A study of plantar stresses underneath metatarsal heads in the human foot 5

117









CHAPTER 5
CONCLUSIONS



~ There is a wisdom of the head, and a wisdom of the heart~
Charles Dickens


118
5.1 Conclusion
This study explored a sophisticated computational model of human foot using
musculoskeletal FE modeling, to investigate forefoot plantar stresses underneath
MTHs. A 3-D human foot FE model with detailed anatomy was constructed to
study the foot mechanism of muscular control, internal joint movements, and
plantar stress distributions in three-dimensions. Mechanical responses of the
foot’s soft tissue were specifically collected by using an instrumented tissue
tester and the material hyperelasticity were determined for the plantar soft tissue
under the MTHs. Experimental validation was conducted by a novel gait platform
system that is capable of measuring the vertical and shear force components

acting at local MTHs during walking. It was found that, with accurately quantified
tissue property, muscular loading characteristics, and foot’s geometric
positioning, a realistic stress response during foot-ground interactions can be
reproduced.
The unique joint-angle-dependent tissue responses obtained from
underneath an individual MTH provide more accurate and realistic mechanical
characterization of the plantar soft tissue property in the forefoot. The force-
displacement curve and force relaxation behavior produced, pertinent to the foot
geometry and loading regime applied to the foot model, has the potential to
provide insights into the mechanical behavior of forefoot tissues. An important
feature of this tissue tester is that the tester alone is generally applicable for
exploration of (joint-angle-dependent) tissue property as an indicator of disease
states and risks, such as ulcer formation at the MTH region in neuropathic

119
diabetic feet. These data collected was used in this study for extraction of the
hyperelastic material constants of the sub-MTH soft tissue. By transforming the
uni-axial stress (σ)–stretch (λ) equations of the Ogden model into contact
equations, the material constants can be directly extracted. The material
properties of the sub-MTH tissue determined presently were in between those
published results of the skin and fat tissue of human heel pad. This could be
explained by the fact that only single a homogeneous lumped tissue volume was
used to model such tissue, and that material difference between the skin and fat
pad had been weighted. And the use of such patient-specific lumped tissue
property input for the 3-D foot FE model appears to produce reasonable accurate
plantar contact stresses.
The dynamic in vivo plantar forces obtained underneath MTHs during gait
allows regional interfacial contact stresses to be calculated between the foot and
its support surface. The peak sub-MTH shear stresses were quantified. The
shear readings correlate well with existing data in the literature. The three-

dimensional contact stresses predicted at sub-MTH areas of forefoot by the
model that interacts with a highly deformable foam pad is in agreement the
present measurements as well as previous observations in terms of its
magnitude and distributions following heel rise. Based on the simulation results, it
was found that plantar shear stresses varied its distribution throughout the
domain. A model with specific frictional interactions, based on actual calculated
local shear traction ratio, can reproduce the pattern of regional shear distribution
at plantar surface.

120
The model sensitivity study predicted the adaptive changes of the foot
mechanism, in terms of internal joint configurations and plantar loads
distributions, due to reduced muscle effectiveness in G-S complex. The
simulation results correspond well with clinical observations in diabetic patients
following tendo-Achilles lengthening procedures. Pressure reductions at
individual MTHs could be site-specific and possibly dependent on foot structures,
such as intrinsic alignment of the metatarsals. These highlighted the clinical
relevance of the model in analyzing the foot mechanism.
Inclusion of a metatarsal support into the foam pad for enhanced forefoot
(i.e. sub-MTH areas) plantar stress relief requires more technical efforts. Plantar
pressure distributions were very sensitive to the metatarsal support’s placement
as well as its material selection. Based on the simulation results, an additional
soft metatarsal support placed at 12 mm proximal to the 2
nd
MTH helped to
reduce local peak pressure compared to using soft foam pad alone. However,
placing a stiffer metatarsal support just underneath the 2
nd
MTH could cause
local increase in peak pressure at forefoot plantar surface. The current FE model

provides an efficient computational tool to investigate the efficacy of certain
design variables used in therapeutic footwear, and also, the model has potential
for three-dimensional contact stress analysis at foot-shoe interface.

5.2 Original Contributions
The results of this present study may have significant impact on both
understanding the three-dimensional plantar stresses tensor, and providing a

121
useful tool for effective evaluation of existing or the development of new ‘Off-
loading’ techniques at the foot-support interface for sub-MTH stress relief.
This thesis examines both plantar pressure and shear stress distributions at
plantar surface of the foot experimentally and analytically. In terms of
experimental work, an instrumented indentation device was developed to
quantify material characteristics of the sub-MTH soft tissue. Also, an integrated
gait platform system, incorporating a customized sensor array and high-speed
photogrammetry, was designed, fabricated, and calibrated to measure the
dynamic interfacial stresses underneath individual metatarsal heads during gait.
This provided valuable data for verification of such a sophisticated
musculoskeletal foot FE model. A series of parametric modeling analysis was
conducted to highlight potential implications of specific muscle force variations on
forefoot stress redistribution. The validated model was further demonstrated as a
basic tool for possible applications in studying the influence of therapeutic
interventions using metatarsal support on plantar forefoot stress redistributions.

5.3 Future directions
 The analysis was performed in the model with a homogeneous mass of
plantar soft-tissue underneath MTHs. The tissue stress values were only
extracted from the soft-tissue boundary, i.e. contact stress distributions at
the plantar surface. A more adequate integration of the “true” internal

structures of the plantar soft-tissue, such as the one recently presented in
the 54th Orthopaedic Research Society (ORS) by Cavanagh et al., (2009),

122
with layered sections of skin, plantar fat pad, and muscle, might help in the
future to clarify for example internal tissue trauma among different levels
more precisely.
 During walking, the time-dependent properties of the soft tissue may affect
the stress response of the foot for various phases during gait. Advanced
material models that include hyperelasticity and viscoelasticity may be used
in the future dedicated to improving the current model. Such models could
potentially be calibrated separately using the current instrumented
indentation device, or other possible techniques that address both tissue
elastic as well as time-dependent behaviors.
 The current model only considered the flexor muscle forces for Tibialis
posterior, Flexor hallucis longus, Flexor digitorum longus, Peroneus brevis,
and Peroneus longus, other than the Achilles tendon. For stance phase gait
simulation (i.e. heel strike to toe off), modeling of the extensors was
necessary. Future simulation studies should consider using both flexors and
extensors when applying muscular loads.

Fig. 5.3.1.1. FE mesh of a
musculoskeletal foot model with
flexors and extensors for stance
phase gait simulation

123
 The model was validated against the novel gait platform system at patient-
specific level. However, the system is subjected to limitations such as the
relatively small sensor array size, which requires targeted walking in order

to take specific measurements at each MTHs. Future studies may consider
expanding the number of sensors used in the gait platform that should allow
large scale experimental investigations to be performed. The schematic
diagram of such ‘next generation’ force platform system is in Fig. 5.3.1.2.

Fig. 5.3.1.2. A future gait platform system with a larger sensor array can be
installed in a typical gait lab for large-scale experiments

 Application of the current model in design of therapeutic footwear is only
preliminary. Analysis of other design factors such as custom-molded
insole, arch profiles, wedged shoe soles and variable-stiffness shoe soles
may also equally applicable.

Sensors
Mounting
Plate
Customized force
plate
Pit cove
r
Heel Strike
Push-off
Walkway

124
Appendix A: Finite element analysis (FEA)
Finite element analysis is a numerical based method; the basic idea is that rather
than obtaining an exact algebraic solution of the governing partial differential
equations throughout the domain of interest, one instead numerically solves a
system of simultaneous equations that arise from enforcing those governing

equations for an array of discrete simplified sub-domains (known as elements).
Within these individual elements, specific interpolation basis functions (usually
polynomials) are assumed, from which continuous internal variables (e.g., strains)
are piecewise approximated on the basis of corresponding parameters (e.g.,
displacements, in the case of strain) evaluated at a discrete number of
characteristic local points (known as nodes).
Although the theory of FEA is rather complicated, below is a code written
in MATLAB to solve the 2-D Helmholtz equation using the Galerkin finite element
method. The basic concept of FEA as numerical approximation techniques was
demonstrated. The following MATLAB code is provided:
1. “Main.m” contains code to generate meshes at different resolutions and
to generate the boundary conditions described above along with some necessary
initializations (including Gaussian quadrature points and weights). The mesh
resolution can be changed through the “elemsPerSide” variable.
2. “psi.m” contains code to evaluate bilinear Lagrange basis functions and
their derivatives, and should not need modification.
3. “dxiIdxJ.m” is designed to calculate the transformation Jacobian and the
term gu(i,j) =(dxi_i/dx_k)*(dxi_j/dx_k).
%======================================================

125
% Program to solve 2D Helmholtz equation using Galerkin
% FEM
% by CHEN Wenming
%======================================================

clear all;
clc;

%======================================================

% Set up the mesh
%
% numNodes = total number of global nodes
% nodePos = each row is (x,y), row number is the node number
% numElems = total number of global elements
% elemNode = each row gives the four nodes for each element
% in order, the row number is the element number
% BCs = boundary conditions (type,value).
% Type 1 = essential (displacement)
% Type 2 = natural (gradient)
%======================================================

ksq = 1;
elemsPerSide = 32;

sideLength = 8;
nodesPerSide = elemsPerSide+1;
incr = sideLength/elemsPerSide;
nodesPerElement = 4;

numNodes = nodesPerSide * nodesPerSide;
numElems = elemsPerSide * elemsPerSide;

nodePos(1:numNodes,2) = 0;
elemNode(1:numElems,4) = 0;
BCs(1:numNodes,2) = 0;

% This is the u matrix initialization ( k*u = f)
u(numNodes) = 0;

%======================================================
% Generate nodes
%======================================================
n = 1;
for j = 1:nodesPerSide
for i = 1:nodesPerSide
nodePos(n,1) = (i-1)*incr;
nodePos(n,2) = (j-1)*incr;
if( i == 1 )
BCs(n,1) = 1;
BCs(n,2) = 1;
u(n) = BCs(n,2);
elseif( i == nodesPerSide )
BCs(n,1) = 1;
BCs(n,2) = 0;
u(n) = BCs(n,2);

126
else
BCs(n,1) = 2;
BCs(n,2) = 0;
end
n = n + 1;
end
end

%======================================================
% Generate elements
%======================================================
e = 1;

for j = 1:elemsPerSide
for i = 1:elemsPerSide
elemNode(e,1) = (j-1)*nodesPerSide+i;
elemNode(e,2) = elemNode(e,1)+1;
elemNode(e,3) = elemNode(e,1)+nodesPerSide;
elemNode(e,4) = elemNode(e,1)+nodesPerSide+1;
e = e + 1;
end
end

%======================================================
% This initialization is for the Stiffness matrix and
% pi, dpibydxi1 and dpibydxi2.
%======================================================

% This is the pi matrix.
pi = 0;

% This is the Dpi/Dxi1 matrix.
pi1 = 0;

% This is the Dpi/Dxi2 matrix.
pi2 = 0;

% This is the matrix corresponding to first part of LHS in derivation.
stiff1(1:4,1:4) = 0;

% This is the matrix corresponding to second part of LHS in derivation.
stiff2(1:4,1:4) = 0;

% This is the matrix that calculates Dxi1/Dx, Dxi2/Dx, Dxi1/Dy and
Dxi2/Dy.

result(1:2,1:2) = 0;

% This is the Jacobian matrix
Jacobi = 0;

% This is the Element Stiffness matrix.
ES(1:4,1:4) = 0;

% This matrix is to calculate section 2 in first part of LHS in
derivation

gu(1:4) = 0;

127

% This is the matrix that contains the function PSI value for each of
the

% Gauss points.
value(:,:) = 0;

% This corresponds to the four basis functions.
numerator = 4;
denominator = 4;

% This is the Global stiffness Matrix.
GM(1:numNodes,1:numNodes) = 0.0;

% This is the Element Stiffness matrix.
ES(1:nodesPerElement,1:nodesPerElement) = 0.0;


%======================================================
% Set up the numerical integration points 'gaussPos'
% and weights 'gaussWei' (2x2)
%======================================================

numGaussPoints = 4; % 2x2

gp1 = 0.5 - ( 1 / ( 2 * sqrt( 3 )));
gp2 = 0.5 + ( 1 / ( 2 * sqrt( 3 )));

gaussPos = [ gp1 gp1;
gp2 gp1;
gp1 gp2;
gp2 gp2 ];

gaussWei = [ 0.25;
0.25;
0.25;
0.25 ];

%======================================================
% Initialise the global matrix system K*u=f
%======================================================

K(1:numNodes,1:numNodes) = 0.0;

f(1:numNodes) = 0.0;
f = f';

% THIS IS TO CALCULATE PI(N) , DPIBYDXI1 AND DPIBYDXI2.
% These matrices are 4x4 in nature. These matrices contains the values
for

% the 4 basis points at the four Gaussian points.

for j = 1:numerator
for h = 1:numGaussPoints
der = 0;
num = j;

128
xi1 = gaussPos(h,1);
xi2 = gaussPos(h,2);
pi(j,h) = psi (num, der, xi1, xi2 );
der = 1;
pi1(j,h) = psi (num, der, xi1, xi2 );
der = 2;
pi2(j,h) = psi (num, der, xi1, xi2 );
end
end

%====================================
% Create the element stiffness matrix
%====================================

%*** Q. Write code here to make the

%*** element stiffness matrix, ES
% Initialise element stiffness (ES)


% Loop over elements

for elem = 1:numElems


ES(:,:) = 0;
stiff1(:,:) = 0;
stiff2(:,:) = 0;

% This step is to populate x and y values.

for k = 1:4

v = elemNode(elem,k);
dx(k) = nodePos(v,1);
dy(k) = nodePos(v,2);

end

% This paragraph calculates the Jacobian and the gu matrix.

for p = 1:numGaussPoints

xi1 = gaussPos(p,1);
xi2 = gaussPos(p,2);
value1 = 0;

value2 =0;
result(:,:) = 0;

% This step calculates the values of dx/dxi1, dx/dxi2, dy/dxi1
and

% dy/dxi2 for the four Gauss points.

for q = 1:numerator

129

value1 = psi(q,1,xi1,xi2);
value2 = psi(q,2,xi1,xi2);

result(1,1) = result(1,1) + ( value1*dx(q) );
result(1,2) = result(1,2) + ( value2*dx(q) );
result(2,1) = result(2,1) + ( value1*dy(q) );
result(2,2) = result(2,2) + ( value2*dy(q) );

end

% This step calculates Jacobian value.

Jacobi = det(result);

% This step calculates gu.

ku = inv(result);
kt = (ku)';

gu = ku * kt;

% This is to calculate the Element Stiffness matrix.

for x = 1:numerator
for y = 1:denominator

First = pi1(x,p)*pi1(y,p)*gu(1,1);
Second = pi1(x,p)*pi2(y,p)*gu(1,2);
Third = pi2(x,p)*pi1(y,p)*gu(2,1);
Four = pi2(x,p)*pi2(y,p)*gu(2,2);
stiff1(x,y) = stiff1(x,y) + 0.25 * Jacobi * (First
+ Second + Third + Four);

stiff2(x,y) = stiff2(x,y) + 0.25 * (pi(x,p)*pi(y,p))
* Jacobi;


end
end
end

stiff3 = ksq*stiff2;
ES = stiff1 - stiff3;

%==========================================
% Assemble into the global stiffness matrix
%==========================================

%*** Q. Write code here to assemble this ES

%*** into the global stiffness matrix, GM


a = elemNode(elem,1);
b = elemNode(elem,2);
c = elemNode(elem,3);
d = elemNode(elem,4);

130

GM(a,a) = GM(a,a) + ES(1,1);
GM(a,b) = GM(a,b) + ES(1,2);
GM(a,c) = GM(a,c) + ES(1,3);
GM(a,d) = GM(a,d) + ES(1,4);
GM(b,a) = GM(b,a) + ES(2,1);
GM(b,b) = GM(b,b) + ES(2,2);
GM(b,c) = GM(b,c) + ES(2,3);
GM(b,d) = GM(b,d) + ES(2,4);
GM(c,a) = GM(c,a) + ES(3,1);
GM(c,b) = GM(c,b) + ES(3,2);
GM(c,c) = GM(c,c) + ES(3,3);
GM(c,d) = GM(c,d) + ES(3,4);
GM(d,a) = GM(d,a) + ES(4,1);
GM(d,b) = GM(d,b) + ES(4,2);
GM(d,c) = GM(d,c) + ES(4,3);
GM(d,d) = GM(d,d) + ES(4,4);

end %elements

%==========================

% Apply boundary conditions
%==========================

%*** Q. Write code here to apply the boundary
%*** conditions to K. You can use any method
%*** but may find overwriting the rows of K easiest.

%*** This method finds rows that has known boundary conditions and
%*** adjusts the values.

for n = 1:numNodes
if BCs(n,1) == 1
GM(n,1:numNodes) = 0;
GM(n,n) = 1;
f(n) = BCs(n,2);
end
end

%*** This method adjusts the value of load vector f.

for n = 1:numNodes
if BCs(n,1) == 1
for z = 1:numNodes
f(z) = f(z) - GM(z,n)*BCs(n,2);
end
f(n) = BCs(n,2);
end
end

%*** This method removes the rows and columns passing the daigonal

%*** elements with known values.

for n = numNodes:-1:1
if BCs(n,1) == 1

131
GM(n,:) = [];
GM(:,n) = [];
f(n) = [];
u(n) = [];
end
end

%======================================================
% Solve
%======================================================
u = GM \ f;
u1(1:numNodes) = 0;

% This method is to properly populate te u1 vector.

n = 1;
t = 1;
for j = 1:nodesPerSide
for i = 1:nodesPerSide
if( i == 1 )
BCs(n,1) = 1;
BCs(n,2) = 1;
u1(n) = BCs(n,2);
elseif( i == nodesPerSide )

BCs(n,1) = 1;
BCs(n,2) = 0;
u1(n) = BCs(n,2);
else
u1(n) = u(t);
t = t + 1;
end
n = n + 1;
end
end

% Rearrange solution as 2D array for surface plot
n = 1;
for j = 1:nodesPerSide
for i = 1:nodesPerSide
soln(i,j) = u1(n);
n = n + 1;
end
end
surf(soln)

% Point to look at is x=1.6 y=2.4
x = 1.6;
y = 2.4;

% This step calculates the reminder of dividing x and y by incr.

x1 = mod(x,incr);
y1 = mod(y,incr);

132
% This step is to calculate element in which given values of x & y
are

% calculated.

x2 = (x - x1);
y2 = (y - y1);

node(1:4) = 0;

for n = 1:numNodes
if (nodePos(n,1) == x2 && nodePos(n,2) == y2)
break;
end
end

for e = 1:numElems
if ( elemNode(e,1) == n)
node(1) = elemNode(e,1);
node(2) = elemNode(e,2);
node(3) = elemNode(e,3);
node(4) = elemNode(e,4);
break;
end
end

x3(1:4) = 0;
y3(1:4) = 0;

for g = 1:4

r = 0;

r = node(g);


x3(g) = nodePos(r,1);
y3(g) = nodePos(r,2);

end

xi_1 = ( x - x3(1) )/( x3(2) - x3(1) );
xi_2 = ( y - y3(1) )/( y3(3) - y3(1) );

% This step is to calculate the interpolated value at x=1.6, y=2.4.

final = 0;
si = 0;
val = 0;

for num = 1:4
si = psi (num, 0, xi_1, xi_2 );
val = u(node(num));
final = final + si * val ;
end

133
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141
Vita
CHEN Wenming received his double Bachelor's degrees in Optomechatronics
and Bioengineering from Huazhong University of Science & Technology and
Central China Normal University, in 2004, respectively. He studied further and
obtained his Master degree in Orthopedic Biomechanics from INJE University of
Korea in 2006 before he came to National University of Singapore for Ph.D.
study. His current research work is mainly focused on biomechanical studies on
load-bearing soft tissue associated with musculoskeletal disorders.

Publications arising from the thesis
 Original Research Articles

1. Chen WM
, Shim V, Park SB, Lee T, An instrumented tissue tester for
measuring soft tissue property under the metatarsal heads in relation to
metatarsophalangeal joint angle. J Biomech, 2011.
[doi:10.1016/j.jbiomech.2011.03.031]

2. Chen WM, Lee PVS, Park SB, Lee SJ, Lee T, A novel gait platform to
measure isolated plantar metatarsal forces during walking. J Biomech,

43(10): 2017–2021, 2010.

3. Chen WM, Lee T, Lee PVS, Lee JW, Lee SJ, Effects of internal stress
concentrations in plantar soft-tissue—A preliminary three-dimensional
finite element analysis. Med Eng & Phys, 32(4): 324–331, 2010.

4. Chen WM, Lee PVS, Lee T, Plantar soft-tissue stress states in standing: a
three-dimensional finite element foot modeling study, Korean J Sport
Biomech, 19(2):197-204, 2009.

5. Chen WM
, Lee PVS, Shim V, Lee T, Direct determination of toe flexor
muscle forces based on sub-metatarsal/toe pad load sharing by using
finite element method. Clin Biomech
(submitted)

6. Chen WM
, Shim V, Park SB, Lee SJ, Lee T, Three-dimensional finite
element analysis of the musculoskeletal foot mechanism – role of the
gastrocnemius-soleus muscle complex. J Biomech, (submitted)



 Podium Presentations at International Conferences

1. Chen WM, Lee PVS, Lee T: Finite Element Model of the Human Foot-
Ankle Joint Complex Validated with Patient-Specific Data. 54th Annual
Meeting of the Orthopaedic Research Society, San Francisco, USA, 2008.