FractaLPK — Methodology

Population PK — Auto-Diagnose

Population NLME (adaptive 1/2-CMT) · single-profile 11-model search · AIC-ranked

Sampling needed before any fractional claim. A pre-registered study (3 360 simulated profiles, Sep 2026) measured what a design must have for each question: one disposition phase or two — 8 sampling points with the last one at 24 h; classical or fractional in a single phase — 8 points with the last one at 48 h, or 12 points to 24 h; classical or fractional in two phases — 24 points with the last one at 168 h, that is seven days. 8 of the 11 structure pairs did not separate under any design tested, up to 32 points and a 168 h window. The report states which of these your profile meets before it fits anything.

The engine adapts to your data. For population data (multiple subjects) it fits a mixed-effects model (nlmixr2 FOCE-I), selecting adaptively between 1-compartment and 2-compartment structures — the simpler one first, escalating only when it is unreliable — plus a fractional-order screen. For a single averaged / sparse profile it runs the broader search below: 3 baseline candidates plus an 8-model multi-compartment structural set. In both cases you receive the AIC ranking of the structures actually evaluated, 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
FractaLPK Fractional	monofractional	C(t) = A·E_α(−k·t^α), α ∈ (0.1, 1.0]

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.

Identifiability (validated 2026-08-16). With rich single-profile sampling (~14 points, ≤15% residual), the engine reliably recovers 1-compartment (classical/fractional) and 2-compartment classical structures; it does not reliably distinguish fractional from classical order, nor monofractional from multifractional — it reports these as statistically indistinguishable rather than crowning one. With sparse sampling it may crown incorrectly, so a fractional verdict requires adequate sampling density.

Tumor Growth

4 candidate models · same ranking pipeline

Fitted on per-time mean volume (mm³). Hahnfeldt-class models include explicit vascular dynamics.

Fractional model not reported: model-selection criterion under review (false positives 15 % with 6 measurements, study FFCD-0). Only the classical fits are shown.

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)

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, 1.0]

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

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

Model-risk report aligned with the vocabulary of ICH M15, with identifiability and sampling adequacy measured in pre-registered studies.

What a single concentration profile can answer

Eleven structure pairs · measured, not assumed

Every row below comes from a pre-registered study: 3360 simulated datasets across 21 sampling designs (up to 32 points and a 168 h window), plus a population Fisher-information bound. No figure on this page is typed by hand — the page is generated from validation/diseno_muestreo.json and validation/i1_fim_poblacional.json.

Simpler structure	Richer structure	Can it be answered?
1-CMT classical	2-CMT classical	answerable from a single profile with 8 sampling points and the last one at 24 h.
1-CMT monofractional	2-CMT monofractional	The information bound says a single profile carries enough information, but our pre-registered separation criterion does not classify it as answerable — precision and decision are not the same thing. No claim is made for this pair.
2-CMT classical	3-CMT classical	Not answerable from a single profile. The information bound does not hold for this pair (measured), so no number of subjects is claimed until it is measured directly.
2-CMT monofractional	3-CMT monofractional	Not answerable from a single profile. A population study of about 21 subjects would carry enough information — but that study has not been run, so this is a bound, not a result. This figure is an information bound, calibrated in one structure pair only: 2 subjects separated in 9 of 12 replicates. A pre-registered extension then measured it directly: 7 subjects separated in 30 of 30 replicates (95% CI 0.88-1.00, p = 0.0012), so in the one pair measured the bound understates the requirement by about 3.5 times. Whether that factor holds for other pairs has not been measured. Whether 2 subjects would have sufficed is not established: settling that would take 433 replicates and was not run.
2-CMT multifractional	3-CMT multifractional	not answerable, and a population study of up to 50 subjects would not answer it either — the bound puts it at 7746 subjects. This figure is an information bound, calibrated in one structure pair only: 2 subjects separated in 9 of 12 replicates. A pre-registered extension then measured it directly: 7 subjects separated in 30 of 30 replicates (95% CI 0.88-1.00, p = 0.0012), so in the one pair measured the bound understates the requirement by about 3.5 times. Whether that factor holds for other pairs has not been measured. Whether 2 subjects would have sufficed is not established: settling that would take 433 replicates and was not run. The bound is optimistic, so the real requirement is larger and this pair stays out of reach.
1-CMT classical	1-CMT monofractional	answerable from a single profile with 8 sampling points and the last one at 48 h (equivalently 12 points to 24 h).
2-CMT classical	2-CMT monofractional	answerable from a single profile with 24 sampling points and the last one at 168 h.
3-CMT classical	3-CMT monofractional	Not answerable from a single profile. A population study of about 7 subjects would carry enough information — but that study has not been run, so this is a bound, not a result. This figure is an information bound, calibrated in one structure pair only: 2 subjects separated in 9 of 12 replicates. A pre-registered extension then measured it directly: 7 subjects separated in 30 of 30 replicates (95% CI 0.88-1.00, p = 0.0012), so in the one pair measured the bound understates the requirement by about 3.5 times. Whether that factor holds for other pairs has not been measured. Whether 2 subjects would have sufficed is not established: settling that would take 433 replicates and was not run.
2-CMT monofractional	2-CMT multifractional	not answerable, and a population study of up to 50 subjects would not answer it either — the bound puts it at 73 subjects. This figure is an information bound, calibrated in one structure pair only: 2 subjects separated in 9 of 12 replicates. A pre-registered extension then measured it directly: 7 subjects separated in 30 of 30 replicates (95% CI 0.88-1.00, p = 0.0012), so in the one pair measured the bound understates the requirement by about 3.5 times. Whether that factor holds for other pairs has not been measured. Whether 2 subjects would have sufficed is not established: settling that would take 433 replicates and was not run. The bound is optimistic, so the real requirement is larger and this pair stays out of reach.
3-CMT monofractional	3-CMT multifractional	not answerable, and a population study of up to 50 subjects would not answer it either — the bound puts it at 1750 subjects. This figure is an information bound, calibrated in one structure pair only: 2 subjects separated in 9 of 12 replicates. A pre-registered extension then measured it directly: 7 subjects separated in 30 of 30 replicates (95% CI 0.88-1.00, p = 0.0012), so in the one pair measured the bound understates the requirement by about 3.5 times. Whether that factor holds for other pairs has not been measured. Whether 2 subjects would have sufficed is not established: settling that would take 433 replicates and was not run. The bound is optimistic, so the real requirement is larger and this pair stays out of reach.
1-CMT monofractional	2-CMT multifractional	Not answerable from a single profile, and not measured for population data (not nested in a single scalar).

The design criteria above are established results for classical models. Their extension to Mittag-Leffler (fractional) kinetics is our own, and it is pre-registered: the criteria and the thresholds were committed before any data were generated, and both the simulation study and the Fisher-information route are reproducible from validation/ in the repository.

References

Macheras, P. & Iliadis, A. Modeling in Biopharmaceutics, Pharmacokinetics and Pharmacodynamics: Homogeneous and Heterogeneous Approaches, 2nd ed., Springer (2016). doi:10.1007/978-3-319-27598-7
Sopasakis, P., Sarimveis, H., Macheras, P. & Dokoumetzidis, A. «Fractional calculus in pharmacokinetics». J Pharmacokinet Pharmacodyn 45, 107–125 (2018). doi:10.1007/s10928-017-9547-8
Morales-Delgado, V.F., Taneco-Hernández, M.A., Vargas-De-León, C. & Gómez-Aguilar, J.F. «Exact solutions to fractional pharmacokinetic models using multivariate Mittag-Leffler functions». Chaos, Solitons & Fractals 168, 113164 (2023). doi:10.1016/j.chaos.2023.113164
Atkinson, A.C. & Fedorov, V.V. «The design of experiments for discriminating between two rival models». Biometrika 62, 57–70 (1975) — T-optimality. Population Fisher information as implemented in PFIM and PopED. doi:10.1093/biomet/62.1.57