Electrochemical Treatment of Tumors

1. Electrochemical Treatment of Tumors Speaker: Hao-Kai Ken

2. Motivation • The electrochemical treatment of tumors implies that diseased tissue is treated with direct current through the use of metallic electrodes inserted in the tumor. • When tissue is electrolyzed, two competing reactions take place at the anode: oxygen evolution and chlorine production.

3. Model definition

4. This simplified model considers only a 1D model of the transport between two points, that is, between the two electrodes. The material balance for the species i is given by where ci is the concentration (mol/m3), Di give the diffusivities (m2/s), zi equals the charge, Umi represents the mobility ((mol·m2)/(J· s)), and Ri is the production term for species i (mol/(m3· s)), F denotes Faraday’s constant (As/mol), and V is the potential (V).

5. The mobility, umi, can be expressed in terms of Di, R and T as • The conservation of electric charge is obtained through the divergence of the current density:

6. At the electrode surface (r = ra) you specify the fluxes for the ionic species that are included in the electrode reactions, H+ and Cl-. • For the inert ionic species, Na+, the transport through the electrode surface equals zero. The expression for molar fluxes at the boundary for the reacting species is where Ni is the flux, νij represents the stoichiometric coefficient for the ionic species i in reaction j, and nj is the number of electrons in reaction j.

7. Introducing dimensionless pressure, P = p/pb, and concentration, C=c/cb, (where b denotes the reference concentration), you can express the current density for the two reactions. For the oxygen evolution reaction it is where j0,I is the exchange current density (A/m2) and Eeq,I is the standard electrode potential (V). The chlorine evolution reaction is given by the expression

8. Using the input values nI = nI = 1, νH,I = 1, and νCl,I = –1 gives the fluxes at the electrode surface

9. At the exterior boundary, assume the concentration is constant, ci = ci0, and the potential is set to: • The initial concentration is constant according to ci = ci0. You obtain the initial potential profile from the solution of the domain equations and boundary conditions at t = 0, yielding where t denotes time. where V0,ra is the potential satisfying jI + jII = j0 and κ0 is the conductivity at t = 0.

10. where V0,ra is the potential satisfying jI + jII = j0 and κ0 is the conductivity at t = 0.

11. Model Navigator • Start COMSOL Multiphysics. • In the Model Navigator select Axial symmetry (1D) from the Space dimension list. • Select the application mode Chemical Engineering Module>Mass balance>Nernst-Planck>Transient analysis. • Locate the Dependent variables edit field and enter V cNa cH cCl. Click OK.

13. Constant

14. Scalar expression

15. Geometry modeling • Select Draw>Specify Objects>Line. • Type 1e-3 6e-2 in the r edit field, then click OK. • Click the Zoom Extents button on the Main toolbar to zoom the geometry

17. Subdomain setting

20. Jtot_o*ra

22. Boundary Conditions

25. -0.4977-0.2567*log(t+100)

28. Mesh generation • From the Mesh menu open the Free Mesh Parameters dialog box. • Click the Custom mesh size button, go to the Global page and set the Maximum element size to 5e-3. • Click the Boundary tab. • Select Boundary 1 and type 1e-5 in the Maximum element size edit field. • Select Boundary 2 and type 1e-4 in the Maximum element size edit field. • Click Remesh, then click OK.

32. Solution

33. Postprocessing and Visualization • Open the Domain Plot Parameters dialog box from the Postprocessing menu. • On the General page, select Interpolated times and type 0:600:3600 in the Times edit field. • Click the Line/Extrusion tab and type the expression for pH in the Expression edit field: -log10(cH*1e-3). • Click the Line Settings button, click to select the Legend check box, and click OK. • Click Apply to generate the plot.