact amount is the integral of concentration over volume; using \(CV\) then assumes \(C\) is the appropriate volume-average concentration.

$$ A(0)=C(0)V $$

Units provide a quick check. For example, a concentration in mg/L multiplied by a volume in L gives an amount in mg. A concentration reported per litre of plasma should not be multiplied by a whole-blood volume unless it is first converted to a whole-blood concentration, or the calculation is otherwise explicitly defined on a plasma-volume basis.

Let \(A_T\), \(A_A\), and \(A_V\) denote amounts in tissue, arterial blood, and venous blood, with corresponding concentrations \(C_T\), \(C_A\), and \(C_V\), and volumes \(V_T\), \(V_A\), and \(V_V\). In the examples below, these are separate, nonoverlapping pools. A model may instead use a single blood state, or use a tissue amount that includes some vascular content. The initialization must follow the states actually defined in that model; the labels alone do not determine what has been counted.

Assumptions for the simple conversion: The concentration is measured or specified at the same initial time as the model state; its basis matches the selected volume; the amount is represented by a well-defined compartment; and the volume is known and expressed in compatible units. If any of these conditions fails, \(CV\) may not represent the intended initial amount.

Worked Initializations

Worked Example: Initializing Separate Tissue and Blood States

Hypothetical data. At \(t=0\), a model has three distinct states. Arterial blood has \(C_A=1.80\) mg/L and \(V_A=1.20\) L; venous blood has \(C_V=1.00\) mg/L and \(V_V=0.800\) L; and tissue has \(C_T=0.350\) mg/L and \(V_T=12.0\) L. Assume the tissue amount excludes the separately represented blood pools.

Convert each concentration using the volume for that same state:

\begin{aligned} A_A(0)&=C_A(0)V_A=(1.80\ \text{mg/L})(1.20\ \text{L})=2.16\ \text{mg},\\ A_V(0)&=C_V(0)V_V=(1.00\ \text{mg/L})(0.800\ \text{L})=0.800\ \text{mg},\\ A_T(0)&=C_T(0)V_T=(0.350\ \text{mg/L})(12.0\ \text{L})=4.20\ \text{mg}. \end{aligned}

Because the three pools are disjoint, their initial total is \(A_{\mathrm{total}}(0)=2.16+0.800+4.20=7.16\) mg. This total is the amount represented by these three states; it is not automatically the whole-body amount if other states are present.

Worked Example: Converting Plasma Concentrations for Blood States

Hypothetical data. Suppose the model requires whole-blood amounts, but the initial arterial and venous measurements are plasma concentrations. Let the blood-to-plasma concentration ratio be \(R_B=C_{\mathrm{blood}}/C_{\mathrm{plasma}}=0.850\) for both pools. The arterial plasma concentration is \(2.40\) mg/L with blood volume \(0.500\) L; the venous plasma concentration is \(1.50\) mg/L with blood volume \(0.700\) L. A separate tissue state has \(C_T=0.200\) mg/L and \(V_T=8.00\) L, and excludes these blood pools.

First convert each plasma concentration to whole-blood concentration, then multiply by the whole-blood volume:

\begin{aligned} C_{A,\mathrm{blood}}(0)&=R_B C_{A,\mathrm{plasma}}(0) =(0.850)(2.40\ \text{mg/L})=2.04\ \text{mg/L},\\ A_A(0)&=(2.04\ \text{mg/L})(0.500\ \text{L})=1.02\ \text{mg},\\ C_{V,\mathrm{blood}}(0)&=(0.850)(1.50\ \text{mg/L})=1.275\ \text{mg/L},\\ A_V(0)&=(1.275\ \text{mg/L})(0.700\ \text{L})=0.8925\ \text{mg},\\ A_T(0)&=(0.200\ \text{mg/L})(8.00\ \text{L})=1.60\ \text{mg}. \end{aligned}

The total is \(1.02+0.8925+1.60=3.5125\) mg, or \(3.513\) mg to four significant figures. This calculation assumes the stated ratio applies to both blood pools at the initial time. If the model instead defines a plasma amount state, the appropriate plasma volume and plasma concentration should be used directly; converting to whole blood would then be inappropriate.

Worked Example: Avoiding Double Counting in a Tissue Region

Hypothetical data. A reported tissue-region concentration is \(0.500\) mg/L, averaged over a gross region volume of \(1.20\) L. This reported amount includes blood within the region. The vascular blood volume inside that region is \(0.0800\) L, and its blood concentration is \(1.00\) mg/L. The model uses separate vascular and nonvascular tissue states.

The total drug in the measured region is \(C_{\mathrm{region}}V_{\mathrm{region}}=(0.500\ \text{mg/L})(1.20\ \text{L})=0.600\) mg. The amount already assigned to the region's blood state is \(A_{\mathrm{blood,region}}=(1.00\ \text{mg/L})(0.0800\ \text{L})=0.0800\) mg. Therefore, the nonvascular tissue amount is the regional total minus that vascular amount:

A_{\mathrm{tissue}}(0)= C_{\mathrm{region}}(0)V_{\mathrm{region}} -C_{\mathrm{blood}}(0)V_{\mathrm{blood,region}} =0.600\ \text{mg}-0.0800\ \text{mg} =0.520\ \text{mg}.

The two state amounts sum to \(0.520+0.0800=0.600\) mg, matching the measured regional amount. Assigning \(0.600\) mg to tissue and then adding \(0.0800\) mg to blood would give \(0.680\) mg and count the vascular drug twice. This subtraction is valid here because the given regional concentration is explicitly an average over the gross region and includes the specified vascular blood.

Summing States and Checking the Initial Total

A total amount is the sum of amounts in mutually exclusive pools. If the model has tissue states indexed by \(i\) and distinct blood states, the initial total is:

A_{\mathrm{total}}(0)=\sum_i A_{T_i}(0)+A_A(0)+A_V(0)+\sum_j A_j(0)

The final sum includes any other disjoint states in the model. It must not include the same physical blood or tissue drug under two labels. Conversely, omitting a modeled pool understates the total. A useful justification is simply to partition the modeled drug into disjoint pools: the total amount in a union of disjoint pools equals the sum of the amount in each pool. If the pools overlap, that sum counts the overlap more than once and is not the amount in their union.

When dose history is available, compare the calculated total with the amount expected to remain in the modeled system at \(t=0\). That expectation may differ from the administered dose if drug was eliminated, remains outside the modeled system, or entered or left the system before the chosen initial time. A dose is an input to the system, not an additional state amount to add after that drug has already been assigned to states.

List the states and their physical pools.
Record whether each state represents tissue, arterial blood, venous blood, or another pool, and identify any overlap in how the pools are defined.

Match each concentration to a volume.
Check the time, units, and basis of the concentration and volume. Convert between plasma and whole-blood concentration only when the state definition requires it.

Calculate amounts and sum distinct states.
Use \(A(0)=C(0)V\) where its assumptions hold. If a measured region includes a separately modeled pool, remove that pool from the regional amount before assigning the remainder.

Check plausibility and the total.
Confirm that amounts are nonnegative, units are consistent, and the summed amount agrees with available dose and mass-balance information at the chosen starting time.

Diagnostics, Alternatives, and Limitations

Several discrepancies can reveal an initialization problem. A negative amount after subtracting vascular content suggests incompatible concentration bases, volumes, or measurements, or an incorrect overlap assumption. A total larger than the amount plausibly present can indicate double counting, such as adding a separate blood amount to a tissue measurement that already includes it. A total that is too small can indicate an omitted state or use of a concentration measured in one phase with the volume of another. These checks identify a problem to investigate; they do not by themselves establish which input is wrong.

There are several legitimate ways to specify starting states, but they answer different questions:

Approach	What is supplied	Main check
Concentration-based initialization	Concentration and matching compartment volume; calculate amount as \(CV\).	Confirm the concentration basis, volume, and compartment boundaries match.
Amount-based initialization	Amount for each model state directly.	Confirm the amounts refer to distinct pools and sum to a plausible total.
Dose-allocation initialization	A dose amount and fractions assigned to states.	Confirm the fractions describe the amount present at the initial time and sum to one if they allocate the entire dose.

An amount-based specification avoids a concentration-to-volume conversion, but it does not remove the need to define what each state contains. Dose fractions can be useful when initial concentrations are unavailable, but they are assumptions about distribution, not measurements of it. Likewise, an initial condition need not imply that tissue and blood are at equilibrium. The specified values define the starting state; whether they are biologically plausible must be judged against the data and purpose of the model.

A common mistake is to reason, “The measured tissue concentration describes the tissue, so I should multiply it by tissue volume and then add all the blood amounts.” The correction is to check exactly what the tissue measurement includes. If it is an extravascular tissue concentration and the state volume excludes blood, the separate blood amount can be added. If it is a gross-region average that already includes vascular drug, the vascular contribution must not be counted again. If the measurement basis is unknown, the assignment cannot be resolved by arithmetic alone; the measurement definition must be clarified.

Key takeaway: Initialize amounts from concentrations only after matching each concentration to its physical pool and volume. Sum only disjoint state amounts, and use the initial total as a consistency check rather than as a substitute for defining the states.

Check Your Understanding

Answer each question, then compare with the brief solutions.

A tissue concentration is \(0.250\) mg/L in a \(4.00\)-L tissue compartment that excludes blood. What is the initial tissue amount?
A blood state requires whole-blood concentration. The measured plasma concentration is \(1.60\) mg/L, \(R_B=0.900\), and blood volume is \(0.500\) L. Find the initial blood amount.
A gross tissue-region amount is \(0.900\) mg and includes \(0.120\) mg in its vascular blood. What nonvascular tissue amount should be assigned if the model has separate states for these pools?
Why should an administered dose not automatically be added to the sum of initial state amounts?

Solutions: (1) \(A_T(0)=(0.250\ \text{mg/L})(4.00\ \text{L})=1.00\) mg. (2) First, \(C_{\mathrm{blood}}=(0.900)(1.60\ \text{mg/L})=1.44\) mg/L; then \(A_{\mathrm{blood}}=(1.44\ \text{mg/L})(0.500\ \text{L})=0.720\) mg. (3) \(0.900-0.120=0.780\) mg. (4) The administered dose is an input, not a separate pool of drug. If its drug is already represented in the initial states, adding the dose again counts it twice.