Why a Failed Fit Needs a Numerical Trace
An optimizer can report a convergence failure even when the model is scientifically reasonable. The reverse can also happen: an optimizer may stop successfully after following an objective function that is distorted by an inaccurately integrated ODE. A useful diagnosis therefore traces the computation in order: identify the first failed parameter evaluation, check the event schedule, inspect the solver’s integration diagnostics, and determine whether changes in numerical accuracy materially alter predictions or the objective.
This worked case uses a one-compartment model with a constant-rate infusion. It is deliberately simple enough to provide an analytic benchmark. The diagnostic logic also applies to more complex pharmacometric ODE systems, although those systems may not have a closed-form solution.
Model, Events, and the Objective
Let \(A(t)\) be the amount of drug in the compartment, measured in mg. During an infusion, the input rate is \(R\), in mg/h; outside the infusion it is zero. The elimination rate constant is \(k\), in h\(^{-1}\), and the volume is \(V\), in L. Concentration is \(C(t)=A(t)/V\), in mg/L. The model is
For the hypothetical case, a 100 mg dose is infused from time 0 to time 1 h at \(R=100\) mg/h. Take \(k=0.20\) h\(^{-1}\), \(V=20\) L, and \(A(0)=0\). The rate changes discontinuously at the infusion stop. The amount itself does not jump at that event: only the input rate changes. A numerical implementation should therefore integrate up to the event, apply the new rate, and continue from the event state.
The infusion model has a useful analytic solution between events. For a constant infusion beginning from amount \(A(0)=0\),
After the infusion stops, the input is zero and the amount decays from its value at the event:
If the observed concentrations are \(y_i\), the model predictions are \(C(t_i)\), and \(\sigma\) is fixed, the Gaussian negative twice log-likelihood with constants omitted is
An optimizer’s objective value is meaningful only if the predictions used to calculate it are sufficiently accurate and refer to the intended event history.
Worked Example: Establishing a Prediction and Objective Benchmark
Suppose hypothetical observations are \(2.40\) mg/L at 0.5 h, \(4.08\) mg/L at 1.5 h, and \(3.73\) mg/L at 2.0 h. First calculate the analytic predictions at the stated parameter values. During infusion, \(R/k=100/0.20=500\) mg, so
At the infusion stop,
The later concentrations are therefore
Using observed minus predicted residuals, the three residuals are \(0.0209\), \(-0.0205\), and \(0.0197\) mg/L, to four significant figures. With \(\sigma=0.05\) mg/L,
The small value here is a consequence of these particular hypothetical observations and assumed error scale. It does not establish that the model is adequate. Its diagnostic use is that an ODE implementation with the same events and parameters should reproduce predictions near the analytic values, within a justified numerical tolerance.
Tracing the Failure from the Optimizer Message
Consider a hypothetical fit log with several successful iterations followed by a message that an ODE integration failed near the infusion stop, after which the optimizer reports a non-finite objective or gradient. The first task is to locate the earliest failed parameter evaluation, not to interpret the final optimizer message in isolation. Record the parameter values at that evaluation, the objective status, the solver’s last successful time, and any reported step-size or error-test information.
A non-finite gradient following an integration failure is often a downstream symptom: a finite-difference calculation may need objective evaluations at nearby parameter values, and one or more of those evaluations may not have produced a valid solution. It does not, by itself, show that the likelihood is nonsmooth or that the parameters are intrinsically unidentifiable. Conversely, if all ODE evaluations succeed but objective or gradient values vary erratically under small parameter changes, investigate other causes rather than attributing every stall to the event.
Identify the exact trial parameters, event record, observation times, failure time, and solver message. Distinguish an integration failure from an optimizer stopping rule or a non-finite objective calculated after integration.
Verify event time units, ordering, dose amount, infusion rate and duration, and whether observation times are intended to refer to pre-event or post-event states when a state-changing event occurs.
Hold the parameters and input data fixed while checking event splitting, solver status, tolerances, and maximum step size. Compare predictions with an analytic solution where possible.
Determine whether a reliable, repeatable objective can be evaluated in the relevant parameter region before interpreting optimizer convergence or changing the model.
Event Handling: A Discontinuity Can Masquerade as a Fit Problem
At \(t=1\) h, the right-hand side changes from \(100-0.20A(t)\) to \(-0.20A(t)\). An integration algorithm that treats this interval as if the input rate were constant across an unsplit step can accumulate substantial error. For illustration, suppose a crude fixed-step calculation incorrectly continues the infusion to \(t=2\) h. It would predict
The correctly event-split calculation gives \(A(1)=90.6346\) mg and then
This extreme discrepancy illustrates why an event-crossing calculation must respect the changed input. It is not a claim that every adaptive solver will make this error: a properly configured solver may locate or be explicitly stopped at the event. The diagnostic question is whether the implementation actually does so, rather than whether it nominally uses an adaptive method.
For this infusion, the event should divide the integration into \([0,1]\) and \([1,t]\). Check that the duration and rate imply the intended total dose: \(100\) mg/h multiplied by 1 h equals 100 mg. Check also that time is consistently expressed in hours. If event records are sorted incorrectly, duplicated, assigned inconsistent units, or silently omitted, tightening solver tolerances will not repair the event history.
Worked Example: Testing a Prediction Near the Infusion Stop
Suppose the solver returns \(C(2)=3.782\) mg/L at the stated parameters using a loose setting, \(3.7108\) mg/L after tightening accuracy, and \(3.7103\) mg/L with a still tighter setting. The analytic benchmark is \(3.7103\) mg/L. The loose-setting discrepancy is
That discrepancy is not negligible relative to the assumed observation error of \(0.05\) mg/L. For the hypothetical observation \(y=3.73\) mg/L at 2 h, the objective contribution under the loose prediction is
Using \(3.7108\) mg/L gives \((0.0192/0.05)^2=0.1475\); using \(3.7103\) mg/L gives \((0.0197/0.05)^2=0.1552\). The tighter two settings are close in prediction, while the loose result changes this observation’s objective contribution materially. This comparison supports investigating numerical accuracy; it does not by itself identify whether the original discrepancy came from event handling, solver tolerance, or another implementation detail.
Solver Diagnostics and How to Interpret Them
A solver failure can arise because local error tests cannot be satisfied, the required step becomes too small, the right-hand side returns invalid values, or the system is difficult for the selected method. A solver’s termination message and last successful time help distinguish these possibilities. Inspect state values and derivatives near the failure, not only the final likelihood. If the state becomes non-finite immediately after an event, verify the event update and model equations. If step sizes collapse while states remain finite near a sharp change, check event segmentation and whether the solver is appropriate for the system.
Tolerance tests should be controlled comparisons. Keep data, parameters, event records, and objective definition fixed; then tighten tolerances or reduce the maximum step size and compare predictions at the observation times. A smaller step or tighter tolerance is a diagnostic, not an automatic solution. If predictions do not stabilize, investigate event handling, model implementation, or solver suitability. If they do stabilize but the optimizer still fails, the remaining problem may lie in the parameter search or objective evaluation rather than the ODE integration.
Worked Example: Separating Solver Error from Optimizer Behavior
At a fixed trial parameter vector, suppose a fit reports an integration failure at \(t=0.997\) h, while the infusion stop is encoded at \(t=1.00\) h. The optimizer then reports a failed gradient. Repeat the forward solve at the same parameters after explicitly dividing integration at 1.00 h, without changing the data or parameters.
If the segmented solve reaches every observation time and its predictions agree with the analytic benchmark, the first failure is localized to handling the event or to integration across its neighborhood. The gradient message is then plausibly downstream of the failed evaluation. If the segmented solve still fails before 1.00 h, examine the state, derivative, units, and solver diagnostics at \(t=0.997\); event splitting alone has not resolved the failure. If the segmented solve succeeds but its predictions change substantially when tolerances are tightened, numerical accuracy remains unresolved.
This example gives a diagnostic decision, not a fabricated numerical result: without the solver outputs, one cannot calculate a prediction or claim that the event split fixed the fit. The controlled comparison is valuable precisely because it changes one numerical feature while holding the trial parameters and model inputs fixed.
Choosing Among Diagnostic Approaches
| Approach | Best use | What it cannot establish alone |
|---|---|---|
| Analytic benchmark | Checking event logic and predictions in a simplified case with a known solution | Accuracy of a more complex model that has different dynamics or events |
| Explicit event splitting | Ensuring integration does not treat a known discontinuity as a smooth interval | That tolerances are adequate or that the model equations are correct |
| Tighter tolerances or smaller maximum step | Testing whether predictions and objective values stabilize numerically | Correct event records; excessive computation can still give a wrong model result |
| Alternative solver suited to the dynamics | Checking whether solver behavior is driving a failure, especially when stiffness is suspected | That a result is correct merely because another solver completed |
Use the evidence to choose the next test. A prediction that disagrees with an analytic solution points first to implementation, event handling, or inadequate integration. A solver that repeatedly fails with finite states and shrinking steps may warrant evaluation with a method appropriate to the system’s stiffness. A stable forward solve paired with optimizer stalls instead shifts attention toward the estimation problem. In every case, compare results at the same parameter values before drawing conclusions about the fit.
Common Pitfalls and Limits
- Treating the final optimizer message as the root cause. A gradient or convergence warning may follow an earlier failed ODE evaluation. Find the first failure in the trace.
- Changing many settings at once. Simultaneously changing tolerances, solver, parameter bounds, and model code prevents a clear attribution. Make controlled changes and retain the corresponding outputs.
- Assuming tighter tolerances fix an incorrect event. Numerical accuracy cannot correct a missing, duplicated, mistimed, or mis-scaled dose event.
- Calling any successful solve accurate. Completion means the solver met its own stopping criteria, not that predictions are accurate enough for the likelihood. Compare predictions across settings and against benchmarks where available.
- Overinterpreting a simplified benchmark. The one-compartment analytic solution validates only the matching equations and event schedule. It cannot verify every part of a larger model.
These diagnostics localize a numerical source; they do not prove parameter identifiability, model adequacy, or scientific validity. A stable ODE calculation is necessary for interpreting the optimization, but it is not sufficient to establish that the fitted model is appropriate.
Check Your Understanding
Answer each question, then compare with the solutions.
- In the infusion case, which quantity changes discontinuously at 1 h: the amount \(A(t)\), the input rate \(R(t)\), or both?
- Why does a non-finite gradient after an integration failure not, by itself, establish that the likelihood is nonsmooth?
- At \(t=2\) h, what concentration follows from the analytic event-split solution, to three significant figures?
- If tightening tolerances stabilizes predictions but the optimizer still stalls, what broad diagnostic conclusion is justified?
- Why is reducing the maximum step size not a substitute for checking event records?
Solutions. (1) The input rate changes from 100 mg/h to zero; the amount is continuous because stopping the infusion does not instantaneously add or remove drug. (2) A gradient calculation can depend on nearby objective evaluations, and a failed ODE solve can make one of those evaluations invalid; the gradient warning may therefore be downstream of the integration failure. (3) \(C(2)=4.5317e^{-0.20}=3.7103\) mg/L, or \(3.71\) mg/L to three significant figures. (4) The forward numerical solution appears stable under that test, so further investigation should include the optimizer and objective evaluation rather than assuming the ODE solver remains the cause. (5) A smaller step may reduce integration error across a correctly represented discontinuity, but it cannot supply an omitted event or correct its time, units, or dose amount.