Finite Element Modeling of the Human Foot and Footwear Man Cheung - - PowerPoint PPT Presentation

Finite Element Modeling of the Human Foot and Footwear

Jason Jason Tak Tak-

• Man Cheung

Man Cheung1,2

1,2, Ph.D.

, Ph.D. Ming Zhang Ming Zhang1

1, Ph.D.

, Ph.D.

1Department of Health Technology & Informatics,

The Hong Kong Polytechnic University, Hong Kong, China

2Human Performance Laboratory,

University of Calgary, Calgary, Alberta, Canada

Department of Health Technology and Informatics 醫療科技及資訊學系

Common Foot Problems

Calluses Corns http://www.foot.com Bunions Hammertoe Claw Toe Mallet Toe Metatarsalgia Achilles Tendonitis Plantar Fasciitis Heel Spurs Calluses Corns http://www.foot.com Bunions Hammertoe Claw Toe Mallet Toe Metatarsalgia Achilles Tendonitis Plantar Fasciitis Heel Spurs

Why Finite Element (FE) Approach?

• Experimental measurements of the biomechanical

variables such as joint motion and load distribution are costly and difficult for the ankle-foot complex.

• Finite element method allows

– predictions of joint motion, load distribution between the foot and supports and in bony and soft tissue structures. – efficient parametrical analyses of loading conditions, structural and material variables.

Summary on FE Analysis on Foot & Footwear

Previous FE foot models

• have shown the contributions to the understanding of

biomechanics of the foot and footwear

• were developed under certain simplifications

(Simplified or partial foot structures, assumptions of linear material properties, simplified loading and boundary conditions).

Bandak et al (2001), Camacho et al (2002), Chen et al (2003), Chu et al (1995), Erdemir et al (2005), Gefen et al (2000), Goske et al (2005), Jacob & Patil (1999), Lemmon et al (1997), Shiang (1997).

Objectives

• To develop a comprehensive 3D FE model to

quantify the biomechanical response of the human foot and ankle (joint motion, load distribution of bony and soft tissue structures and foot-support interface).

• To provide a systematic tool for the parametric

analyses of different foot structures, surgical and footwear performances.

Development of the Finite Element Model

• Coronal MR images of 2mm

intervals obtained from the right foot of a healthy male subject in unloaded, neutral position

3D Reconstruction of Foot Structures

Boundaries for Foot Bones Boundary for Soft Tissue

Segmentation (Mimics v7.10, Materialise.) Surface Model Solid Model

(SolidWorks v2001, SolidWorks Corp.)

Finite Element Mesh of Bony and Soft Tissue Structures

Automatic mesh creation in ABAQUS v6.4, HKS.

Anatomical References of the Ligaments

Interactive Foot & Ankle, Ver.1.0.0, Primal Picture Ltd. Interactive Foot & Ankle, Ver.1.0.0, Primal Picture Ltd.

Structural Components of the FE Model

• 28 bones embedded in a volume of soft tissue

(Tetrahedral elements)

• 72

associated ligaments (excluding the ligaments between the toes) and the plantar fascia (Tension-only truss elements)

Joint Articulations of the Model

• The phalanges were connected together using

2 mm thick structural elements to simulate the connections.

• The

interaction between the metatarsals, cuneiforms, cuboid, navicular, talus, calcaneus, tibia and fibula were defined by contact surfaces with a prescribed contacting stiffness of articular cartilage to allow relative bone movement.

Material Properties of Ankle-Foot Model

Encapsulated soft tissue (Hyperelastic) Bony & ligamentous structures (Homogeneous, Linearly elastic)

Component Element Type Young’s Modulus E (MPa) Poisson’s Ratio ν Cross-sectional Area (mm2) Bony Structures 3D-Tetrahedra 7,300 0.3

• Soft Tissue

3D-Tetrahedra Hyperelastic

• Cartilage

3D-Tetrahedra 1 0.4

• Ligaments

Tension-only Truss 260

• 18.4

Fascia Tension-only Truss 350

• 58.6

Nakamura et al., 1981 (Bone); Lemmon et al., 1997 (soft tissue); Athanasiou et al., 1998 (Cartilage); Siegler et al., 1988 (ligaments); Wright and Rennels, 1964 (Plantar Fascia).

Hyperelastic Material Model for Soft Tissue

2 i e l 2 1 i i j i 2 1 j i i j

( J D ) I ( I ( C ) 1 1 3 ) 3 U

2 __ 1 __

− + − − =

∑ ∑

= = +

where U is the second-order strain energy per unit of reference volume; Cij and Di are material parameters;

1 __

I

2 __

I

and are the first and second deviatoric strain invariants:

I

2 3 __ 2 2 __ 2 1 __ 1 __

λ + λ + λ = I

) 2 ( 3 __ ) 2 ( 2 __ ) 2 ( 1 __ 2 __ − − −

λ + λ + λ =

with the deviatoric stretches

i __

λ

= Jel

• 1/3 λi ;

Jel and λi are the elastic volume ratio & the principal stretches.

ABAQUS v6.4, Hibbitt, Karlsson & Sorensen, Inc.

Application of Loading and Boundary Conditions

Fixed Surfaces Connector Elements for Muscles Force Application Moving Support for Foot-Insole Interface and Ground Reaction Force Application

References for Muscular Insertion Points

Interactive Foot & Ankle, Ver.1.0.0, Primal Picture Ltd., UK, 1999

Muscles and Ground Reaction Forces for Standing and Midstance Simulation

Tendon/External Forces Standing Midstance Achilles 175N

• Reaction of Lateral Retinaculum
• 50N

Reaction of Medial Retinaculum

• 60N

350N 750 Tibialis Posterior 70N Flexor Hallucis Longus 35N Flexor Digitorum Longus 40N Peroneus Brevis 40N Peroneus Longus 35N Vertical Ground Reaction 550N

The active extrinsic muscles forces during midstance were estimated from normalized EMG data using a constant muscle gain and cross-sectional area relationship (Dul, 1983; Kim et al., 2001; Perry, 1992).

Simulation of Midstance Contact

10 Degrees

Plantar Pressure

F-scan Measurement FE Prediction

MPa MPa Contact Area 68.8 cm2 Contact Area 68.3 cm2

Predicted Von Mises Stress of Bony and Ligamentous Structures

MPa

Plantar View Dorsal View

Parametrical Studies

• Effect of plantar fascia stiffness (E = 0 to 700 MPa).
• Effect of plantar soft tissue stiffness.
• Effect of Achilles tendon loading.
• Effect of posterior tibial tendon dysfunction.
• Effect of different parametrical design of foot orthoses.

The Plantar Fascia and Plantar Ligaments

Plantar fascia Long plantar lig. Short plantar lig. Spring lig.

Effect of varying Young’s modulus of fascia

• n arch height and arch length

37 38 39 40 41 42 43 44 175 350 525 700 Young's Modulus of Fascia, MPa Arch Height, mm Arch Height 141 142 143 144 145 146 147 148 149 175 350 525 700 Young's Modulus of Fascia, MPa Arch Length, mm Arch Length

Deformed Arch Height (42.5 mm) FE (44 mm) Measured Unloaded Arch Height (52.5 mm)

Effect of varying Young’s modulus of fascia

• n the tensions of the ligamentous structures

Plantar fascia – Major arch-supporting ligamentous structure

sustaining tension ~45% of applied body weight short plantar lig. > long plantar lig. > spring lig. Tension of plantar ligaments

50 100 150 200 175 350 525 700 Young's Modulus of Fascia, MPa Fascia Tension, N Total Tension

50 100 150 175 350 525 700 Young's Modulus of Fascia, MPa Ligament Tension, N Long Plantar Lig. Short Plantar Lig. Spring Lig.

With Fasciotomy

Clinical Implications

• The plantar fascia is one of the major stabilizers of the

longitudinal arch of the foot.

• Laceration or surgical dissection of plantar fascia may

induce excessive loading in the ligamentous and bony structures.

• Surgical release of the plantar fascia should be well-

planned to minimize the effect on its structural integrity to reduce the risk of possible post-operative complications.

Parametrical Studies

• Effect of plantar fascia stiffness.
• Effect of plantar soft tissue stiffening (Up to 5 times).
• Effect of Achilles tendon loading.
• Effect of posterior tibial tendon dysfunction.
• Effect of different parametrical design of foot orthoses.

Simulation of Stiffened Soft Tissue

0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 Strain Stress (MPa)

F5 F3 F2 Normal

Nonlinear compressive stress-strain response of plantar soft tissue was adopted from the in-vivo measurements (Lemmon et al., 2002). F2, F3 and F5 correspond to simulations of two, three and five times the stiffness of normal tissue. Pathologically stiffened tissue with increasing stages of diabetic neuropathy (Klaesner et al., 2002; Gefen et al., 2001).

Effect of Soft Tissue Stiffening on Plantar Pressure Distribution

5 Times 3 Times MPa MPa 2 Times Normal MPa MPa Peak 0.230 MPa Peak 0.263 MPa Peak 0.291 MPa Peak 0.306 MPa

Increasing Soft Tissue Stiffness

Effect of Soft Tissue Stiffening on Peak Plantar Pressure and Contact Area

20 40 60 80 1 2 3 4 5 Factor of Soft Tissue Stiffening Contact Area (cm

2)

ForeFoot MidFoot RearFoot WholeFoot

0.1 0.2 0.3 0.4 1 2 3 4 5 Factor of Soft Tissue Stiffening Peak Pressure (MPa)

ForeFoot MidFoot RearFoot

Five times Heel (33%), Forefoot (35%) 47% Soft tissue stiffness Peak Plantar Pressure Contact Area

Clinical Implications

• Stiffening of plantar soft tissue may induce excessive

pressure in the plantar foot – possible link to tissue breakdown and foot ulceration.

• The percentage increase in peak plantar pressure is less

pronounced than the increase in soft tissue stiffness.

• Screening of plantar soft tissue stiffness can be a viable

method in addition to plantar pressure measurement for routine identification of diabetic feet at risk of ulceration.

Parametrical Studies

• Effect of plantar fascia stiffness.
• Effect of plantar soft tissue stiffening.
• Effect of Achilles tendon loading (0 to 700 N).
• Effect of posterior tibial tendon dysfunction.
• Effect of different parametrical design of foot orthoses.

Simulated Conditions

(1) Pure Compression –

Vertical compression up to 700 N.

(2) Compression with Achilles tendon loading –

Vertical compression preload of 350 N with an increasing Achilles tendon tension up to 700 N.

Six nonpaired fresh cadaveric ankle-foot specimens

– Middle-aged male donors – Unknown body masses – Average foot length: 24.2 cm – Average foot width: 9.4 cm – Kept under -20 0C before experiment

Cadaveric Experiment

Specimen Preparation

After thawing at room temperature

• Skin, subcutaneous tissues and muscles above the ankle joint

level dissected with all muscular tendons left intact

• Distal fibula and tibia potted in acrylic resin

Compression Test of Cadaveric Foot

F-scan pressure sensor (Tekscan, Inc.) Implanted displacement transducer (Microstrain, Inc.) Load cell (MTS Systems)

Cadaveric Foot under Vertical Compression up to 700 N

Vertical Deformation and Plantar Fascia Strain under Vertical Compression

2 4 6 8 10 100 200 300 400 500 600 700 Vertical compression, N Vertical deformation, mm

Specimen_1 Specimen_2 Specimen_3 Specimen_4 Specimen_5 Specimen_6 FE

1 2 3 4 100 200 300 400 500 600 700 Vertical compression, N Strain of plantar fascia, %

Specimen_2 Specimen_3 Specimen_4 Specimen_5 Specimen_6 FE_Average FE_Max

Displacement Fascia Strain Fascia strain (>100N) ICC (Consistency) : 0.892 0.880 0.994

Effects of Vertical Compression and Achilles Tendon Loading on the Plantar Fascia Tension

50 100 150 200 250 300 350 100 200 300 400 500 600 700 Vertical compressive/Achilles tendon forces, N Total fascia forces, N

Vertical compressive forces (0-700N) 350N compression preload + Achilles tendon forces (0-700N)

Clinical Implications

• Achilles tendon loading produces a greater straining

effect on plantar fascia than the weight on the foot.

• Overstretching of the Achilles tendon is plausible

mechanical factors for overloading the plantar fascia.

• Lengthening or tension relief of the Achilles tendon

especially in subjects with tight calf muscles and Achilles tendon may be beneficial in terms of plantar fascia stress relief.

Parametrical Studies

• Effect of plantar fascia stiffness, partial and total plantar

fascia release.

• Effect of plantar soft tissue stiffness.
• Effect of Achilles tendon loading.
• Effect of posterior tibial tendon dysfunction.
• Effect of different parametrical design of foot orthoses.

Simulated Conditions

Intact PTTD PTTD + Fasciotomy Fasciotomy

Simulations of Fasciotomy

Intact Fasciotomy

Experimental Setup

F-scan Pressure Sensor (Plantar foot pressure) Bone Marker (Joint movement) Displacement Transducer (Microstrain, Inc.) (Fascia strain) Tendon Clamp (Muscle forces application)

Stance Phase Simulation

3D Laser Scanner Deadweights Marker Scanning

(Realscan USB 200, 3D Digital Corp.)

Predicted Changes in Arch Height

3.7 3.8 3.9 4 4.1 4.2 4.3 Intact PTTD PFR PTTD+PFR

Simulated Conditions Arch Height, cm

Effect of PTTD on Plantar Fascia Strain

0.5 1 1.5 2 2.5 Intact PTTD Intact PTTD

FE Prediction Measurement Fascia Strain, %

Predicted Changes in Fascia & Ligaments Tension

50 100 150 200 250 300 350 400 450 Intact PTTD PFR PTTD+PFR

Simulated Conditions Tension, N

Plantar Fascia Long Plantar Lig. Short Plantar Lig. Spring Lig.

Prediction of Joint Motion

Effect of PTTD & PFR on Joint Motion

Relative Bones Intact with PTTD Intact with PFR PFR with PTTD Talus to Tibia Plantar Flexion Eversion External Rotation Plantar Flexion Eversion Internal Rotation Eversion External Rotation External Rotation External Rotation Internal Rotation Eversion Inversion Plantar Flexion Eversion Internal Rotation Calcaneus to Talus Dorsi Flexion Eversion External Rotation Inversion Plantar Flexion Dorsi Flexion Dorsi Flexion Dorsi Flexion Dorsi Flexion Inversion External Rotation Navicular to Talus Dorsi Flexion Eversion Internal Rotation Dorsi Flexion Eversion External Rotation 1st Metatarsal to Navicular Plantar Flexion Inversion Internal Rotation Dorsi Flexion Eversion Internal Rotation 1st Metatarsal to Talus Dorsi Flexion Eversion External Rotation Dorsi Flexion Eversion External Rotation

FE Prediction

67% 78% 44% 22% 56%

Green: Agreement with cadaveric studies Red: Disagreement agreement with cadaveric studies Percentage of Agreement (%) Sagittal plane Coronal Plane Transverse Plane 73% 60% 27%

Clinical Implications

• Both PFR and PTTD decreased the arch height and

resulted in foot pronation.

• PFR in general have a greater arch flattening effect than

PTTD.

• The lack of foot arch support with PFR and PTTD may lead

to attenuation of surrounding soft tissue structures and progressive elongation and flattening of foot arch.

Parametrical Studies

• Effect of plantar fascia stiffness, partial and total plantar

fascia release.

• Effect of plantar soft tissue stiffness.
• Effect of Achilles tendon loading.
• Effect of posterior tibial tendon dysfunction.
• Effect of different parametrical design of foot orthoses.

Geometry of Foot Orthosis

Laser scanning during balanced standing

INFOOT Laser Scanner, I-Ware Laboratory Co. Ltd.

Geometrical Model of Foot Orthosis

(MATLAB, The MathWorks, Inc)

Foot Surface Model Solid Model of Foot Orthosis

(SolidWorks 2001, SolidWorks Corporation)

Orthosis Surface Model

Finite Element Model of the Foot Support

Insole (Polyurethane forms, Poron) Midsole (Ethylene Vinyle Acetate, Nora SL) Outsole (Ethylene Vinyle Acetate, Nora AL)

Component Element Type Thickness Insole (Poron) 3D-Brick 3mm, 6mm. 12mm, 24mm 3mm (base), 30mm (arch) 12mm Midsole (Nora SL) 3D-Brick Outsole (Nora AL) 3D-Brick

Compression Test of Insole Material

Hounsfield material testing machine (Model H10KM), Hounsfield Test Equipment, UK

Hyperfoam Material Model for Orthotic Material

∑

= −

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − + λ + λ + λ α =

2 1 α α α α 2

3 2

i β e l i i i

i i i i i

( J β

• μ

) 1 1 ˆ ˆ ˆ U

3 2 1

where U is the second order strain energy per unit of reference volume;

i

λ ˆ are principal stretches;

e l

J = λ λ λ

3 2 1

ˆ ˆ ˆ

ABAQUS v6.4, HKS, Inc.

µi, αi and βi are material parameters with µi related to the initial shear modulus, µ0, by

∑

=

=

2 1 i i

μ μ

and the initial bulk modulus, K0 defined by

) 1 ( 2

i i i

β μ K + = ∑

=

3

2 1

The coefficient βi determines the degree of compressibility, which is related to the Poisson's ratio, νi , by

i i i

ν ν β 2

• 1

=

Jel is the elastic volume ratio with

Simulation of Midstance

Effect of Insole Thickness on Plantar Pressure

Shod Insole3 Insole6 Insole12 Insole24

MPa

Increasing Insole Thickness

Effect of Insole Thickness on Plantar Pressure

0.05 0.1 0.15 0.2 3 6 9 12 15 Insole Thickness Peak Plantar Pressure, MPa

Forefoot Rearfoot

2 4 6 8 10 12 14 16 18 20 Shod Insole3 Insole6 Insole12 Insole24 Foot Support Bone Stress (Von Mises), MPa

ForeFoot MidFoot RearFoot

Effect of Insole Support on Bone Stress

Design Factors & Levels of Taguchi Method

Level Design factor Level 1 Level 2 Level 3 Level 4 Arch Type F FWB HWB NWB Insole Thickness (mm) 3 6 9 12 Midsole Thickness (mm) 3 6 9 12 Insole Material (Hardness) 10 20 30 40 Midsole Material (Hardness) 20 30 40 50 F: Flat, FWB: Full-weight-bearing, HWB: Half-weight-bearing, NWB: Non- weight-bearing. Hardness values of 10, 20, 30, 40 and 50 correspond to Poron_L24, Poron_L32, Nora_SLW, Nora_SL, Nora_AL, respectively.

Taguchi Experimental Design

Experiment No. Arch Height Insole Thickness Midsole Thickness Insole Stiffness Midsole Stiffness 1 1 1 1 1 2 3 4 3 4 1 2 4 3 2 1 2 1 4 3 2 1 2 2 1 2 3 4 4 3 2 1 2 1 4 3 3 4 1 3 1 3 3 4 1 4 4 5 2 1 2 6 2 2 1 7 2 3 4 8 2 4 3 9 3 1 3 10 3 2 4 11 3 3 1 12 3 4 2 13 4 1 4 2 14 4 2 3 15 4 3 2 16 4 4 1

Example of an L Example of an L16

16 Orthogonal Array

Orthogonal Array

Robust Simulation = 16 < Full Factorial Simulation = 45 = 1024

0.06 0.07 0.08 0.09 1 2 3 4 Level Midfoot Plantar Pressure, MPa Arch Type Insole Thickness Midsole Thickness Insole Stiffness Midsole Stiffness

Mean Effects of Design Factors at Each Level on the Predicted Peak Plantar Pressure

0.1 0.15 0.2 0.25 1 2 3 4 Level Forefoot Plantar Pressure, MPa Arch Type Insole Thickness Midsole Thickness Insole Stiffness Midsole Stiffness 0.1 0.125 0.15 0.175 1 2 3 4 Level Rearfoot Plantar Pressure, MPa Arch Type Insole Thickness Midsole Thickness Insole Stiffness Midsole Stiffness

Fabrication of Foot Orthosis

Computerized Numerical Control (CNC) Machining

LeadWell CNC Machines MFG. CORP.

Insole Milling

Fabrication of Foot Orthosis

Plantar Pressure Measurement

F-scan in-shoe sensors F-scan System, Tekscan, Inc. Video capture of foot-shank position Sensor calibration by single-leg standing

Plantar Pressure & Foot-Shank Position Measurement during Normal Walking

Normal walking with self-selected pace (~1.15s cycle time) Synchronization of pressure and video data

Predicted and Measured Plantar Pressure Distributions during Midstance

MPa MPa

Flat Arch supported Flat Arch supported

F-scan measured mean peak plantar pressure with different configurations of foot orthosis

Configurations of foot orthosis F-scan measurement, MPa Trial Number Arch Type Insole (Poron_L32) Thickness, mm Midsole (Nora_SL) Thickness, mm Forefoot Midfoot Rearfoot 1 F 3 0.133 0.077 0.100 2 F 3 3 0.120 0.070 0.087 3 F 6 3 0.113 0.073 0.090 4 FWB 3 0.117 0.073 0.070 5 FWB 3 3 0.097 0.053 0.060 6 FWB 6 3 0.110 0.047 0.060 7 HWB 3 0.103 0.06 0.070 8 HWB 3 3 0.090 0.057 0.057 9 HWB 6 3 0.100 0.060 0.060 10 NWB 3 0.083 0.063 0.047 11 NWB 3 3 0.073 0.047 0.047 12 NWB 6 3 0.087 0.050 0.043 F: Flat, FWB: Full-weight-bearing, HWB: Half-weight-bearing, NWB: Non-weight-bearing

Design Guidelines on Pressure-relieving Foot Orthoses Among five design factors

(arch type, insole material, insole thickness, midsole material and midsole thickness)

• Use of an arch-conforming foot orthosis;
• Soft insole material;
• Increase thickness of Insole;
• Soft midsole material;
• Increase thickness of midsole.

Other Parametrical Analysis

Shape Design Material Design

Custom-molded Shape Heel Elevation Forefoot Region Number of Layers & Thickness Insole Body Metatarsal Padding Heel Region Shank & Arch Profile

Incorporation of Foot Incorporation of Foot-

• Shoe Interface

Shoe Interface

Simulations of Stance Phases of Gait

Extension to Knee-Ankle-Foot FE Model

Tissue Testing

Plantar Heel Pad Plantar Heel Pad -

• Compression Test

Compression Test Fascia and Ligaments Fascia and Ligaments – – Tensile Test Tensile Test

Conclusions

The developed finite element ankle-foot model

• Allow efficient parametric evaluations of different design

parameters of orthoses without the prerequisite of fabricated orthoses and replicating patient trials.

• Contribute to the knowledge base for the design of
• ptimal foot orthoses or footwear in terms of pressure

redistribution, foot arch support or bone and ligament stress relief.

Acknowledgements

• Dr. Ameersing Luximon, Dr. Terry Koo, Research Students & Colleagues

Department of Health Technology & Informatics, The Hong Kong Polytechnic University, Hong Kong.

• Prof. Kai-Nan An and Colleagues

Biomechanics Laboratory, Department of Orthopedic Surgery, Mayo Clinic, Rochester, Minnesota, USA.

• Dr. Jun Auyeung and Colleagues

Institute of Clinical Anatomy, The Southern Medical University, Guangzhou, China for facilitating the cadaveric experiment. Financial support from the Hong Kong Jockey Club endowment, research grant from The Hong Kong Polytechnic University and the Research Grant Council of Hong Kong. (Project No. PolyU 5249/04E, PolyU 5317/05E)

Cheung JT, Zhang M, 2006. Consequences of partial and total plantar fascia release – a finite element study. Foot and Ankle International. 27, 125-132. Dai XQ, Li Y, Zhang M, Cheung JT, 2006. Effect of sock on biomechanical responses of foot during walking. Clinical Biomechanics. 21, 314-321. Cheung JT, Zhang M, An KN, 2006. Effect of Achilles tendon loading on plantar fascia tension in the standing foot. Clinical Biomechanics. 21, 194-203. Cheung JT, Zhang M, 2006. A serrated jaw clamp for tendon gripping. Medical Engineering and Physics. 28, 379-382. Cheung JT, Zhang M, Leung AK, Fan YB, 2005. Three-dimensional finite element analysis of the foot during standing – A material sensitivity study. Journal of Biomechanics. 38, 1045- 1054. Cheung JT, Zhang M, 2005. A 3-dimensional finite element model of the human foot and ankle for insole design. Archives of Physical Medicine and Rehabilitation. 86, 353-358. Cheung JT, Zhang M, An KN, 2004. Effects of plantar fascia stiffness on the biomechanical responses of the ankle-foot complex, Clinical Biomechanics. 19, 839-846. Cheung JT, Luximon A, Zhang M, 2006. Parametrical design of pressure-relieving foot

• rthoses using statistical-based finite element method, Journal of Biomechanics, submitted.

Peer-reviewed Journal Publications

Department of Health Technology and Informatics 醫療科技及資訊學系