FractaLPK — Methodology

Population PK — Auto-Diagnose

11 candidate models · AIC-ranked · diagnostic-flag gated

Each upload is fitted against 3 baseline candidates plus an 8-model multi-compartment structural search. The fitter runs models in parallel; you receive the AIC ranking, the winning equation with parameters, and a verdict statement.

Baseline candidates

Name	Kinetic class	Form
Classical 1-CMT	classical	Bateman: C(t) = A·e^−kt
PBFTPK (Panos ODE)	finite-time PK	ODE system with finite absorption / elimination windows (Macheras)
FractaLPK Fractional	monofractional	C(t) = A·E_α(−k·t^α), α ∈ (0.1, 2.0)

Multi-CMT structural search (8 models)

Name	Class	Form
1-CMT classical	classical	A·e^−kt
2-CMT classical	classical	A₁·e^−k₁t + A₂·e^−k₂t
3-CMT classical	classical	Σ_i=1..3 Aᵢ·e^−kᵢt
1-CMT monofractional	monofractional	A·E_α(−k·t^α), shared α
2-CMT monofractional	monofractional	Σ Aᵢ·E_α(−kᵢ·t^α), shared α
3-CMT monofractional	monofractional	Σ Aᵢ·E_α(−kᵢ·t^α), shared α
2-CMT multifractional	multifractional	Σ Aᵢ·E_αᵢ(−kᵢ·t^αᵢ), independent αᵢ
3-CMT multifractional	multifractional	Σ Aᵢ·E_αᵢ(−kᵢ·t^αᵢ), independent αᵢ

E_α is the single-parameter Mittag-Leffler function. multifractional models allow each compartment its own fractional order αᵢ — useful when central and peripheral compartments exhibit different memory effects, something monofractional and classical models cannot capture.

Tumor Growth

5 candidate models · same ranking pipeline

Fitted on per-time mean volume (mm³). Hahnfeldt-class models include explicit vascular dynamics; the fractional variant uses a Caputo derivative on the same Hahnfeldt structure to capture sub-/super-diffusive growth memory.

Name	Class	Form
Exponential	2-parameter	V(t) = V₀·e^kt
Logistic	3-parameter	dV/dt = r·V·(1 − V/K)
Gompertz	3-parameter	dV/dt = λ·V·ln(K/V)
Hahnfeldt classical	4-parameter	Tumor + vasculature ODE pair (integer order)
Hahnfeldt fractional	4-parameter + α	Caputo^α(V) = same RHS, α ∈ (0.1, 2.0)

Drug Release / Dissolution

5 candidate models · cumulative-release fraction in [0,1]

For dissolution and in-vitro release profiles. The fractional Mittag-Leffler model is parsimony-gated: it must beat the runner-up by ΔAIC ≥ 8 to win, because lower-α drift can otherwise mimic Korsmeyer-Peppas / Weibull tails spuriously.

Name	Class	Form
First-order	1-parameter	F(t) = 1 − e^−kt
Higuchi	1-parameter	F(t) = k_H·√t
Korsmeyer-Peppas	2-parameter	F(t) = k·tⁿ, with mechanism band from n
Weibull	2-parameter	F(t) = 1 − e^{−(t/τ)^β}
Mittag-Leffler fractional	2-parameter	F(t) = 1 − E_α(−k·t^α), α ∈ (0.1, 2.0)

Verdict logic

AIC ranking + parsimony + diagnostic flags

All models are ranked by Akaike Information Criterion (AIC). The reported winner is not always the lowest-AIC model — if a simpler model is within ΔAIC < 4 of the leader, the simpler one is picked under parsimony. When any diagnostic flag fires, the verdict is downgraded.

AIC ranking. Compute AIC = OFV + 2·k for every candidate that converged.
Parsimony rule. If two models are within ΔAIC < 4 (statistically equivalent), the model with fewer parameters wins. For drug-release, the fractional model needs ΔAIC ≥ 8 over runner-up to be reported as winner (stricter threshold to avoid spurious fractional verdicts on noisy tails).
Diagnostic flags. If any flag fires, the verdict line shifts from the model-specific statement to "FIT QUESTIONABLE — see diagnostics". AIC, R², ΔAIC and winner name remain visible — you still need the evidence to understand why the flag fired.

Diagnostic flags

BOUND_HIT

Fitted α landed on the optimisation boundary (0.1 or 2.0). Indicates the data does not support a meaningful fractional order — the optimiser pushed against the wall.

COMPARTMENT_DEAD

In a multi-compartment fit, one of the amplitudes Aᵢ is below 10% of the total. Suggests the dataset does not actually support that many compartments and a simpler model is more honest.

Key references

Macheras, P. & Iliadis, A. (2016). Modeling in Biopharmaceutics, Pharmacokinetics and Pharmacodynamics, 2nd ed., Springer. — Mittag-Leffler PK, finite-time absorption (PBFTPK).
Hahnfeldt, P. et al. (1999). Tumor development under angiogenic signaling. Cancer Res. 59(19):4770–5. — Vascular tumor ODE used as the Hahnfeldt baseline.
Korsmeyer, R.W. et al. (1983). Mechanisms of solute release from porous hydrophilic polymers. Int. J. Pharm. 15(1):25–35. — Korsmeyer-Peppas release law.
Higuchi, T. (1961). Rate of release of medicaments from ointment bases. J. Pharm. Sci. 50:874–875. — Higuchi √t model.
Boeckmann, A.J. et al. (1994). NONMEM Users Guide. — Theophylline 12-subject benchmark used in our validation samples.

Full per-engine implementation notes and software references live in the engineering documentation accompanying each report.

See it in action

Sample PDFs use public benchmark datasets and the same pipeline a paying client gets.

PopPK sample (PDF) Tumor sample (PDF) Drug-release sample (PDF)