Table 1.

Parameters used to solve Equations (5)–(8). The carrying capacities of normal and tumour cells (K_1,2 respectively), growth rates (r_1,2), tumour diffusion (D₂), and vascular evacuation (d_v) parameters were taken from a previous acid-invasion modelling and experimental study by Gatenby et al. (2006). The diffusion of protons occurs in association with mobile buffer species to ensure charge balance, and the effective diffusion coefficient value used was estimated in Gatenby et al. (2006). From these values, Gatenby et al. (2006) calculate the tumour acid production rate (r₃), and a range of values for the normal cell death rate from acid (d₁) from which we select. The blood convection rate (U) is determined from the experimental values found in venules (Jain, 2001, Torchilin, 2006). The blood buffering and proportional delivery of buffer from the bloodstream to the tissue is estimated from extensive systemic buffering simulations (Martin et al., A Mathematical Model of Tumour and Blood pH, Submitted). For the increased vascular permeability simulations, we use ν₂ which is increased two-fold from the baseline value of vessel permeability, ν₁.

Parameter	Definition	Dimensions	Value	Source
K1	N1 carrying capacity	cells/cm3	5 × 108	Gatenby et al. (2006)
K2	N2 carrying capacity	cells/cm3	5 × 108	Gatenby et al. (2006)
r1	N1 growth rate	1/s	2 × 10−6	Gatenby et al. (2006)
r2	N2 growth rate	1/s	3.2 × 10−6	Gatenby et al. (2006)
D2	N2 diffusion	cm2/s	2 × 10−10	Gatenby et al. (2006)
D3	Lactic acid diffusion	cm2/s	5 × 10−6	Gatenby et al. (2006)
r3	H+ ion production rate	M · cm3/[s · cell]	1.1 × 10−17	fit to Gatenby et al. (2006)
dv	Normal vascular evacuation	1/s	1.1 × 10−4 –2.4 × 10−2	Gatenby and Gawlinski (1996), Gatenby et al. (2006)
d1	N1 death rate from acid	cells/[M· s]	0 – 10	Gatenby and Gawlinski (1996)
U	blood advection	cm/s	5 × 10−2	Jain (2001), Torchilin (2006)
κ2	Proportional delivery buffer	-	0.25	estimated
δ6	g2/r1	-	8 × 104	estimated
ρ1	$d_1{r}_3{K}_2/r_1^2$	-	3 × 104	selected from range above
δ2	r2/r1	-	1.6	from above
α2	D2/D3	-	4 × 10−5	from above
σ	$U\sqrt{r_1}/r_1\sqrt{D_3}$	-	1.6 × 104	from above
ν1	dv/r1	-	1.2 × 103	from above
ν2	dv/r1	-	1.2 × 105	increased two-fold from normal