Short answer

• Proliferation is the net rate at which tumor cells are created by division (mitosis) minus the rate they are lost (death). It increases total cell number and, under normal biology, total tumor mass.
• Diffusion (in this context) is a mathematical way to describe random cell motility that causes cells to spread from regions of high to low density. It redistributes cells in space but does not create or destroy them.

Details and how they relate to “tumor growth”

• Proliferation

  • Biology: Cells grow (synthesize biomass) and then divide; they do not just “subdivide” existing mass. So, unless exactly balanced by cell death, proliferation increases total mass and cell number.
  • Modeling: Appears as a local “reaction” term, often logistic: ρ c (1 − c/K), where c is cell density, ρ is the proliferation rate (units 1/time), and K is the carrying capacity (maximum packing density). Doubling time ≈ ln(2)/ρ.

• Diffusion (invasion/motility)

  • Biology: Captures undirected, random motility of cells (not molecular diffusion). In brain tumor models, D can be anisotropic (cells move more easily along white matter tracts).
  • Modeling: Appears as a “diffusion” term ∇·(D∇c), where D has units of length^2/time. This term spreads cells out. With no sources/sinks and no-flux (or whole-space) boundaries, it conserves the total number of cells: it changes spatial distribution, not the total.

• Key implications

  • “Positive diffusion” is not equivalent to tumor growth (in total mass). It increases the spatial extent and lowers local density; total number of cells remains the same unless proliferation (or death) is present.
  • Proliferation drives increases in total tumor burden; diffusion drives how infiltrative versus compact the tumor appears.
  • In the classic reaction–diffusion (Fisher–KPP) model, ∂c/∂t = ∇·(D∇c) + ρ c (1 − c/K), the tumor’s invasion front has a characteristic speed v ≈ 2√(Dρ) (for uniform D and logistic growth). Thus:
    • Larger ρ (faster proliferation) and/or larger D (greater motility) both increase how fast the visible radius expands.
    • The “diffuseness” of the margin scales with √(D/ρ): high D/ρ yields a more infiltrative/blurred edge; low D/ρ yields a more nodular/compact tumor.

• Practical interpretation

  • Anti-mitotic therapies primarily lower ρ (slower net growth).
  • Anti-migratory therapies would lower D (less spread).
  • Imaging-based parameter estimation tries to infer D and ρ so the model reproduces the observed pattern and pace of expansion.

So, proliferation increases the number of tumor cells (and mass) locally, while diffusion spreads those cells through surrounding tissue. Clinically observed “tumor growth” (e.g., increasing diameter on imaging) is typically the combined effect: proliferation supplies cells; diffusion/invasion distributes them.