Here is how “proliferation” and “diffusion” differ in most reaction–diffusion models of tumor growth, and how each feeds into overall expansion of the tumor:

1. Proliferation (“reaction” term)
   – What it is
   • The net rate at which cells are born minus die.
   • In simple models you get ∂c/∂t|reaction = ρ·c (exponential) or ρ·c·(1–c/K) (logistic, with carrying capacity K).
   – Units
   • ρ is in [time⁻¹], e.g. per day.
   – Biological meaning
   • Drives local increase in cell number and mass.
   • Measured (in vitro or in vivo) by doubling time, Ki-67 index, BrdU labeling, etc.

2. Diffusion (“spatial” or “motility” term)
   – What it is
   • A Fickian term ∂c/∂t|diffusion = ∇·(D∇c).
   • Cells move (randomly or along small gradients) from high-density to low-density regions.
   – Units
   • D is in [length²/time], e.g. mm²/day.
   – Biological meaning
   • Captures invasive spread or infiltration of tumor cells into surrounding tissue.
   • Measured by in vitro motility assays or estimated from imaging (though note that clinical “ADC” maps from DWI MRI are water diffusion, not cell motility—be careful not to mix these).

3. What each term does to tumor density and size
   – Pure proliferation (D=0)
   • Local density c(t) grows (exponentially or to K).
   • The tumor volume doesn’t “spread out” spatially unless you add movement.
   – Pure diffusion (ρ=0)
   • Total cell number (integral of c over space) stays constant.
   • A high-density “blob” will flatten and spread out, so peak c drops but radius grows.
   • No net increase in mass—just redistribution.
   – Both together (ρ>0, D>0)
   • Locally cells divide, pushing density up to K, then diffusion smooths out the high-density region by moving cells into low-density surroundings.
   • You get an expanding “invasion front.”
   • In the simplest Fisher–KPP model the front speed ≃ 2·√(D·ρ).
   • Tumor volume and total cell number both increase over time.

4. Key takeaway
   – Proliferation controls how fast cells are born and thus how fast total mass goes up.
   – Diffusion controls how fast and how far cells migrate away from where they were born.
   – Tumor growth in volume or radius requires both: you need proliferation to create new cells (mass) and diffusion (or other motility mechanisms) to push them out into new tissue.
   – Positive diffusion alone only spreads existing cells; it does not create new ones.

So in your reading of the paper: when they estimate D they are quantifying the cell’s invasive motility; when they estimate ρ they are quantifying net cell birth. Both parameters together give you a predictive model of how quickly and how invasively a tumor will grow.