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Modeling electrical power absorption and thermally-induced biological tissue damage

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Biomechanics and Modeling in Mechanobiology Aims and scope Submit manuscript

Abstract

This work develops a model for thermally induced damage from high current flow through biological tissue. Using the first law of thermodynamics, the balance of energy produced by the current and the energy absorbed by the tissue are investigated. The tissue damage is correlated with an evolution law that is activated upon exceeding a temperature threshold. As an example, the Fung material model is used. For certain parameter choices, the Fung material law has the ability to absorb relatively significant amounts of energy, due to its inherent exponential response character, thus, to some extent, mitigating possible tissue damage. Numerical examples are provided to illustrate the model’s behavior.

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Notes

  1. The Second Piola-Kirchhoff stress is related to the Cauchy stress (\({{\varvec{T}}}\)) via \({{\varvec{T}}}=\frac{1}{J}{{\varvec{F}}}\cdot {{\varvec{S}}}\cdot {{\varvec{F}}}^T\). Although we will employ referential formulations in the analysis to follow, we note that in the current configuration we have \(\displaystyle {\rho \dot{w}-{{\varvec{T}}}:\nabla _x\dot{{{\varvec{v}}}}+\nabla _x \cdot {{\varvec{q}}}-Z=0}\), where \({{\varvec{T}}}\) is the Cauchy stress, \({{\varvec{v}}}\) is the material velocity, \(\rho \) is the density, and \({{\varvec{q}}}\) is the heat flux.

    Fig. 1
    figure 1

    Electrical flow through bio-tissue

    Full size image

  2. In the case of isotropy, Hooke’s law can be written in terms of the bulk \(\kappa \) and shear moduli \(\mu \) as \({\varvec{\sigma }}={{\varvec{I}}}\!{{\varvec{E}}}:{\varvec{\epsilon }}=3\kappa \frac{tr({{\varvec{\epsilon }}})}{3}\mathbf{1}+2\mu {{\varvec{\epsilon }}}^{\prime }\), where \({{\varvec{\epsilon }}}^{\prime }={\varvec{\epsilon }}-\frac{tr({{\varvec{\epsilon }}})}{3}\mathbf{1}\).

  3. For further details on these types of phenomenological (damage) formulations, the interested reader is referred to the seminal work of Kachanov (1986).

  4. For an in depth mathematical analysis of coupled thermal, diffusive and chemical effects in solids, in particular instabilities, we refer the reader to Markenscoff (2001a, b, 2003).

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of California, Berkeley, CA, 94720-1740, USA

    T. I. Zohdi

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  1. T. I. Zohdi
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Correspondence to T. I. Zohdi.

Appendix: Joule heating

Appendix: Joule heating

Joule heating can be explicitly identified by considering Faraday’s Law

$$\begin{aligned} \nabla \times {\varvec{\mathcal{E }}}=-\frac{\partial {\varvec{\mathcal{B }}}}{\partial t} \end{aligned}$$
(7.1)

and Ampere’s Law

$$\begin{aligned} \nabla \times {\varvec{\mathcal{H }}}=\frac{\partial {\varvec{\mathcal{D }}}}{\partial t}+{\varvec{\mathcal{J }}}\end{aligned}$$
(7.2)

where we recall that \({\varvec{\mathcal{E }}}\) is the electric field, \({\varvec{\mathcal{D }}}\) is the electric field flux, \({\varvec{\mathcal{J }}}\) is the electric current, \({\varvec{\mathcal{H }}}\) is the magnetic field, and \({\varvec{\mathcal{B }}}\) is the magnetic field flux. Joule heating can be motivated by forming the inner product of the magnetic field with Faraday’s law and the inner product of the electric field with Ampere’s law and forming the difference to yield

$$\begin{aligned} \underbrace{{\varvec{\mathcal{E }}}\cdot (\nabla \times {\varvec{\mathcal{H }}})-{\varvec{\mathcal{H }}}\cdot (\nabla \times {\varvec{\mathcal{E }}})}_{-\nabla \cdot ({\varvec{\mathcal{E }}}\times {\varvec{\mathcal{H }}})=-\nabla \cdot {{\varvec{S}}}} \!=\!{\varvec{\mathcal{E }}}\cdot {\varvec{\mathcal{J }}}\!+\!\underbrace{{\varvec{\mathcal{E }}}\cdot \frac{\partial {\varvec{\mathcal{D }}}}{\partial t}\!+\!{\varvec{\mathcal{H }}}\cdot \frac{\partial {\varvec{\mathcal{B }}}}{\partial t}}_{=\frac{\partial \mathcal{W}}{\partial t}},\nonumber \\ \end{aligned}$$
(7.3)

where \(\mathcal{W}=\frac{1}{2}({\varvec{\mathcal{E }}}\cdot {\varvec{\mathcal{D }}}+{\varvec{\mathcal{H }}}\cdot {\varvec{\mathcal{B }}})\) is the electromagnetic energy and where \({\varvec{\mathcal{S }}}={\varvec{\mathcal{E }}}\times {\varvec{\mathcal{H }}}\) is the Poynting vector. Thus,

$$\begin{aligned} \frac{\partial \mathcal{W}}{\partial t}+\nabla \cdot {\varvec{\mathcal{S }}}=-{\varvec{\mathcal{J }}}\cdot {\varvec{\mathcal{E }}}\end{aligned}$$
(7.4)

Equation 7.4 is usually referred to as Poynting’s theorem and can be interpreted as stating that the rate of change of electromagnetic energy within a volume, plus the energy flowing out through the material, is equal to the negative of the total work done by the fields on the sources and electrical conduction. We consider the absorbed energy that is available for heating to be proportional to the energy associated with current flow (\({\varvec{\mathcal{J }}}\cdot {\varvec{\mathcal{E }}}\)) in Eq. 7.4.

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Zohdi, T.I. Modeling electrical power absorption and thermally-induced biological tissue damage. Biomech Model Mechanobiol 13, 115–121 (2014). https://doi.org/10.1007/s10237-013-0489-9

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