Baseline Target Equilibrium and TMDD Initial Conditions

Why Initial Conditions Matter in TMDD

In a target-mediated drug disposition (TMDD) model, a simulation does not begin with a single concentration. It begins with values for the model’s state variables, including free drug, free target, and drug–target complex. Those values affect the early dynamics: an incorrect starting complex concentration can create a large artificial transient that may be mistaken for a genuine response to dosing.

The right initialization depends on the biological situation being represented. Before treatment, if drug is truly absent and the target has reached its drug-free steady state, free target is determined by its production and turnover rates and complex is zero. If drug is already present and target binding has equilibrated, the target and complex must instead satisfy the binding and target-turnover balance at that drug concentration. That conditional equilibrium is not automatically a steady state of the entire drug–target system.

Definition: Let \(D\) denote free-drug concentration, \(R\) free-target concentration, and \(C\) drug–target-complex concentration. A drug-free baseline equilibrium has no drug or complex and has target production balanced by free-target loss. A conditional target-binding equilibrium has free target and complex at equilibrium for a specified, held-fixed value of \(D\). These are different initial-state assumptions.

The Full State Equations and Their Assumptions

Use the following mass-action TMDD system. All three states are concentrations expressed in compatible units. The input \(u_D(t)\) is a drug input rate expressed as concentration per time; for example, it can represent an infusion rate divided by the relevant volume. The parameters \(k_{\mathrm{in}}\) and \(k_{\mathrm{out}}\) describe target production and first-order loss of free target; \(k_{\mathrm{on}}\) and \(k_{\mathrm{off}}\) describe binding and dissociation; \(k_{\mathrm{int}}\) is the first-order removal rate of complex; and \(k_{\mathrm{el}}\) is the first-order elimination rate of free drug.

\begin{aligned} \frac{dD}{dt} &= u_D(t)-k_{\mathrm{el}}D-k_{\mathrm{on}}DR+k_{\mathrm{off}}C,\\ \frac{dR}{dt} &= k_{\mathrm{in}}-k_{\mathrm{out}}R-k_{\mathrm{on}}DR+k_{\mathrm{off}}C,\\ \frac{dC}{dt} &= k_{\mathrm{on}}DR-(k_{\mathrm{off}}+k_{\mathrm{int}})C. \end{aligned}

The equations assume nonnegative states and rates, constant parameters over the initialization interval, and a well-mixed compartment in which the mass-action binding terms apply. Binding removes free drug and free target at rate \(k_{\mathrm{on}}DR\), while dissociation returns each at rate \(k_{\mathrm{off}}C\). Complex internalization removes complex at rate \(k_{\mathrm{int}}C\); in this model, that process removes both bound drug and target. The target’s baseline production is constant, and free target is lost at rate \(k_{\mathrm{out}}R\).

An equilibrium condition means that the relevant derivatives are zero under the specified conditions. It does not mean that every concentration is zero, nor does it mean that an equilibrium computed for one drug concentration remains valid after the drug concentration changes.

Drug-Free Baseline Target

When drug is absent, \(D=0\). If there is also no pre-existing complex, \(C=0\), and the free-target equation reduces to production minus first-order loss. At steady state, production and loss must balance.

0=k_{\mathrm{in}}-k_{\mathrm{out}}R_0 \qquad\Longrightarrow\qquad R_0=\frac{k_{\mathrm{in}}}{k_{\mathrm{out}}}.

This result requires \(k_{\mathrm{out}}>0\) and assumes the drug-free target system has had enough time to reach steady state. It has the correct units: \(k_{\mathrm{in}}\) is concentration per time and \(k_{\mathrm{out}}\) is inverse time, so their ratio is concentration. The drug-free steady-state initial conditions are therefore \(D(0)=0\), \(R(0)=R_0\), and \(C(0)=0\).

Worked Example: Calculating the Drug-Free Target Baseline

Suppose a hypothetical target model has \(k_{\mathrm{in}}=12\) nM/day and \(k_{\mathrm{out}}=0.200\) day\(^{-1}\). The individual is untreated, and the target is assumed to have reached its drug-free steady state. Calculate all three initial concentrations.

The free-target steady state follows from \(R_0=k_{\mathrm{in}}/k_{\mathrm{out}}\):

R_0=\frac{12\ \text{nM/day}}{0.200\ \text{day}^{-1}} =60.0\ \text{nM}.

Because drug is absent, \(D(0)=0\) nM. With no drug to bind, the drug-free equilibrium has \(C(0)=0\) nM. Thus the consistent initial state is \(D(0)=0\) nM, \(R(0)=60.0\) nM, and \(C(0)=0\) nM. Substitution into the equations gives \(dR/dt=12-0.200(60)=0\) nM/day and \(dC/dt=0\), as required.

Conditional Equilibrium When Drug Is Present

Suppose instead that free drug is held at a specified concentration \(D^*\), and the target and complex have equilibrated under that condition. This means setting \(dR/dt=0\) and \(dC/dt=0\), while treating \(D=D^*\) as fixed. The complex equation first gives the equilibrium relationship between \(C\) and \(R\):

C^*=\frac{k_{\mathrm{on}}D^*R^*}{k_{\mathrm{off}}+k_{\mathrm{int}}}.

For the free-target balance, substitute this result into \(dR/dt=0\). The binding terms do not cancel completely: complex internalization removes target from the system, so only the portion of binding that is not returned by dissociation remains as a net loss of free target.

\begin{aligned} 0 &=k_{\mathrm{in}}-k_{\mathrm{out}}R^* -k_{\mathrm{on}}D^*R^*+k_{\mathrm{off}}C^*\\ &=k_{\mathrm{in}}-k_{\mathrm{out}}R^* -k_{\mathrm{on}}D^*R^* +\frac{k_{\mathrm{off}}k_{\mathrm{on}}D^*R^*} {k_{\mathrm{off}}+k_{\mathrm{int}}}\\ &=k_{\mathrm{in}}- \left(k_{\mathrm{out}}+ \frac{k_{\mathrm{on}}D^*k_{\mathrm{int}}} {k_{\mathrm{off}}+k_{\mathrm{int}}}\right)R^*. \end{aligned}

Solving the final balance and then using the complex equation gives the conditional equilibrium concentrations:

R^*= \frac{k_{\mathrm{in}}} {k_{\mathrm{out}}+ \dfrac{k_{\mathrm{on}}D^*k_{\mathrm{int}}} {k_{\mathrm{off}}+k_{\mathrm{int}}}}, \qquad C^*= \frac{k_{\mathrm{on}}D^*R^*}{k_{\mathrm{off}}+k_{\mathrm{int}}}.

For nonnegative rates with \(k_{\mathrm{out}}>0\) and \(k_{\mathrm{off}}+k_{\mathrm{int}}>0\), these expressions are finite and nonnegative. They also recover the drug-free baseline when \(D^*=0\): \(R^*=k_{\mathrm{in}}/k_{\mathrm{out}}\) and \(C^*=0\). At positive \(D^*\), the free target is generally below the drug-free baseline if \(k_{\mathrm{int}}>0\), because complex removal creates an additional route of target loss.

Worked Example: Target and Complex at a Held Drug Concentration

Consider hypothetical parameters \(k_{\mathrm{in}}=12\) nM/day, \(k_{\mathrm{out}}=0.200\) day\(^{-1}\), \(k_{\mathrm{on}}=0.100\) nM\(^{-1}\)day\(^{-1}\), \(k_{\mathrm{off}}=0.400\) day\(^{-1}\), and \(k_{\mathrm{int}}=0.600\) day\(^{-1}\). Suppose free drug is held at \(D^*=10.0\) nM while target binding equilibrates. Find \(R^*\) and \(C^*\), and verify the target and complex balances.

First, \(k_{\mathrm{off}}+k_{\mathrm{int}}=1.00\) day\(^{-1}\). The effective loss term in the denominator for \(R^*\) is \(0.200+(0.100)(10.0)(0.600)/(1.00)=0.800\) day\(^{-1}\). Therefore,

R^*=\frac{12\ \text{nM/day}}{0.800\ \text{day}^{-1}} =15.0\ \text{nM}, \qquad C^*=\frac{(0.100)(10.0)(15.0)}{1.00} =15.0\ \text{nM}.

The complex formation rate is \(k_{\mathrm{on}}D^*R^*=(0.100)(10.0)(15.0)=15.0\) nM/day. Its loss rate is \((k_{\mathrm{off}}+k_{\mathrm{int}})C^*=(1.00)(15.0)=15.0\) nM/day, so \(dC/dt=0\). For target, production is 12.0 nM/day; free-target loss is \((0.200)(15.0)=3.00\) nM/day; binding loss is 15.0 nM/day; and dissociation return is \((0.400)(15.0)=6.00\) nM/day. Thus \(dR/dt=12.0-3.00-15.0+6.00=0\) nM/day. The conditional target and complex balances are satisfied.

Conditional Equilibrium Is Not Necessarily a Whole-Model Steady State

In the preceding calculation, \(D^*\) was held fixed. In a full simulation, free drug is usually a state that changes as a result of input, elimination, binding, and dissociation. A positive \(D^*\) is not a whole-model steady state merely because \(R^*\) and \(C^*\) satisfy their own equilibrium equations.

For the complete system to remain at \(D^*\), the drug equation must also balance. At the conditional target and complex equilibrium, \(k_{\mathrm{on}}D^*R^*-k_{\mathrm{off}}C^*=k_{\mathrm{int}}C^*\), by the complex balance. Consequently, the input needed to maintain the drug concentration is:

u_D^*=k_{\mathrm{el}}D^*+k_{\mathrm{int}}C^*.

The first term replaces free drug removed by linear elimination; the second replaces drug removed when complex is internalized. If no input is supplied, \(D^*\) will generally change, and the conditional target equilibrium will no longer be maintained. This distinction is useful when constructing simulation starts: a model may begin with target binding equilibrated at the initial drug concentration, then evolve immediately because the drug concentration is not being held constant.

Worked Example: Initializing a Drug-Maintained Equilibrium

Use the conditional equilibrium from the previous example, \(D^*=10.0\) nM and \(C^*=15.0\) nM. Suppose \(k_{\mathrm{el}}=0.0500\) day\(^{-1}\). Calculate the constant drug input required to keep the complete system at equilibrium.

The maintenance input is

u_D^*=k_{\mathrm{el}}D^*+k_{\mathrm{int}}C^* =(0.0500)(10.0)+(0.600)(15.0) =0.500+9.00 =9.50\ \text{nM/day}.

With this input, the free-drug equation is also balanced. The input replaces 0.500 nM/day of free-drug elimination and 9.00 nM/day of drug removed through complex internalization. If the input were zero instead, the initial drug derivative would be \(-9.50\) nM/day, even though the target and complex derivatives are initially zero. The three states would therefore not constitute a whole-model steady state.

Choosing Initial Conditions for a Simulation

Initial states should represent the history immediately before the simulation begins. For a drug-free baseline, use \(D(0)=0\), \(R(0)=k_{\mathrm{in}}/k_{\mathrm{out}}\), and \(C(0)=0\) only when the target is plausibly at its untreated steady state. If a simulation begins after drug has been held at a known concentration long enough for target and complex to equilibrate, use \(D(0)=D^*\), \(R(0)=R^*\), and \(C(0)=C^*\). Whether the drug input is also set to its maintenance value depends on the intended dosing history after time zero.

A bolus dose introduces a different situation. In the usual model, a bolus changes the drug state instantaneously but does not instantaneously change free target or complex. If the system was drug-free just before the bolus, the post-bolus state has the new drug concentration but retains the pre-bolus target and complex values. The resulting derivatives describe the start of binding and target response. Replacing those target states with conditional equilibrium values would represent a different history: target and complex would already have equilibrated at the post-bolus drug concentration.

Worked Example: A Bolus Does Not Instantly Equilibrate Target

Suppose the drug-free baseline from the first example is \(R=60.0\) nM and \(C=0\) nM. A hypothetical bolus instantaneously sets free drug to \(D=10.0\) nM. Use \(k_{\mathrm{on}}=0.100\) nM\(^{-1}\)day\(^{-1}\), \(k_{\mathrm{off}}=0.400\) day\(^{-1}\), and \(k_{\mathrm{int}}=0.600\) day\(^{-1}\) to calculate the immediate post-bolus derivatives of target and complex.

The target and complex concentrations do not jump at the bolus, so their immediate post-bolus values remain \(R=60.0\) nM and \(C=0\) nM. The complex derivative is

\frac{dC}{dt}=(0.100)(10.0)(60.0)-(1.00)(0) =60.0\ \text{nM/day}.

The free-target derivative is

\frac{dR}{dt}=12-(0.200)(60.0)-(0.100)(10.0)(60.0)+(0.400)(0) =-60.0\ \text{nM/day}.

These nonzero derivatives are the expected response to a new drug exposure. Starting instead at \(R=15.0\) nM and \(C=15.0\) nM would erase that initial binding transient by assuming that the target had already equilibrated at 10.0 nM drug. The appropriate choice depends on the dosing and pretreatment history, not just on the drug concentration recorded at time zero.

Diagnostics, Alternatives, and Limitations

A practical check is to substitute proposed initial values into the differential equations under the actual time-zero input. If a state is claimed to be at equilibrium, its derivative should be zero under those conditions, up to numerical rounding. For a drug-free equilibrium, verify both \(k_{\mathrm{in}}-k_{\mathrm{out}}R_0=0\) and \(dC/dt=0\). For a conditional target equilibrium, verify \(dR/dt=0\) and \(dC/dt=0\); for a whole-model equilibrium, verify \(dD/dt=0\) as well.

Unexpected early transients can indicate an initialization mismatch. Examples include starting with \(C>0\) while asserting that all drug is absent, using a drug-free target baseline despite long-standing drug exposure, or initializing target and complex at conditional equilibrium while setting the drug input to a value that immediately moves \(D\). In a model with these equations, \(C>0\) represents bound drug; if free drug is zero but complex remains, dissociation can produce free drug. That may be a valid post-treatment washout state, but it is not a state in which drug is entirely absent.

There are several defensible alternatives when the history is uncertain. A drug-free steady-state calculation is transparent and reproducible when untreated baseline conditions are credible. A conditional-equilibrium initialization is appropriate when drug exposure has been stable long enough for target and complex to equilibrate at the specified concentration. If neither assumption matches the data, use measured or estimated initial states, or simulate the known treatment history before the observation window. An observed baseline total target \(R+C\) alone does not generally identify both \(R\) and \(C\); further information or assumptions are needed to split the total into its components.

These formulas also rely on the stated turnover structure. A model with different target production, complex loss, or target recycling will have different equilibrium balances. The equilibrium calculation is an initialization result for the specified full state equations; it does not by itself justify replacing the differential equations with a reduced model or assuming that binding is fast.

Common pitfall: Setting \(C=0\) and \(R=k_{\mathrm{in}}/k_{\mathrm{out}}\) for every simulation start because these are the drug-free baseline values. They are valid only when the target is at its drug-free steady state and no complex is present. Under ongoing drug exposure, check the binding and turnover balances at the relevant drug concentration; after a bolus, preserve target and complex states unless the assumed pretreatment history supports a different start.

Specify the time-zero history.
Determine whether drug is absent, maintained at a concentration, or introduced by a dose at time zero.

Choose the corresponding equilibrium assumption.
Use the drug-free baseline only for an untreated steady state; use conditional target equilibrium only when binding and target turnover are assumed equilibrated at the specified drug concentration.

Check every relevant derivative.
Substitute the proposed states and input into the model equations. Do not call the whole state a steady state unless all state derivatives are zero.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

If \(k_{\mathrm{in}}=8.0\) nM/day and \(k_{\mathrm{out}}=0.25\) day\(^{-1}\), what is the drug-free steady-state target concentration?
At a conditional equilibrium with fixed positive drug concentration, what two derivatives must be zero to solve for free target and complex? What additional derivative must be zero for a whole-model steady state?
Why can a drug-free target baseline be inconsistent with a simulation that starts after a long period of drug exposure?
Immediately after a bolus, which states change instantaneously in this model, and why?

Solutions. (1) \(R_0=8.0/0.25=32\) nM. (2) Set \(dR/dt=0\) and \(dC/dt=0\); for a whole-model steady state also require \(dD/dt=0\). (3) Drug exposure can alter free target and create complex, so the drug-free equilibrium values need not satisfy the target and complex balances under the exposed conditions. (4) The drug state \(D\) changes instantaneously in the bolus representation; \(R\) and \(C\) remain continuous and then evolve according to their derivatives.