Two different biological processes – one that creates new tumour cells and one that redistributes the cells that already exist – are wrapped up in the words “proliferation” and “diffusion”. Mathematically they are represented by two different terms of the reaction-diffusion equation that is often used to model solid‐tumour growth. Keeping the biology and the mathematics separate helps to see why they are not the same thing.

1. What is meant by “cell proliferation”?
   • Biology: Proliferation is mitotic division minus cell loss (apoptosis, necrosis, shedding). If the net balance is positive the number of cells, the total mass and, eventually, the macroscopic volume of the tumour all increase.
   • Units: ρ (rho) in the literature, with units 1 / time (e.g. day⁻¹).
   • Laboratory proxies: BrdU uptake, Ki-67 index, PHH3 staining, mitotic index, later MRI-based “cell proliferation” maps, etc.
   • Mathematical term (“reaction” term) in the equation

   ∂c/∂t = … + ρ c (1 – c/K)
   

   where c(x,t) is the cell density and K is the local carrying capacity.

2. What is meant by “cell diffusion”?
   • Biology: Individual tumour cells migrate, crawl along fibres, slip between host cells, or follow chemokine gradients. When you view a dense population at a coarse spatial scale these random, undirected motions look like Fickian diffusion. No new cells are produced; the existing ones simply change places.
   • Units: D, with units length² / time (e.g. mm² year⁻¹).
   • Laboratory proxies: In vitro scratch-wound assays, time-lapse microscopy of random walks, diffusion‐weighted MRI estimating “apparent diffusion coefficient (ADC)” of water as a surrogate, etc.
   • Mathematical term

   ∂c/∂t = ∇·(D ∇c) + …
   

   If the boundary is closed (no flux), ∫_Ω ∇·(D ∇c) dV = 0, so the total number of cells is conserved; only their spatial distribution changes.

3. How the two processes interact in a tumour.
   • Proliferation alone (ρ > 0, D = 0): The region occupied by the tumour becomes denser and denser until it reaches the carrying capacity K, but its physical footprint does not expand.
   • Diffusion alone (D > 0, ρ = 0): Cells spread out; the occupied region grows, but the integrated cell number stays the same. Density necessarily falls in most places.
   • Both present (ρ > 0 and D > 0): This is the realistic case for invasive solid tumours. New cells are produced locally and, at the same time, migration smooths the spatial gradients. The coupled effect gives rise to travelling-wave solutions whose speed is approximately

   v ≈ 2 √(ρ D)
   

   (Fisher–Kolmogorov result). In other words, proliferation determines how many new cells appear, diffusion determines how fast the leading edge moves, and the two together set how rapidly the tumour expands through tissue.

4. Why a daughter cell does add mass.
   You were right that a cell can divide before either daughter has regained the size of the original mother, so immediately after cytokinesis the summed mass is roughly conserved. But each daughter then grows to full size before its next division. In tumours with positive net proliferation, this growth phase out-paces cell loss, so total biomass increases.

5. In the paper you cite.
   Wang et al. fit patient MRI data to a reaction-diffusion model of glioblastoma.
   • D estimates the glioma cells’ random motility through brain parenchyma.
   • ρ estimates the net mitotic rate.
   • Having both parameters lets them forecast spatial spread (controlled largely by D) and volumetric growth (controlled largely by ρ) simultaneously.

Key points to remember.
• Proliferation changes the number (and therefore mass/volume) of tumour cells; diffusion redistributes those cells in space.
• A positive diffusion coefficient does not, by itself, cause tumour growth; only positive net proliferation does that.
• In practical modelling, both parameters are needed because tumours both grow and infiltrate surrounding tissue.