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How to use
Choose a preset or tune particle size, pore size, tortuosity, uptake kinetics and dosing. The colour map shows concentration relative to the medium dose C0; contour rings mark 50%, 25% and 10%. Watch the front advance into the spheroid, then compare the profile with the analytic steady state in the first graph and the Thiele modulus graph. Space pauses, R restarts. Switch to a bolus to see the concentration wash out of the rim before it reaches the core.
The science
A multicellular tumour spheroid is a standard 3D culture model: drugs and nanoparticles must diffuse in from the medium while cells take them up, so the core can be shielded. With spherical symmetry and uptake rate q(C) the concentration obeys
dC/dt = D (d2C/dr2 + (2/r) dC/dr) - q(C), C(R, t) = Cs(t), dC/dr = 0 at r = 0
With the substitution u = r C this becomes a 1D equation, solved here by implicit finite differences (120 nodes, 2 s step; for the saturable case the rate coefficient is taken from the previous step, a semi-implicit scheme). For first-order uptake q = k C and a constant surface concentration the steady state is exact:
C(r) / Cs = (R/r) sinh(phi r / R) / sinh(phi), phi = R sqrt(k / D) (Thiele modulus)
The centre then sees C(0)/Cs = phi / sinh(phi): about 0.55 for phi = 2 and 0.005 for phi = 8. For large phi the drug only penetrates about sqrt(D/k) from the surface. Diffusivity comes from Stokes-Einstein in water, D0 = kT / (3 pi eta d) (310 K, eta = 0.69 mPa s), reduced by the Renkin hindrance H(lambda) = (1-lambda)^2 (1 - 2.104 lambda + 2.09 lambda^3 - 0.95 lambda^5) for a sphere of diameter d in a pore of diameter d/lambda, and by a tortuosity factor 1/theta^2. Cell kill is modelled as survival S(r,t) = exp(-kappa integral max(C - Ct, 0) dt).
Checks. (1) The numerical steady profile matches the sinh profile (relative L2 error below 1%). (2) The centre/surface ratio at phi = 2 matches phi/sinh(phi). (3) With no uptake the centre concentration at t = 0.12 R2/D matches the exact series 1 + 2 sum (-1)^n exp(-n2 pi2 D t / R2). (4) Mass balance: drug content plus cumulative uptake equals the integrated flux through the surface (within 3%, limited by first-order time stepping). (5) For phi = 30 the measured depth to Cs/e matches sqrt(D/k). (6) Stokes-Einstein scaling D ~ 1/d.
Honest limits. The model has no binding or unbinding, no cell-cycle or hypoxia effects, no necrotic core or interstitial pressure, no flow and a single uniform tissue. Real spheroid penetration also depends on binding affinity (the binding-site barrier), charge and protein corona, so the numbers are indicative scalings, not predictions for a specific drug. The kill model is a simple threshold dose-response with an arbitrary coefficient. The particles drawn in the medium are illustrative only.