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Anatomically based geometric modelling of the musculo-skeletal system and other organs

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Abstract

Anatomically based finite element geometries are becoming increasingly popular in physiological modelling, owing to the demand for modelling that links organ function to spatially distributed properties at the protein, cell and tissue level. We present a collection of anatomically based finite element geometries of the musculo-skeletal system and other organs suitable for use in continuum analysis. These meshes are derived from the widely used Visible Human (VH) dataset and constitute a contribution to the world wide International Union of Physiological Sciences (IUPS) Physiome Project (www.physiome.org.nz). The method of mesh generation and fitting of tricubic Hermite volume meshes to a given dataset is illustrated using a least-squares algorithm that is modified with smoothing (Sobolev) constraints via the penalty method to account for sparse and scattered data. A technique (“host mesh” fitting) based on “free-form” deformation (FFD) is used to customise the fitted (generic) geometry. Lung lobes, the rectus femoris muscle and the lower limb bones are used as examples to illustrate these methods. Geometries of the lower limb, knee joint, forearm and neck are also presented. Finally, the issues and limitations of the methods are discussed.

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Acknowledgements

The authors would like to acknowledge the important input to this project from the members of the Bioengineering Institute, and the financial assistance from the Foundation for Research, Science and Technology New Zealand, and Marsden grant 01-UOA-070.

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Correspondence to J. W. Fernandez.

Appendix A

Appendix A

Differentiating Eq. 22 with respect to η 2 gives,

$$ \begin{array}{*{20}l} {{\frac{{\partial ^{2} {\mathbf{u}}}} {{\partial \eta _{1} \partial \eta _{2} }}} \hfill} & { = \hfill} & {{\frac{{\partial {\mathbf{u}}}} {{\partial \xi _{1} }}\frac{{\partial ^{2} \xi _{1} }} {{\underline{{\partial \eta _{1} \partial \eta _{2} }} }} + \frac{{\partial \xi _{1} }} {{\partial \eta _{1} }}\frac{{\partial ^{2} {\mathbf{u}}}} {{\underline{\underline {\partial \xi _{1} \partial \eta _{2} }} }} + } \hfill} \\ {{} \hfill} & {{} \hfill} & {{\frac{{\partial {\mathbf{u}}}} {{\partial \xi _{2} }}\frac{{\partial ^{2} \xi _{2} }} {{\underline{{\partial \eta _{1} \partial \eta _{2} }} }} + \frac{{\partial \xi _{2} }} {{\partial \eta _{1} }}\frac{{\partial ^{2} {\mathbf{u}}}} {{\underline{\underline {\partial \xi _{2} \partial \eta _{2} }} }} + } \hfill} \\ {{} \hfill} & {{} \hfill} & {{\frac{{\partial {\mathbf{u}}}} {{\partial \xi _{3} }}\frac{{\partial ^{2} \xi _{3} }} {{\underline{{\partial \eta _{1} \partial \eta _{2} }} }} + \frac{{\partial \xi _{3} }} {{\partial \eta _{1} }}\frac{{\partial ^{2} {\mathbf{u}}}} {{\underline{\underline {\partial \xi _{3} \partial \eta _{2} }} }}.} \hfill} \\ \end{array} $$
(27)

In Eq. 27, the terms on the RHS underlined with a single line are the unknowns needed to be solved. The terms with double under lines are however “mixed derivatives” and are not desirable in the current form. They can further be manipulated in the following manner using the chain rule:

$$ \begin{array}{*{20}l} {{\frac{{\partial ^{2} {\mathbf{u}}}} {{\partial \xi _{1} \partial \eta _{2} }}} \hfill} & { = \hfill} & {{\frac{\partial } {{\partial \eta _{2} }}{\left( {\frac{{\partial {\mathbf{u}}}} {{\partial \xi _{1} }}} \right)}} \hfill} \\ {{} \hfill} & { = \hfill} & {{\frac{{\partial ^{2} {\mathbf{u}}}} {{\partial \xi ^{2}_{1} }}\frac{{\partial \xi _{1} }} {{\partial \eta _{2} }} + \frac{{\partial ^{2} {\mathbf{u}}}} {{\partial \xi _{1} \partial \xi _{2} }}\frac{{\partial \xi _{2} }} {{\partial \eta _{2} }} + \frac{{\partial ^{2} {\mathbf{u}}}} {{\partial \xi _{1} \partial \xi _{3} }}\frac{{\partial \xi _{3} }} {{\partial \eta _{2} }},} \hfill} \\ \end{array} $$
(28)
$$ \begin{array}{*{20}l} {{\frac{{\partial ^{2} {\mathbf{u}}}} {{\partial \xi _{2} \partial \eta _{2} }}} \hfill} & { = \hfill} & {{\frac{\partial } {{\partial \eta _{2} }}{\left( {\frac{{\partial {\mathbf{u}}}} {{\partial \xi _{2} }}} \right)}} \hfill} \\ {{} \hfill} & { = \hfill} & {{\frac{{\partial ^{2} {\mathbf{u}}}} {{\partial \xi ^{2}_{2} }}\frac{{\partial \xi _{2} }} {{\partial \eta _{2} }} + \frac{{\partial ^{2} {\mathbf{u}}}} {{\partial \xi _{1} \partial \xi _{2} }}\frac{{\partial \xi _{1} }} {{\partial \eta _{2} }} + \frac{{\partial ^{2} {\mathbf{u}}}} {{\partial \xi _{2} \partial \xi _{3} }}\frac{{\partial \xi _{3} }} {{\partial \eta _{2} }},} \hfill} \\ \end{array} $$
(29)
$$ \begin{array}{*{20}l} {{\frac{{\partial ^{2} {\mathbf{u}}}} {{\partial \xi _{3} \partial \eta _{2} }}} \hfill} & { = \hfill} & {{\frac{\partial } {{\partial \eta _{2} }}{\left( {\frac{{\partial {\mathbf{u}}}} {{\partial \xi _{3} }}} \right)}} \hfill} \\ {{} \hfill} & { = \hfill} & {{\frac{{\partial ^{2} {\mathbf{u}}}} {{\partial \xi ^{2}_{3} }}\frac{{\partial \xi _{3} }} {{\partial \eta _{2} }} + \frac{{\partial ^{2} {\mathbf{u}}}} {{\partial \xi _{1} \partial \xi _{3} }}\frac{{\partial \xi _{1} }} {{\partial \eta _{2} }} + \frac{{\partial ^{2} {\mathbf{u}}}} {{\partial \xi _{2} \partial \xi _{3} }}\frac{{\partial \xi _{2} }} {{\partial \eta _{2} }}.} \hfill} \\ \end{array} $$
(30)

All the terms on the RHS of Eqs. 28, 29 and 30 are known. For instance, terms like \( \frac{{\partial \xi _{i} }} {{\partial \eta _{2} }} \) (i=1, ..., 3) can be evaluated using Eq. 25 and the other terms such as \( \frac{{\partial ^{2} {\mathbf{u}}}} {{\partial \xi _{i} \partial \xi _{j} }} \) (i=1, ..., 3, j=1, ..., 3, ij) are readily available either from undeformed or deformed host geometry. Expanding Eqs. 28, 29 and 30 and putting into matrix form gives,

$$ {\left[ {\frac{{\partial {\left( {x,y,z} \right)}}} {{\partial {\left( {\xi _{1} ,\xi _{2} ,\xi _{3} } \right)}}}} \right]}{\left[ {\begin{array}{*{20}c} {{\frac{{\partial ^{2} \xi _{1} }} {{\partial \eta _{1} \partial \eta _{2} }}}} \\ {{\frac{{\partial ^{2} \xi _{2} }} {{\partial \eta _{1} \partial \eta _{2} }}}} \\ {{\frac{{\partial ^{2} \xi _{3} }} {{\partial \eta _{1} \partial \eta _{2} }}}} \\ \end{array} } \right]} = {\left[ R \right]}, $$
(31)

where

$$ {\left[ R \right]} = {\left[ {\begin{array}{*{20}l} {{\frac{{\partial ^{2} x}} {{\partial \eta _{1} \partial \eta _{2} }}} \hfill} & { - \hfill} & {{\frac{{\partial \xi _{1} }} {{\partial \eta _{1} }}\frac{{\partial ^{2} x}} {{\partial \xi _{1} \partial \eta _{2} }}} \hfill} & { - \hfill} & {{\frac{{\partial \xi _{2} }} {{\partial \eta _{1} }}\frac{{\partial ^{2} x}} {{\partial \xi _{2} \partial \eta _{2} }}} \hfill} & { - \hfill} & {{\frac{{\partial \xi _{3} }} {{\partial \eta _{1} }}\frac{{\partial ^{2} x}} {{\partial \xi _{3} \partial \eta _{2} }}} \hfill} \\ {{\frac{{\partial ^{2} y}} {{\partial \eta _{1} \partial \eta _{2} }}} \hfill} & { - \hfill} & {{\frac{{\partial \xi _{1} }} {{\partial \eta _{1} }}\frac{{\partial ^{2} y}} {{\partial \xi _{1} \partial \eta _{2} }}} \hfill} & { - \hfill} & {{\frac{{\partial \xi _{2} }} {{\partial \eta _{1} }}\frac{{\partial ^{2} y}} {{\partial \xi _{2} \partial \eta _{2} }}} \hfill} & { - \hfill} & {{\frac{{\partial \xi _{3} }} {{\partial \eta _{1} }}\frac{{\partial ^{2} y}} {{\partial \xi _{3} \partial \eta _{2} }}} \hfill} \\ {{\frac{{\partial ^{2} z}} {{\partial \eta _{1} \partial \eta _{2} }}} \hfill} & { - \hfill} & {{\frac{{\partial \xi _{1} }} {{\partial \eta _{1} }}\frac{{\partial ^{2} z}} {{\partial \xi _{1} \partial \eta _{2} }}} \hfill} & { - \hfill} & {{\frac{{\partial \xi _{2} }} {{\partial \eta _{1} }}\frac{{\partial ^{2} z}} {{\partial \xi _{2} \partial \eta _{2} }}} \hfill} & { - \hfill} & {{\frac{{\partial \xi _{3} }} {{\partial \eta _{1} }}\frac{{\partial ^{2} z}} {{\partial \xi _{3} \partial \eta _{2} }}} \hfill} \\ \end{array} } \right]} $$
(32)

Equations similar to those given in Eq. 31 can be derived for \( \frac{{\partial ^{2} \xi _{i} }} {{\partial \eta _{2} \partial \eta _{3} }} \) and \( \frac{{\partial ^{2} \xi _{i} }} {{\partial \eta _{3} \partial \eta _{1} }} \) (i=1, ..., 3). Solving Eq. 31 with the Jacobian matrix of the deformed host geometry, all second order derivatives of the transformed slave mesh can be determined.

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Fernandez, J.W., Mithraratne, P., Thrupp, S.F. et al. Anatomically based geometric modelling of the musculo-skeletal system and other organs. Biomech Model Mechanobiol 2, 139–155 (2004). https://doi.org/10.1007/s10237-003-0036-1

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