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Smoothed particle hydrodynamics simulation of biphasic soft tissue and its medical applications

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Abstract

Modeling the coupled fluid and elastic mechanics of blood perfused soft tissues is important for medical applications. In particular, the current study aims to capture the effect of tissue swelling and the transport of blood through damaged tissue under bleeding or hemorrhaging conditions. The soft tissue is considered a dynamic poro-hyperelastic material with blood-filled voids. A biphasic formulation—effectively, a generalization of Darcy’s law—is utilized, treating the phases as occupying fractions of the same volume. A Stokes-like friction force and a pressure that penalizes deviations from volume fractions summing to unity serve as the interaction force between solid and liquid phases. The resulting equations for both phases are discretized with the method of smoothed particle hydrodynamics (SPH). The solver is validated separately on each phase and demonstrates good agreement with exact solutions in test problems. Simulations of oozing, hysteresis, swelling, drying and shrinkage, and tissue fracturing and hemorrhage are shown in the paper.

In the paper, a new methodology for the numerical simulation of the full dynamic response of blood-perfused soft tissues was developed.

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Funding

We gratefully acknowledge the support of the US Army Medical Research Acquisition Activity (grant no. W81XWH-15-1-0147) and the US Office of Naval Research (grant no. N00014-13-C-0357).

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Authors and Affiliations

  1. Mechanical and Aerospace Engineering Department, University of California, Los Angeles, CA, USA

    Yi-Jui Chang & Jeff D. Eldredge

  2. Department of Surgery, University of California, Los Angeles, CA, USA

    Peyman Benharash & Erik P. Dutson

  3. Center for Advanced Surgical and Interventional Technology, University of California, Los Angeles, CA, USA

    Erik P. Dutson

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  1. Yi-Jui Chang
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Correspondence to Jeff D. Eldredge.

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Appendix: Kernel functions

Appendix: Kernel functions

The cubic spline kernel

$$ W(r,\kappa h)=\alpha_{0} \begin{cases} 1-\frac{3}{2}(r/h)^{2}+\frac{3}{4}(r/h)^{3}, & 0\leq r/h<1\\ \frac{1}{4}(2-r/h)^{3}, & 1\leq r/h<2\\ 0, & r/h\geq2\\ \end{cases} $$
(47)

where κ = 2 and α0 = 2/(3h), 10/(7πh2), 1/(πh3) in 1, 2, and 3 dimensions, respectively.

The third-degree polynomial kernel

$$ \begin{array}{@{}rcl@{}} &&{}W(r,\kappa h)\\&&{}=\alpha_{0} \begin{cases} -\frac{1}{2}(r/h)^{3}+(r/h)^{2}+\frac{1}{2}(r/h)-1, & 0\leq r/h<1\\ 0, & r/h\geq1\\ \end{cases} \end{array} $$
(48)

where κ = 1 and α0 = 10/(3πh2), 15/(2πh3) in 2 and 3 dimensions, respectively.

The sixth-degree polynomial kernel

$$ W(r,\kappa h)=\alpha_{0} \begin{cases} (1-(r/h)^{2})^{3}, & 0\leq r/h<1\\ 0, & r/h\geq1\\ \end{cases} $$
(49)

where κ = 1 and α0 = 35/(16h), 4/(πh2), 315/(64πh3) in 1, 2, and 3 dimensions, respectively.

The spiky kernel:

$$ W(r,\kappa h)=\alpha_{0} \begin{cases} (1-r/h)^{3}, & 0\leq r/h<1\\ 0, & r/h\geq1\\ \end{cases} $$
(50)

where κ = 1 and α0 = 4/h, 10/(πh2), 15/(πh3) in 1, 2, and 3 dimensions, respectively.

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Chang, YJ., Benharash, P., Dutson, E.P. et al. Smoothed particle hydrodynamics simulation of biphasic soft tissue and its medical applications. Med Biol Eng Comput 59, 227–242 (2021). https://doi.org/10.1007/s11517-020-02283-w

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