Tutorials › Pharmacometrics › Conservation Laws in QSP
Pharmacokinetics · QSP Foundations

Conservation Laws in QSP

Learn how conservation laws provide the mathematical backbone of quantitative systems pharmacology models by enforcing mass balance, species conservation, compartmental accounting, and mechanistic consistency across biological systems.

Intermediate QSP Foundations Mechanistic Modeling Mass Balance
01 · The big picture

1. What Are Conservation Laws in QSP?

Conservation laws are mathematical statements that describe how quantities such as mass, chemical species, cells, or other modeled entities are created, transported, transformed, and removed within a system.

In quantitative systems pharmacology (QSP), conservation laws are especially important because QSP models attempt to represent biological mechanisms rather than simply fit empirical curves. A model must therefore account consistently for where a quantity comes from, where it goes, and what processes can change it.

The basic accounting principle is:

$$ \text{Rate of accumulation} = \text{inputs} - \text{outputs} + \text{generation} - \text{consumption} $$

For a conserved quantity that is neither generated nor destroyed internally, this simplifies to:

$$ \text{Rate of accumulation} = \text{inputs} - \text{outputs} $$
Core idea: conservation laws make a QSP model obey an accounting system. If a molecule leaves one compartment, it must either appear somewhere else in the model or be explicitly removed through a modeled process.
02 · Why conservation matters

2. Why Are Conservation Laws Important in QSP?

QSP models frequently connect multiple biological scales. A drug may move between plasma and tissue, bind to a receptor, undergo metabolism, alter a signaling pathway, and ultimately change a downstream biomarker or cell population.

Conservation relationships help ensure that these connected mechanisms remain internally consistent.

Modeling problem Conservation principle What it provides
Drug distribution Mass balance Tracks drug entering, leaving, and accumulating in compartments
Receptor binding Species conservation Links free receptor, free ligand, and bound complex
Enzyme systems Pool conservation Tracks substrate, enzyme, complex, and product quantities
Cell populations Population balance Accounts for proliferation, differentiation, migration, and death
Physiological compartments Mass balance Connects flows between blood, tissues, organs, and other spaces
Biomolecular networks Stoichiometric constraints Ensures reaction equations are chemically consistent

Conservation laws are therefore not an optional mathematical decoration. They often determine the structure of the differential equations themselves.

03 · The accounting equation

3. The General Conservation Equation

Let \(X(t)\) represent the amount of a quantity in a compartment. The most general accounting form is:

$$ \frac{dX}{dt} = \text{Input} - \text{Output} + \text{Production} - \text{Consumption} $$

For example, if drug enters a tissue through blood flow, leaves through venous flow, and is metabolized within the tissue, one possible balance is:

$$ \frac{dA_{\mathrm{tissue}}}{dt} = R_{\mathrm{in}} - R_{\mathrm{out}} - R_{\mathrm{met}} $$

Every term corresponds to a physical or biological process. This connection between equations and mechanisms is a defining feature of mechanistic QSP modeling.

Biological pool amount or concentration changes according to a balance Input Output Accumulation = inputs − outputs + generation − consumption

A conservation equation tracks how a quantity changes by accounting explicitly for processes that add, remove, generate, or consume it.

04 · Amount versus concentration

4. Why QSP Models Often Use Amounts

Conservation laws are naturally expressed in terms of amount. If \(A(t)\) is the amount of drug in a compartment, an input of 10 mg adds 10 mg to the system regardless of the compartment volume.

Concentration is obtained by dividing amount by an appropriate volume:

$$ C(t)=\frac{A(t)}{V} $$

This distinction becomes important when physiological volumes differ among compartments or when fluid moves between compartments.

Modeling principle: conservation is fundamentally about accounting for quantities. Concentrations are often useful observables, but the underlying mass balance is usually easiest to formulate in terms of amounts.

For example, if a tissue contains 5 mg of drug in a volume of 2 L, the concentration is:

$$ C=\frac{5\text{ mg}}{2\text{ L}}=2.5\text{ mg/L} $$

If the amount changes or the volume changes, concentration can change even when the other quantity remains constant.

05 · Compartments

5. Conservation Laws Across Compartments

A QSP model commonly divides the biological system into compartments or well-mixed spaces. These may represent plasma, tissues, extracellular fluid, intracellular fluid, organs, or other modeled spaces.

Consider two compartments with amounts \(A_1\) and \(A_2\). If material moves from compartment 1 to compartment 2 at rate \(R_{12}\), then:

$$ \frac{dA_1}{dt}=-R_{12} $$
$$ \frac{dA_2}{dt}=+R_{12} $$

Adding the two equations gives:

$$ \frac{d}{dt}(A_1+A_2)=0 $$

Thus, internal transfer changes the distribution of the quantity without changing its total amount.

Process Compartment 1 Compartment 2 Total system
Internal transfer −\(R\) +\(R\) 0
External input +\(R_{\mathrm{in}}\) 0 +\(R_{\mathrm{in}}\)
Systemic elimination −\(R_{\mathrm{out}}\) 0 −\(R_{\mathrm{out}}\)

This simple accounting structure becomes increasingly valuable as QSP models include many physiological compartments and interacting mechanisms.

06 · Species conservation

6. Conservation of Chemical and Biological Species

QSP models often include several forms of the same underlying species. A receptor, for example, might exist as free receptor \(R\), ligand-bound receptor \(RL\), and perhaps receptor in another molecular state.

If no receptor is synthesized or degraded, the total receptor pool is:

$$ R_{\mathrm{tot}}=R+RL $$

Therefore:

$$ \frac{dR_{\mathrm{tot}}}{dt}=0 $$

If receptor synthesis and degradation occur, the balance becomes:

$$ \frac{dR_{\mathrm{tot}}}{dt} = R_{\mathrm{syn}} - R_{\mathrm{deg}} $$

The conservation relationship remains useful even when the total pool itself is dynamic.

Important distinction: a conserved pool does not necessarily mean that each individual species is constant. Free and bound forms can change substantially while their total pool remains conserved.
07 · Binding systems

7. Conservation Laws in Receptor-Ligand Binding

Consider the reversible binding reaction:

$$ L+R \underset{k_{\mathrm{off}}}{\overset{k_{\mathrm{on}}}{\rightleftharpoons}} LR $$

Here \(L\) is free ligand, \(R\) is free receptor, and \(LR\) is the ligand- receptor complex.

If ligand and receptor are not otherwise produced or removed, conservation relationships are:

$$ L_{\mathrm{tot}}=L+LR $$
$$ R_{\mathrm{tot}}=R+LR $$

The complex changes according to:

$$ \frac{dLR}{dt} = k_{\mathrm{on}}LR? $$

More precisely, because binding consumes one free ligand and one free receptor:

$$ \frac{dLR}{dt} = k_{\mathrm{on}}LR_{\mathrm{free}} - k_{\mathrm{off}}LR $$

Using conventional notation for the free species:

$$ \frac{d[LR]}{dt} = k_{\mathrm{on}}[L][R] - k_{\mathrm{off}}[LR] $$

The corresponding equations for free ligand and receptor contain the opposite reaction terms, ensuring that the binding reaction does not create or destroy molecules.

08 · Reaction networks

8. Stoichiometric Conservation in QSP Models

For larger biochemical networks, conservation can be represented compactly using a stoichiometric matrix.

Let \(x\) be a vector containing the amounts or concentrations of modeled species and \(v(x)\) be a vector of reaction rates. The dynamic system can be written as:

$$ \frac{dx}{dt}=Sv(x) $$

where \(S\) is the stoichiometric matrix. Each column of \(S\) describes how one reaction changes the modeled species.

For a reaction such as:

$$ A+B\rightarrow C $$

the corresponding stoichiometric change is:

$$ \begin{bmatrix} -1\\ -1\\ +1 \end{bmatrix} $$

This means that each reaction event consumes one unit of \(A\), consumes one unit of \(B\), and produces one unit of \(C\).

Linear combinations of species can remain conserved even while individual species change. These conserved quantities correspond to the left-null space of the stoichiometric matrix.

$$ \ell^{T}S=0 $$

For such a vector \(\ell\), the quantity \(\ell^Tx\) remains constant in the absence of external inputs, outputs, or other mechanisms that alter the pool.

09 · Conservation with turnover

9. Conservation Laws Do Not Mean Everything Is Constant

One of the most common misconceptions is that a conservation law means a quantity cannot change. In mechanistic models, conservation usually means that all changes have an accounting explanation.

Consider a biomarker or protein \(P\) with synthesis and degradation:

$$ \frac{dP}{dt}=k_{\mathrm{syn}}-k_{\mathrm{deg}}P $$

The total amount of \(P\) is not conserved because molecules can be created through synthesis and removed through degradation.

At steady state:

$$ 0=k_{\mathrm{syn}}-k_{\mathrm{deg}}P_{\mathrm{ss}} $$

so:

$$ P_{\mathrm{ss}}=\frac{k_{\mathrm{syn}}}{k_{\mathrm{deg}}} $$

This is a mass-balance equation with turnover. The model conserves accounting consistency even though the molecular pool itself is dynamic.

10 · Cell populations

10. Conservation and Cell-Population Models

Conservation ideas extend beyond molecules. QSP models frequently represent cell populations such as tumor cells, immune cells, target cells, or infected cells.

For a cell population \(N(t)\), a simple balance is:

$$ \frac{dN}{dt} = R_{\mathrm{prolif}} - R_{\mathrm{death}} $$

If drug exposure changes the death rate:

$$ \frac{dN}{dt} = R_{\mathrm{prolif}}(N) - R_{\mathrm{drug}}(C,N) - R_{\mathrm{natural}}(N) $$

The same accounting principle applies: every increase or decrease in the population is associated with a modeled process.

For multiple cell types, conservation can become a population balance in which cells move between phenotypic states:

$$ \frac{dN_i}{dt} = \sum_j R_{j\rightarrow i} - \sum_j R_{i\rightarrow j} + R_{\mathrm{birth},i} - R_{\mathrm{death},i} $$

This structure is useful for modeling differentiation, activation, trafficking, proliferation, and cell death.

11 · Physiological systems

11. Conservation Laws and Physiological Flows

Physiological QSP models often describe organs and tissues connected by blood flow. Conservation equations provide the link between organ-level physiology and molecular kinetics.

For a well-mixed tissue compartment, a generic balance may be:

$$ V_T\frac{dC_T}{dt} = Q(C_{\mathrm{in}}-C_T) - R_{\mathrm{met}} $$

where \(V_T\) is tissue volume, \(Q\) is blood flow, \(C_{\mathrm{in}}\) is the incoming concentration, \(C_T\) is tissue concentration, and \(R_{\mathrm{met}}\) represents metabolic removal.

The flow term appears with opposite signs in connected compartments. What leaves one compartment enters another, unless an explicit extraction or elimination process removes it from the modeled system.

Blood incoming concentration Tissue distribution + metabolism Q · Cin Q · Cout Additional removal may occur through metabolism

Organ-level QSP models use conservation equations to connect blood flow, tissue distribution, and biological elimination.

12 · Drug mass balance

12. A Simple Drug Mass-Balance Model

Consider a drug administered into a central compartment and distributed to a peripheral compartment. Let \(A_C\) and \(A_P\) denote the drug amounts in the central and peripheral compartments.

Suppose drug moves from central to peripheral at rate \(R_{CP}\), returns at rate \(R_{PC}\), and is eliminated from the central compartment at rate \(R_{\mathrm{elim}}\).

The balance equations are:

$$ \frac{dA_C}{dt} = R_{\mathrm{in}} - R_{CP} + R_{PC} - R_{\mathrm{elim}} $$
$$ \frac{dA_P}{dt} = R_{CP} - R_{PC} $$

Adding the equations gives:

$$ \frac{d(A_C+A_P)}{dt} = R_{\mathrm{in}} - R_{\mathrm{elim}} $$

Notice what happened: the internal distribution terms canceled. Distribution moves drug within the system, but does not itself eliminate drug.

Key modeling insight: when equations for connected compartments are added together, internal transfer terms should cancel. This is one of the most useful checks for detecting errors in a mechanistic model.
13 · Dimensional consistency

13. Conservation Laws and Units

A conservation equation must also be dimensionally consistent. If \(A\) is an amount measured in mg, then:

$$ \frac{dA}{dt} $$

has units of mg/time.

Every term on the right-hand side must therefore also have units of mg/time. For example:

$$ \frac{dA}{dt} = R_{\mathrm{in}} - R_{\mathrm{out}} - R_{\mathrm{met}} $$

requires each rate to have units of amount per time.

Quantity Example units Typical interpretation
Amount mg, nmol, cells Total quantity in a compartment or pool
Concentration mg/L, nM Amount normalized by volume
Flow L/h Physiological transport rate
Reaction rate mg/h, nmol/h Rate of production, consumption, or transfer
Rate constant 1/h Fractional change per unit time

Dimensional analysis is therefore an important practical tool for checking QSP equations before numerical simulation.

14 · Steady state

14. Conservation Laws and Steady State

A steady state occurs when the quantities in the model are no longer changing, so that:

$$ \frac{dx}{dt}=0 $$

For a simple production-elimination system:

$$ \frac{dX}{dt} = R_{\mathrm{in}} - kX $$

the steady-state value satisfies:

$$ 0=R_{\mathrm{in}}-kX_{\mathrm{ss}} $$

and therefore:

$$ X_{\mathrm{ss}}=\frac{R_{\mathrm{in}}}{k} $$

In larger QSP systems, steady state can involve hundreds or thousands of simultaneous balance relationships. Solving for steady state can therefore provide a useful way to initialize simulations or investigate baseline physiology.

15 · Worked example

15. Worked Example: A Two-Compartment Mass Balance

Consider a hypothetical drug administered into a central compartment. At a particular instant, suppose:

  • Central amount: \(A_C=100\) mg
  • Peripheral amount: \(A_P=50\) mg
  • Central-to-peripheral transfer rate: \(R_{CP}=8\) mg/h
  • Peripheral-to-central transfer rate: \(R_{PC}=3\) mg/h
  • Elimination rate: \(R_{\mathrm{elim}}=5\) mg/h
  • No external input at this instant: \(R_{\mathrm{in}}=0\)

Step 1: Central compartment

The central balance is:

$$ \frac{dA_C}{dt} = -R_{CP}+R_{PC}-R_{\mathrm{elim}} $$

Substituting the values:

$$ \frac{dA_C}{dt} = -8+3-5 = -10\text{ mg/h} $$

The central amount is therefore decreasing at 10 mg/h.

Step 2: Peripheral compartment

$$ \frac{dA_P}{dt} = R_{CP}-R_{PC} $$
$$ \frac{dA_P}{dt} = 8-3 = 5\text{ mg/h} $$

The peripheral amount is increasing at 5 mg/h.

Step 3: Total amount

The total drug amount is:

$$ A_{\mathrm{tot}}=A_C+A_P=100+50=150\text{ mg} $$

Its rate of change is:

$$ \frac{dA_{\mathrm{tot}}}{dt} = -10+5 = -5\text{ mg/h} $$

This equals the systemic elimination rate:

$$ -\;R_{\mathrm{elim}}=-5\text{ mg/h} $$
What the example demonstrates: internal distribution changes the amount in each compartment, but the total amount changes only because drug enters or leaves the overall system. This cancellation of internal transfer terms is a fundamental conservation check.
16 · Algebraic constraints

16. Conservation Laws as Model Constraints

Not every conservation relationship needs to appear as an additional differential equation. Sometimes it can be represented as an algebraic constraint.

For example, if total receptor is fixed:

$$ R_{\mathrm{tot}}=R+RL $$

then one variable can be calculated from the others:

$$ R=R_{\mathrm{tot}}-RL $$

This can reduce the number of differential equations that must be solved. However, care is needed when simplifying a model because removing a state variable can affect numerical behavior, parameter interpretation, or the ability to represent additional mechanisms later.

In more complex systems, conservation relationships can therefore function as structural constraints that define the allowable state space of the model.

17 · Biological feasibility

17. Conservation and Physically Meaningful States

A mechanistic QSP model should generally preserve biologically meaningful states. Amounts and concentrations cannot ordinarily become negative.

For example, a model for a molecular amount \(A\) should not predict:

$$ A(t)<0 $$

unless the quantity has a mathematical interpretation that allows signed values.

Conservation laws help, but conservation alone does not guarantee positivity. A poorly constructed reaction rate or numerical implementation can still produce physically impossible states.

Consequently, model checking should consider both:

  • Conservation: are quantities accounted for correctly?
  • Positivity: do amounts and concentrations remain physically meaningful?
  • Units: are all equations dimensionally consistent?
  • Bounds: do physiological quantities remain within plausible ranges?
  • Stoichiometry: do reactions consume and produce the correct species?
18 · Model checking

18. Using Conservation Laws to Debug QSP Models

Conservation relationships provide powerful model-development checks. If a quantity should be conserved but the simulation shows unexplained gain or loss, there may be an error in the equations or implementation.

A practical debugging workflow is:

  1. List the modeled pools. Identify the amount or concentration associated with each compartment or species.
  2. List every process. Record each input, output, reaction, transport process, synthesis term, and degradation term.
  3. Check stoichiometry. Confirm that every process changes the correct species with the correct sign.
  4. Add connected balances. Internal transfer terms should cancel when the appropriate system total is calculated.
  5. Check units. Every term in a differential equation must have compatible dimensions.
  6. Check limiting cases. Set rates to zero or remove mechanisms and verify that the expected simplified behavior occurs.
  7. Monitor conserved quantities numerically. Track totals during simulation to detect unexpected drift.
Practical rule: before asking whether a QSP model fits data, first ask whether the model obeys its own biological accounting rules.
19 · Interpretation

19. What Conservation Laws Do Not Tell Us Automatically

Conservation laws are powerful constraints, but they do not determine the entire biological model.

  • Conservation does not identify mechanisms. Several different mechanisms can satisfy the same overall mass balance.
  • Conservation does not guarantee correct parameter values. A structurally consistent model can still be poorly estimated.
  • Conservation does not prove biological truth. The modeled pools and reactions are still abstractions of biology.
  • Conservation does not guarantee identifiability. Multiple parameter combinations may produce similar observations.
  • Conservation depends on the modeled system boundary. A quantity may be conserved inside one subsystem but enter or leave the larger physiological system.
  • Numerical conservation can differ from analytical conservation. Numerical solvers may introduce small errors, especially in difficult systems.

The modeler must therefore distinguish between a mathematical conservation relationship and a biological claim about what is actually conserved.

20 · Practical workflow

20. A Practical Conservation-Law Workflow for QSP

  1. Define the system boundary. Decide which compartments, species, and processes belong to the modeled system.
  2. Identify conserved pools. Determine which molecules, atoms, cells, or other quantities have meaningful total-pool relationships.
  3. Draw the process network. Show inputs, outputs, transfers, reactions, synthesis, degradation, and transformations.
  4. Write the balance equations. Translate each process into a rate term with the correct sign.
  5. Check stoichiometry. Confirm that every reaction consumes and produces the intended quantities.
  6. Check dimensions. Verify that every equation is dimensionally consistent.
  7. Derive system totals. Add relevant equations and verify that internal transfers cancel appropriately.
  8. Check limiting cases. Confirm that the equations reduce correctly when particular mechanisms are removed.
  9. Implement and simulate. Monitor conserved quantities and physical bounds throughout the simulation.
  10. Connect to observations. Only after the mechanistic structure is internally coherent should model predictions be compared with experimental data.

21. Key Takeaways

  • Conservation laws provide the accounting framework underlying many quantitative systems pharmacology models.
  • The basic balance is accumulation = inputs − outputs + generation − consumption.
  • Mass balance is naturally expressed using amounts, while concentrations are obtained by normalizing amounts by volume.
  • Internal transport redistributes material but should not create or destroy it unless an explicit mechanism does so.
  • Species conservation can link free, bound, and transformed molecular forms within a mechanistic model.
  • Stoichiometric matrices provide a compact way to represent conservation relationships in biochemical reaction networks.
  • Turnover systems can change dynamically while remaining fully consistent with mass-balance accounting.
  • Cell populations can be represented using the same framework through proliferation, differentiation, migration, and death terms.
  • Conservation equations are useful for checking units, stoichiometry, model structure, and numerical implementation.
  • Conservation laws can be expressed through differential equations, algebraic constraints, or both.
  • A conservation relationship is always defined relative to a modeled system boundary; quantities can enter or leave that boundary.
  • Conservation provides an essential structural constraint, but it does not by itself establish that a biological mechanism or parameterization is correct.
Next step

Where to Go Next

A natural progression is to study mass-balance equations in mechanistic models, followed by ordinary differential equations, compartmental systems, reaction networks, receptor binding, turnover models, physiological flow models, and QSP model construction.

The next tutorial can build directly on conservation laws by showing how a biological process diagram is translated systematically into a system of ordinary differential equations, with each equation derived from explicit mass-balance assumptions.

Related Tutorials

Clinical Biostats

← Back to Pharmacokinetics Tutorials