Multi-Start: Michaelis-Menten Elimination

Michaelis-Menten (saturable) elimination is the canonical example of a local-minimum problem in NLME modelling. The Vmax/Km pair is only weakly identifiable from sparse PK data: many (Vmax, Km) combinations produce similar predicted concentrations, so the OFV surface contains a narrow ridge. A single-start FOCEI run is sensitive to the initial values — starting far from the true optimum, the optimizer often settles on an inflated Vmax/Km pair with a higher OFV than the true optimum.

The problem

For the model below with true parameters Vmax ≈ 3.5 mg/h and Km ≈ 5.5 mg/L, starting with Vmax = 12 and Km = 20 (both roughly 3× too high) a single FOCEI run may converge to a local minimum rather than the global one. The two solutions predict similar concentration-time profiles on the observed time grid but differ in OFV by several units.

The model file

{{#include ../../../examples/mm_multistart.ferx}}

What multi-start does

With n_starts = 8 and start_sigma = 0.5:

Start 0 uses the exact initial values from [parameters] (Vmax=12, Km=20).
Starts 1–7 multiply each log-packed theta by exp(N(0, 0.5)), giving a spread of roughly ×0.6–×1.6 around the initial values. With start_sigma = 0.5 this is wider than the default 0.3 — appropriate for a ridge-shaped surface.
All 8 runs execute in parallel. On an 8-core machine wall-clock time equals a single run.
The converged run with the lowest OFV is returned. If start 0 was already at the global optimum, nothing changes. If one of the perturbed starts found a better solution, a warning on FitResult.warnings reports which start won and its OFV.

Running it

# Simulate data first
cargo run --release -- examples/mm_multistart.ferx --simulate

# Fit with multi-start
cargo run --release -- examples/mm_multistart.ferx --data mm_multistart-sim.csv

When to use multi-start

Model feature	Risk of local minima	Recommended n_starts
Linear 1-cpt / 2-cpt	Very low	1 (default)
Michaelis-Menten elimination	High	4–8
Full-block omega (≥ 3 etas)	Moderate	4
Many correlated covariates	Moderate	4–8

start_sigma = 0.3 (the default, ≈ 30% CV) is appropriate for most models. For ridge-shaped surfaces like Vmax/Km, start_sigma = 0.4–0.5 explores a wider neighbourhood.