Fiber Tractography for Finite-Element Modeling of Transversely Isotropic Biological Tissues of Arbitrary Shape Using Computational Fluid Dynamics Joshua Inouye University of Virginia Charlottesville, VA, USA [email protected]Geoffrey Handsfield University of Virginia Charlottesville, VA, USA [email protected]Silvia Blemker* University of Virginia Charlottesville, VA, USA [email protected]*Corresponding author ABSTRACT Fiber tractography is useful for studying a variety of bio- logical phenomena associated with transversely isotropic tis- sues, in which fibers serve to provide functional strength along a specific axis. One useful application of fiber trac- tography is finite-element analysis (FEA) studies. Here, we present a method utilizing computational fluid dynam- ics (CFD) for efficiently determining fiber trajectories in a transversely isotropic material with arbitrary structures of any complexity (such as those determined from biomedical imag- ing). We demonstrate assignment of fiber directions to FEA mesh by registration with the CFD mesh. Sensitivity analysis on various solver settings, flow characteristics, and material parameters shows less than 2 degrees of average deviation from the nominal fiber vectors if the Reynolds number is <1 and the flow is laminar and incompressible with our nominal fluid properties (viscosity of 1Pa-s and density of 1g/cm 3 ). Flow guides can be used to help match fiber trajectories to experimental or anatomical observations, such as twisting in the Achilles tendon. This method also provides an elegant solution to determining fiber tracts in muscles that intertwine with each other, such as in the soft palate complex. For FEA studies, this method enables efficient determination and as- signment of fiber directions to any finite-element mesh. Author Keywords Finite-element analysis; tissue mechanics; computational fluid dynamics ACM Classification Keywords I.6.0 SIMULATION AND MODELING: General 1. INTRODUCTION As the orientation of the fibers in transversely isotropic bio- logical tissues such as tendon and muscle is critical to their function and adaptation to mechanical stimuli, finite-element models of these tissues require parametrization of the com- plex trajectories of the fibers. Finite-element models of tis- sues such as tendon [33], skeletal muscle [4, 24, 7, 8, 30, 32, 14], cardiac muscle [11, 6], ligament [18, 9], vocal cords [1], SCSC 2015, July 26-29, 2015, Chicago, IL, USA c 2015 Society for Modeling & Simulation International (SCS) cartilage [2], heart valves [20], and meniscus [27] must spec- ify a fiber direction vector at each element or node throughout the tissue. Very simple geometries such as rectangles or cylinders can be modeled with global- or circumferentially-defined fiber di- rections. However, complex geometries are more physiologi- cally relevant and the fiber trajectories are consequently more complicated. This necessitates methods for determining these fiber trajectories in finite-element tissue studies. Several experimental and computational methods have been previously developed to determine fiber trajectories in trans- versely isotropic biological tissues, predominantly muscle. Cadaver dissections have been used to reconstruct muscle fiber morphology [16]. This allows direct visualization and determination of fiber trajectories, but the dissection disturbs the in vivo placement of the muscle and is time-consuming. Furthermore, cadaveric muscles do not represent healthy sub- jects well [12]. Another experimental method to determine fiber trajectories in muscles is MRI diffusion tensor imaging combined with tractography [15]. While this method has the advantage of allowing non-invasive in vivo determination of fiber tracts, the process requires manual selection of regions of interest and is sensitive to noise in the MRI data, resulting in some unrealistic fiber tracts. Another experimental method is ultrasound imaging. This has the advantage of in vivo use, but 2D imaging of muscle [19] cannot reproduce 3D trajec- tories and 3D imaging [17, 23] can suffer from low spatial resolution of fiber vector maps. One computational method uses interpolated cubic splines to determine fiber trajectories [3]. A template with interpolated fiber arrangements origi- nates as a simple geometry, such as a cube, and is then mor- phed into physiologically realistic geometries, such as those obtained from MRI data. This method produces realistic fiber trajectories, but requires manual specification of a fiber tem- plate and morphing of the template. Another computational method uses a rotation and divergence-free (Laplacian) vector field to automatically determine muscle fascicle tracts in arbi- trary muscle shapes [5]. However, the solution is unique, pre- venting adjustment of the trajectories based on experimental data, and the associated rotation free condition prevents the reproduction of twisting trajectories. A similar technique that formulates a boundary value problem has also been proposed previously [10].
Transcript
Fiber Tractography for Finite-Element Modeling ofTransversely Isotropic Biological Tissues of Arbitrary
Shape Using Computational Fluid DynamicsJoshua Inouye
ABSTRACTFiber tractography is useful for studying a variety of bio-logical phenomena associated with transversely isotropic tis-sues, in which fibers serve to provide functional strengthalong a specific axis. One useful application of fiber trac-tography is finite-element analysis (FEA) studies. Here,we present a method utilizing computational fluid dynam-ics (CFD) for efficiently determining fiber trajectories in atransversely isotropic material with arbitrary structures of anycomplexity (such as those determined from biomedical imag-ing). We demonstrate assignment of fiber directions to FEAmesh by registration with the CFD mesh. Sensitivity analysison various solver settings, flow characteristics, and materialparameters shows less than 2 degrees of average deviationfrom the nominal fiber vectors if the Reynolds number is <1and the flow is laminar and incompressible with our nominalfluid properties (viscosity of 1Pa-s and density of 1g/cm3).Flow guides can be used to help match fiber trajectories toexperimental or anatomical observations, such as twisting inthe Achilles tendon. This method also provides an elegantsolution to determining fiber tracts in muscles that intertwinewith each other, such as in the soft palate complex. For FEAstudies, this method enables efficient determination and as-signment of fiber directions to any finite-element mesh.
ACM Classification KeywordsI.6.0 SIMULATION AND MODELING: General
1. INTRODUCTIONAs the orientation of the fibers in transversely isotropic bio-logical tissues such as tendon and muscle is critical to theirfunction and adaptation to mechanical stimuli, finite-elementmodels of these tissues require parametrization of the com-plex trajectories of the fibers. Finite-element models of tis-sues such as tendon [33], skeletal muscle [4, 24, 7, 8, 30, 32,14], cardiac muscle [11, 6], ligament [18, 9], vocal cords [1],
cartilage [2], heart valves [20], and meniscus [27] must spec-ify a fiber direction vector at each element or node throughoutthe tissue.
Very simple geometries such as rectangles or cylinders canbe modeled with global- or circumferentially-defined fiber di-rections. However, complex geometries are more physiologi-cally relevant and the fiber trajectories are consequently morecomplicated. This necessitates methods for determining thesefiber trajectories in finite-element tissue studies.
Several experimental and computational methods have beenpreviously developed to determine fiber trajectories in trans-versely isotropic biological tissues, predominantly muscle.Cadaver dissections have been used to reconstruct musclefiber morphology [16]. This allows direct visualization anddetermination of fiber trajectories, but the dissection disturbsthe in vivo placement of the muscle and is time-consuming.Furthermore, cadaveric muscles do not represent healthy sub-jects well [12]. Another experimental method to determinefiber trajectories in muscles is MRI diffusion tensor imagingcombined with tractography [15]. While this method has theadvantage of allowing non-invasive in vivo determination offiber tracts, the process requires manual selection of regionsof interest and is sensitive to noise in the MRI data, resultingin some unrealistic fiber tracts. Another experimental methodis ultrasound imaging. This has the advantage of in vivo use,but 2D imaging of muscle [19] cannot reproduce 3D trajec-tories and 3D imaging [17, 23] can suffer from low spatialresolution of fiber vector maps. One computational methoduses interpolated cubic splines to determine fiber trajectories[3]. A template with interpolated fiber arrangements origi-nates as a simple geometry, such as a cube, and is then mor-phed into physiologically realistic geometries, such as thoseobtained from MRI data. This method produces realistic fibertrajectories, but requires manual specification of a fiber tem-plate and morphing of the template. Another computationalmethod uses a rotation and divergence-free (Laplacian) vectorfield to automatically determine muscle fascicle tracts in arbi-trary muscle shapes [5]. However, the solution is unique, pre-venting adjustment of the trajectories based on experimentaldata, and the associated rotation free condition prevents thereproduction of twisting trajectories. A similar technique thatformulates a boundary value problem has also been proposedpreviously [10].
Figure 1. Methods. (a) We use computational fluid dynamics to solve for flow fields within an image-based solid model. Photographic image adapted fromcadaveric iliacus (hip flexor) muscle in Ward et al., 2009. (b) The CFD solution enables fiber mapping of a separate FEA mesh by nearest-neighbor registration.
Here, we present a method to determine 3D fiber trajecto-ries in arbitrary structures using computational fluid dynam-ics. The characteristics of viscous, incompressible fluid flowwith the proper boundary conditions help satisfy the observa-tions about fiber trajectories in biological tissues [5]: i) theyare coaxially aligned and do not cross each other ii) they donot branch iii) they will not reverse their directions abruptlyand iv) they must connect between attachment points. Thismethod is simple and intuitive to implement and enables ef-ficient, robust, and reproducible calculations of fiber trajecto-ries for FEA studies. Another advantage of this method is thatthe trajectories can be arbitrarily adjusted with flow guides toincorporate experimental data and observations such as twist-ing. We focus on muscle and tendon tissue, but the methodis generalizable to other transversely isotropic tissues such asligaments or cartilage.
2. METHODS
2.1 Computational fluid dynamics (Figure 1a)Solid model creationA variety of methods can be used to reconstruct complexshapes of biological tissues. Common in vivo methods in-clude MRI and ultrasound, while common ex vivo methodsinclude photographs, sectioning, or slicing. These imagesmust be converted to solid models.
Flow guides can be prescribed in the model so that the flowmatches specific observations (Figure 2). These surfaces orchannels can be specified mathematically or manually withina solid modeling environment to subdivide the solid into sep-arate flow channels.
Boundary conditions and fluid propertiesIt is necessary to specify where the fibers originate andterminate (Figure 1a). For muscles, this is at the tendinousaponeuroses, which can be located via imaging, cadavericdata, or knowledge of the anatomy. For tendons and liga-ments, the origin and termination regions are typically atthe ends of their simpler geometries. One region of fiberorigin or termination is set to an inlet surface condition (e.g.,
Figure 2. Methods. Flow guides enable CFD solution to match experimentaldata or observations. Solid created from cadaveric data in Ward et al., 2009.
positive gage pressure, normal flow velocity, or volume flowrate). The other surface is an outlet surface condition witha gage pressure of 0Pa. The other surfaces are specifiedas slip surfaces that the fluid cannot penetrate. Nominally,we arbitrarily set the inlet pressure to 1Pa and the fluid tobe incompressible with a viscosity of 1Pa-s and density of1g/cm3. This resulted in very low Reynolds numbers (<1)and good convergence.
CFD solverIn the CFD environment, incompressible, laminar, viscous,and steady-state flow are nominally prescribed. These con-ditions help satisfy the observations about fiber trajectoriesin biological tissues [5]: i) they are coaxially aligned and do
not cross each other (viscous flow) ii) they do not branch (in-compressible flow) iii) they will not reverse their directionsabruptly (laminar flow) and iv) they must connect between at-tachment points (boundary conditions). The simulation is runand the steady-state flow velocity vectors for each mesh nodeare computed. Fluid streamlines can be visualized within theCFD package.
2.2 FEA fiber mapping (Figure 1b)Generally, the same solid shape generated from the medicalimages will be used for both CFD and FEA. Depending onthe solid finite element formulation and shape (e.g., tetrahe-dral, hexahedral, linear, quadratic, cubic), it will be requiredto specify fiber directions for either single elements [4] ornodes (if the fiber directions need to be interpolated through-out the element [32]). We use a nearest-neighbor algorithmto transfer the fiber directions from the CFD mesh to the FEAmesh (as the meshes may not be identical). That is, for eachelement or node from the FEA mesh needing a fiber direction,the nearest CFD mesh node is determined and that velocityvector is assigned to the FEA node or element center as thefiber direction vector. 1
2.3 Sensitivity analysisWe performed sensitivity analysis on different inlet condi-tions, fluid viscosities, and compressibility. We calculated theaverage angle deviation of the fiber direction vector from thefiber direction vectors found from our nominal CFD param-eter values (1Pa inlet gage pressure, 1Pa-s viscosity, 1g/cm3
density; laminar and incompressible flow) for each node inthe FEA mesh.
3. RESULTS
3.1 Iliacus (Figure 2)The iliacus is a hip flexor muscle that fans broadly. We imple-mented one curved flow guide and two straight flow guides.The results showed slightly different flow patterns that fol-lowed the guides accurately.
3.2 Soft palate muscle complex (Figure 3a)The soft palate contains several muscles that move and de-form the soft palate in speech and swallowing tasks. Somefibers of two of the muscles, the levator veli palatini and thepalatopharyngeus, blend inside the soft palate [25]. The re-sults demonstrate these blended fiber trajectories.
3.3 Achilles tendon (Figure 3b)The Achilles tendon is composed of three distinct fascicles,connecting to the medial gastrocnemius, lateral gastrocne-mius, and the soleus muscles. Interestingly, these tendonfascicles exhibit approximately a 90 degree twist from ori-gin to insertion [26, 31]. Flow guides subdivide the wholeAchilles tendon image (obtained from MRI) into three twist-ing compartments. CFD results reproduced the twist for allthree compartments.1It can be noted that other least-squares algorithms [13] could alsobe applied to transfer the fiber directions, but if the CFD mesh isspecified to be much finer than the FEA mesh, the errors can bereduced to any desired tolerance using nearest-neighbor methods.
3.4 Biceps femoris longhead (Figure 3c)The biceps femoris longhead is a hamstring muscle with acomplex three-dimensional morphology. We use an MRI-derived solid model of the muscle [24] and use CFD to calcu-late realistic fiber trajectories that start at the proximal tendonand end at the distal tendon and these flow patterns are trans-ferred to the FEA mesh. Sensitivity analysis on various solversettings, flow characteristics, and material parameters (Table1) shows less than 2 degrees of average deviation from thenominal fiber vectors in the FEA mesh if the Reynolds num-ber is <1 and the flow is laminar and incompressible withour nominal fluid properties (viscosity of 1Pa-s and density of1g/cm3). Furthermore, the highest average angle deviation re-sults from turbulent, compressible airflow with inlet pressureof 10,000Pa and a Reynolds number around 1,000,000. Thehigh Reynolds number combined with the intricate curvatureof the muscle causes turbulent eddies, resulting in unrealisticfiber trajectories for muscle (Figure 3d).
3.5 Adductor Brevis (Figure 4)The adductor brevis is a thigh adductor that has a smaller ori-gin than the insertion. The fiber trajectories fan out from theorigin. A reference vector was reproduced from a previouscadaveric study [29]. In the cadaveric study, the average an-gle of the fibers from the reference vector (also termed “pen-nation angle”) was calculated as 6.1 degrees with a standarddeviation of 3.1 degrees averaged across 21 cadavers. Theaverage angle of the FEA fiber direction from this referencevector in our study was 6.9 degrees. This represents a closeagreement of experimental results with our computational re-sults for fiber direction.
4. DISCUSSIONWe have proposed a method using computational fluid dy-namics to compute fiber trajectories in transversely isotropicmaterials of arbitrary complexity and shape. The methodleverages the power of CFD solvers and enables efficient fibermapping of FEA meshes derived from geometries determinedwith in vivo or ex vivo techniques. For certain tissues withcomplex fiber trajectories that cannot be feasibly be deter-mined from our procedure, flow guides can easily be imple-mented to reproduce realistic trajectories (e.g., those deter-mined by cadaveric dissection or diffusion tensor imaging).
Furthermore, this method provides an elegant solution to de-termining fiber tracts in muscles that have complex attach-ments, multiple heads, or intertwine with each other. For lowReynolds numbers, the CFD solutions are robust to simula-tion parameter variations as measured by average angle devi-ation.
Our quantitative results from the angle of the fibers fromthe reference vector (pennation angle) in the adductor bre-vis compared favorably with a cadaveric study [29]. This wasa 2D calculation for that study as well as in our study. Manystudies report 2D pennation angles, but our study enables thecalculation of 3D pennation angles which may prove usefulfor muscle architecture studies. Future studies can provideadditional validation of these measurements in more complexmuscles.
Figure 3. Results. (a) Soft palate muscle complex. Some fibers of two of the muscles, the levator veli palatini and the palatopharyngeus, blend inside the softpalate. The results demonstrate these blended fiber trajectories. (b) Twisting 3-fascicle Achilles tendon results. CFD results reproduced the twist for all threecompartments. 5,000 out of 31,125 mesh point vectors shown. (c) Biceps femoris longhead. The CFD solution easily calculates realistic fiber trajectories thatstart at the proximal tendon and end at the distal tendon and these flow patterns are transferred to the FEA mesh. (d) Laminar, incompressible fluid flow with lowReynolds numbers produces realistic fiber trajectories, while turbulent, compressible airflow with high Reynolds numbers causes turbulent eddies, resulting inunrealistic fiber trajectories.
Table 1. Sensitivity analysis of a variety of CFD conditions. There are less than 2 degrees of average angle deviation from the nominal fiber vectors in the FEAmesh if the Reynolds number is <1 and the flow is laminar and incompressible with our nominal fluid properties (viscosity of 1Pa-s and density of 1g/cm3).Furthermore, the highest average angle deviation results from turbulent, compressible airflow with inlet pressure of 10,000Pa and a Reynolds number around1,000,000 (Figure 3d).
Figure 4. Results for the adductor brevis. Using the CFD solution, the average angle between the reference vector and the fiber direction vectors (pennationangle) was calculated. Model created from cadaveric data in Ward et al., 2009.
In spite of the advantages of this method over previous meth-ods, some limitations can be recognized. Validation or cali-bration with experimental observations of very complex 3Dtissue architectures may be difficult. Study-specific sensitiv-ity analyses on boundary conditions, flow guides, and simula-tion parameters may be desirable in certain cases. Discretiza-tion and nearest-neighbor procedures can result in small er-rors.
Supplementary information, including code, models, demon-stration videos, and data, are available online [28].
5. CONCLUSIONAs demonstrated by the numerous examples in this study, thismethod has great potential to enhance and empower futureFEA studies on complex, transversely isotropic biological tis-sues. Future work will also enable the use of this method toconduct in vivo studies of muscle architecture.
ACKNOWLEDGMENTSThis work was supported by a grant from The Hartwell Foun-dation to S. Blemker. The authors wish to thank Dr. TimWu and Dr. Jacob Barhak for their thoughtful reviews of thiswork [22, 21].
REFERENCES1. Alipour, F., Berry, D. A., and Titze, I. R. A
finite-element model of vocal-fold vibration. TheJournal of the Acoustical Society of America 108, 6(2000), 3003–3012.
2. Almeida, E. S., and Spilker, R. L. Finite elementformulations for hyperelastic transversely isotropic
biphasic soft tissues. Computer Methods in AppliedMechanics and Engineering 151, 3 (1998), 513–538.
3. Blemker, S. S., and Delp, S. L. Three-dimensionalrepresentation of complex muscle architectures andgeometries. Annals of Biomedical Engineering 33, 5(2005), 661–673.
4. Blemker, S. S., Pinsky, P. M., and Delp, S. L. A 3Dmodel of muscle reveals the causes of nonuniformstrains in the biceps brachii. Journal of Biomechanics38, 4 (Apr. 2005), 657–665.
5. Choi, H. F., and Blemker, S. S. Skeletal muscle fasciclearrangements can be reconstructed using a laplacianvector field simulation. PloS One 8, 10 (2013), e77576.
6. Costa, K. D., Holmes, J. W., and McCulloch, A. D.Modelling cardiac mechanical properties in threedimensions. Philosophical Transactions of the RoyalSociety of London A: Mathematical, Physical andEngineering Sciences 359, 1783 (2001), 1233–1250.
7. Fiorentino, N. M., and Blemker, S. S. Musculotendonvariability influences tissue strains experienced by thebiceps femoris long head muscle during high-speedrunning. Journal of Biomechanics 47, 13 (2014),3325–3333.
8. Fiorentino, N. M., Rehorn, M. R., Chumanov, E. S.,Thelen, D. G., and Blemker, S. S. Computational modelspredict larger muscle tissue strains at faster sprintingspeeds. Medicine and Science in Sports and Exercise 46,4 (2014), 776–786.
9. Gardiner, J. C., and Weiss, J. A. Subject-specific finiteelement analysis of the human medial collateralligament during valgus knee loading. Journal ofOrthopaedic Research 21, 6 (2003), 1098–1106.
10. Gladilin, E., Zachow, S., Deuflhard, P., and Hege, H.-C.A biomechanical model for soft tissue simulation incraniofacial surgery. In Medical Imaging andAugmented Reality, 2001. Proceedings. InternationalWorkshop on, IEEE (2001), 137–141.
11. Guccione, J. M., Costa, K. D., and McCulloch, A. D.Finite element stress analysis of left ventricularmechanics in the beating dog heart. Journal ofBiomechanics 28, 10 (1995), 1167–1177.
12. Handsfield, G. G., Meyer, C. H., Hart, J. M., Abel,M. F., and Blemker, S. S. Relationships of 35 lower limbmuscles to height and body mass quantified using MRI.Journal of Biomechanics 47, 3 (2014), 631–638.
13. Hung, A. P. L., Wu, T., Hunter, P., and Mithraratne, K. Aframework for generating anatomically detailedsubject-specific human facial models for biomechanicalsimulations. The Visual Computer (2014), 1–13.
14. Inouye, J. M., Pelland, K., Lin, K. Y., Borowitz, K., andBlemker, S. S. A Computational Model ofVelopharyngeal Closure for Simulating Cleft PalateRepair. Journal of Craniofacial Surgery 26, 3 (2015),658–662.
15. Kermarrec, E., Budzik, J.-F., Khalil, C., Le Thuc, V.,Hancart-Destee, C., and Cotten, A. In vivo diffusiontensor imaging and tractography of human thighmuscles in healthy subjects. American Journal ofRoentgenology 195, 5 (2010), W352—-W356.
16. Kim, S. Y., Boynton, E. L., Ravichandiran, K., Fung,L. Y., Bleakney, R., and Agur, A. M. Three-dimensionalstudy of the musculotendinous architecture ofsupraspinatus and its functional correlations. ClinicalAnatomy 20, 6 (2007), 648–655.
17. Kurihara, T., Oda, T., Chino, K., Kanehisa, H.,Fukunaga, T., and Kawakami, Y. Use ofthree-dimensional ultrasonography for the analysis ofthe fascicle length of human gastrocnemius muscleduring contractions. International Journal of Sport andHealth Science 3, Special Issue 2005 (2005), 226–234.
18. Limbert, G., Taylor, M., and Middleton, J.Three-dimensional finite element modelling of thehuman ACL: simulation of passive knee flexion with astressed and stress-free ACL. Journal of Biomechanics37, 11 (2004), 1723–1731.
19. Muramatsu, T., Muraoka, T., Kawakami, Y., Shibayama,A., and Fukunaga, T. In vivo determination of fasciclecurvature in contracting human skeletal muscles.Journal of Applied Physiology 92, 1 (2002), 129–134.
20. Prot, V., Skallerud, B., and Holzapfel, G. A.Transversely isotropic membrane shells with application
to mitral valve mechanics. Constitutive modelling andfinite element implementation. International Journal forNumerical Methods in Engineering 71, 8 (2007),987–1008.
21. Public scientific review by Dr. Jacob Barhak.https://goo.gl/VabLSF, Last accessed May 2015.
22. Public scientific review by Dr. Tim Wu.https://goo.gl/FUhPnb, Last accessed May 2015.
23. Rana, M., and Wakeling, J. M. In-vivo determination of3D muscle architecture of human muscle using freehand ultrasound. Journal of Biomechanics 44, 11(2011), 2129–2135.
24. Rehorn, M. R., and Blemker, S. S. The effects ofaponeurosis geometry on strain injury susceptibilityexplored with a 3D muscle model. Journal ofBiomechanics 43, 13 (2010), 2574–2581.
25. Sakamoto, Y. Spatial relationship between thepalatopharyngeus and the superior constrictor of thepharynx. Surgical and Radiologic Anatomy (2015), 1–7.
26. Slane, L. C., and Thelen, D. G. Non-uniformdisplacements within the Achilles tendon observedduring passive and eccentric loading. Journal ofBiomechanics 47, 12 (2014), 2831–2835.
27. Spilker, R. L., Donzelli, P. S., and Mow, V. C. Atransversely isotropic biphasic finite element model ofthe meniscus. Journal of Biomechanics 25, 9 (1992),1027–1045.
28. Supplementary information.http://simtk.org/home/m3lab cfd4fea, Last accessedMay 2015.
29. Ward, S. R., Eng, C. M., Smallwood, L. H., and Lieber,R. L. Are current measurements of lower extremitymuscle architecture accurate? Clinical Orthopaedicsand Related Research 467, 4 (2009), 1074–1082.
30. Webb, J. D., Blemker, S. S., and Delp, S. L. 3D finiteelement models of shoulder muscles for computing linesof actions and moment arms. Computer Methods inBiomechanics and Biomedical Engineering 17, 8(2014), 829–837.
31. White, J. W. Torsion of the Achilles tendon: its surgicalsignificance. Archives of Surgery 46, 5 (1943), 784.
32. Wu, T., Hung, A. P. L., Hunter, P., and Mithraratne, K.Modelling facial expressions: A framework forsimulating nonlinear soft tissue deformations usingembedded 3D muscles. Finite Elements in Analysis andDesign 76 (Nov. 2013), 63–70.
33. Yin, L., and Elliott, D. M. A biphasic and transverselyisotropic mechanical model for tendon:: application tomouse tail fascicles in uniaxial tension. Journal ofBiomechanics 37, 6 (2004), 907–916.
