Mathematical model of the post-ablation enhancement zone as a tissue-level oedematic response

Figures & data

Figure 1. Schematic diagram of model of cell oedema post-ablation.

Table 1. Gating variables at quasi-steady state (m_∞, h_∞ and n_∞).

x	αx	βx	x∞
m
h
n

Figure 2. Temperature profile imposed on the system.

Table 2. Parameter values.

Name	Value and unit	Description	Reference
R	8314 J/kmol/K	Universal gas constant
e	1.6*10^−19 C	Elementary charge
F	96485 C/mol	Faraday’s constant
T0	310 K	Physiological body temperature
Cm	47pF	Membrane capacitance	[19]
As	7.67*10^−5 cm^2	Membrane surface area	[16]
vc^0	3.8*10^−8 cm^3	Initial volume of cell	[16]
Nc	0.2105*10^8 1/cm^3	Tumour cell density
Ai^0	80.35 mM	Initial concentration of intracellular non-permeable molecules	Calculated*
gNa^l	4.21*10^−6 mC/V/min	Leak conductance of Na^+	Calculated*
gK^l	7*10^−6 mC/V/min	Leak conductance of K^+	Calculated*
gNa^g	8.174*10^−4 mC/V/min	Maximum gated conductance of Na^+	Calculated*
gK^g	9.123*10^−4 mC/V/min	Maximum gated conductance of K^+	Calculated*
gCl	2.81*10^−4 mC/V/min	Leak conductance of Cl^-	Calculated*
Km, Nai	10 mM	[Na]i at half maximum pump current	[17]
Km, Ke	1.5 mM	[K]e at half maximum pump current	[17]
Pmax	73.97pA	Max. pump current	[17]
Lp	10^−9 m/Pa/s	Membrane permeability to water flow	[16]
	10.5 cm^2/min	Na^+ diffusion coefficient	[12]
	11.76 cm^2/min	K^+ diffusion coefficient	[12]
	12.18 cm^2/min	Cl^− diffusion coefficient	[12]
kf	3.33*10^−3 1/s	Scaling constant for forward rate	[11]
kb	7.77*10^−3 1/s	Backward rate constant	[11]
Tk	40.5 °C	Calibration temperature	[11]
ks	0.316*10^−3 1/s	Scaling factor for slow-death rate	[11]
Dτ	0.208	Cell viability threshold	[11]
L	3 cm	Length of computational domain	[18]

Table 3. Initial values of variables.

Name	Value and unit	Description	Reference
[Na]e	120 mM	Extracellular Na^+ concentration	[20]
[Na]i	27 mM	Intracellular Na^+ concentration	[20]
[K]e	4 mM	Extracellular K^+ concentration	[20]
[K]i	130.99 mM	Intracellular K^+ concentration	[20]
[Cl]e	124 mM	Extracellular Cl^− concentration	[20]
[Cl]i	9.66 mM	Intracellular Cl^− concentration	[20]
[∅]i	0.8	Intracellular volume fraction	[21]
Em	−68 mV	Membrane potential	Calculated*
A	0.9999	Fraction of cells alive	[11]
D	0	Fractions of dead cells	[1]

Figure 3. Cell viability in response to the imposed temperature profile.

Figure 4. Simulation results of ion concentrations: top panels Na_e and Na_i; middle panels K_e and K_i; bottom panels Cl_e and Cl_i.

Figure 5. Simulation result of volume fraction of intracellular space.

Figure 6. Dependence of the ablation volume and r_90% upon variations in: (a) ΔT; (b) t_s; and (c) r_s.

Figure 7. Dependence of r_90% upon ion diffusion coefficients.

Figure 8. Dependence of intracellular volume fraction on its initial level: (a) φ_{i, 0} = 0.8; and (b) φ_{i, 0} = 0.6.

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