Initial Conditions and Baseline Signals in Cascades

Why Initial Conditions Matter

A cascade does not respond only to the input at the current time. Its output also reflects the signal already stored in each intermediate stage when observation begins. Consequently, a model can have an appropriate cascade structure and plausible rate constants yet predict the wrong early response if its initial states do not represent the system before the modeled interval.

This is especially important when a time course begins after a period of ongoing drug exposure, endogenous stimulation, or other background input. The measured baseline is not necessarily zero, and the states of upstream and downstream intermediates need not be equal if the system has not reached steady state. Initializing every stage to zero, or setting every stage to the observed output without checking the model’s meaning, can introduce a transient that is an artifact of the initialization rather than a response to the modeled input change.

States, Equilibrium, and Baseline Output

Consider a two-stage linear cascade with input \(u(t)\), intermediate signals \(x_1(t)\) and \(x_2(t)\), and positive first-order rate constants \(k_1\) and \(k_2\). The output is \(x_2(t)\). The model is

Each stage moves toward its driving signal: the first toward the input and the second toward the first stage. A reported biomarker may be a scaled and offset version of the final state, for example \(y(t)=b+g x_2(t)\), where \(b\) is the measurement offset and \(g\) is a scale factor. These are distinct quantities: \(x_2\) is a model state, while \(y\) is the reported signal.

The equilibrium result follows by setting both derivatives to zero. Since \(k_1\) and \(k_2\) are positive, \(k_1(u_0-x_1)=0\) implies \(x_1=u_0\); then \(k_2(x_1-x_2)=0\) implies \(x_2=x_1=u_0\). Thus, under this model, the baseline output is \(y_0=b+g u_0\). A measured baseline value alone does not generally identify \(b\), \(g\), or all the individual cascade states; their interpretation depends on the model and measurement definition.

Non-Equilibrium Initial States

For a constant input \(u_0\) over the prediction interval, define deviations from the equilibrium values by \(z_1(t)=x_1(t)-u_0\) and \(z_2(t)=x_2(t)-u_0\). Let \(z_{10}=x_1(0)-u_0\) and \(z_{20}=x_2(0)-u_0\). The deviations satisfy

The first equation gives \(z_1(t)=z_{10}e^{-k_1t}\). To obtain the second-stage state, multiply its equation by the integrating factor \(e^{k_2t}\). Then

For unequal rates, integrate from 0 to \(t\), divide by \(e^{k_2t}\), and substitute the initial value:

This result has two contributions. The first is the decaying effect of the second-stage starting deviation; the second is the first-stage deviation as it propagates through the second stage. If \(k_1=k_2=k\), the integral instead gives \(z_2(t)=e^{-kt}(z_{20}+kz_{10}t)\). This equal-rate expression is also the limit of the unequal-rate result as \(k_2\) approaches \(k_1\); the apparent zero denominator in the unequal-rate formula is removable, not a reason to set the propagated contribution to zero.

Initialization After an Input Change

When input changes at time zero, initialize the states using the conditions immediately before that change, then solve forward using the new input. If the system was at equilibrium under \(u_0\) before the change, both states start at \(u_0\), even if the new input is \(u_1\). The states do not jump to \(u_1\) in this first-order model; they move toward it over time.

For a step from \(u_0\) to \(u_1\), define \(\Delta u=u_1-u_0\). The initial deviations from the new equilibrium are \(z_{10}=z_{20}=-\Delta u\). Substitution into the unequal-rate solution yields

The bracketed term equals 1 at \(t=0\), so the output state starts at \(u_0\); it tends to 0 as time increases, so the state approaches \(u_1\). These checks are useful for detecting a sign or rate-order error in a step-response calculation.

Checking Initial Conditions and Common Failure Modes

Before interpreting an early response, check that the initial states, input history, and measurement scale are consistent. For a claimed equilibrium baseline under constant input, substitute the proposed states into every differential equation: each derivative must be zero. If the stated initial states do not satisfy this condition, the model predicts a baseline transient even before any new drug effect is considered.

• Check the initial output. Evaluate the observation equation at time zero. If the model says \(y=b+gx_2\), compare \(b+gx_2(0)\), not \(x_2(0)\), with the measured baseline.
• Check direction and bounds. Under the linear model with positive rates and a positive input step from equilibrium, each stage should move toward the new equilibrium. A calculated value outside the range of the starting and new equilibrium states signals a likely algebra, sign, or setup error.
• Check time units. The product \(k_it\) must be dimensionless. A rate in per hour cannot be combined with a time in minutes without conversion.
• Check the assumed prehistory. “Baseline” does not establish that the cascade was at equilibrium. If input varied before observation began, the current upstream and downstream states may differ and need to be specified or supported by the observed history.

A common mistake is to set every state equal to the observed output at time zero. This is wrong when the observation has an offset or scale factor, and it may also be wrong when the output reports only the final stage. A second mistake is to initialize all stages at zero simply because the modeled drug concentration is zero at baseline. The drug input and the downstream signal states are not interchangeable: endogenous or background input can maintain nonzero baseline signals.

The main alternative is to assume a steady-state baseline and derive all starting states from the model’s equilibrium equations. That is simple and often appropriate when the pre-dose input was stable long enough. Explicit non-equilibrium initial states are preferable when prior input varied, the system is recovering, or baseline observations show a transient. When prehistory is known, one can instead simulate the model through that history to obtain the states at the analysis start. These approaches are not interchangeable without assumptions: steady-state initialization discards transient history, while free initial states require data or constraints sufficient to support them. Neither approach repairs an incorrect cascade structure or an inappropriate observation model.

For nonlinear stage transfers, equilibrium values are found by setting each derivative to zero and solving the resulting steady-state equations; they need not equal the input or one another. The linear formulas above then do not apply automatically. Initial states must still satisfy the model’s state definitions, and any proposed baseline must be checked against the actual nonlinear equations.