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Finite element implementation of a multiscale model of the human lens capsule

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Abstract

An axisymmetric finite element implementation of a previously described structural constitutive model for the human lens capsule (Burd in Biomech Model Mechanobiol 8(3):217–231, 2009) is presented. This constitutive model is based on a hyperelastic approach in which the network of collagen IV within the capsule is represented by an irregular hexagonal planar network of hyperelastic bars, embedded in a hyperelastic matrix. The paper gives a detailed specification of the model and the periodic boundary conditions adopted for the network component. Momentum balance equations for the network are derived in variational form. These balance equations are used to develop a nonlinear solution scheme to enable the equilibrium configuration of the network to be computed. The constitutive model is implemented within a macroscopic finite element framework to give a multiscale model of the lens capsule. The possibility of capsule wrinkling is included in the formulation. To achieve this implementation, values of the first and second derivatives of the strain energy density with respect to the in-plane stretch ratios need to be computed at the local, constitutive model, level. Procedures to determine these strain energy derivatives at equilibrium configurations of the network are described. The multiscale model is calibrated against previously published experimental data on isolated inflation and uniaxial stretching of ex vivo human capsule samples. Two independent example lens capsule inflation analyses are presented.

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Acknowledgments

The mesh for the in situ capsule inflation analysis was generated by GS Wilde. The authors acknowledge the assistance provided by RI Barraquer, S Krag, H Martin and R Michael in providing numerical values of previously published experimental data. RAR gratefully acknowledges funding from the US Army Medical Research and Materiel Command (USAMRMC) grant W81XWH-10-1-1036, the US–UK Fulbright Commission, and the Royal Society International Exchanges Scheme.

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Authors and Affiliations

  1. Department of Engineering Science, Oxford University, Oxford, UK

    H. J. Burd

  2. Department of Civil, Environmental, and Architectural Engineering, University of Colorado Boulder, Boulder, CO, USA

    R. A. Regueiro

Authors
  1. H. J. Burd
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  2. R. A. Regueiro
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Corresponding author

Correspondence to H. J. Burd.

Appendices

Appendix 1: Table of joint coordinates

See Table 4.

Table 4 Initial joint coordinates for the network shown in Fig. 1 \(A_X/L_0 = 6\), \(A_Y/L_0 = 3 \sqrt{3}\)
Full size table

Appendix 2: Capsule thickness data from Barraquer et al. (2006)

See Table 5 and Fig. 12.

Table 5 Numerical values of capsule thickness data (units \(\upmu \)m) from Fig. 7 of Barraquer et al. (2006)
Full size table
Fig. 12
figure 12

a Data on lens capsule thickness re-plotted from Fig. 7 of Barraquer et al. (2006). Group A, mean age \(=\) 36 years; Group B, mean age \(=\) 65 years; Group C, mean age \(=\) 92 years. \(R_{d}\) is normalized position, measured along the lens meridian from the anterior pole. Normalized position is determined such that at the anterior pole \(R_d=0\), at the equator \(R_d=100\) and at the posterior pole \(R_d=200\) (see b)

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Burd, H.J., Regueiro, R.A. Finite element implementation of a multiscale model of the human lens capsule. Biomech Model Mechanobiol 14, 1363–1378 (2015). https://doi.org/10.1007/s10237-015-0680-2

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  • DOI: https://doi.org/10.1007/s10237-015-0680-2

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