PRODUCCIÓN DE BIODIESEL DE MICROALGAS

The aim of this section is to derive the equations used in the model of excitation in the myometrium. This derivation is based on the formulation developed for cardiac mod- elling [26], coupled with the FitzHugh-Nagumo model of cell membrane conductance [82]. Excitation is defined as a substantial temporary excursion of the membrane potential of a cell, away from its resting value. Consider an anisotropically conductive medium with conductivity tensor G, which represents the conductance of the medium in all directions

I

I

I

Cell membrane

Figure 1: cc.

Figure 4.1: The relationship between intra-cellular current Ii, extra-cellular current Ie, and

transmembrane currentIm. The currentImis governed by the conductive elements present in the cell membrane.

at any given point. The current density, J, is given by Ohm’s law to be

J=−G∇φ, (4.1)

whereφ is the electric potential.

The present model accounts for these quantities in both intra- and extra-cellular media, which are denoted with subscript i and e respectively. Applying this notation to the equation (4.1), the intra- and extra-cellular current densities are given by

Ji=−Gi∇φi

Je=−Ge∇φe.

1

The membrane current Im is split into three parallel currents: the capacitive current, given by Cm∂Vm/∂t; the current through the tunnel diode, given by F(Vm); and the current through the resistor-inductor circuit, denotedIL.

This current is the only local source/sink of current in both media. Accordingly, Kirch- hoff’s first law gives

∇ ·Ji =−Im (4.2)

∇ ·Je=Im, (4.3)

where Im is the transmembrane current per unit volume. Summing these two equations

gives the following relationship:

∇ ·(Ji+Je) = 0. (4.4)

The FitzHugh-Nagumo model is used to simulate the behaviour of the transmembrane current [82]. This model splits the transmembrane current into various contributions: a capacitive component, simulating the capacitive nature of the cell membrane; a voltage- dependent resistor, such as a tunnel diode; and a resistor-inductor circuit, as illustrated in Figure 4.2. The latter two of these components model the involvement of ion channels in conducting the transmembrane current. In this model, the transmembrane current is

A

F(V

)

B

f(v)

V

v

0 ↵ 1

Figure 1: cc.

1

Figure 4.3: The current through the tunnel diode as a function of potential difference A: An

example curveF(Vm) defining the current through the tunnel diode as a function of membrane volt-

age. B: The nondimensional functionf(v), which is a rescaled and translated version of−F(Vm)

such thatf(0) = 0 andf(1) = 0.

the sum of the currents in each of these parallel components, which is given by

Im =βm Cm ∂Vm ∂t +F(Vm) +IL , (4.5)

whereβmis the surface area to volume ratio of a cell,Cmis the membrane capacitance per

unit area, F(Vm) is the current through the voltage-dependent resistor per unit area, IL

is the current through the resistor-inductor circuit per unit area, andVmis the membrane

voltage, given by

Vm=φi−φe. (4.6)

The precise mathematical specification of the function F is not essential for the correct behaviour of this model, provided that the general shape is as illustrated in Figure 4.3A. In the simulations a following cubic is used, given in Equation 4.13 below.

Kirchhoff’s second law on the resistor-inductor circuit yields

L∂IL

∂t =Vm−VL−RLIL, (4.7)

v= Vm V1 , u= φe V1 , w= ARLIL V1 , τ = t ACmRL , f(v) = −ARLF(V1v) V1 .

Where V1 is the peak voltage of the system and A is the surface area of a cell. In terms

of these scaled quantities, the equations take on the following form: ∂v ∂t =∇ ·(Di∇v) +∇ ·(Di∇u) +f(v)−w (4.10) ∇ ·(Di+e∇u) =−∇ ·(Di∇v) (4.11) ∂w ∂t =(v−γw+v0), (4.12) whereDi =ARLGi/βm,Di+e =ARL(Gi+Ge)/βm,=ACmR2_L/L,γ = 1/A,v0 =VL/V1,

and thas been substituted for τ for ease of notation. The cubic function f is defined by

f(v) =Av(1−v)(v−α). (4.13)

whereA and α are positive constants with 0< α <1. This function is shown graphically in Figure 4.3B. Equations (4.10)–(4.12) are known as thebidomain equations [26]. These equations can be simplified by approximatingGe by a constant multiple ofGi:

This yields the equations

∂v

∂t =∇ ·(D∇v) +f(v)−w (4.15)

∂w

∂t =(v−γw+V0), (4.16)

whereD=λARLGi/((1 +λ)βm) and all other constants are as before.

Equations (4.15) and (4.16) are known as themonodomainequations [26]. The variable v is called thefast variable and the variablew is called theslow variable in this system.

The final step in the set up of the electrodiffusion equations is the boundary condition. The boundary of the system is the edge of the tissue, and therefore no current will flow through the boundary, giving the boundary condition

∂v

∂x ·n= 0,

wheren is the direction normal to the boundary.

These equations form the mathematical basis for excitation simulation for the remain- der of this chapter. The fast variable v represents membrane potential, and thus the following sections will present a methodology to approximate the value of v numerically.