INITIALISING SIMULATION ENGINE
PDEOutreach ↗
v0.6
PDEOncology

A browser-based platform for simulating tumor drug penetration using reaction-diffusion-convection PDEs. v0.6 added IFP convection, experimental validation, and parameter sweep. v0.6 adds lymphatic sink, vascular normalisation window, metronomic/MTD dosing, and retardation factor r_f.

∂C/∂t = ∇·(D(x,y)∇C) − v·∇C − λC − k·ρ(x,y)·C − λ_L·δ∂Ω·C

v0.6: convection term v·∇C from IFP-driven interstitial flow (Jain 1987). v0.6 adds lymphatic sink, vascular normalisation window, metronomic/MTD dosing, and retardation factor r_f — features conceptually motivated by Nikmaneshi, Jain & Munn (PLoS Comput Biol 2023 ↗). Validated against Thurber et al. 2008 →

FDM · FTCS + upwind 80 × 80 grid IFP convection validated vs spheroid data parameter sweep Python export LaTeX snippet Claude API lymphatic sink vascular normalisation metronomic dosing

What's new in v0.6

⊘

Lymphatic Sink

Boundary-localised extra degradation term λ_L·δ∂Ω·C at the tumour periphery ring. Models dysfunctional lymphatic drainage that enforces a low-concentration periphery. Tunable λ_L slider in Simulation tab.

↷

Vascular Normalisation Window

Time-windowed IFP reduction modelling anti-VEGF–induced normalisation (Jain 2001). Configurable window start, width, and IFP reduction depth. Conceptually motivated by Nikmaneshi, Jain & Munn (PLoS Comput Biol 2023).

⊛

Metronomic / MTD Dosing + r_f

Five boundary PK profiles including low-dose metronomic cycles and MTD bolus schedules. Retardation factor r_f (Nikmaneshi et al. 2023) scales convective transport for large MW drugs and nanoparticles.

IFP model — scientific basis

In solid tumors, elevated interstitial fluid pressure (IFP) creates an outward radial fluid flow that opposes convective drug delivery. This is modelled using Darcy's law for a homogeneous spherical tumor (Jain 1987):

v(r) = −κ/μ · dP/dr · r̂ ≈ ifpMag · (r/R) · r̂   [inside tumor, r ≤ R]

Where κ is hydraulic conductivity, μ fluid viscosity, and R the tumor radius. The linear radial profile (v ∝ r) is the standard result for uniform interstitial conductivity (Jain 1987, Stylianopoulos et al. 2012). This opposes drug diffusion toward the core — explaining why high-IFP tumors (pancreatic, TNBC) are especially resistant to chemotherapy.

The convection term uses an upwind finite difference scheme for numerical stability. CFL condition extended to: dt ≤ min(dx²/4D, dx/max|v|).

References

Link
Jain RK (1987)
Nugent LJ & Jain RK (1984)
Thurber GM et al. (2008)
Chauhan VP et al. (2011)
Tannock IF et al. (2002)
Minchinton AI & Tannock IF (2006)
Kyle AH et al. (2007)
Stylianopoulos T et al. (2012)
Nikmaneshi MR, Jain RK, Munn LL (2023)
Parameter Guide & Units

Biophysical interpretation of each slider (normalised ranges) and the cited literature links.

SymbolMeaningNormalised RangeReal-World Units / Typical ValueLiterature
D Diffusion coefficient 0.005–0.25 ~10-8 – 10-6 cm2/s (ECM effective) Jain RK (1987)
λ Degradation rate 0.001–0.05 1/time (metabolic clearance; model-scaled; typical half-life ~hours) Chauhan et al. (2011)
k Cellular uptake rate 0.01–0.15 Uptake/retention (receptor-dependent; model-scaled; timescale ~hours) Thurber et al. (2008)
r Tumor radius 10–38 px ≈0.125–0.475 cm on the 1 cm tissue section Tannock et al. (2002)
v₀ IFP magnitude (IFP-driven convection) 0.00–0.15 Scaled from typical tumor IFP 5–30 mmHg via Darcy’s law Stylianopoulos et al. (2012)
n Time steps 100–800 Numerical integration length (explicit FDM resolution) Nugent & Jain (1984)
⚙

Parameters

Boundary C = 1 held constant throughout simulation.
D · diffusion0.08
D (diffusion coefficient): 0.005–0.25 (normalised vs free diffusion in water ≈ 10−5 cm2/s). Small molecules (e.g. doxorubicin) ≈ 0.08; large antibodies ≈ 0.01.
λ · decay0.005
λ (degradation rate): 0.001–0.05 (normalised 1/time). Reflects metabolic clearance.
k · uptake0.06
k (cellular uptake rate): 0.01–0.15 (normalised). Depends on receptor/transporter expression.
r · radius22 px
Tumor radius: 10–38 px (≈ 0.125–0.475 cm on the 1 cm tissue section).
n · steps400
Steps: 100–800 (time steps; higher = more accurate but slower).
OFF Jain 1987 · Darcy's law
IFP magnitude 0.00
IFP magnitude: 0.00–0.15 (dimensionless; scaled from typical tumour IFP 5–30 mmHg via Darcy's law). Outward flow opposes inward drug diffusion.
r_f retardation 1.00
r_f: ratio of solute to interstitial fluid velocity (Nikmaneshi, Jain & Munn 2023, Eq. 1). Small molecules ≈ 0.8–1.0; antibodies ≈ 0.1–0.3. Reduces effective convective transport for large MW drugs.

IFP = 0 → standard diffusion. IFP > 0 → outward radial flow opposing drug.

OFF Jain — boundary drainage
λ_lymph 0.00
λ_lymph: first-order degradation rate at tumour boundary ring (±2 px). Values 5–15 produce visible steepening; 20–30 enforces near-zero concentration at periphery. Represents dysfunctional lymphatic drainage (Jain et al. 2007).

Dysfunctional tumour lymphatics are unable to drain interstitial fluid, contributing to elevated IFP and a low-concentration boundary condition at the tumour periphery (Jain et al. 2007).

▸ Advanced Biophysics Features vascular norm · receptor · particle size · 2-compartment
requires IFP > 0
k_eff = k × 1.0×
Expression φ 1.0×
k_eff = k_base × φ. 1× = baseline. Overexpression (HER2+) → 2–3×. Low → 0.1–0.5×. Key for trastuzumab, ADCs, targeted nanoparticles.
Stokes–Einstein
Radius (nm) 0.5
D_eff (Stokes–Einstein): 0.0800 — auto-syncs D slider
D = k_BT/(6πηr), T=310K, η=0.7 mPa·s. Upper-bound estimate — tissue D is 5–20× lower for NPs.

Models encapsulated depot diffusing slowly, releasing active drug at rate κ. Relevant for Abraxane, liposomal formulations.

ready — press Run
1.00.50.0
Tumor avg C
—
Max C
—
50% depth
—
px
Coverage
—
%
📈

Radial penetration curve

🔬

Image → ρ(x,y) field

Experimental / Patient-specific

Upload a grayscale histology slice (PNG/JPG) or MRI slices exported from DICOM viewers (PNG/JPG). For MRI: T1-post generates the contrast-enhancement map (thresholded), and T2/FLAIR generates the oedema/tumor mask (thresholded). The app then auto-converts to a spatially-varying cell density field ρ(x,y) using the same blur + threshold pipeline.

Click or drag to upload
PNG / JPG · grayscale preferred · any size
Threshold (histo / T2 mask)128
Blur σ2
histo preview
ρ(x,y) field

AI Drug Input

Describe a drug and tumor in plain English. Claude extracts biophysically grounded PDE parameters and loads them into the simulator.

🔑

API key

Session-only — sent directly to api.anthropic.com via CORS proxy. Never logged.

◈

Describe your simulation

Claude response
Enter a description above and click Ask Claude.

Drug Compare

Simulate two drugs on the same tumor side-by-side. Compare penetration depth, coverage, and radial concentration profiles.

◉

Shared tumor settings

Note: Compare uses D, λ, k only. IFP convection, receptor expression, and two-compartment nanoparticle release are not applied here — results will differ from Simulation tab for NP formulations. For those drugs, run separate simulations and compare via CSV export.

A

Drug A

D · diffusion0.08
λ · decay0.005
k · uptake0.06
B

Drug B

D · diffusion0.06
λ · decay0.004
k · uptake0.05

Model Validation

Radial penetration curves generated by PDEOncology are compared against approximately digitized experimental data from peer-reviewed spheroid studies. Experimental points extracted from published figures using WebPlotDigitizer methodology (±5–10% digitization error). RMSE computed in normalised concentration units.

Disclaimer: Experimental data points are approximate digitizations from published figures. Quantitative agreement is therefore semi-quantitative. The comparison validates that model dynamics (penetration shape, depth, and drug-class ordering) are physically consistent with published spheroid measurements. Full quantitative validation would require digitization of raw tabular data directly from authors.

⬆

Custom Dataset — CSV Upload

Researcher data

Upload your own radial penetration data. Two columns: distance, concentration (header optional). Distance and concentration are auto-normalised. Tune D, λ, k in the Simulation tab to match your formulation, then run validation to compute RMSE.

Click or drag CSV / TXT
Two columns: distance, concentration
CSV preview

Select dataset

Parameter Sweep

Systematic exploration of the parameter space. Run a 5×5 grid of D (diffusion) vs k (uptake) and visualise how tumor coverage, mean concentration, and penetration depth respond. Identifies which parameter dominates in each tumor type — directly supports a sensitivity analysis section in your paper.

⚙

Sweep configuration

to (5 steps)
to (5 steps)

Results & Report

Run a simulation first, then return here for analytics, manuscript exports, and reproducible code.

No simulation data yet.

Drug Database

Built-in parameter library. Click any row to load into the simulator.

DrugTumorDλkrDifficultyClass

About these parameters

D scaled relative to free diffusion in water. Uptake rates reflect receptor/transporter expression. Degradation accounts for metabolic clearance.

Link
Jain RK (1987)
Nugent LJ & Jain RK (1984)
Thurber GM et al. (2008)
Chauhan VP et al. (2011)
Tannock IF et al. (2002)
Minchinton AI & Tannock IF (2006)
Kyle AH et al. (2007)
Stylianopoulos T et al. (2012)