Cp5 Pharma — Fractional Engine 2D

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

⚠

WEB DEMO — LIMITED COMPUTATION

This is a free-tier demo with a 30-second server timeout. FOCE-I runs with reduced iterations (10 outer / 12 inner) and datasets are capped at 12 subjects. Results are clinically valid but not research-grade precision. For full-precision analysis with large datasets, use the local Python package: pip install fractalpk

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

—

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

🧬

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.

💉

Trastuzumab

Anti-HER2 · Breast Cancer

💉

Bevacizumab

Anti-VEGF · Colorectal

💉

Nivolumab

Anti-PD-1 · Melanoma

💉

Adalimumab

Anti-TNFα · Autoimmune

💉

Pembrolizumab

Anti-PD-1 · NSCLC

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

💊

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

💊

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

📈

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).

📈

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.

🔬

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.

🔬

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.

🔬

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.

💊

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.

💊

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.

🧪

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.

🧪

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.

📐

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.

📐

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.

💊

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.

💊

Anomalous Drug Release via Fractional Calculus

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

🎯

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.

🎯

Precision Dosing with Fractional Sensitivity dC/dn

Select a drug and enter observed TDM levels for dose recommendation

📋

Regulatory Report Generator — FDA/EMA Submission

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

Report Parameters

Drug Name

ka

ke

n (frac)

Dose

📋

Regulatory Submission Package Generator

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

🐾

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)

🐾

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.

💰

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.

💰

Pharmacoeconomics Powered by Fractional PK Precision

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

📐

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.

📐

Design Better Trials with Fractional PK Precision

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

📈

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.

📈

PK-Linked Weibull Survival with Clinical Cure Fraction

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

⚖️

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.

⚖️

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.

🎲

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

📂

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

🎲

Monte Carlo Simulation

Stochastic — result varies between runs

⚡

Cp5 Exact Integration

Deterministic — same result every run

🔢

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.

🔢

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.

📡

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

📁

Drag TSV from GDC portal

📷 Upload CT/MRI Image

📷

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)

📡

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 Dashboard

Validated Modules

Research Modules

Synthetic / Demo

Monoclonal Antibody PK — Fractional vs Classical Model

Precision Dosing — Individualized mAb Schedules

Fractional NCA — Non-Compartmental Analysis

PBFTPK 4-Model Comparison

3-CMT Fractional Mammillary Model

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

Regulatory Report Generator — FDA/EMA Submission

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

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