A study of plantar stresses underneath metatarsal heads in the human foot 2

The bones were modeled as articulated parts enveloped into a bulk soft tissue Two pixel phases were defined based on gray-scale threshold computing: the foot skeletal phase and a lumped

CHAPTER 2 NUMERICAL MODELING OF

HUMAN FOOT

~ If you want to accomplish something in the world, idealism is not enough

- you need to choose a method that works to achieve the goal ~

Richard Stallman

2.1 Finite element model of foot-ankle complex

2.1.1 Finite element method

The FEM is a versatile numerical method which allows stress and strain analyses

of complex structures with irregular geometry and material nonlinearities When a foot structure is loaded, stresses are generated in different materials (i.e tissues) The distribution of these stresses, their magnitudes and orientations throughout the structure, is a result of complex interplay of the foot skeleton, cartilages, muscles, ligaments, fascia, and the external environment that arises from foot-ground interactions In such a model that mimics the real structures to a certain degree of refinement, the structural aspects (geometry, material properties, and loading/boundary conditions) are required to be expressed mathematically

In using the FEM, the model of the human foot as a geometrical entity, has to be firstly defined Modern musculoskeletal imaging techniques such as magnetic resonance imaging (MRI) and computer tomography (CT) can be a source for such complete anatomical structures The geometrically complex foot structures, are then discretized into finite number of relatively simple elements (i.e FE mesh generation), connected by nodal points or nodes Each element can have its own material properties The equations for describing the mechanical behavior of these elements are known The computer program (e.g ABAQUS) can calculate the stiffness matrix of each element, and the stiffness matrix of the whole structure is determined Knowledge of the structural stiffness matrix, of the loads and the boundary conditions allows the response of a model

to any form of external loading to be predicted

2.1.2 3-D reconstruction of foot geometry

The three-dimensional soft tissue-skeletal geometry was created based on the volume reconstruction of the coronal computer tomography (CT) images of the right foot (non-weight-bearing condition) of a male subject (27 years old, height of

169 cm and body weight of 65.1 kg) The CT scan has a resolution of 0.25ｘ0.25

mm2 (pixel size) with a slice spacing of 1 mm This allows a detailed morphological reconstruction of the various anatomical structures of the foot, including bones, cartilages, and a bulk soft tissue boundary (Fig 2.1.2.1.) Model segmentation was performed by MIMICS program (Materialise Inc., Belgium)

Fig 2.1.2.1 Segmentation of a human foot from individual coronal CT slices The bones were modeled as articulated parts enveloped into a bulk soft tissue

Two pixel phases were defined based on gray-scale threshold computing: the foot skeletal phase and a lumped soft-tissue phase The hard foot skeleton and soft tissue were then reconstructed into a solid model (Fig 2.1.2.2.) The

solid model was finally imported into a pre-processor, PATRAN (MacNeal Schwendler Corporation, USA), for FE mesh generation Thirty bony parts, including sesamoids, were created individually and enveloped into the homogenous mass of foot soft-tissue Particular attention was given to the surface geometry of various foot bones (tibia, fibula, talus, calcaneus, navicular, medial, intermediate and lateral cuneiform, and cuboid, the medial, intermediate and lateral cuneiforms, and the cuboid, metatarsal bones and phalanges) that are internally articulated

Fig 2.1.2.2 3-D solid model of foot geometry, including a bulk soft tissue (A) and bones (B) Note that the foot skeletal was stabilized by actually anatomical ligaments

Passive soft-tissue stabilizers of the foot skeleton, including ligaments and plantar fascia, which could not be reconstructed from CT were determined based

on the anatomical descriptions from Primal Pictures 2006 (ANATOMY.TV, Primal Pictures Ltd., London, UK) A total of 134 ligaments (i.e., multiple element modeling for major ligament bundles) and a fan-shaped plantar fascia structure with a 2 mm out-of-plane thickness were incorporated into the model

2.1.3 Model discretization

The reconstructed solid foot model was then used to generate three-dimensional finite element mesh using various types of structural elements; this process is known as model discretization The foot skeleton and the soft-tissue component were meshed by 160,000 and 240,000 tetrahedral elements (C3D4: full integration linear element) Meshes for the bones and the soft-tissue component share same nodes at the interface except those joint space regions where a series of contact conditions were defined between adjacent bones (Fig 2.1.3.1.) The ligament structures were represented by 3-D truss elements (T3D2) with a

“no compression” option, in accordance to their physiological function This would allow these elements to resist tension-producing forces when stabilizing the foot skeleton

Fig 2.1.3.1 Finite element mesh of a human foot with (A) soft tissue and (B) internal bony structures

For the muscular structures, six major extrinsic plantar flexors were included To model the gastrocnemius-soleus (G-S) complex, a three-dimensional geometry of the Achilles tendon was constructed and incorporated into the posterior extreme of the calcaneus This facilitates application of G-S muscle forces through the Achilles tendon-bone junction, and ensure more realistic muscle load transfer compared to those in previous models, which only employed nodal points to apply such forces (Gefen et al., 2000, Cheung et al., 2006) The long tendons of the other five muscles were also inserted into the model, at their corresponding anatomical attachment sites This was done using bar elements based on straight-line approximation, i.e several bar elements stringed together to represent the actual tendon trajectory inside the foot

A mesh sensitivity analysis was performed to ensure that the mesh density used in the FE model was sufficient to reach the converged numerical results Mesh refinement process was carried out in a 2-D plane-strain finite element model based on a sagittal section through the 2nd ray of the foot The total strain energy and displacement served as the convergence criteria, with the tolerance level being set as the change of less than 5%

2.2 Material properties for finite element modeling

2.2.1 Cortical and cancellous bones, ligaments, and cartilages

Mechanical properties of cortical bone have been well documented in the literature Traditional mechanical testing such as uni-axial tensile or compressive testing has revealed that material property of cortical bone may be considered as

having ‘nearly’ linearly elastic behavior Measuring mechanical properties of cancellous bone tissue is far more difficult than measuring those of cortical bone tissue, due to the extremely small dimension of individual trabeculae Because of this difficulty, the reported cancellous bone tissue modulus ranges from 0.76 to

20 GPa (Cowin et al., 1998)

In the current analysis, a quasi-static loading system is applied to the foot model According to Huiskes (1996), in the case of quasi-static loading, both cortical and trabecular bones may be linearly elastic for simplification Consequently, in the present model, bone tissues are assumed to be homogeneous, isotropic, and linear elastic For the foot bones, Young’s modulus

is taken as 7300 N/mm2, a value that was weighted by Nakamura et al (1981) from human cortical and cancellous bone properties and the Poisson ratio is taken as 0.3 The ligaments, following Cheung et al (2005), were considered as non-compressive materials and assigned with a Young’s modulus of 260 MPa, a Poisson’s ratio of 0.3, and a cross-section of 18.4 mm2

2.2.2 Achilles and other flexor tendons

In the literature, mechanical properties have been reported for many tendons (e.g Achilles tendon and anterior cruciate ligament) Modulus values are generally in the range of 500~1,850 MPa (Yamamoto et al., 1992, Danto and Woo, 1993) For the Achilles tendon, uni-axial tension tests have been conducted by Wren et

al (2001) at lower strain rates of 1%/s and relatively higher strain rate of 10%/s The mean moduli found were 816 MPa (± 218) at 1%/s rate and 822 MPa (±211)

at 10%/s rate, respectively Considering the quasi-static loading system used in the current study, as will be discussed in the following sections, the elastic modus obtained at lower strain rate is chosen and a common Poisson’s ratio of 0.3 was used as the material properties for the Achilles tendon in the current foot

FE model

For the flexor tendons, the literature contains the least amount of information pertaining to their mechanical properties It was stated that the stiffness values of flexor hallucis longus (FHL) and peroneus brevis (PB) were 43.3 N/mm (± 14.1) and 43.6 N/mm (± 18.9) (Maffulli et al., 2008), respectively Using the relation, a flexor tendon of the foot with a typical cross-sectional area

of 12.5 mm2 was calculated to have a Young’s modulus of 450 MPa Gonzalez et al., 2009) Thus, the Achilles and other flexor tendons, were idealized as isotropic linear elastic materials with different Young’s moduli of 816 MPa (Wren et al., 2001) and 450 MPa (Garcia-Gonzalez et al., 2009), and a common Poisson’s ratio of 0.3 A summary of material properties to define different tissues for finite element modeling is given in Table 2.2.2.1

(Garcia-Table 2.2.2.1 Summary of FE model listing element type and material properties for

different model entities

Entity

Element type (ABAQUS)

E (MPa), v

section (mm 2 )

Cross-Reference

Bone

4-node tetrahedral continuum

7300, 0.3  Cortical and cancellous bone properties weighted

by Nakamura et al (1981) Cartilage

4-node tetrahedral continuum

1.01, 0.4  Indentation test conducted on 1st MTP joint by

Athanasiou et al (1998) Ligament

2-node tension-only truss

260, 0.4 18.4

Properties obtained from collateral ligaments of the human ankle joint by Siegler et al (1988)

Achilles

tendon

4-node tetrahedral continuum

816, 0.3  Tensile modulus obtained at lower strain rate (Wren et

450, 0.3 12.5

Calculation based on the stiffness values of flexor hallucis longus and peroneus brevis (Garcia-Gonzalez et al., 2009) Plantar

fascia

2-node tension-only truss

350, 0.4 290.7 Wright and Rennels,

(1964))

Plantar

soft-tissue

4-node tetrahedral continuum

Hyperelastic  Stress-strain curve determined experimentally

by Chen et al (2011)

2.2.3 Constitutive model for plantar soft tissue

2.2.3.1 Hyperelastic material model

The plantar soft-tissue is a typical load-bearing soft tissue that undergoes very large strains/deformation (large-strain elasticity) with strongly non-linear stress-strain behavior when subject to high loads during gait Furthermore, the plantar soft-tissue in normal foot is often rich in fluid content, and its material behavior under compression often resembles those of most elastomers (i.e solid rubberlike materials), which exhibit very little compressibility compared to their shear flexibility For these materials, their stress-strain relationship is suitable to

be derived from a ‘strain-energy density function’ This function defines the strain energy stored in the material per unit of reference volume (volume in the initial configuration) as a function of the strain at that point in the material For this reason, these rubberlike materials are usually referred to as hyperelastic materials, as is the case for the plantar soft tissue And various hyperelastic material models have been successfully used to model plantar soft tissue behavior in finite element simulation (Lemmon et al., 1997, Cheung et al., 2005, Erdemir et al., 2006)

There are several forms of strain energy potentials available in ABAQUS

to model approximately incompressible isotropic elastomers: the Arruda-Boyce form, the Marlow form, the Mooney-Rivlin form, the neo-Hookean form, the Ogden form, the polynomial form, the reduced polynomial form, the Yeoh form, and the Van der Waals form (Hibbert and Karssonn, 2006) This study focuses

on the Ogden-form hyperelastic constitutive model that are widely used in the

literature to model the plantar soft tissue material behavior, because of its superior non-linear curve-fitting capacity (Erdemir et al., 2005, Cheung et al.,

2006, Chen et al., 2010a)

Firstly, it is useful to give a quick review on the general concepts of strain elasticity applied on hyperelastic isotropic and incompressible materials (Ogden, 1972) Hyperelastic models are based on the definition of a strain-

large-energy function U For an isotropic and incompressible material U can be

expressed as a function of the principal stretches:

U U1,2,3 2.1 The principal Cauchy stresses are given by:

U p

i i

where p is the hydrostatic part of stress, whereas the first term is relative to the

deviatoric part For incompressible materials (i.e total volume remains constant when subjected to compressive forces), we have:

123 1,   1

2 1

1,   /

 2.4

We can so define a strain-energy function depending only on the one remaining

independent stretch (λ in the principal loading direction):

i i

3 2

1 3

2

where N is a positive integer (normally it can be taken smaller than 3, depending

on its capacity for nonlinear stress-strain curve-fitting), and α i are real parameters (i.e material constants) that describe the behavior of this rubber-like material model These material constants, and α i, can be positive or negative, satisfying the condition that:

12

1    2.8

where is the ground state (i.e initial) shear modulus Introducing Eq (2.7) into

Eq (2.6) and imposing N = 1 (1st-order model), we obtain the form of

hyperelastic Ogden formulation with stress () and stretch () relation given as:

In the literature, the stress-strain curves of the plantar soft tissue are often obtained based on tests of the fat pad under the heel bone (i.e heel pad) (Erdemir et al., 2005, Cheung et al., 2005, Lemmon et al., 1997) Since the primary focus of the current model is the sub-MTH region, the material behavior has to be determined for soft tissues specific to that location For this purpose, an

instrument-driven indentation device was developed for realistic in vivo

mechanical characterization (i.e tissue stiffness and force relaxation behavior) of the forefoot plantar soft tissue The indentation force-displacement curve was directly measured from soft tissue under the metatarsal heads, and was used for extraction of the material constants to model such tissue To determine these material constants, we performed additional simulations of indentation experiments on human plantar soft tissue under the 2nd MTH, based on this customized tissue tester The parameters were chosen such that the reaction forces upon indentation optimally fit the experimental observations; this yielded

= 3.75 ｘ10-2 MPa and α = 5.5 (See chapter 3, section 3.1.4.3, for detailed

descriptions of plantar soft tissue material property characterization.)

2.2.4 Material model for foot-supporting surface

2.2.4.1 Insoles as supporting interface for foot

Current practice in the prevention of neuropathic foot ulcers often involves prescription of accommodative in-shoe orthoses or insoles which reduce plantar pressure levels at locations of bony prominences, particularly under the metatarsal heads (MTHs) (Cavanagh et al., 2000, Lott et al., 2007, Bus et al.,