Research Papers

Optimization of Nonlinear Hyperelastic Coefficients for Foot Tissues Using a Magnetic Resonance Imaging Deformation Experiment

[+] Author and Article Information
Marc Petre

Division of Anesthesiology and Critical Care Medicine,
Cleveland Clinic,
Cleveland, OH 44195

Ahmet Erdemir

Computational Biomodeling (CoBi) Core,
Lerner Research Institute,
Department of Biomedical Engineering,
Cleveland Clinic,
Cleveland, OH 44195

Vassilis P. Panoskaltsis

Department of Civil Engineering,
Demokritos University of Thrace,
Xanthi, 67100 Greece

Peter R. Cavanagh

e-mail: cavanagh@uw.edu
Department of Orthopaedics and Sports Medicine,
University of Washington,
Seattle, WA 98195

1Corresponding author.

Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received November 18, 2011; final manuscript received January 15, 2013; accepted manuscript posted February 19, 2013; published online May 9, 2013. Editor: Victor H. Barocas.

J Biomech Eng 135(6), 061001 (May 09, 2013) (12 pages) Paper No: BIO-11-1481; doi: 10.1115/1.4023695 History: Received November 18, 2011; Revised January 15, 2013; Accepted February 19, 2013

Accurate prediction of plantar shear stress and internal stress in the soft tissue layers of the foot using finite element models would provide valuable insight into the mechanical etiology of neuropathic foot ulcers. Accurate prediction of the internal stress distribution using finite element models requires that realistic descriptions of the material properties of the soft tissues are incorporated into the model. Our investigation focused on the creation of a novel three-dimensional (3D) finite element model of the forefoot with multiple soft tissue layers (skin, fat pad, and muscle) and the development of an inverse finite element procedure that would allow for the optimization of the nonlinear elastic coefficients used to define the material properties of the skin muscle and fat pad tissue layers of the forefoot based on a Ogden hyperelastic constitutive model. Optimization was achieved by comparing deformations predicted by finite element models to those measured during an experiment in which magnetic resonance imaging (MRI) images were acquired while the plantar surface forefoot was compressed. The optimization procedure was performed for both a model incorporating all three soft tissue layers and one in which all soft tissue layers were modeled as a single layer. The results indicated that the inclusion of multiple tissue layers affected the deformation and stresses predicted by the model. Sensitivity analysis performed on the optimized coefficients indicated that small changes in the coefficient values (±10%) can have rather large impacts on the predicted nominal strain (differences up to 14%) in a given tissue layer.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.


Budhabhatti, S. P., Erdemir, A., Petre, M., Sferra, J., Donley, B., and Cavanagh, P. R., 2007, “Finite Element Modeling of the First Ray of the Foot: A Tool for the Design of Interventions,” ASME, J. Biomech. Eng., 129(5), pp. 750–756. [CrossRef]
Cheung, J. T., An, K. N., and Zhang, M., 2007, “Consequences of Partial and Total Plantar Fascia Release: A Finite Element Study,” Foot Ankle Int., 27(2), pp. 125–132. Available at: http://fai.sagepub.com/content/27/2/125.full.pdf
Goske, S., Erdemir, A., Petre, M., Budhabhatti, S., and Cavanagh, P. R., 2006, “Reduction of Plantar Heel Pressures: Insole Design Using Finite Element Analysis,” J. Biomech., 39(13), pp. 2363–2370. [CrossRef] [PubMed]
Erdemir, A., Saucerman, J. J., Lemmon, D., Loppnow, B., Turso, B., Ulbrecht, J. S., and Cavanagh, P. R., 2005, “Local Plantar Pressure Relief in Therapeutic Footwear: Design Guidelines From Finite Element Models,” J. Biomech., 38(9), pp. 1798–1806. [CrossRef] [PubMed]
Mackey, J. R., and Davis, B. L., 2006, “Simultaneous Shear and Pressure Sensor Array for Assessing Pressure and Shear at Foot/Ground Interface,” J. Biomech., 39(15), pp. 2893–2897. [CrossRef] [PubMed]
Yavuz, M., Erdemir, A., Botek, G., Hirschman, G. B., Bardsley, L., and Davis, B. L., 2007, “Peak Plantar Pressure and Shear Locations: Relevance to Diabetic Patients,” Diabetes Care, 30(10), pp. 2643–2645. [CrossRef] [PubMed]
Yavuz, M., Botek, G., and Davis, B. L., 2007, “Plantar Shear Stress Distributions: Comparing Actual and Predicted Frictional Forces at the Foot-Ground Interface,” J. Biomech., 40(13), pp. 3045–3049. [CrossRef] [PubMed]
Reiber, G. E., 2001, “Epidemiology of Foot Ulcers and Amputations in the Diabetic Foot,” In The Diabetic Foot, Mosby, St. Louis, MO, pp. 13–22.
Lemmon, D., Shiang, T. Y., Hashmi, A., Ulbrecht, J. S., and Cavanagh, P. R., 1997, “The Effect of Insoles in Therapeutic Footwear—A Finite Element Approach,” J. Biomech., 30(6), pp. 615–620. [CrossRef] [PubMed]
Chen, W. P., Ju, C. W., and Tang, F. T., 2003, “Effects of Total Contact Insoles on the Plantar Stress Redistribution: A Finite Element Analysis,” Clin. Biomech. Bristol, 18(6), pp. S17–S24. [CrossRef]
Cheung, J. T. M., and Zhang, M., 2005, “A 3-Dimensional Finite Element Model of the Human Foot and Ankle for Insole Design,” Arch. Phys. Med. Rehabil., 86(2), pp. 353–358. [CrossRef] [PubMed]
Jacob, S., and Patil, M. K., 1999, “Stress Analysis in Three-Dimensional Foot Models of Normal and Diabetic Neuropathy,” Front Med. Biol. Eng., 9(3), pp. 211–227. [PubMed]
Spears, I. R., Miller-Young, J. E., Sharma, J., Ker, R. F., and Smith, F. W., 2007, “The Potential Influence of the Heel Counter on Internal Stress During Static Standing: A Combined Finite Element and Positional MRI Investigation,” J. Biomech., 40(12), pp. 2774–2780. [CrossRef] [PubMed]
Actis, R. L., Ventura, L. B., Smith, K. E., Commean, P. K., Lott, D. J., Pilgram, T. K., and Mueller, M. J., 2006, “Numerical Simulation of the Plantar Pressure Distribution in the Diabetic Foot During the Push-Off Stance,” Med. Biol. Eng. Comput., 44(8), pp. 653–663. [CrossRef] [PubMed]
Gefen, A., 2003, “Plantar Soft Tissue Loading Under the Medial Metatarsals in the Standing Diabetic Foot,” Med. Eng. Phys., 25(6), pp. 491–499. [CrossRef] [PubMed]
Ledoux, W. R., and Blevins, J. J., 2007, “The Compressive Material Properties of the Plantar Soft Tissue,” J. Biomech., 40(13), pp. 2975–2981. [CrossRef] [PubMed]
Miller-Young, J. E., Duncan, N. A., and Baroud, G., 2002, “Material Properties of the Human Calcaneal Fat Pad in Compression: Experiment and Theory,” J. Biomech., 35(12), pp. 1523–1531. [CrossRef] [PubMed]
Bosboom, E. M., Hesselink, M. K., Oomens, C. W., Bouten, C. V., Drost, M. R., and Baaijens, F. P., 2001, “Passive Transverse Mechanical Properties of Skeletal Muscle Under In Vivo Compression,” J. Biomech., 34(10), pp. 1365–1368. [CrossRef] [PubMed]
Oomens, C. W., Bressers, O. F., Bosboom, E. M., Bouten, C. V., and Blader, D. L., 2003, “Can Loaded Interface Characteristics Influence Strain Distributions in Muscle Adjacent to Bony Prominences,” Comput. Methods Biomech. Biomed. Eng., 6(3), pp. 171–180. [CrossRef]
Zobitz, M. E., Luo, Z. P., and An, K. N., 2001, “Determination of the Compressive Material Properties of the Suprasinatus Tendon,” ASME, J. Biomech. Eng., 123(1), pp. 47–51. [CrossRef]
Petre, M., Erdemir, A., and Cavanagh, P. R., 2008, “An MRI-Compatible Foot-Loading Device for Assessment of Internal Tissue Deformation,” J. Biomech., 41(2), pp. 470–474. [CrossRef] [PubMed]
Smith, K. E., Commean, P. K., Mueller, M. J., Roberston, D. D., Pilgram, T., and Johnson, J., 2000, “Assessment of the Diabetic Foot Using Spiral Computed Tomography Imaging and Plantar Pressure Measurements; A Technical Report,” J. Rehabil. Res. Dev., 37(1), pp. 31–40. [PubMed]
Cavanagh, P. R., 1999, “Plantar Soft Tissue Thickness During Ground Contact in Walking,” J. Biomech., 32(6), pp. 623–628. [CrossRef] [PubMed]
Gefen, A., Megido-Ravid, M., Azariah, M., Itzchak, Y., and Arcan, M., 2001, “Integration of Plantar Soft Tissue Stiffness Measurements in Routine MRI of the Diabetic Foot,” Clin. Biomech. Bristol, 16(10), pp. 921–925. [CrossRef]
Cheung, Y. Y., Doyley, M., Miller, T. B., Kennedy, F., Lynch, F., Wrobel, J. S., Paulson, K., and Weaver, J., 2006, “Magnetic Resonance Elastography of the Plantar Fat Pads: Preliminary Study in Diabetic Patients and Asymptomatic Volunteers,” J. Comput. Assist Tomogr., 30(2), pp. 321–326. [CrossRef] [PubMed]
Weaver, J. B., Doyley, M., Cheung, Y., Kennedy, F., Madsen, E. L., Van Houten, E. E., and Paulsen, K., 2005, “Imaging the Shear Modulus of the Heel Fat Pads,” Clin. Biomech. Bristol, 20(3), pp. 312–319. [CrossRef]
Bus, S. A., Maas, M., Cavanagh, P. R., Michels, R. P. J., and Levi, M., 2004, “Plantar Fat-Pad Displacement in Neuropathic Diabetic Patients With Toe Deformity: A Magnetic Resonance Imaging Study,” Diabetes Care, 27(10), pp. 2376–2381. [CrossRef] [PubMed]
Erdemir, A., Viveiros, M. L., Ulbrecht, J. S., and Cavanagh, P. R., 2006, “An Inverse Finite-Element Model of Heel-Pad Indentation,” J. Biomech., 39(7), pp. 1279–1286. [CrossRef] [PubMed]
Abaqus 6.10 User Manual http://simulia.com.
Petre, M., Wyllie, J., Sferra, J., Donley, B., Erdemir, A., Doehring, T., Considine, J., Thangudu, P., and Cavangh, P. R., 2006, “Normal Sesamoid Motion During First MTPH Dorisiflexion,” American Academy of Orthopaedic Surgeons Annual Meeting, Chicago, IL.
Ogden, R. W., Non-Linear Elastic Deformations, Ellis Horwood, Chinchester.
Criscione, J. C., 2003, “Rivlin's Representation Formula Is Ill-Conceived for the Determination of Response Functions Via Biaxial Testing,” J. Elasticity, 70, pp. 129–147. [CrossRef]
Belegundu, A. D., and Chandrupatla, T. R., 1999, Optimization Concepts and Applications in Engineering, Prentice Hall, Upper Saddle River, NJ.
Budhabhatti, S. P., Erdemir, A., and Cavangh, P. R., 2005, “Influence of Foot Orientation and Bone Structure on Plantar Pressure Distribution,” XXth Congress of the International Society of Biomechanics/29th Annual Meeting of the American Society of Biomechanics, Cleveland, OH.
Wu, J. Z., Dong, R. G., Smutz, W. P., and Schopper, A. W., 2003, “Nonlinear and Viscoelastic Characteristics of Skin Under Compression; Experiment and Analysis,” Biomed. Mater. Eng., 13(4), pp. 373–385. [PubMed]
Natali, A. N., Fontanella, C. G., and Carniel, E. L., 2012, “A Numerical Model for Investigating the Mechanics of Calcaneal Fat Pad Region,” J. Mech. Behavior Biomed. Mater., 5, pp. 216–223. [CrossRef]
Sharafi, B. and Blemker, S. S., 2010, “A Micromechanical Model of Skeletal Muscle to Explore the Effects of Fiber and Fascicile Geometry,” J. Biomech., 43, pp. 3207–3213. [CrossRef] [PubMed]
Chokhandre, S., Halloran, J. P., van denBogert, A. J., and Erdemir, A., 2012, “A Three-Dimensional Inverse Finite Element Analysis of the Heel Pad,” J. Biomech. Eng., 134, pp. 1–9. [CrossRef]
Veress, A.I., Gullberg, G. T., and Weiss, J. A., 2005, “Measurement of Strain in the Left Ventricle During Diastole With Cine-MRI and Deformable Image Registration,” ASME, J. Biomech. Eng., 127(7), pp. 1195–1207. [CrossRef]
Petre, M. T., Erdemir, A., and Cavanagh, P. R., 2006, “Determination of Elastomeric Foam Parameters for Simulations of Complex Loading,” Comput. Methods Biomech. Biomed. Eng., 9(4), pp. 231–242. [CrossRef]


Grahic Jump Location
Fig. 1

The foot-loading device used during collection of MRI deformation data. For a more thorough description, see Petre et al. [21].

Grahic Jump Location
Fig. 2

Top: MRI image sets collected in the unloaded state and with 12.5% and 50% of the subject's body weight applied to the forefoot. Middle: The 3D nature of the data allows for visualization and analysis of the tissue deformation along any plane. Bottom: Frontal plane sections through the approximate metatarsal head region. Note that most of the deformation has already occurred when only one quarter of the total load (12.5% BW) was applied.

Grahic Jump Location
Fig. 3

(a) The outer surface of the 3D finite element forefoot mesh. (b) The FE mesh with the toes, skin, and muscle removed to display the bones and plantar fat. Note that the phalanges of each toe were fused for this study. (c) Joint definitions used to describe model bone kinematics. Open circles represent the locations of joint centers of rotation. The midfoot node was fixed in space throughout simulation of the loading experiment (see Fig. 1). (d) FE model of the MRI loading experiment incorporating the 3D, layered tissue FE mesh. The midfoot and cast were fixed during the simulation and forces were applied to the rigid plate, as in the MRI experiment.

Grahic Jump Location
Fig. 4

An exemplar slice from the undeformed MRI image set (a). The same slice showing the segmentation of structures; red denoting skin, blue, fat pad, and green, bone, where all remaining tissue was classified as muscle (b). The recolored ternary image based on the segmentation of structures (c). It should be noted that the tissue on the dorsal side of the foot was considered to be either muscle or skin only to ease in the construction of the FE mesh as this simplification did not impact the results of the FEM model prediction of plantar pressure.

Grahic Jump Location
Fig. 5

A schematic description of the mesh overlay process in the region of interest shown at the top. (a) Each node of the undeformed FE mesh is assigned a pixel intensity value corresponding to its location in the undeformed MRI image set. This pixel value is fixed to the node as it undergoes deformation during simulation of the experiment. The deformed mesh can be used to generate a (relatively coarse) mode-predicted deformed image that can be compared to the experimental MRI images on a node by node basis. (b) If the current material coefficients are, for example, stiffer than the real material, then the mesh will not deform as much as the tissue and the node intensity values will fail to correlate with the deformed MRI image set. (c) If the material properties governing node motion correctly match the tissue, then the deformed mesh will correlate well with the deformed image set. Although this process is shown here in 2D with 8 nodes, the technique was applied in 3D allowing for out-of-plane deformations of more than 17,000 nodes induced by nearly incompressible materials.

Grahic Jump Location
Fig. 6

Schematic diagram of the inverse finite element technique used to determine material coefficients. Inputs to the technique (at the left) are deformed and undeformed MRI image sets, undeformed FE mesh, and an initial guess for material coefficients. The mesh overlay process, used to determine the pixel value at each node, is further described in Fig. 5. The error calculation and its inputs are described with Eq. (3). Briefly, each vector In,f contains the corresponding image intensities for each of the n nodes at loading level f. Error is calculated by comparing predicted intensities at each of the loading levels to the intensities in the unloaded state In,0.

Grahic Jump Location
Fig. 7

Qualitative comparison of the deformations predicted in the region of the metatarsal heads when a compressive load equal to 50% body weight was applied to the model versus deformations observed in MRI images when the equivalent load was applied to the imaged forefoot

Grahic Jump Location
Fig. 8

Overall deformation (nominal strain, bottom row) of the forefoot at full 50% BW loading using optimized coefficients for layered tissue representation (best fit, see Table 1 for coefficients), along with hydrostatic pressure (middle row), and von Mises stress distribution (top row) within tissue layers. The loading and boundary conditions were representative of the MRI experimentation. The forefoot model was sectioned at an approximate level of metatarsal heads to emphasize a potential region of clinical interest. Distributions of all strain components illustrate the complex nature of the combined deformation modes in the foot. See text for further discussion.

Grahic Jump Location
Fig. 9

Clinically relevant mechanical variables were not highly sensitive to increased mesh density. While a coarser mesh was used for the identification of material coefficients (left: 26,335 elements, 29,697 nodes), a higher mesh density may be desirable for convergence (right: 39,675 elements, 43,893 nodes). Plantar pressures and von Mises stress distribution are plotted. Both the magnitude and location of peaks were similar between the predictions of coarse and fine meshes as obtained by using the best material coefficient fit (see Table 1 for coefficients). Loading and boundary conditions of these simulations approximated those of peak forefoot loading instant of the push-off phase.

Grahic Jump Location
Fig. 10

Comparison of clinically relevant mechanical variables predicted when tissue layer material properties were incorporated in the model (left) and when lumped material properties were used (right). Top row illustrates similar predicted plantar pressure distributions in both models. However, the lumped material model predicted higher magnitudes of pressure. Middle row illustrates the comparison of von Mises stresses in the region under the metatarsal heads. While stresses are concentrated under the third metatarsal head in both models, stress distribution is more uniform in the lumped model (right) than in the layered model (left) where reduced levels of stress are observed in the highly compliant fat pad tissue layer. Bottom row illustrates the comparison of maximum principal strains in the region under the metatarsal heads. Peak strain values were concentrated under the first metatarsal head in the layered model (Left) and under the third metatarsal head in the lumped model (right). Strain distribution was more uniform in the lumped model (right) than in the layered model where high strain values were observed throughout the highly compliant fat pad layer.



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In