Piecewise-Constant Exposure and Recursive Solutions

Why Recursive Updates Are Useful

In an indirect response model, a drug changes the rate at which a response is produced, lost, or both. The response itself usually does not jump merely because concentration or a drug-effect multiplier changes. Instead, its rate of change changes, and the response then moves toward a new level over time. When exposure is treated as constant between successive times, the response can be updated exactly from one interval to the next.

This approach is useful when concentrations are available at discrete times, when a dosing change creates distinct exposure intervals, or when calculating a predicted response at observation times. It avoids treating each observation as a new initial condition: the endpoint of one interval is the starting response for the next.

Model and Assumptions

Let the interval boundaries be \(t_0,t_1,\ldots,t_J\), with interval duration \(\Delta_j=t_{j+1}-t_j>0\). On interval \(j\), suppose the production multiplier \(m_{\mathrm{in},j}\) and loss multiplier \(m_{\mathrm{loss},j}\) are constant. A general turnover model for response \(R(t)\) is

Here \(k_{\mathrm{in}}\) is the baseline production rate, in response units per time, and \(k_{\mathrm{out}}\) is the baseline fractional loss rate, in inverse time. The multipliers are dimensionless. A multiplier of 1 means that the corresponding rate is at baseline; a multiplier below 1 reduces that rate, while a multiplier above 1 increases it. For example, inhibition of production can be represented by \(m_{\mathrm{in},j}<1\).

The exact update below assumes \(k_{\mathrm{in}}\) and \(k_{\mathrm{out}}\) are constant, the multipliers do not vary inside an interval, and the response follows the stated turnover equation throughout that interval. The multipliers may be computed from a concentration using an effect function, such as a Hill function, but their values must be fixed over the interval for this update to apply exactly. The model also assumes no separate instantaneous jump in response at a dosing or concentration change.

Deriving the Interval Update

On interval \(j\), define the effective loss rate \(\lambda_j=k_{\mathrm{out}}m_{\mathrm{loss},j}\) and, when \(\lambda_j>0\), the interval equilibrium \(R_{\mathrm{eq},j}=k_{\mathrm{in}}m_{\mathrm{in},j}/\lambda_j\). These definitions rewrite the differential equation as

The equilibrium is the response at which production and loss balance for the fixed interval multipliers. To solve the equation, subtract that constant equilibrium from the response. Since \(R_{\mathrm{eq},j}\) is constant on this interval, the difference \(D(t)=R(t)-R_{\mathrm{eq},j}\) obeys \(dD/dt=-\lambda_jD\). Its solution is \(D(t)=D(t_j)e^{-\lambda_j(t-t_j)}\), as can be checked by differentiation and by evaluating at \(t=t_j\). Therefore, if \(R_j=R(t_j)\), then

This is an exact update under the interval assumptions, not an approximation from a small time step. If \(R_j\) is above the interval equilibrium, the exponential term shrinks the difference and the response moves downward; if it is below, the response moves upward. The amount of movement depends on both the interval duration and the effective loss rate.

If \(m_{\mathrm{loss},j}=0\), then \(\lambda_j=0\), so the equilibrium formula involving division by \(\lambda_j\) cannot be used. The differential equation instead has constant slope \(k_{\mathrm{in}}m_{\mathrm{in},j}\), giving \(R_{j+1}=R_j+k_{\mathrm{in}}m_{\mathrm{in},j}\Delta_j\). This is also the limit of the general update as \(\lambda_j\) approaches zero.

Changes in Loss and in Exposure-Driven Effects

The same update handles a change in loss. As long as the loss multiplier remains positive, reducing it while holding positive production fixed raises the interval equilibrium; at zero loss, there is no finite interval equilibrium. Increasing the loss multiplier makes the response approach a lower equilibrium more quickly. These are distinct effects: the multiplier affects both the equilibrium through the production-to-loss ratio and the rate of approach through \(\lambda_j\).

If concentration is constant within each interval, an effect function can provide the multiplier used in the update. For instance, for inhibition of production with maximum fractional inhibition \(I_{\max}\), half-maximal concentration \(IC_{50}\), and Hill coefficient \(\gamma\), one possible multiplier is \(m_{\mathrm{in}}(C)=1-I_{\max}C^\gamma/(IC_{50}^\gamma+C^\gamma)\). Once a concentration \(C_j\) is assigned to interval \(j\), evaluate the effect function to obtain that interval's multiplier, then apply the recurrence. This procedure does not require concentration itself to be a response state in the turnover equation; it only requires the assumed interval multiplier to be known.

Using the Recurrence and Checking Its Behavior

A practical calculation proceeds in order: specify the interval boundaries, assign the constant multipliers for each interval, calculate that interval's effective rate and equilibrium, update the response, and use the unrounded endpoint as the next starting value. This is a recursive calculation because each update depends on the preceding one.

Several checks help detect errors. With nonnegative production and loss multipliers, nonnegative initial response, and positive effective loss, the update is a weighted average of \(R_j\) and \(R_{\mathrm{eq},j}\), because \(0<e^{-\lambda_j\Delta_j}<1\). It must therefore lie between those two values. If it does not, check the exponent sign, units, and interval duration. If all multipliers equal 1 and the response begins at the baseline equilibrium \(k_{\mathrm{in}}/k_{\mathrm{out}}\), the response should remain there. At a switching boundary, a computed response jump usually indicates that the new equilibrium was mistakenly substituted for the carried response.

The piecewise-constant assumption is a modeling choice, not a property guaranteed by having observations at discrete times. If concentration changes materially within an interval, holding its effect multiplier fixed may give a poor representation. One diagnostic is to compare predictions after subdividing intervals or using a more appropriate concentration profile; large changes in predicted responses indicate sensitivity to the interval approximation. The exact update is exact for the assumed constant coefficients, but cannot correct misspecified within-interval exposure.

Alternatives, Limitations, and a Common Pitfall

The recursive method is especially convenient for predictions at interval boundaries and for verifying calculations from a numerical solver under constant coefficients. It provides the response at those boundaries directly, but does not by itself describe a concentration that varies continuously within an interval. It also relies on the chosen turnover model and known multipliers; exact arithmetic under a model does not establish that the model is biologically adequate.

Common mistake: A learner may calculate a new interval equilibrium at a dosing change and set the response immediately to that value. This confuses the balance point under the new rates with the response already present at the switching time. The correction is to keep the preceding interval's endpoint as the next starting value, then use the new equilibrium only in the next exponential update. An instantaneous response jump would require a separate, explicitly modeled mechanism.