Structural PK models: solutions, parameterisations, and the ADVAN map

Always parameterise in clearance and volume

Micro-constants (k10, k12, k21) and macro-constants (A, alpha, B, beta) are outputs, not parameters to estimate. Clearance and volume are the parameters that:

• have physiological meaning and known covariate relationships (CL scales with weight^0.75, V with weight^1.0, CL with renal function);
• are comparable across studies, formulations and populations;
• keep their meaning when a compartment is added — k10 changes when you add a peripheral compartment, CL does not.

_models.py accepts only cl, v1, q, vp. micro_constants() converts for reporting.

Linear mammillary disposition

Any linear model with a central compartment and n peripheral compartments has a unit-bolus impulse response that is a sum of n+1 exponentials:

C(t) = (D) * sum_i  coef_i * exp(lambda_i * t)

_models.disposition() gets lambda_i and coef_i by eigendecomposition of the rate matrix rather than from the textbook quadratic/cubic root formulas. Same answer, no special cases, works for any number of compartments.

Given that impulse response, every input is a convolution with a closed form:

The absorption term has a removable singularity at ka = -lambda_i. The limit is coef_i * t * exp(lambda_i t). This is not an edge case: it is precisely the flip-flop boundary, and an optimiser walking through it returns inf or nan without the branch. _models.py handles it; a hand-written Bateman function usually does not.

Useful identities that hold for every linear model, and make good unit tests:

AUC(0-inf) after an IV bolus = D / CL          (independent of the number of compartments)
Vss = V1 + sum(Vp)
MRT after an IV bolus = Vss / CL

Flip-flop kinetics

When absorption is slower than elimination (ka < lambda_z), the terminal slope of an oral profile reflects absorption, not elimination. The two exponentials are mathematically interchangeable: the fit is identical if you swap ka and k. Consequences:

• t½ from an oral profile is the absorption half-life, and Vz/F is meaningless;
• which root the optimiser lands on is a starting-value accident;
• the assignment cannot be resolved from extravascular data alone. It needs IV data, or a formulation with faster absorption, or an external argument.

fit_compartmental.py reports flip_flop_suspected and raises a finding. Depot formulations, extended-release products and subcutaneous biologics are routinely flip-flop.

Absorption models

The transit model must be evaluated with lgamma, not a literal factorial: fitted n routinely exceeds 20 and the direct form overflows. _models.conc_transit() does this.

A lag time and a transit chain describe the same phenomenon differently. The transit model is continuous and usually estimates better; the lag is easier to explain. Do not fit both.

Nonlinear elimination

Michaelis-Menten: dA/dt = -Vmax * C / (Km + C), with C = A/V1.

• Below Km, clearance is approximately Vmax/Km and the drug looks linear;
• above Km, clearance falls and exposure rises faster than the dose;
• superposition is invalid, so multiple-dose behaviour cannot be derived from a single dose, and the accumulation ratio is dose-dependent.

Phenytoin is the classic example: within the therapeutic range a 10% dose increase can produce a much larger exposure increase. If a dose-proportionality analysis shows AUC rising faster than dose, MM elimination is one explanation; saturable first-pass metabolism and solubility-limited absorption (which produce less than proportional increases) are others.

Combining PK with a delayed effect

An effect compartment and an indirect response model can both fit a hysteresis loop. They differ in what happens when the drug is stopped, and in how the delay scales with dose — an indirect response model's onset is dose-dependent while ke0 is not. See pd-and-exposure-response.md.

NONMEM ADVAN/TRANS map

For translating a model built here into a control stream:

Watch the compartment numbering: in ADVAN4 the central compartment is 2 and V2 is the central volume, whereas in ADVAN3 the central compartment is 1 and V1 is central. Mixing the two conventions when converting a model is a standard source of a silently wrong volume.

Analytical ADVANs are far faster and more numerically stable than $DES and should be used whenever the model is linear. Reach for ADVAN13 only when the structure genuinely is not.

Choosing the number of compartments

Use, in order: the residual pattern (runs of the same sign mean the shape is wrong), the F test for nested models, BIC, and the parameter precision. Do not use AIC alone — its fixed penalty of 2 per parameter frequently selects an extra compartment whose intercompartmental clearance has an RSE above 50%. fit_compartmental.py --compare prints all four side by side.

Adding a compartment is justified when it changes the conclusions — Vss, the terminal half-life, the accumulation ratio, the predicted trough — not merely when it improves the objective function.