Representing the Body as a Connected Organ Network

From Organ Compartments to a Connected Circulation

An organ-resolved PBPK model is more than a list of tissues: it specifies how blood connects those tissues and carries drug between them. This matters because a drug can reach an organ only through its supplying blood pathway, and blood leaving different organs returns to a shared circulation. Representing the body as a network makes those connections explicit, while avoiding the assumption that every tissue is one undifferentiated space.

A useful starting point is a simplified closed circulation. Blood leaving the heart enters the systemic arterial pool. From there, it flows through organ beds arranged largely in parallel, and their outflows join in the systemic venous pool. Blood then passes through the right heart, pulmonary circulation, and left heart before returning to the systemic arterial pool. This is a structural representation, not a claim that each organ has identical anatomy or kinetics.

Definition: A compartment is a model region represented as a unit with a specified contents or state, such as an organ tissue or a blood pool. A flow connection is a directed pathway that carries blood from one compartment to another. A connected organ network is a set of compartments and directed flow connections arranged to represent circulation and organ perfusion. In a perfusion-limited network, blood flow connects an organ’s supplying blood to its tissue and then to an outflow pathway; the network alone does not specify the tissue’s drug-transfer or elimination kinetics.

For a simple systemic network, let \(Q_{\mathrm{CO}}\) denote cardiac output and \(Q_i\) the blood flow through systemic organ bed \(i\). The organ beds are parallel pathways between the arterial and venous pools. Under a steady-flow representation with no unrepresented systemic pathway, the flows must satisfy the conservation relationship shown below.

Q_{\mathrm{CO}} = \sum_{i=1}^{m} Q_i

Here \(m\) is the number of systemic beds represented. If organ flows are expressed as fractions \(f_i\) of cardiac output, then \(Q_i=f_i Q_{\mathrm{CO}}\), and the represented fractions sum to one. This is the same flow-accounting principle used when allocating organ blood flows; the network view emphasizes where each flow goes.

A Schematic Whole-Body Network

The following table gives a compact directed diagram. The organ labels are illustrative. In an actual PBPK model, the set of compartments and connections should match the physiological detail needed for the prediction.

From	Directed connection	To	Role in the network
Left heart	Systemic arterial flow	Arterial blood pool	Supplies systemic organ beds
Arterial blood pool	\(Q_{\mathrm{liver}}\)	Liver bed	One systemic organ pathway
Arterial blood pool	\(Q_{\mathrm{kidney}}\)	Kidney bed	One systemic organ pathway
Arterial blood pool	\(Q_{\mathrm{other}}\)	Other represented beds	Aggregated systemic pathways
Systemic organ beds	Venous outflows	Venous blood pool	Combines systemic return
Venous blood pool	Venous return	Right heart	Returns systemic blood to the heart
Right heart	Pulmonary flow	Lungs	Routes blood through the pulmonary circulation
Lungs	Pulmonary venous return	Left heart	Completes the circulation loop

In this schematic, organs are parallel with respect to systemic blood delivery: each receives blood from the arterial side and returns it to the venous side. The pulmonary circulation, by contrast, is in series with the systemic circulation in the overall loop. At steady state, with no shunt or net storage of blood between the two circuits, pulmonary flow and systemic cardiac output agree. If the model represents a transient or a shunt, the connections and flow accounting must reflect that situation rather than imposing this simplified equality without qualification.

Arterial and venous pools are often represented explicitly because they connect multiple organ beds and provide locations for blood concentrations to be defined. They are not interchangeable with an organ tissue compartment: a blood pool represents circulating blood, while an organ compartment represents tissue. A model may further divide blood into individual vessels or regional pools, but added nodes are useful only if the distinctions matter to the intended prediction.

Worked Example: Checking a Systemic Flow Network

Suppose a hypothetical steady-state model uses a cardiac output of \(5.00\ \mathrm{L/min}\) and represents three systemic pathways: liver, kidney, and a combined group of other tissues. Their assigned flows are \(1.00\), \(0.75\), and \(3.25\ \mathrm{L/min}\), respectively. Check whether the network conserves flow and find each pathway’s fraction of cardiac output.

First sum the pathway flows:

\sum_i Q_i = 1.00 + 0.75 + 3.25 = 5.00\ \mathrm{L/min}

The total equals \(Q_{\mathrm{CO}}=5.00\ \mathrm{L/min}\), so the assigned systemic pathways account for the stated cardiac output under the assumed network. The fractions are each pathway flow divided by cardiac output:

f_{\mathrm{liver}}=\frac{1.00}{5.00}=0.200,\qquad f_{\mathrm{kidney}}=\frac{0.75}{5.00}=0.150,\qquad f_{\mathrm{other}}=\frac{3.25}{5.00}=0.650

Their sum is \(0.200+0.150+0.650=1.000\). Thus, the flows and fractions are internally consistent. This check establishes consistency with the stated whole-body flow; it does not establish that the hypothetical allocation is physiologically appropriate for a particular person or condition.

How Flow Carries Drug Through the Network

A directed blood-flow connection carries drug as well as blood. If blood entering a pathway has drug concentration \(C\) and volumetric flow \(Q\), the drug mass carried across that connection per unit time is \(Q C\). For example, \(\mathrm{L/min}\) multiplied by \(\mathrm{mg/L}\) gives \(\mathrm{mg/min}\). This identifies delivery by blood flow; it does not by itself specify how much drug enters tissue, remains in tissue, is metabolized, or exits in venous blood.

Worked Example: Drug Delivery to a Hypothetical Organ

In a hypothetical model, an organ receives blood at \(Q=1.20\ \mathrm{L/min}\). The concentration in its supplying arterial blood is \(C_a=2.00\ \mathrm{mg/L}\). Calculate the drug mass entering the organ in the blood per minute.

The carried mass rate is flow multiplied by concentration:

Q C_a = (1.20\ \mathrm{L/min})(2.00\ \mathrm{mg/L}) = 2.40\ \mathrm{mg/min}

Thus, \(2.40\ \mathrm{mg/min}\) is delivered to the organ boundary in the incoming blood under the stated conditions. It is not automatically the rate of tissue uptake or elimination. Those quantities require a model of what happens within and across the organ, beyond the blood-flow connection.

When blood from several organ pathways joins a common venous pool, the resulting concentration depends on the amount of drug carried by each flow. Under the assumption of complete mixing at the junction, concentration is total incoming drug mass rate divided by total incoming blood-flow rate. This follows directly from conservation at the mixing junction: the sum of the incoming mass rates must equal the mass rate in the mixed outflow, provided there is no drug gain, loss, or storage at that junction.

C_{\mathrm{mix}} = \frac{\sum_{i=1}^{m} Q_i C_i}{\sum_{i=1}^{m} Q_i}

This is a flow-weighted average, not generally the arithmetic average of the concentrations. Its assumptions are local to the junction: the listed flows enter the same pool, the drug is conserved during mixing, and the resulting pool is sufficiently mixed to be represented by one concentration.

Worked Example: Mixing Two Venous Outflows

Suppose two hypothetical organ pathways return blood to the same venous pool. Pathway 1 has \(Q_1=1.00\ \mathrm{L/min}\) and \(C_1=4.00\ \mathrm{mg/L}\); pathway 2 has \(Q_2=3.00\ \mathrm{L/min}\) and \(C_2=1.00\ \mathrm{mg/L}\). Assume both flows enter a single, completely mixed pool, with no drug loss or storage at the junction. Find the mixed concentration.

The incoming drug mass rates are \(Q_1C_1=(1.00)(4.00)=4.00\ \mathrm{mg/min}\) and \(Q_2C_2=(3.00)(1.00)=3.00\ \mathrm{mg/min}\). The total incoming flow is \(1.00+3.00=4.00\ \mathrm{L/min}\). Therefore:

C_{\mathrm{mix}} = \frac{(1.00)(4.00)+(3.00)(1.00)}{1.00+3.00} = \frac{7.00\ \mathrm{mg/min}}{4.00\ \mathrm{L/min}} = 1.75\ \mathrm{mg/L}

The \(1.75\ \mathrm{mg/L}\) result is closer to \(1.00\ \mathrm{mg/L}\) than to \(4.00\ \mathrm{mg/L}\), because the lower-concentration pathway contributes three times as much flow. The simple arithmetic mean, \(2.50\ \mathrm{mg/L}\), would incorrectly give equal weight to unequal flows.

Building and Checking the Network

A practical network specification begins by naming the compartments and then drawing each connection with its direction and flow. Before adding drug-specific behavior, check whether the structure and flow assignments agree with the intended circulation. The following sequence is useful when translating physiology into a model diagram.

Choose the represented regions.
List the blood pools and organ beds needed for the prediction. Keep blood and tissue compartments conceptually distinct.

Draw directed connections.
Show where arterial blood enters each pathway and where its venous return goes. Include the pulmonary route in a whole-body circulation loop.

Assign and reconcile flows.
Use the relevant cardiac output and organ-flow allocation, then check that represented branches account for the flow entering their shared junctions.

State what each pool means.
Specify whether a node is a mixed blood pool, an organ tissue, or an aggregate of tissues. Do not infer mixing or tissue kinetics from connectivity alone.

The network can be contrasted with a single lumped tissue compartment. A lumped representation combines regions and may be suitable when their separate routes and concentrations do not matter for the prediction. An organ-resolved network preserves distinct pathways and allows organ-specific blood flows and concentrations to be represented. Neither representation is automatically superior: a more detailed network has value only when its distinctions are supported and relevant.

Several diagnostics help identify structural or bookkeeping problems. A mismatch between the sum of systemic branch flows and the flow supplied to their junction can indicate an omitted pathway, a duplicated flow, or inconsistent inputs. A branch drawn in the wrong direction can send drug through the wrong sequence of compartments. A mixed concentration outside the range of its incoming concentrations signals an arithmetic or assumption error when all flows are nonnegative and drug is conserved. These checks test internal consistency, not whether the model’s physiological assumptions are true.

Common Mistakes and Limitations

Counting the same cardiac output more than once. Organ pathways are parallel allocations of the systemic flow, not separate full cardiac outputs. Their flows are summed to check the total.
Treating a connection as a kinetic mechanism. A blood-flow arrow says where blood goes. It does not define tissue partitioning, uptake, metabolism, or elimination.
Averaging concentrations without flow weights. A low-flow branch and a high-flow branch contribute different mass rates. Use the flow-weighted formula when complete mixing and drug conservation at the junction are assumed.
Assuming all circulation details are represented. A simplified network may omit shunts, regional subdivisions, or specialized organ supplies. If an omitted route affects the prediction, the diagram must be expanded or its limitation acknowledged.
Applying steady-state flow equalities indiscriminately. The simple equality between pulmonary and systemic flow is appropriate for a closed, steady circulation without shunts or net storage; transient conditions may require a more careful representation.

Key takeaway: A whole-body PBPK network makes organ pathways and blood-pool connections explicit. Flow conservation checks the circulation, while flow multiplied by concentration describes drug carried in blood. Neither calculation alone determines what happens to drug inside an organ.

Check Your Understanding

Answer the questions, then compare with the solutions.

A hypothetical network has systemic organ flows of \(0.80\), \(1.20\), and \(2.00\ \mathrm{L/min}\). What cardiac output is implied if these are the only systemic pathways?
A pathway carries \(0.50\ \mathrm{L/min}\) of blood at \(3.00\ \mathrm{mg/L}\). What drug mass rate enters the pathway in blood?
Two venous flows are \(2.00\ \mathrm{L/min}\) at \(1.00\ \mathrm{mg/L}\) and \(1.00\ \mathrm{L/min}\) at \(4.00\ \mathrm{mg/L}\). Under complete mixing with no loss or storage, calculate the mixed concentration.
Why does a flow diagram alone not establish the rate of drug elimination by an organ?

Solutions. (1) The implied cardiac output is \(0.80+1.20+2.00=4.00\ \mathrm{L/min}\), assuming the listed pathways account for all systemic flow. (2) The incoming mass rate is \((0.50\ \mathrm{L/min})(3.00\ \mathrm{mg/L})=1.50\ \mathrm{mg/min}\). (3) The mixed concentration is \(\{(2.00)(1.00)+(1.00)(4.00)\}/(2.00+1.00)=6.00/3.00=2.00\ \mathrm{mg/L}\). (4) The diagram specifies blood connections and flow, but elimination requires additional information or a kinetic model for drug disposition within the organ.