3D non-linear analysis of the acetabular construct following impaction grafting.

Abstract

The study investigates the short-term behaviour of the acetabular construct following revision hip arthroplasty, carried out using the Slooff-Ling impaction grafting technique; using 3D finite element analyses. An elasto-plastic material model is used to describe the constitutive behaviour of morsellised cortico-cancellous bone (MCB) graft, since it has been shown that MCB undergoes significant plastic deformation under normal physiological loads. Based on previous experimental studies carried out by the authors and others, MCB is modelled using non-linear elasticity and Drucker Prager Cap (DPC) plasticity. Loading associated with walking, sitting down, and standing up is applied to the acetabular cup through a femoral head using smooth sliding surfaces. The analyses yield distinctive patterns of migration and rotation due to different activities. These are found to be similar to those observed in the clinical setting.

Full text

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3D non-linear analysis of the acetabular construct
following impaction grafting
A.T.M. PHILLIPS, P. PANKAJ, C.R. HOWIE, A.S. USMANI, A.H.R.W. SIMPSON
This is a preprint of an article published in:
COMPUTER METHODS IN BIOMECHANICS AND BIOMEDICAL ENGINEERING
c ? 2006 copyright Taylor & Francis
COMPUTER METHODS IN BIOMECHANICS AND BIOMEDICAL ENGINEERING
is available online at:
http://www.journalsonline.tandf.co.uk/openurl.asp?genre=journal&issn=1025-5842
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3D non-linear analysis of the acetabular construct
following impaction grafting
A.T.M. PHILLIPS1,2⋆, P. PANKAJ2, C.R. HOWIE3, A.S. USMANI2,
A.H.R.W. SIMPSON1,3
Affiliations:
1Edinburgh Orthopaedic Engineering Centre
The University of Edinburgh, Edinburgh, Scotland, UK
2School of Engineering and Electronics
The University of Edinburgh, Edinburgh, Scotland, UK
3Edinburgh Royal Infirmary and Department of Orthopaedics
The University of Edinburgh, Edinburgh, Scotland, UK
Corresponding author:
⋆Dr. Andrew Phillips
The University of Edinburgh
Edinburgh Orthopaedic Engineering Centre
Room FU.413
Chancellor’s Building
49 Little France Crescent
Edinburgh EH16 4SB
Scotland, UK
Email: andrew.phillips@ed.ac.uk
Telephone: +44 (0)131 242 6295
Original Article
Keywords: finite element, plasticity, hip arthroplasty, impaction grafting
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3D non-linear analysis of the acetabular construct
following impaction grafting
A.T.M. PHILLIPS, P. PANKAJ, C.R. HOWIE, A.S. USMANI, A.H.R.W. SIMPSON
10th July 2006
Abstract
The study investigates the short-term behaviour of the acetabular construct fol-
lowing revision hip arthroplasty, carried out using the Slooff-Ling impaction graft-
ing technique; using 3D finite element analyses. An elasto-plastic material model is
used to describe the constitutive behaviour of morsellised cortico-cancellous bone
(MCB) graft, since it has been shown that MCB undergoes significant plastic defor-
mation under normal physiological loads. Based on previous experimental studies
carried out by the authors and others MCB is modelled using non-linear elasticity
and Drucker Prager Cap plasticity. Loading associated with walking, sitting down,
and standing up is applied to the acetabular cup through a femoral head using
smooth sliding surfaces. The analyses yield distinctive patterns of migration and
rotation due to different activities. These are found to be similar to those observed
in the clinical setting.
1Introduction
Hip arthroplasty is one of the most successful operations carried out by orthopaedic
surgeons, resulting in significant improvement in quality of life for the patient. How-
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ever, primary operations are subject to failure due to the production of wear par-
ticles at prosthesis-prosthesis and prosthesis-bone interfaces (Howie 1997), causing
a biological response leading to degradation of host bone around prosthetic com-
ponents (Learmonth et al. 1995). Thus skeletal structures that were adequate at
the time of surgery, deteriorate to such an extent that revision surgery is required.
The number of revision operations is increasing, with patients undergoing primary
arthroplasty being younger and more active than in the past (Toms et al. 2004).
The Slooff-Ling impaction grafting technique has found favour with orthopaedic
surgeons for contained femoral and acetabular defects (Gie et al. 1993, Slooff
et al. 1984). The technique uses morsellised cortico-cancellous bone allograft, which
is impacted into the skeletal defects to form a bed onto which new prosthetic com-
ponents are cemented. If short-term stability of the components is achieved, over a
period of months the bone graft bed is incorporated into the host skeleton, ensuring
long-term stability of the components. Migration and rotation of the acetabular
cup within the bone graft bed often leads to aseptic failure (Iida et al. 2000, Inao &
Matsuno 2000). Experimental studies have been carried out with a view to quan-
tifying and improving the material behaviour of the bone graft bed (Bavadekar
et al. 2001, Brewster et al. 1999, Dunlop et al. 2003, Giesen et al. 1999, Phillips
et al. 2005, Phillips et al. 2006, Verdonschot et al. 2001, Voor et al. 2000). How-
ever, there has been little research into the movement of the acetabular cup, and
the behaviour of MCB within the acetabular construct. Phillips et al. (Phillips
et al. 2004a) presented a plane-strain model of the acetabular construct. However
they did not consider the consolidation of MCB in their material model, and were
not able to apply realistic 3D loading patterns due to the 2D nature of their model.
This study investigates the behaviour of the acetabular construct using a 3D
finite element model, subjected to load cases representative of walking, standing
up, and sitting down. The behaviour of the bone graft, the most complex of the
materials in the acetabular construct is considered in detail.
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2Model definition
There have been 3D numerical studies examining the behaviour of the natural hip
joint, and the hip joint following primary arthroplasty (Anderson et al. 2004, Dalstra
& Huiskes 1995, Garcia et al. 2000, Goel et al. 1978, Oonishi et al. 1983, Siggelkow
et al. 2004). However the literature does not include 3D numerical studies of the hip
joint following revision arthroplasty, carried out using impaction grafting; perhaps
due to the complexities of modelling the material properties of bone graft.
The geometry, material properties, and loading cases associated with the finite
element analyses presented are discussed below.
2.1 Geometric definition
The geometry of a patient’s left hand side hemi pelvis was described based on a
Sawbones anatomic model. Laser topography was carried out using a 3D laser
scanner, with an accuracy of around 0.1 mm, consisting of a laser strip sensor,
positioning arm, and desktop PC. The generated point cloud was converted to
form a triangular surface mesh. Interpolative repairs were made to the surface
mesh in a few areas were laser topography had failed to describe the surface of the
anatomic model fully. This surface mesh was used to produce cross sections along
an axis running from the connection between the ischium and the pubis, through
the acetabulum, to the iliac crest. In total 50 cross sections were produced at a
spacing of around 4.5 mm.
Each of the sections was then edited to define the boundaries of the various
materials comprising the acetabular construct. The shape of the bone graft bed was
defined so as to represent a cavitory defect, in which the acetabular wall remained
intact. Previous 3D finite element studies of the natural hip joint (Anderson et al.
2004, Dalstra & Huiskes 1995, Garcia et al. 2000, Goel et al. 1978, Oonishi et al.
1983, Siggelkow et al. 2004) used either membrane or shell elements to represent
the cortical bone of the pelvis. In general this was to avoid problems associated
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with excessive aspect ratios that could occur using solid elements. In the models
presented here solid elements were used to represent cortical bone.The mesh
density adopted ensured that the element aspect ratios were appropriate. Previous
studies have used uniform cortical bone thicknesses in the range 1-2 mm (Anderson
et al. 2004, Dalstra & Huiskes 1995, Garcia et al. 2000, Goel et al. 1978, Oonishi
et al. 1983, Phillips et al. 2004b, Siggelkow et al. 2004, Thompson et al. 2002).
Subject specific studies in which the cortical bone thickness was varied based on
computed tomography (CT) scans found a range of around 0.5−5.0 mm (Anderson
et al. 2004, Dalstra & Huiskes 1995). The thickness of cortical bone close to the
acetabulum was found to be around 2 mm. In the models presented here the
thickness of the cortical bone was assumed to be around 2 mm throughout. The
model was orientated in 3D space based on the pelvis co-ordinate system, in the
anatomic position described in Bergmann et. al (Bergmann et al. 2001), with
the x axis running between the centres of the femoral heads, the y axis pointing
forwards, and the z axis pointing upwards (running from the centre of the femoral
head, through the L5S1 disc in the lateral view). Figure 1 shows a cross section
of the acetabular construct in the transverse (x − y) plane, showing a graft bed
thickness of around 20 mm (representing a relatively large defect).
Following generation of the solid geometry, four noded tetrahedral elements
were used in Abaqus/CAE to generate unstructured meshes of the hemi pelvis and
acetabular construct. The average element edge length used was 2 mm. Figure 2
shows the MCB mesh with the outlines of the trabecular bone and cortical bone
also shown. Figure 3 shows meshes of the bone cement, acetabular cup, and femoral
head, with the outline of the MCB bed also shown. A cement thickness of 2 mm was
assumed. An acetabular cup, with an inside radius of 9 mm, and an outside radius
of 18 mm, including a 2 mm flange was used. Continuity was assumed between the
materials comprising the acetabular construct. Fixed boundary conditions were
applied at the sacro-iliac and pubic joints. Loading was applied at the centre of a
near rigid femoral head. Smooth sliding surface interactions were assumed between
the femoral head and the acetabular cup. In order to provide better resolution
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cortical?bone
trabecular?bone
MCB?graft?bed
bone?cement
acetabular?cup
Figure 1: Transverse cross section through the acetabulum
of the stresses the average element edge length was reduced to 0.5 mm at the
interface between the surfaces. The average element edge length in the femoral
head was reduced to 1.0 mm to provide better distribution of the stresses due to
the application of point loading acting at its centre. In total the developed hemi
pelvis model (excluding the femoral head) contained 439305 elements.
2.2Materials definition
Following revision hip arthroplasty, carried out using impaction grafting, the ac-
etabular construct is composed of a number of synthetic and natural materials.
Synthetic materials are bone cement made from poly-methyl-methacrylate (PM-
MA) and the acetabular cup made from ultra high molecular weight poly-ethylene
(UHMWPE). Natural materials can be identified as cortical bone, trabecular bone
and impacted morsellised bone graft. Each of these materials must be assigned
a constitutive driver. The short term stability of the acetabular construct is for
the most part dependent on the behaviour of the bone graft. This is because the
bone graft undergoes significant plastic deformation under normal physiological
loads (Phillips et al. 2004a, Phillips et al. 2006, Phillips 2005). This is not the case
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Figure 2: 3D mesh of the MCB bed, following revision hip arthroplasty
(a) Lateral view(b) Frontal view
Figure 3: 3D mesh of the bone cement, acetabular cup, and femoral head, following
revision hip arthroplasty
for the other materials comprising the acetabular construct, or the metal femoral
head. UHMWPE, PMMA, trabecular bone, cortical bone and the metal femoral
head were assumed to be isotropic linear elastic materials. Dalstra et. al (Dalstra
et al. 1993) found that pelvic trabecular bone was not significantly anisotropic,
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and could be modelled as an isotropic material. The adopted values of Young’s
modulus, E and Poisson’s ratio, ν used for these materials are given in Table 1.
Table 1: Values of E and ν used in the finite element model
MaterialEν
(N/mm2)
0.9 × 103
1.8 × 103
1.5 × 102
1.8 × 104
2.0 × 105
UHMWPE†
PMMA‡
Trabecular bone
Cortical bone
Metal
0.4
0.4
0.2
0.3
0.3
†Ultra High Molecular Weight Poly-ethylene
‡Poly-Methyl-Methacrylate
Previous investigations on morsellised cortico-cancellous bone (MCB) by the au-
thors and others indicate that it is an isotropic elasto-plastic material (Bavadekar
et al. 2001, Blom et al. 2002, Bolder et al. 2003, Brewster et al. 1999, Brodt
et al. 1998, Dunlop et al. 2003, Fosse et al. 2004, Frei et al. 2005, Giesen et al.
1999, Heiner & Brown 2001, Heiner et al. 2005, Phillips et al. 2005, Ullmark
& Nilsson 1999, Ullmark 2000, Verdonschot et al. 2001, Voor et al. 2000, Voor
et al. 2004). Phillips et. al (Phillips et al. 2006) and Phillips (Phillips 2005) per-
formed confined compaction tests on MCB and developed constitutive models to
describe its material behaviour. It was found that the elastic behaviour of MCB
could be defined using a linear pressure dependent relationship.
E = c1+ c2p
(1)
where E is the elastic modulus, p is the pressure, and c1and c2are constants set
to 5 N/mm2and 35, when E and p are expressed in units of N/mm2.
Previous studies (Brewster et al. 1999, Dunlop et al. 2003, Grimm 2003, Phillips
et al. 2005, Verdonschot et al. 2001, Phillips et al. 2006, Phillips 2005) have shown
that MCB experiences irrecoverable plastic strains due to yielding in shear and
compression. The Drucker Prager yield criterion was chosen as a pressure sensitive
yield criterion capable of modelling the shear behaviour of MCB. The addition of
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a cap to the Drucker Prager yield criterion also allows the consolidation behaviour
of MCB under compressive stress to be modelled. A detailed description of the
Drucker Prager Cap (DPC) yield criterion is given by Phillips et. al (Phillips
et al. 2006). In brief the the profile of the yield surface of the DPC model in the
meridional (p − q) stress plane is defined by a line representing yielding in shear,
and an arc representing yielding in compression. In principal stress space, the yield
surface resembles an ice cream cone, with the shear yield surface representing the
cone, and the compression yield surface representing the ice cream.
Figure 4 shows the DPC yield surface in the meridional stress plane. The shear
surface is defined by:
Fs= q − ptanβ − d = 0 (2)
where Fsis the shear yield surface, d represents the cohesion or particle interlocking,
β is the DPC friction angle, p is the pressure, and q is the von Mises stress.
The compression surface is defined by:
Fc=
?
(p − pa)2+
?q
2
?2
−d + patanβ
2
= 0 (3)
where Fcis the cap yield surface, and pais a cap configuration parameter.
The hardening behaviour (position) of the cap is defined by the evolution of pb,
the yield stress in hydrostatic compression, with ǫp
vol, the plastic volumetric strain.
Confined compaction tests carried out by the authors (Phillips et al. 2006, Phillips
2005) found that pbcould be related to ǫp
volusing:
pb= c3(ec4ǫp
vol− 1)(4)
where c3and c4are constants set to 0.5 and 5.0, when pbis expressed in units of
N/mm2.
The initial value of pb(cap position) was assumed to be 1 N/mm2. Lab exper-
iments showed that this value was consistent with the test conducted by surgeons
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p
q
d
?
pa
pb
0.5 d p
(??+???tan )
a
Fs
Fc
hardening
?
Figure 4: Drucker Prager Cap yield surface
during impaction grafting; that bone graft should bounce back (remain elastic)
under thumb pressure.
Based on previous studies (Brewster et al. 1999, Dunlop et al. 2003, Grimm
2003) the DPC friction angle, β was taken to be 40◦and the cohesion d was taken
to be 0.05 N/mm2(close to zero).
Preliminary analyses found that the DPC plasticity model failed to obtain con-
verged solutions in areas where the bone graft bed was unconfined, such as between
the flange of the acetabular cup and the cortical bone. In practice bone graft can
be removed from these areas of the acetabular construct. In the analysis presented
here this was addressed by overlaying the bone graft elements with a layer of elastic
elements with a very low value of Young’s modulus, less than 5% of the lowest val-
ue taken for the bone graft. This artifice provided computational stability without
altering the overall behaviour of the model.
2.3 Loading definition
The model of the hemi pelvis following revision hip arthroplasty was subjected to
three 3D load histories, representative of 5 cycles of normal walking, sitting down,
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and standing up, with the resultant force, R acting at the centre of the femoral head.
The aim of these load histories was to investigate the development of plastic strains
in the bone graft bed, leading to migration and rotation of the acetabular cup, from
its virgin position, due to various activities. Complete Rx, Ry, Rzamplitude curves
were applied, based on Bergmann et. al (Bergmann et al. 2001), who reported
resultant force values in terms of percentage body weight. Figure 5 shows the
applied loading histories, assuming body weight to be 1000 N.
3Results and discussion
Figures 6 and 7 show the displacement, and rotation of the acetabular cup, within
the acetabular construct for the three loading histories.
Comparing the displacement of the acetabular cup over the 5 walking cycles
(Figure 6(a)) with the applied resultant force (Figure 5) it is observed that the
largest displacement occurs in the superior direction, following Rz. After the first
load cycle a large amount of the displacement is not recovered due to the develop-
ment of plastic strains in the bone graft, resulting in migration of the acetabular
cup. The extent of migration is seen to increase gradually with each load cycle,
appearing to tend towards a constant value. This is consistent with the findings of
Phillips et. al (Phillips et al. 2004a) using a plane strain model of the acetabular
construct. Comparing the displacement of the acetabular cup after sitting down
and standing up (Figures 6(b) and 6(c)) with the applied resultant force (Figure 5)
it is observed the largest displacement occurs in the posterior direction, following
Ry. The migration of the acetabular cup due to sitting down and standing up is
similar, and these two activities give rise to a different pattern of migration than
walking.
Examining the rotation of the acetabular cup over the 5 walking cycles (Fig-
ure 7(a)) it is observed that large rotations occur about the y axis, associated with
high values of Rz. This is illustrated in Figure 8 which shows the movement of the
acetabular cup in the frontal plane following 5 normal walking cycles.
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00112233445566
−1000 −1000
−500 −500
00
500500
1000 1000
1500 1500
2000 2000
25002500
Time (Secs)
Force (N)
Rx
Ry
Rz
R
(a) 5 walking cycles
000.50.511 1.51.5222.5 2.533 3.53.544
−1500−1500
−1000−1000
−500−500
00
500500
1000 1000
15001500
20002000
Time (Secs)
Force (N)
Rx
Ry
Rz
R
(b) sitting down
000.50.5111.51.5222.52.5
−1500−1500
−1000 −1000
−500−500
00
500 500
1000 1000
15001500
20002000
Time (Secs)
Force (N)
Rx
Ry
Rz
R
(c) standing up
Figure 5: Rx, Ry, Rzamplitude curves for various activities
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001122334455
−4 −4
−2 −2
00
22
44
66
88
10 10
12 12
14 14
Normalised step time
(a) 5 walking cycles
Displacement (mm)
Ux
Uy
Uz
U
000.10.1 0.20.20.30.30.40.40.50.50.60.60.70.70.8 0.80.90.911
−8−8
−6 −6
−4−4
−2−2
00
22
44
66
88
Normalised step time
(b) sitting down
Displacement (mm)
Ux
Uy
Uz
U
000.10.10.20.20.30.30.40.4 0.50.5 0.6 0.60.70.70.8 0.80.90.911
−12 −12
−10−10
−8 −8
−6−6
−4 −4
−2−2
00
22
44
66
88
10 10
1212
Normalised step time
(c) standing up
Displacement (mm)
Ux
Uy
Uz
U
Figure 6: Displacement of the acetabular cup for various activities
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001122334455
−5 −5
00
55
10 10
1515
2020
Normalised step time
(a) 5 walking cycles
Anticlockwise rotation (degrees)
ωx
ωy
ωz
000.10.10.20.20.30.3 0.40.40.50.50.60.60.70.70.80.80.90.911
−20−20
−15−15
−10−10
−5−5
00
55
1010
1515
2020
Normalised step time
(b) sitting up
Anticlockwise rotation (degrees)
ωx
ωy
ωz
00 0.10.1 0.20.20.30.30.4 0.40.50.50.60.60.70.70.80.80.9 0.911
−30 −30
−25−25
−20−20
−15 −15
−10−10
−5 −5
00
55
1010
1515
20 20
2525
3030
Normalised step time
(c) standing up
Anticlockwise rotation (degrees)
ωx
ωy
ωz
Figure 7: Rotation of the acetabular cup for various activities
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