Cp5 Pharma — Fractional Engine 2D

FractaLPK Services

Fractional calculus applied to drug development. Mechanistic models that detect what classical methods miss.

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DRUG RELEASE ANALYSIS

Fractional diffusion model for your formulation

48-72h

What we do

Fractional diffusion model for tablets, PLGA, implants, and extended release formulations.
Method: Mittag-Leffler + cylindrical Bessel-Caputo PDE solution vs classical Weibull/first-order.

You get

▸ α parameter per formulation batch
▸ α shift between batches = matrix structure change detected
▸ Prediction of late-stage release beyond dissolution study window
▸ Mechanistic model (not empirical curve fitting)

Value: Identify formulation issues before costly bioequivalence trials — Data needed: Dissolution profile (time vs % released)

Request Proposal

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BIOEQUIVALENCE PREDICTION

Pre-trial prediction — will your generic pass BE?

1 week

What we do

Compare α_generic vs α_reference using fractional diffusion.
Method: Fractional dissolution mechanism comparison (not just f2 similarity).

You get

▸ Go/no-go recommendation before committing to clinical trial
▸ Quantitative comparison of dissolution mechanisms
▸ Risk assessment based on fractional order difference

Value: Avoid failed BE trials and costly reformulation cycles — Data needed: Dissolution profiles of test vs reference product

Request Proposal

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POPULATION PK — FRACTIONAL ANALYSIS

Detect anomalous diffusion hidden in your drug's pharmacokinetics

1-2 weeks

What we do

Individual n_frac estimation per subject using Cp5 engine.
Method: Fractional compartment model vs NONMEM population fixed-order.

You get

▸ ΔOFV comparison (fractional vs classical)
▸ % of subjects with n_frac < 0.95 (anomalous subpopulation)
▸ Individual parameter estimates with mechanistic interpretation
▸ Discovery of hidden variability your current model cannot explain

Value: Understand why 30-40% of patients respond differently — Data needed: Concentration-time data (CSV, under NDA)
Validated: Abacavir pediatric (ΔOFV=342, 40% anomalous), Niclosamide (ΔOFV=260)

Request Proposal

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TUMOR DYNAMICS & TREATMENT RESPONSE

Fractional tumor growth model with treatment coupling

2 weeks

What we do

Fractional tumor growth model with treatment coupling.
Method: RPSM (Residual Power Series Method) + Cp5 Levin-U acceleration.

You get

▸ α estimation per patient/tumor from serial measurements
▸ Treatment response prediction (validated: α drops 0.83→0.49 post-radiation)
▸ Coupled PK-PD: drug fractional diffusion + tumor fractional dynamics
▸ Patient stratification by fractional order
▸ DLA invasion front with fractal dimension

Value: Predict who will respond to treatment and who won't — before the trial ends
Data needed: Serial tumor measurements (imaging, caliper, biomarkers)
Reference: Gündoğdu & Joshi, Mathematics 2025, 13(3), 536 — our method extends this with Cp5 acceleration

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FORMULATION QbD CHARACTERIZATION

Fractional order α as a Quality-by-Design process parameter

1 week/batch

What we do

Track α across production batches using fractional dissolution model.
Method: Fractional order monitoring per batch as QbD parameter.

You get

▸ α per batch — early detection of manufacturing drift
▸ α shift signals structural changes in polymer matrix
▸ Without changing existing dissolution specifications
▸ Quantitative link between microstructure and release kinetics

Value: Catch manufacturing issues before they reach stability or clinical studies — Data needed: Dissolution profiles from multiple batches

Request Proposal

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FIRST-IN-HUMAN DOSE PREDICTION

Fractional PK model for safer FIH dose selection

1-2 weeks

What we do

Fractional compartment model with anomalous diffusion correction for safer FIH dose selection.
Method: Fractional PK + tissue heterogeneity correction vs classical allometric scaling.

You get

▸ Dose prediction accounting for tissue heterogeneity
▸ Safety margin calculation with fractional tail behavior
▸ Comparison vs classical allometric scaling

Value: Safer dose selection for Phase I — especially for drugs with complex tissue distribution — Data needed: Preclinical PK data (animal studies)

Request Proposal

HOW WE WORK

✓ All analyses performed locally — your data never leaves our system

✓ Full NDA before any data transfer

✓ Signed PDF report delivered within agreed timeline

✓ Independent analysis — not affiliated with any CRO or pharma company

PLATFORM: fractalpk.es

DEMO: fractalpk.es/dictionary (13 drugs validated)

CONTACT: cperapa@fractalpk.es

ENGINE: Cp5 Fractional Calculus — Dyadic Pochhammer + Levin-Padé acceleration

PBFTPK₁ — First-Order Truncated Absorption

Macheras & Chryssafidis 2020 · Bateman kernel during [0, τ], pure k_e decay after

Parameters

Stages

τ₁ [h]

F·D/V_d (stage 1)

k_a,1 [1/h]

k_e [1/h]

t_max [h]

Model. Single stage: C(t) = (F·D·k_a/(V(k_a-k_e)))(e^-k_et-e^-k_at) for 0 ≤ t ≤ τ; pure decay from C(τ) after. 2-stage superposes two consecutive first-order inputs with distinct k_a. L'Hôpital branch activates when |k_a - k_e| < 10^-6.

Bioequivalence under the F.A.T. Concept

Macheras & Tsekouras 2024 · Lévy-Macheras plot + F_rel(AUC_0→τ) with bootstrap 90% CI

Test formulation (T)

F·D/V

τ_T [h]

Reference formulation (R)

F·D/V

τ_R [h]

Shared

k_e [1/h]

t_max [h]

N subjects

CV% (inter-subj)

Method. Top chart: simulated individual profiles for Test (blue) and Reference (orange). Bottom chart: Lévy-Macheras cumulative ratio R(t) = cumAUC_T(t) / cumAUC_R(t). Plateau ≈ F_rel. Right-hand box reports F_rel geometric-mean with bootstrap 90% CI and the FDA 80-125 verdict (server-side be_utils.F_rel_with_CI).

PK/PD Models (21)

Cp5 ENGINE ⭐

PK

PD

PK-PD

Clinical

⚙️ TCM (Cp5 1D)

🤖 IA FARMA

Hello! I am the AI Pharma assistant. Run a model and ask me about the results. I can:
🧪 Interpret PK/PD
💊 Recommend dosing
⚠️ Safety alerts
📊 Compare models

📐 Model Fitting — Reference Drug Library

Automated curve fitting (Diff. Evolution) to published PK profiles + FDA/EMA goodness-of-fit metrics

📂 Fit Your Own Data

Upload CSV with time + concentration columns (max 50 points on Starter plan)

📄

Drop CSV here or click to browse

Format: time,conc (header row required)

Or load an example dataset:

Dose (mg): Route:

⚡ Benchmark: FractaLPK vs Classical Methods

Compare Cp5 automatic convergence against classical fractional PK fitting with generic initial parameters.

📊 Fractional PopPK — SAEM + FOCE-I with Cp5 Analytical Gradients

Population PK with fractional TCM (n ∈ ℝ) — SAEM + FOCE-I (analytical Cp5 Mittag-Leffler gradients) + GOF diagnostics

Structural Model

Subjects Dose

SAEM Iterations

Estimation Method

📁 Upload Patient CSV

📁

Click or drag CSV file
(ID, TIME, DV, AMT)

Population Parameters (θ_pop)

Run SAEM to see results...

IIV & Residual Error

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Theophylline Validation — FOCE-I vs SAEM

FOCE-I OFV

252.2

SAEM OFV

263.4

FOCE-I Iterations

17

SAEM Iterations

500

ka: 1.46 vs 1.80

CL: 2.94 vs 3.43 L/h

V: 3.15 vs 41.0 L

12 subjects, 119 observations. FOCE-I uses analytical Cp5 Mittag-Leffler gradients (dE_alpha/dz exact, no finite differences). Dataset: Boeckmann, Sheiner & Beal (1994).

Indomethacin IV — Real Data Validation

INTERMEDIATE CASE

Cp5 engine auto-detects when kinetics are anomalous (n_frac<0.90) vs classical (n_frac=1.0).
In Indomethacin: 2/6 subjects show real subdiffusion (n_frac ≈ 0.87). Classical model wins overall OFV.

Mean n_frac

0.947

Subdiffusive

2/6

R² Classical

0.977

R² Fractional

0.976

Subject	n_frac	R² Classical	R² Fractional	ΔOFV
1	1.000	0.995	0.994	-1.1
2 *	0.865	0.960	0.958	-0.4
3	0.947	0.944	0.957	+2.9
4	1.000	0.995	0.995	-0.5
5	1.000	0.991	0.965	-14.6
6 *	0.870	0.977	0.989	+7.6

6 real subjects, 66 observations. Kwan et al. (1976). * Subjects with n_frac < 0.90 (clear subdiffusion). Classical model wins total OFV (-154.4 vs -148.3) — fractional only improves subjects 3 and 6.

IPRED vs Observed (GOF)

CWRES vs Time

SAEM Convergence

ETA Distributions

FractaLPK vs Classical Model vs Simcyp — Technical Comparison

Classical Model is the regulatory gold standard (40+ years). FractaLPK offers unique mathematical capabilities not yet FDA-validated.

Feature	FractaLPK	Classical Model	Simcyp
Fractional Compartments (n ∈ ℝ)	✅ Cp5 exact	❌ integer only	❌ integer only
Solution for n ∈ ℝ	Analytic (Cp5)	N/A	N/A
↑ FractaLPK supports fractional compartments (n ∈ ℝ) with exact analytical evaluation. Standard tools round to the nearest integer (up to 34% error).
TCM 1D (oral transit, n ∈ ℝ)	✅ exact	⚠️ chain n∈ℤ	⚠️ chain n∈ℤ
Mammillary 2D (n ∈ ℝ)	✅ exact	❌	❌
Numerical Precision (n ∈ ℝ models)	~10⁻¹⁰ (analytic)	~10⁻⁶ (RK45)	~10⁻⁶ (ODE)
* For integer-compartment models, all solvers achieve adequate clinical precision. FractaLPK advantage is specific to fractional n ∈ ℝ.
SAEM Estimation	✅ (OFV=263.4*)	✅	❌
FOCE-I (Cp5 gradients)	✅ OFV=252.2*	✅ (gold std)	❌
* Validated on 2 real datasets: Theophylline (12 subjects, Boeckmann 1994) + Indomethacin IV (6 subjects, Kwan 1976). FOCE-I converges in 17 iter with analytical Cp5 Mittag-Leffler gradients. Cp5 auto-detects subdiffusion (n_frac<0.90) in 2/6 Indomethacin subjects.
MCMC Bayesian (NUTS)	✅	✅	❌
Parallel Computing	✅	✅	⚠️
GOF Diagnostics	✅	✅	⚠️
VPC / Covariate SCM	✅	✅ (PsN)	❌
Drug Library Fitting	✅ (8 ref drugs)	✅ (gold standard)	⚠️
Interactive Dashboard	✅	❌ (CLI)	⚠️
AI Assistant	✅ (prototype)	❌	❌
Regulatory Acceptance	Pending	✅ FDA/EMA	✅ FDA
Price (target)	TBD	$5-20K/yr	$50-150K/yr

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Monoclonal Antibody PK — Fractional vs Classical Model

2-Comp Fractional (Cp5) vs 2-Comp + Michaelis-Menten vs 2-Comp Linear · 5 FDA-approved mAbs · Published Pop-PK parameters

Benchmark Settings

Select mAb

Models compared:
A) 2-Comp Linear (4 params)
B) 2-Comp + Michaelis-Menten (6 params) — Classical Model standard
C) 2-Comp Fractional n∈ℝ (5 params) — FractaLPK innovation

WHAT n_frac MEANS (THEORETICAL)

n < 1 → Subdiffusion: 150 kDa molecules diffuse slowly through tissue. Common in IgG antibodies.
n = 1 → Standard: Normal first-order kinetics (classical 2-comp).
n > 1 → Superdiffusion: Enhanced transport via FcRn recycling.
⚠ n_frac is a theoretical parameter from fractional calculus. Not yet validated in clinical trials. Precision dosing results below are exploratory simulations, not clinical recommendations.

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Trastuzumab

Anti-HER2 · Breast Cancer

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Bevacizumab

Anti-VEGF · Colorectal

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Nivolumab

Anti-PD-1 · Melanoma

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Adalimumab

Anti-TNFα · Autoimmune

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Pembrolizumab

Anti-PD-1 · NSCLC

Click RUN mAb BENCHMARK to compare Fractional Cp5 vs Classical Model Michaelis-Menten on published PK data

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Precision Dosing — Individualized mAb Schedules

Virtual patient population · n_frac-driven dose optimization · Cost savings analysis

Drug

Patients

Weight range

HOW IT WORKS

1. Generate virtual patients with IIV on CL, V, n_frac
2. Simulate fractional PK profile per patient
3. Find when C(t) drops below trough target
4. Calculate optimal interval per patient
5. Compare standard vs individualized dosing

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Select a drug and click RUN PRECISION DOSING to generate individualized dosing recommendations based on each patient's fractional PK profile.

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Fractional Tumor-Immune Dynamics (RPSM + Cp5)

Caputo ODE system solved by Residual Power Series + Levin-U acceleration · 4 coupled equations · DLA morphology

Parameters

α (fractional order)

0.3 (subdiffusion)0.751.0 (classical)

ρ (tumor growth rate)

1.5

s (immune recruitment)

0.3

Include drug treatment

System (Caputo Dα):

Dα N = r1·N(1-b1·N) - c1·NT - a1·NC
Dα T = r2·T(1-b2·T) - c2·IT - a2·TC
Dα I = s + ρ·IT/(η+T) - c3·IT - d1·I + a3·C
Dα C = -d2·C + v(t)

Solved by RPSM + Levin-U (Cp5)

Cell Populations

Drug + Tumor Response

DLA Invasion Front

Click "Generate DLA" to simulate

Levin-U vs Raw Series

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Fractional NCA — Non-Compartmental Analysis

Non-compartmental analysis using industry-standard methods with automatic fractional kinetics detection.

VALIDATED

Data Input

Route

Dose (mg)

CSV Data (TIME, CONC — optional ID column for multi-subject)

Or upload CSV file

METHODS

Linear-up/log-down trapezoidal AUC, best-fit terminal slope (adjusted R²).

References: Gibaldi & Perrier 1982, Purves 1992, Boeckmann 1994.

VALIDATION

Validated against published NCA standards. 0.0000% deviation on 12-subject reference dataset (Boeckmann 1994).

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Non-Compartmental Analysis with Fractional Kinetics Detection

Upload a CSV with TIME and CONC columns, or click "Load Theophylline Demo" to see the validated results.
Automatically detects anomalous (power-law) terminal phases and computes fractional NCA parameters.

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PBFTPK 4-Model Comparison

Bateman Classical vs PBFTPK (Macheras) vs Fractional (FractaLPK) vs Hybrid PBFTPK+Fractional

VALIDATED

Data Input

CSV Data (ID, TIME, DV, AMT, EVID, MDV, WGHT)

Or upload CSV file

MODELS

1. Bateman: C = (D·ka)/(V·(ka-ke))·[e^-ke·t - e^-ka·t]
2. PBFTPK: Zero-order input (τ) + first-order elimination
3. Fractional: Classical absorption + Mittag-Leffler elimination
4. Hybrid: ZO input (τ) + ML elimination (n_frac)

⚠ LOCAL ONLY

Full analysis may take 5-15 min for 100+ subjects. Demo loads instantly.

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PBFTPK 4-Model Comparison

Upload PK data or click "Load Abacavir Demo" to see the 4-model comparison.
Compares Bateman, PBFTPK (Macheras), Fractional (FractaLPK), and Hybrid models.

🧫

3-CMT Fractional Mammillary Model

Central + rapid tissue + deep tissue — each with independent fractional order (α₁, α₂, α₃)

RESEARCH

Rate Constants (h⁻¹)

k₁₂ (central→rapid) 6.84

k₂₁ (rapid→central) 3.30

k₁₃ (central→deep) 2.51

k₃₁ (deep→central) 0.198

k₁₀ (elimination) 7.14

Fractional Orders

α₁ (central) 1.00

α₂ (rapid tissue) 0.85

α₃ (deep tissue) 0.60

Dosing

Dose (mg)

V₁ (L)

🧫

3-Compartment Fractional Mammillary Model

Use manual sliders, upload PK data for auto-fitting, or load a precalculated demo.
Each compartment can have a different fractional order — detecting anomalous diffusion in specific tissues.

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Multi-Compartmental Fractional (Macheras 2018)

Two-compartment IV bolus with Caputo fractional derivative — peripheral return as anomalous diffusion

NEW

Model (Eqs. 36a-b):


      dq₁/dt = -(k₁₂ + k₁₀)·q₁(t) + k₂₁,f · ᶜD^1-α[q₂(t)]

      dq₂/dt =  k₁₂·q₁(t)          - k₂₁,f · ᶜD^1-α[q₂(t)]

Inverted from the closed-form Laplace domain (Eqs. 38a-b) via mpmath Talbot NILT.
Reference: Dokoumetzidis, Macheras. J Pharmacokinet Pharmacodyn (2018) 45:107-125, DOI 10.1007/s10928-017-9547-8

Fractional Order

α (Caputo order on q₂) 0.587

Rate Constants

k₁₂ (central → peripheral, 1/day) 2.95

k₁₀ (elimination, 1/day) 0.004

k₂₁,f (fractional return, 1/day^α) 0.48

Dosing & Observation

Dose q₁(0) (mg)

V (L)

t_max (days)

n points

Presets

When α = 1 the model reduces to the classical linear 2-cmt IV bolus.
α < 1 introduces a power-law tail in the peripheral return — a hallmark of anomalous diffusion in deep tissue (Macheras, Fig. 4).

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Macheras 2018 two-compartment fractional IV bolus

Set α, k₁₂, k₁₀, k₂₁,f, dose and volume — then click RUN.
Try the Amiodarona preset to reproduce Fig. 4 of the paper.

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Fractional PBFTPK — Oral Absorption (Macheras 2020 + Caputo)

Zero-order finite input over [0, τ_i], Caputo-fractional elimination (α<1)

NEW

Classical PBFTPK (Macheras 2020, Eqs. 9 & 11):


      0 ≤ t ≤ τ_i :   C_b(t) = D/(τ_i·V_d·k_el) · (1 − exp(−k_el·t))

          t > τ_i  :   C_b(t) = C_b(τ_i) · exp(−k_el·(t − τ_i))

Fractional extension (new):


      ᶜD^α C(t) = u(t)/V_d − k_el,f·C(t),   u(t)=D/τ_i on [0, τ_i]

      C(s) = D/(τ_i·V_d) · (1 − e^−τ_is) / [s · (s^α + k_el,f)]

Inverted via mpmath Talbot NILT (same method as Multi-CMT Fractional).
Reference: Macheras & Chryssafidis, Pharm Res (2020), DOI 10.1007/s11095-020-02894-w. Companion: Dokoumetzidis & Macheras J Pharmacokinet Pharmacodyn (2018) 45:107-125, Eq. 33.

BCS Drug Class

Fractional Order

α (Caputo order on C) 0.75

Kinetic Parameters

k_el (elimination, 1/h^α) 0.97

τ_i (input duration, h) 0.75

Dosing & Observation

Dose D (mg)

V_d (L)

t_max (h)

n points

Presets (Macheras 2020)

α = 1 → classical PBFTPK (Eqs. 9 & 11).
α < 1 → Caputo elimination produces a heavy-tail, non-exponential decay after τ_i.

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Fractional PBFTPK (oral) — Macheras 2020 + Caputo elimination

Pick a BCS class, set α, k_el, τ_i, dose and V_d — then click RUN.
Try a preset: Cephradine, Ibuprofen, Flurbiprofen (Macheras 2020 Table).

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Fractional Hahnfeldt Tumor-Vascular (Can 2026)

Two coupled Caputo fractional ODEs for tumor volume V(t) and carrying capacity K(t), solved by Adams-Bashforth-Moulton

NEW

Temporal Hahnfeldt (Eq. 8 with D_V=D_K=0):


      ᶜD^α V(t) = λ·V(t)·ln(K(t)/V(t)) − d·V(t) − u₁(t)·V(t) − u₂(t)·V(t)

      ᶜD^α K(t) = b·V(t) − d·K(t)

V = tumor volume density, K = vascular carrying capacity, u₁ = immunotherapy, u₂ = chemotherapy.
Solver: Diethelm Adams-Bashforth-Moulton Predictor-Corrector (α∈(0,1]); at α=1 we also match scipy RK45 within 1e-5.
Reference: Can E (2026) Sci Rep 16:12549, DOI 10.1038/s41598-026-41810-x. Note: optimal control (FBSM) not yet implemented — forward simulation only.

Fractional Order

α (Caputo order) 0.70

Kinetic Parameters

λ (Gompertz growth) 0.100

b (angiogenic stim.) 0.100

d (natural decay) 0.050

Initial Conditions & Horizon

V₀

K₀

t_max (d)

n steps

Immunotherapy u₁ schedule

startendintensity

Chemotherapy u₂ schedule

startendintensity

Presets (Can 2026)

α = 1 → classical Hahnfeldt (matches scipy RK45).
α < 1 → Caputo memory produces slower / heavy-tail dynamics.

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Fractional Hahnfeldt tumor-vascular model (Can 2026)

Set α, λ, b, d plus V₀, K₀ and optional u₁/u₂ schedules — then click RUN.
Start with the Can 2026 nominal preset.

🧠

fPINN — Population PK (physics-informed, small data)

Neural net (numpy, no PyTorch) constrained by Caputo-fractional 1-cmt oral PK residual (Grünwald-Letnikov)

RESEARCH PROTOTYPE

Research prototype — not clinically validated. This PINN learns a population-level fractional PK model from sparse multi-patient C(t) data. Results demonstrate algorithmic behaviour, not clinical utility.

Loss:


      L = L_data + λ_phys·L_physics + λ_prior·L_prior

      L_physics = ‖ᶜD^αC(t) + k_el·C(t) − (k_a·D/V)·e^−k_at‖²

Architecture: Feedforward (2 → 32 → 16 → 1), tanh + softplus output, L-BFGS-B optimizer.
Reference: Raissi et al., J. Comput. Phys. 378 (2019) 686-707.

Multi-patient data (CSV: id,time,dose,conc[,weight,age])

Dose column: fill only on first row per patient. weight/age columns are optional — when present the engine activates allometric scaling + individual η random effects (NONMEM-style).

Use covariates (WT/AGE) if present — allometric kel~WT^0.75, V~WT^1.0 + η ~ N(0, ω²)

Fractional order prior

α (Caputo) prior 0.90

Loss weights

λ_physics

λ_prior

iterations

seed

Presets

α = 1 → integer-order classical PK baseline.
α < 1 → Caputo memory kernel; helps in sparse or heavy-tail regimes.

🧠

Physics-informed PopPK for small / sparse data

Paste multi-patient (id,time,dose,conc) CSV or load the Theophylline preset, then click RUN.
Try Small data to see the few-shot regime (3 patients).

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Mechanistic Basis of α

D^dT/dt^d = ∇²_cyl T → C(t) → α_fit ≈ d

VALIDATED

Fractional order d 0.90

Tissue radius R₂ 0.20

Precision medium

Ready. Select parameters and run.

d (PDE)

—

α_fit (ML)

—

Error |α−d|

—

R² fit

—

📋 Upload Patient C(t) Data

Upload TIME,CONC data to estimate α from your patient and compare with PDE geometry prediction.

Upload data or load demo to estimate α from patient C(t)

Tissue → α Prediction (all routes)

Find d* such that α_fit(PDE, d*) ≈ α_clinical for Niclosamide IM, Diazepam IV, Abacavir oral.

Step 4 — R₂ variability → inter-individual variability in α

Fixed d* per tissue (from Step 3). Varying tissue thickness R₂ across the biologically plausible range. Shows that variability in α between patients comes from variability in tissue geometry.

R₂ points

Step 5 — Covariates (Weight) → R₂ → α predicted

Validates the full mechanistic chain: patient weight → allometric R₂ → PDE-predicted α. Uses Abacavir pediatric data (169 children, Zhao 2012).

Precision

Theory

The fractional order α estimated by FractaLPK from clinical PK data is not empirical — it corresponds to the fractional order d of the diffusion PDE governing drug transport in tissue.

PDE: ∂ᵈT/∂tᵈ = (1/r)∂T/∂r + ∂²T/∂r²   (Caputo, cylindrical coords)
BC:  T(R₁,t) = 1 (drug source)  |  ∂T/∂r(R₂,t) = 0 (impermeable boundary)

Collapsing: C(t) = 2/(R₂²-R₁²) ∫ T(t,r)·r dr
Fitting:    C(t) = A · [1 - Eα(-(k·t)α)]

Result: α_fit ≈ d with error < 5%. This validates that FractaLPK recovers the true tissue geometry parameter from observational C(t) data alone.

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IVIVC — Fractional GI Transit Correlation

In Vitro / In Vivo Correlation — Modified-release formulations require non-integer GI transit modeling for accurate Level A correlation.

Formulation Settings

Formulation

CLINICAL ADVANTAGE

Modified-release tablets dissolve through a GI transit chain with non-integer compartments.

Classical Model: Must round n to integer → poor IVIVC Level A.
FractaLPK: Fits n ∈ ℝ exactly → better correlation → FDA accepts Level A.

This is a real regulatory advantage for 505(b)(2) and ANDA submissions.

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In Vitro / In Vivo Correlation for Modified-Release

Fractional GI transit (n ∈ ℝ) gives tighter Level A correlation than integer-compartment models.
Demo uses synthetic data based on published formulation profiles. Upload real dissolution + PK data for actual analysis.

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First-In-Human Dose Prediction — Fractional Allometry

Animal → Human PK scaling with n_frac as species-specific tissue architecture parameter. Standard allometry cannot capture this.

Compound Selection

Compound

FRACTIONAL ALLOMETRY

Standard: CL = a · BW^b, n rounded to integer.
FractaLPK: n_frac scales with BW too — captures how GI transit complexity increases with body size.

Mouse n≈2 → Dog n≈3.5 → Human n≈4+

This extra dimension improves Cmax and AUC prediction for FIH dose selection.

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From Animal PK to Human Dose — with Fractional Tissue Architecture

Standard allometry uses CL = a·BW^b with integer compartments. FractaLPK adds n_frac scaling to capture species-specific GI transit complexity.
Demo uses published PK parameters. Results are exploratory — not a substitute for formal nonclinical PK assessment.

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D-Optimal Sampling — Fractional PK Design

Optimal PK sampling times change when n is a free parameter. Neither PFIM nor PopED support fractional compartments.

Design Settings

Number of samples

n (fractional)

ka (h-1)

ke (h-1)

V (L)

WHY THIS MATTERS

When the fractional order is a free parameter, optimal blood sampling times shift to capture absorption variability.

This means fewer PK samples can achieve the same statistical power — reducing patient burden and trial costs.

Standard optimal design tools (PFIM / PopED) cannot account for fractional compartments.

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Optimal Sampling Design for Fractional PK Models

When n is free, optimal blood sampling times shift. Compute the D-optimal design that no other tool can calculate.
Optimal sampling times account for fractional compartment sensitivity — unique to FractaLPK.

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Fractional Drug Release — Anomalous Diffusion

Anomalous release kinetics · Multilayer fractional · pH-dependent dissolution · Burst+Sustained

Release Parameters

Model

alpha

tau (h)

WHY FRACTIONAL: Real drug release follows anomalous diffusion that cannot be captured by integer-order models. Fractional exponents match observed release profiles more accurately, especially for multilayer and matrix formulations.

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Anomalous Drug Release via Fractional Calculus

5 models: Fick, Erosion-Diffusion, Multilayer, Burst+Sustained, pH-Dependent

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Adaptive Precision Dosing — dC/dn Sensitivity

Vancomycin · Tacrolimus · Aminoglycosides · Warfarin · Methotrexate

Dosing Parameters

Drug

Obs Level

Obs Time (h)

Current Dose (mg)

FRACTIONAL SENSITIVITY: How does plasma concentration change when the absorption pathway varies between patients? This sensitivity metric enables precision dose adjustments based on individual GI transit characteristics.

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Precision Dosing with Fractional Sensitivity dC/dn

Select a drug and enter observed TDM levels for dose recommendation

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Internal Decision Report Generator

Auto-generate PK Summary · Exposure-Response · Dosing Justification · Model Comparison

Report Parameters

Drug Name

ka

ke

n (frac)

Dose

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Regulatory Submission Package Generator

Section 2.7.2 Clinical Pharmacology · Exposure-Response · Dosing Justification · Model Comparison

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Non-Animal PK Prediction — In-Vitro to Human

Caco-2 Permeability · Microsomal CLint · QSPR for n_frac · 6 Reference Compounds

Compound Properties

MW

logP

Caco-2 Papp (x10-6)

CLint (uL/min/mg)

fu (plasma)

Dose (mg)

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Predict Human PK from In-Vitro Data Only

n_frac predicted from Caco-2, logP, MW via QSPR. Correlation R=0.91 vs known human data.

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Health Economics — Cost-Effectiveness with Fractional PK

Markov CEA · Budget Impact · Value-Based Pricing · 4 Reference Drugs

HEOR Parameters

Drug

Analysis

Horizon (yr)

IMPACT: Fractional PK changes ICER by ~8% on average. This can flip cost-effectiveness decisions at WTP thresholds ($50K-$100K/QALY). Precision dosing saves an average of 3.8% per patient.

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Pharmacoeconomics Powered by Fractional PK Precision

When exposure prediction is more accurate, ICER calculations change. Average 8% ICER difference vs integer models.

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Fractional Trial Design — Sample Size & Dose Optimization

Sample Size · Dose Arms · Bioequivalence · Pediatric Extrapolation

Trial Parameters

Analysis Type

n (frac)

Effect Size

ka

ke

SAVINGS: Fractional PK models reduce unexplained variability, leading to 8.5% smaller trials on average. For antibiotics (n=5.7), reduction reaches 15.5% — saving ~$1.1M per trial.

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Design Better Trials with Fractional PK Precision

Reduce sample size, optimize dose arms, design BE studies, extrapolate to pediatrics

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Fractional Weibull Survival — PK-Linked CCF

GBM · NSCLC · Breast · CRC · Melanoma · Cure Fraction · dS/dn Sensitivity

Tumor & Patient

Tumor Type

Patient n

Dose (mg)

KEY INSIGHT: Integer PK rounding biases AUC, which propagates to survival prediction. A 0.4% AUC bias can shift median OS by weeks. dS/dn quantifies this — impossible in Classical Model/Standard Interpolation.

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PK-Linked Weibull Survival with Clinical Cure Fraction

Integer PK bias propagates to survival curves. Fractional modeling eliminates this bias. See the difference.

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PLGA Profile — Release-kinetics Classifier

FractaLPK ML stretched exponential vs Weibull, Higuchi, Korsmeyer-Peppas. AIC ranking + PDF report. No interpretation.

INPUT — release profile

Times (hours) — comma or whitespace

Cumulative released fraction [0..1]

Fit burst term (ML & Weibull)

No interpretation. This module classifies which structural release model fits best by AIC. Pharmacological and formulation decisions are the responsibility of the client's expert team.

Load the demo dataset or paste your own profile, then click FIT 4 MODELS.
Models: FractaLPK ML stretched exp · Weibull · Higuchi · Korsmeyer-Peppas. Best by AIC wins.

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Bioequivalence Analysis — Cp5 AUC Engine

FDA 21 CFR 320 · EMA/CHMP/EWP/QWP/1401/98 · Westlake 90% CI · Bootstrap 2000

Upload PK Data

Reference CSV

Drop CSV or click to browse

Test CSV

Drop CSV or click to browse

Precision

Format: CSV with time, concentration columns

ENGINE: AUC computed via Cp5 dyadic correction — NOT trapz/Simpson. Independent 1D integrals for AUC_ref and AUC_test. Bootstrap parametric CI + Westlake symmetric 90% interval.

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Bioequivalence Analysis via Cp5 Dyadic AUC

Upload Reference and Test PK profiles (CSV). The engine fits 1-CMT oral models, computes AUC via Cp5 dyadic correction (not trapezoidal), and evaluates 80-125% bioequivalence criteria with Westlake 90% CI.

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Dosing Simulation — Monte Carlo vs Cp5 Exact

Cp5 exact integration vs Monte Carlo approximation | P(Ctrough > target) | IIV on CL, V

Auto-Fit from Data

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Upload CSV (time, concentration)

Drag & drop or click | Auto-fits 1-CMT model

Dose (mg): (0 = auto-estimate)

PK Parameters & IIV

CL (L/h)

V (L)

ka (1/h)

Dose (mg)

Tau (h)

omega CL

omega V

Target Ctrough (mg/L)

MC Simulations

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Monte Carlo Simulation

Stochastic — result varies between runs

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Cp5 Exact Integration

Deterministic — same result every run

∫

AUC Multidose — Closed-Form Fractional

Exact AUC(0,T) via 2-parameter Mittag-Leffler. No ODE solver, no quadrature. Identity: ∫₀ˢ E_α(−k·uᵃ) du = s · E_α,2(−k·sᵃ)

Regimen:

Bolus multidose Continuous IV infusion

PK parameters & regimen

Dose D (mg)

V (L)

ke (1/hᵃ)

α (frac order)

τ (h)

N doses

T (h)

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CEPARAPA 2D — Fractional Engine

Proprietary fractional 2D engine | n₁,n₂ ∈ ℝ | PK-Tumor diffusion coupled

PK-Tumor Mode

Tumor

n_frac

r_max (cm)

nn (accel. depth)

Generic f(x,y)

f(x,y) =

a

x

c

y

CEPARAPA 2D engine:
Evaluates F(x,y) with both limits fractional simultaneously (n₁,n₂ ∈ ℝ) using a proprietary fractional 2D extension.

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CEPARAPA 2D — Proprietary Fractional Engine

Simultaneous fractional PK absorption and tumor diffusion modeling.
Captures anomalous drug penetration in solid tumors.

Default: GBM, n_frac=0.694 (subdiffusion)

Fractional Dosing Simulation


          C(t) = Σ (F·D/V) · E_α(-(CL/V)·(t-t_n)^α)

Part A — Population simulation (QD vs BID vs TID)

Simulates C(t) for multiple patients using their individual α (from weight via allometric model). Compares fractional vs classical model. Shows Time In Range (TIR) for each regimen.

Patients

Days

Part B — Individual optimal regimen

Find optimal dose + interval for a specific patient (α, weight). Maximizes Time In Range using differential evolution.

α (fractional order) 0.80

Weight (kg) 20 kg

Simulation days

Amiodarone IV — Classic Anomalous Kinetics


          Haffajee et al. 1983 / Dokoumetzidis 2009 — 3-model comparison

Amiodarone IV (400 mg bolus) exhibits heavy power-law tails that classical mono-exponential models cannot capture. This demo compares three models: mono-exponential (2 params), bicompartimental (4 params), and FractaLPK fractional (3 params, Mittag-Leffler). The fractional model captures the anomalous tail with fewer parameters.

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CT/MRI → D → n_frac → Dosing (THEORETICAL)

Box-counting + Lacunarity + Perimeter-Area analysis · D → n via anomalous diffusion physics

Imaging Analysis

Tumor morphology (target D)

📁 Upload TCGA Clinical TSV

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Drag TSV from GDC portal

📷 Upload CT/MRI Image

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Click or drag PNG/JPG

D → n_frac formula:
n = 2/(D+1) [O'Shaughnessy-Procaccia]
D=1.0 → n=1.00 (normal diffusion)
D=1.5 → n=0.80 (subdiffusion)
D=2.0 → n=0.67 (strong subdiffusion)

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From Tumor Image to Personalized Dosing

Select tumor morphology and click ANALYZE to see how fractal dimension determines the optimal drug schedule.
Future: Upload real CT/MRI DICOM scans for automatic analysis

Clinical Evidence — When Classical Models Fail

Published datasets · FractaLPK vs NONMEM/Classical comparison

Automatic Model Router

CSV → Diagnostics → Model Selection → Fractional ODE → Fit → Validation

Model

M1

α

0.603

β

—

R²

0.991

ΔOFV

58

Router decided: M1 — 1-CMT Fractional Disposition
Reason: IV bolus + adipose tissue → single-phase power-law disposition
Geometric validation: α=0.603 vs d*=0.581 → 3.6% error ✓

Power-law kinetics confirmed. α=0.603 from adipose tissue geometry (d*=0.581, error=3.6%).

Model Selection Decision Tree

        Route = IV?

          └ Active metabolite? → M4 (non-commensurate α≠β)

          └ Two phases (log-log)? → M3 (2-CMT commensurate)

          └ Single phase → M1 (1-CMT fractional)

        Route = SC?

          └ Absorption detected? → M5 (SC fractional α≠β)

        Route = IM/Oral?

          └ → M2 (fractional absorption)

        Saturable PK?

          └ → M6 (Michaelis-Menten fractional)

FractaLPK Engine

PopPK Auto-Diagnose

Oncology Suite

Formulation Suite

PBFTPK1 — First-Order Truncated Absorption

Parameters

Bioequivalence under the F.A.T. Concept

Test formulation (T)

Reference formulation (R)

Shared

Monoclonal Antibody PK — Fractional vs Classical Model

Precision Dosing — Individualized mAb Schedules

Fractional Tumor-Immune Dynamics (RPSM + Cp5)

Fractional NCA — Non-Compartmental Analysis

PBFTPK 4-Model Comparison

3-CMT Fractional Mammillary Model

Multi-Compartmental Fractional (Macheras 2018)

Fractional PBFTPK — Oral Absorption (Macheras 2020 + Caputo)

Fractional Hahnfeldt Tumor-Vascular (Can 2026)

fPINN — Population PK (physics-informed, small data)

Mechanistic Basis of α

Step 4 — R₂ variability → inter-individual variability in α

Step 5 — Covariates (Weight) → R₂ → α predicted

IVIVC — Fractional GI Transit Correlation

First-In-Human Dose Prediction — Fractional Allometry

D-Optimal Sampling — Fractional PK Design

Fractional Drug Release — Anomalous Diffusion

Adaptive Precision Dosing — dC/dn Sensitivity

Internal Decision Report Generator

Non-Animal PK Prediction — In-Vitro to Human

Health Economics — Cost-Effectiveness with Fractional PK

Fractional Trial Design — Sample Size & Dose Optimization

Fractional Weibull Survival — PK-Linked CCF

Bioequivalence Analysis — Cp5 AUC Engine

Dosing Simulation — Monte Carlo vs Cp5 Exact

AUC Multidose — Closed-Form Fractional

CEPARAPA 2D — Fractional Engine

Fractional Dosing Simulation

Part A — Population simulation (QD vs BID vs TID)

Part B — Individual optimal regimen

Amiodarone IV — Classic Anomalous Kinetics

Model Comparison Table

CT/MRI → D → n_frac → Dosing (THEORETICAL)

Transit Compartment Model (TCM)

Fractional Mammillary 2D Model

Clinical Evidence — When Classical Models Fail

Automatic Model Router

Model Selection Decision Tree

PBFTPK₁ — First-Order Truncated Absorption