Modelling Electrical Stimulation of Tissue
This chapter describes the theory and techniques for modelling the electrical activity of excitable cells and tissues, along with their electrical simulation, using COMSOL. It begins with a summary of Maxwell’s equations, before moving on to electrostatic
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Modelling Electrical Stimulation of Tissue
6.1 Electrical Stimulation Matter is composed of atoms consisting of positively-charged nuclei and negativelycharged electrons. These positive and negative charges produce and respond to electromagnetic fields, which are completely described by Maxwell’s equations.1
6.1.1 Maxwell’s Equations Maxwell’s equations, expressed in so-called “macroscopic” form which characterize electromagnetic fields in materials, are: ∂B ∂t ∂D ∇ ×H = J+ ∂t ∇ · D = ρv ∇ ×E = −
∇ ·B = 0
(6.1) (6.2) (6.3) (6.4)
where E denotes the electric field (SI units: V m−1 ), B the magnetic field (SI units: T), D the electric displacement (SI units: C m−2 ), H the magnetization (SI units: A m−1 ), J the applied current density (SI units: A m−2 ) and ρv the volume charge density (SI units: C m−3 ). Equation 6.1 is also known as Faraday’s Law of Induction and states that a time-varying change in the local magnetic field will produce an electric field. Equation 6.2 represents an extension to Ampère’s Law, which states that a steady electric current produces a magnetic field. Maxwell’s contribution was 1 James
Clerk Maxwell (1831–1879), Scottish mathematical physicist whose contributions to physics, along with those of Einstein and Newton, are regarded as greatest in the history of science. © Springer-Verlag Berlin Heidelberg 2017 S. Dokos, Modelling Organs, Tissues, Cells and Devices, Lecture Notes in Bioengineering, DOI 10.1007/978-3-642-54801-7_6
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6 Modelling Electrical Stimulation of Tissue
the addition of a “displacement current” term ∂D/∂t, such that the magnetization field is also produced by a time-varying electric field in addition to a steady current. Equation 6.3 is known as Gauss’ Law, and states that an electric field is produced by electric charge. Equation 6.4 is also known as Gauss’ Law for Magnetism, and effectively states that there are no analogues of electric charge for magnetic fields (so-called “magnetic charges”). For homogeneous linear isotropic materials, the electric displacement and magnetization fields satisfy D = εE with
B = μH 1 με = 2 c
where ε is the permittivity, μ is the permeability, and c is the speed of light within the material.2 Denoting the permittivity and permeability of free space by ε0 and μ0 , we can express ε and μ for any such material relative to their free-space values as ε = ε0 εr μ = μ0 μr where εr , μr respectively denote the relative permittivity and permeability. Numerical values for ε0 and μ0 are: ε0 = 8.854 × 10−12 F m−1 μ0 = 4π × 10−7 H m−1 where H denotes the Henry, the unit of electrical inductance. Substituting the above expressions for D and H into Maxwell’s equations, we can express these equations in terms of fields E and B as follows:
∇ ×E = −
∂B ∂t
∇ × B = μr μ0 J + ∇ · (εr ε0 E) = ρv ∇ ·B = 0
2 The
(6.5) 1 ∂E c2 ∂t
(6.6) (6.7) (6.8)
relationship between D and E is known as the electric constitutive relation of the material.
6.1 Electrical Stimulation
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6.1.2 Electrostatic Formulations If the electric and magnetic f
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