Mathematical Methods
1. Diffusion-Reaction and Hypoxia of Cellular Spheroids
The concentration of a reactant undergoing diffusion and a first-order irreversible reaction (or consumption by cells) in a spherical catalyst is described by the reaction- diffusion equation
a. Render this equation dimensionless and show that there is only one dimensionless parameter---the (squared) Thiele modulus Φ2 = ka2 /D.
b. Show that the solution for the dimensionless concentration is
c. By doing a Taylor expansion of the dimensionless solution with respect to Φ in b, show that for small Thiele modulus Φ ≪ 1, the concentration is described by
C(r)~ 1 − (1 − r2)Φ2 /6
Note that the concentration is finite at the center of the sphere r=0 at small Thiele modulus.
For large spheres, fast reaction or small diffusivity, such that the Thiele modulus is large Φ >> 1, expand the exact solution from r=1 and large Φ to show the concentration decays exponentially to zero rapidly from the surface and the concentration approaches zero at the center.
d. (Ref: Murphy et al, J. of Royal Society Interface, 14: 20160851 (2017))
Cancer cells are often aggregated into a large sphere called spheroids for drug testing. It has been found that cells in the center of larger spheroids tend to die because oxygen cannot diffuse into the middle due to uptake by cells closer to the surface of the spheroids. You will see the decay of the oxygen concentration from the surface in figures c to e below, corresponding to numerical solution of the reaction-diffusion equation and actual measured data. (Cr is the concentration at a particular r position and C0 is the oxygen concentration at the surface (our C∞ ) of about 267 μM. You can divide the K/D value in fig f by the surface oxygen concentration to get an estimated average value of k/D. (The scale bar in g is 250 microns and be careful with the difference in K and k.)
Use the measured concentration at the spheroid center (r=0) in fig e to determine k/D for each spheroid, given that the radii of the spheroids with different numbers of cells can be measured from fig g ? Compare your results to theirs from fig. f.
e. If cells only die by hypoxia if the oxygen concentration around them is below Cc ,
derive an explicit expression for the critical spheroid size ac D/k, Cc) below which all cells will survive. If Cc is 253 μM for hypoxia, what is the actual dimensional critical spheroid size ac, using the average k/D you have estimated ?
2. Cyclic Voltammetry Consider atypical cyclic voltammetry data shown below with different scan rates. For the peak current of cyclic voltammetry, the possible physical parameters for the problem are
Peak current Ip in C/s
Charge transferred to the electrode nF in C/mol, where n is the valency
Thermal energy RT in J/mol
Surface area of the electrode A in m2
Bulk concentration of the reactant C∞ in mol/m3
Voltagescan (sweep) rate S in V/s
Diffusivity D in m2/s
a. Use Buckingham pi theorem to determine that the peak current is
Ip = AnFC∞(nFDS/RT)1/2
The square root scaling with respect to the scan rate S is shown on the left plot.
You will need to add C (Coulombs) and mole to the three fundamental units L, M and time.
b. Explain qualitatively why this peak current corresponds to a Damkholer number of unity for the surface Faradaic electrode reaction and why it occurs at a voltage roughly equal to RT/nF (~25.6mV) above the oxidation potential, which corresponds to the thermal energy and barrier of most Faradaic reactions.
3. Exact Solution for Cyclic Voltammetry
If the redox reaction of a cyclic voltammetry experiment becomes fast enough, such that the Damkohler number is larger than 1, then the ion concentration vanishes at the electrode at x=0 and a diffusion front begins to expand away from the electrode:
a. Solve the above PDE by self-similar solution and show that the flux at the electrode is
Such that the current is
I = AnFJ
b. Determine how the current decreases with Voltage V by replacing t with the scan rate S, S = V/t.
c. The peak current occurs when the voltage exceeds the thermal voltage (oxidation potential) V~RT/nF at sometime t~tp, hence at Ip,
Recover the scaling result of Problem 2 to get how the peak current depends on the scan rate S and the diffusivity D, modulo an arbitrary constant.