failure mechanism - Siegmund, kolagen2
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ARTICLE IN PRESS
Journal of Biomechanics 41 (2008) 1427–1435
www.JBiomech.com
Failure of mineralized collagen fibrils: Modeling the role of
collagen cross-linking
Thomas Siegmund
a,
, Matthew R. Allen
b
, David B. Burr
b,c,d
a
School of Mechanical Engineering, Purdue University, USA
b
Department of Anatomy and Cell Biology, USA
c
Department of Orthopaedic Surgery, Indiana University School of Medicine, USA
d
Department of Biomedical Engineering, Indiana University—Purdue University at Indianapolis, USA
Accepted 18 February 2008
Abstract
Experimental evidence demonstrates that collagen cross-linking in bone tissue significantly influences its deformation and failure
behavior yet difficulties exist in determining the independent biomechanical effects of collagen cross-linking using in vitro and in vivo
experiments. The aim of this study is to use a nano-scale composite material model of mineral and collagen to determine the independent
roles of enzymatic and non-enzymatic cross-linking on the mechanical behavior of a mineralized collagen fibril. Stress–strain curves were
obtained under tensile loading conditions without any collagen cross-links, with only enzymatic cross-links (modeled by cross-linking the
end terminal position of each collagen domain), or with only non-enzymatic cross-links (modeled by random placement of cross-links
within the collagen–collagen interfaces). Our results show enzymatic collagen cross-links have minimal effect on the predicted
stress–strain curve and produce a ductile material that fails through debonding of the mineral–collagen interface. Conversely, non-
enzymatic cross-links significantly alter the predicted stress–strain response by inhibiting collagen sliding. This inhibition leads to greater
load transfer to the mineral, which minimally affects the predicted stress, increases modulus and decreases post-yield strain and
toughness. As a consequence the toughness of bone that has more non-enzymatically mediated collagen cross-links will be drastically
reduced.
r
2008 Elsevier Ltd. All rights reserved.
Keywords: Bone; Computational mechanics; Fibril; Collagen; Cross-linking
1. Introduction
mineralized collagen fibrils (MCF) can be considered as a
composite of mineral, collagen, and water. While the
mineral dictates much of the tissue stiffness, collagen has a
profound effect on bone’s post-yield properties (e.g. energy
absorption) (
Burr, 2002
).
Biomechanical effects of collagen depend largely on
cross-linking (
Knott and Bailey, 1998
;
Viguet-Carrin et al.,
2006a
;
Wang et al., 2003
). Cross-linking is either enzyma-
tically or non-enzymatically mediated (
Bailey et al., 1998
;
Eyre et al., 1988
;
Viguet-Carrin et al., 2006a
). The
enzymatic process, mediated by lysyl oxidase, results in
the trivalent collagen cross-links pyridinoline (PYD) and
deoxypyridinoline (DPD). Non-enzymatic collagen cross-
linking (producing advanced glycation end products such
as pentosidine) occurs via spontaneous condensation of
arginine, lysine, and free sugars (
Monnier, 1989
;
Bailey
The mechanical behavior of bone is significantly
influenced by bone mass, mineral density, and micro-
architecture (
Gibson, 2005
;
Bevill et al., 2006
), but
contributions of collagen to bone’s pre- and post-yield
properties are less well understood (
Burr, 2002
). Bone is a
hierarchical material system (
Weiner and Traub, 1992
;
Weiner and Wagner, 1998
) that can be influenced
mechanically by alterations at any level (
Ballarini et al.,
2005
). Collagen, however, appears to play a major role on
the bone nano-scale where collagen fibrils (CF) and
Corresponding author. 585 Purdue Mall, West Lafayette, IN 47907,
USA. Tel.: +1 765 494 9766.
E-mail address:
0021-9290/$ - see front matter
r
2008 Elsevier Ltd. All rights reserved.
doi:
ARTICLE IN PRESS
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T. Siegmund et al. / Journal of Biomechanics 41 (2008) 1427–1435
et al., 1998
). Experiments in vitro (
Vashishth et al., 2004
;
Wu et al., 2003
;
Catanese et al., 1999
) and in vivo
(
Boxberger and Vashishth, 2004
;
Tang et al., 2005
;
Wang
et al., 2002
;
Viguet-Carrin et al., 2006b
;
Allen et al., 2007
)
have documented that increases in cross-linking are
associated with enhancement of some mechanical proper-
ties (strength and stiffness) and reductions of others
(energy absorption). These experimental data are limited
in their ability to define individual biomechanical effects of
altered cross-linking as concomitant changes associated
with aging (increased mineralization and increased micro-
damage) or pharmacological treatment also contribute,
either positively or negatively, to mechanical properties
(
Allen and Burr, 2007
). Given difficulties in separating
these factors experimentally, computational models pro-
vide a necessary tool for examining specific contributions
of collagen cross-linking to mechanical properties.
Several past investigations considered mechanistic mod-
els for the mineral–collagen composite underlying bone.
Hellmich et al. (2004)
,
Fritsch and Hellmich (2007)
, as part
of hierarchical micromechanical models, employed the
equivalent inclusion method to study contributions of
water and non-collagenous proteins on stiffness. Using a
shear lag model to predictions of bone fibril stiffness have
been made based on mineral content and spacing between
mineral platelets (
Ja¨ ger and Fratzl, 2000
;
Kotha and
Guzelsu, 2003
). In
Kotha and Guzelsu (2003)
, a yield
condition for the collagen phase was included.
Wang and
Qian (2006)
showed that increased non-enzymatic glyca-
tion reduces the collagen nonlinearity and makes crack
coalescence more likely by allowing stress concentrations
to develop around existing cracks. In combination this
would make the tissue more brittle, and accelerate fracture.
Arnoux et al. (2002)
employed a Mohr–Columb model to
account for failure but did not distinguish between mineral
and collagen as the underlying components of bone.
Ji and
Gao (2004)
employed the virtual internal bond method and
concluded that bone strength hinges upon optimization of
the tensile strength of the mineral phase.
A mechanistic model of a MCF is presented with
detailed consideration of the influence of collagen cross-
links on deformation and failure. Using an enhanced
continuum model, we determine the roles of enzymatic and
non-enzymatic cross-linking. We hypothesize that pre-
dicted stress–strain curves under conditions with no
collagen cross-links, with only enzymatic cross-links
(modeled by cross-links at collagen end terminals), or with
additional non-enzymatic cross-linking (modeled by ran-
domly located cross-links) would exhibit significantly
different characteristics.
2. Methods
2.1. Model geometry
Mineral and collagen phases are distributed following the model of
Petruska and Hodge (1964)
. The mineral phase is located within gaps
created by the collagen assembly (
Lees, 1987
,
Landis, 1995
) at least for
CFs during early mineralization (
Fig. 1
a). The off-set staggered array
model is replaced by a simpler periodicity model,
Ja¨ ger and Fratzl (2000)
,
(
Fig. 1
b). Then, the mechanical analysis can be performed considering a
periodic unit cell. Such a fundamental entity contains two quarters of
mineral phase embedded in collagen (
Figs. 1
b and c). The analysis domain
can be reduced in size employing antisymmetry boundary conditions to
the subdomain given in
Fig. 1
d. Uniaxial tensile loading is considered
along the y-axis,
Fig. 1
b. Boundary conditions are given in the
a/2
p
L/2
t/2
n.d
t/2
y
x
A
Fig. 1. (a) A two-dimensional representation of the staggered array model for minerialized collagen fibril comprised of mineral and collagen, (b)
representation of model (a) with simplified periodicity of collagen (white) and mineral (gray), (c) unit cell underlying the model of (b): mineral platelet
width t, platelet length L, collagen helix diameter d, number of collagen helixes n, (d) computational unit cell with actual geometric proportions.
ARTICLE IN PRESS
T. Siegmund et al. / Journal of Biomechanics 41 (2008) 1427–1435
1429
supplementary website. Collagen is composed of individual collagen
molecules aligned parallel to the main faces of mineral platelets. The
collagen phase is divided into subdomains each corresponding to an
individual collagen molecule. The plate-shape geometry of the hydro-
xyapatite motivates the use of a two-dimensional plane strain model
corresponding to the mid-plane of the mineral–collagen assembly. The
model is characterized by periodicity p (
Ja¨ ger and Fratzl, 2000
):
1.5
1
0.5
p ¼
L þ a
2
¼ 67 nm;
(1)
0
mineral platelet length L and thickness t, the distance between the short
faces of the mineral platelets a, and the distance between the long faces of
mineral platelets b. The mineral volume fraction V
V
is:
-0.5
lt
ðL þ aÞðb þ tÞ
.
-1
V
V
¼
(2)
-1.5
Given V
V
, phase arrangements can be constructed choosing two of the
parameters t, L, a, and b. The space between mineral platelets is a multiple
of the diameter of a collagen helix b ¼ nd. We assume V
V
¼ 0:3(
Currey,
1984
,
Fritsch and Hellmich, 2007
), t ¼ 2.5 nm (
Gourrier et al., 2007
), and
b ¼ 3d ¼ 31.5 nm ¼ 4.5 nm (three collagen domains between mineral
platelets). Consequently, L ¼ 113.9 nm and a ¼ 20.1 nm.
Division of collagen into subdomains is undertaken only for the area
between mineral platelets. Adjacent to the platelet ends shear deformation
is not relevant for the loading conditions considered (
Ja¨ ger and Fratzl,
2000
).
-2
-2.5
-1
0
1
2
3
4
5
6
Δ
n
/
δ
n
1.5
2.2. Elasticity and interfaces
1
Hydroxyapatite is an elastic isotropic solid, E
m
¼ 100GPa and
n
m
¼ 0.28 (
Katz and Ukraincik, 1971
;
Viswanath et al., 2007
). Each
collagen domain represents a collagen triple helix in a wet environment,
and is described as homogeneous elastic solid with E
c
¼ 5GPa and
n
c
¼ 0.2. Considering that a single collagen molecule would not exhibit
shear deformation, collagen subdomains are assigned a high shear
modulus, G
c
¼ 50GPa. Then each collagen domain deforms only in
tension, similar to the situation present in bead models of collagen
(
Bu¨ hler, 2006
).
Elastic model domains are connected across interfaces S
int
with
mechanical interactions described by a cohesive law connecting tractions
T
CZ
to material separation D. The interface state changes in response to
loading. T
CZ
and D possesses normal (n) and tangential (t) components,
T
CZ
¼ T
n
n+T
t
t, D ¼ D
n
n+D
t
t. Material separation D is calculated from
displacements u
+
and u
on corresponding locations on opposing internal
surfaces, + and ,asD ¼ u
+
u
. The constitutive relationship between
T
CZ
and D follows from a potential function f (
Xu and Needleman, 1994
):
0.5
0
-0.5
-1
-1.5
-3
-2
-1
0
1
2
3
Δ
t
/
δ
t
fðD
n
; D
t
Þ¼f
0
f
0
exp
D
n
d
0
D
n
d
0
þ exp
D
t
d
0
.
(3)
Fig. 2. Constitutive relations for interfaces: (a) normal traction T
n
—
normal separation D
n
response for D
t
¼ 0, and (b) tangential traction T
t
—
tangential separation D
t
response for D
n
¼ 0.
Tractions are derivatives of f (
Fig. 2
):
1 þ
"
þ exp
D
t
d
0
!
#
T
n
¼
qf
qD
n
¼s
max
e exp
D
n
D
n
d
0
d
0
max
; d
m2
0
; f
m2
0
are
obtained based on the following: d ¼ 1.5 nm, a rise of the collagen helix
per turn of 0.3 nm, hydrogen bonds present at every two out of three turns
of the collagen helix (a complete layer of structural water between mineral
and collagen), and a bond energy of 20 kJ/mol for hydrogen bonds
(
Steiner, 2002
). Molecular interface bond energies (J/mol) are translated
into the continuum framework of Eq. (4) with cohesive zone energies given
in J/m
2
. With a projected area of the collagen helix onto the mineral
surface (dL) one obtains f
m2
0
¼ 1:5 10
7
J=mm
2
. The equilibrium
bond length of hydrogen bonds determines the cohesive length,
d
m2
0
¼ 0:2 nm, and consequentlys
m2c
.
exp
D
t
d
0
T
t
¼
qf
qD
t
¼2s
max
e
D
t
d
0
exp
D
n
d
0
(4)
p
s
max
¼ 2:33s
max
. The cohesive energy is f
0
¼ s
max
d
0
e.
The model response considering elasticity and interface behavior is
solved with the finite element (FE) method. Details are given on the
supplementary website.
As uncertainty exists on the exact potentials for bonds present in the
MCF, the generic form of potential (Eq. (4)) is employed. Parameters
s
max
; d
0
; f
0
depend on the specific bond acting for a particular interface.
2e
max
¼ 270MPa.
Weak interactions between collagen molecules occur due to hydrogen
bonds, non-collageneous proteins, and electrostatic interaction and are
described by s
c2c
max
; d
c2
0
; f
c2
0
. The shear yield stress is associated with t
c2c
max
.
Cohesion of the mineral–collagen interface is due to a layer of
structural water (
Wilson et al., 2005
). Values of s
m2c
The constitutive parameters are the cohesive strength s
max
—the maximum
traction, and the cohesive length d
0
—the material separation D
n
required
for T
n
to reach s
max
for D
t
¼ 0. The maximum of T
t
at D
n
¼ 0is
t
max
¼
ARTICLE IN PRESS
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T. Siegmund et al. / Journal of Biomechanics 41 (2008) 1427–1435
max
=2:33 12:9MPa, a lower
bound value of the shear yield stress (10–19MPa) of the collagen matrix in
plexiform bone (
Kotha and Guzelsu, 2003
). Following (
Bu¨ hler, 2006
) the
cohesive length of weak collagen–collagen interaction is d
c2c
max
¼ 30MPa it follows that t
c2c
max
¼ s
c2c
0.6
0
¼ 1 nm.
Collagen cross-links provide local strong interactions between collagen
molecules. Properties are represented as s
max
; d
0
; f
0
. Cross-links consist
predominantly of C–C and C–N bonds within a single molecule (
Knott
and Bailey, 1998
). Relevant computations (
Beyer, 2000
) and experiments
(
Garnier et al., 2000
) indicate ultimate loads at rupture of several nN. A
representative value of F
u
¼ 1.5 nN is employed. Each cross-link site
corresponds to an interaction within single a cohesive element, and F
u
is
translated into s
max
. The influence area of a single cross-link equals that of
a cohesive element in the model plane (67 nm/200) times the diameter of
the collagen helix. The cohesive strength follows as
s
max
¼ F
u
=½ðp=200Þd¼3000MPa. The elongation of a C–C bond to
rupture is 0.1 nm (
Beyer, 2000
). In cross-links several C–C bonds exist in
series. For a sequence of 10 C–C bonds thus d
0
¼ 1 nm. Cross-link types
are differentiated by spatial position. Enzymatic cross-links are located at
the collagen overlap position (
Eyre and Wu, 2005
), i.e. the end terminal
position of collagen molecules. Enzymatic cross-links are thus placed
beyond end faces of mineral platelets. Non-enzymatic cross-links appear to
possess no specific spatial arrangement. They are placed randomly
between collagen–collagen domains.
A potential fracture site is introduced at the center of the mineral
(y ¼ 0). Parameter values for mineral fracture (s
max
; d
0
; f
0
) follow from
brittle ceramics. The strength s
max
is a fraction of the elastic modulus,
s
max
¼ 3000MPa ¼ð3=100ÞE
m
MPa. The theoretical fracture toughness
of a perfect covalent crystal is 1–4 J/m
2
. We employ f
0
¼ 0:73 J=m
2
, and
consequently, d
0
¼ 0:09 nm.
0.5
mineralized fibril
perfect interfaces
0.4
0.3
mineralized fibril
weak interactions only
0.2
0.1
collagen only
0
0
0.01
0.02
0.03
0.04
0.05
ε
y
Fig. 3. Predicted stress–strain, (S
y
/E
c
)–e
y
, response for a model with
collagen only, a model fibril with all interfaces perfectly bonded, a model
fibril with weak collagen–collagen interaction (no cross-links). Stresses are
normalized by the extensional modulus assumed for a single collagen
molecule.
Interface debonding
3. Results
Mineral
Collagen
First, a demineralized and a mineralized model fibril
with perfectly bonded interfaces are considered. Stress–-
strain curves are depicted in
Fig. 3
. The demineralized
model fibril behaves nearly linear. The modulus is slightly
lower than E
c
ð1 V
V
Þ as some sliding between collagen
helices enhance deformation. The model MCF with
perfectly bonded interfaces also possesses a linear mechan-
ical response. The modulus is high and approximately
E
c
ð1 V
V
ÞþE
m
V
V
. Deformation occurs through elastic
stretching only, and the reinforcement effect of the mineral
is fully realized.
Fig. 3
compares the two reference solutions
to a MCF including weak collagen–collagen interaction.
Here only the initial mechanical response is linear with the
modulus between that of the two reference cases. The
response is nonlinear for e40.02, and the stress reaches a
maximum value followed by a rapid but stable drop.
Fig. 4
shows a detail of the deformed FE grid. Two mechanisms
are responsible for the nonlinearity. The initial nonlinear
elastic deformation response is dominated by sliding
between collagen helices in overlap zones, visualized by
the offset between elements. Only insignificant sliding
between collagen domains is present outside of the overlap
zones. Failure is due to debonding of the mineral–collagen
interface starting at the short platelet face and progressing
along the long-face of the mineral.
Fig. 5
visualizes the
deformation modes through contour plots of the displace-
ment component in the load direction (u
y
). The deminer-
alized fibril (
Fig. 5
a) is mostly deformed through
homogeneous stretching of collagen (indicated by con-
Collagen
½ Collagen
Sliding of collagen molecules
Fig. 4. Detail of deformed computational mesh for the model fibril
without cross-links. Sliding along the collagen–collagen interfaces is
visualized by the offset in the finite element meshed in individual domains.
Interfaces in the computational model are indicated with symbol ‘‘
K
’’.
tinuity of contours) but sliding (indicated by discontinuity
of contours) close to collagen end positions also is active.
For the model with perfect interfaces (
Fig. 5
b), tensile
deformation is due to stretching alone. In the model with
weak collagen interaction, the elastic property mismatch
between collagen and mineral induces significant sliding
between collagen domains in overlap regions. Outside that
region deformation is due to stretching of collagen together
with some smaller amount of sliding.
Fig. 6
depicts the response of model MCFs with only
weak collagen–collagen interaction in comparison to that
of models containing enzymatic cross-links. The additional
enzymatic cross-links do not alter the characteristics of the
mechanical response. Changes in the displacement field are
minor (
Fig. 5
d). This is not surprising as sliding is
insignificant outside of overlap zones even in the absence
of enzymatic cross-links (
Fig. 5
c).
Fig. 6
shows stress–strain
With s
c2c
ARTICLE IN PRESS
T. Siegmund et al. / Journal of Biomechanics 41 (2008) 1427–1435
1431
u
y
+1.401e−03
+1.168e−03
+9.342e−04
+7.006e−04
+4.671e−04
+2.335e−04
+2.910e−11
−2.335e−04
−4.671e−04
−7.006e−04
−9.342e−04
−1.168e−03
−1.401e−03
u
y
+1.405e−03
+1.171e−03
+9.370e−04
+7.027e−04
+4.685e−04
+2.342e−04
+0.000e+00
−2.342e−04
−4.685e−04
−7.027e−04
−9.370e−04
−1.171e−03
−1.405e−03
u
y
+1.367e−03
+1.139e−03
+9.116e−04
+6.837e−04
+4.558e−04
+2.279e−04
+0.000e+00
−2.279e−04
−4.558e−04
−6.837e−04
−9.116e−04
−1.139e−03
−1.367e−03
u
y
+1.522e−03
+1.268e−03
+1.015e−03
+7.610e−04
+5.073e−04
+2.537e−04
−5.821e−11
−2.537e−04
−5.073e−04
−7.610e−04
−1.015e−03
−1.268e−03
−1.522e−03
u
y
+1.560e−03
+1.300e−03
+1.040e−03
+7.800e−04
+5.200e−04
+2.600e−04
+0.000e+00
−2.600e−04
−5.200e−04
−7.800e−04
−1.040e−03
−1.300e−03
−1.560e−03
u
y
+1.418e−03
+1.182e−03
+9.456e−04
+7.092e−04
+4.728e−04
+2.364e−04
+0.000e+00
−2.364e−04
−4.728e−04
−7.092e−04
−9.456e−04
−1.182e−03
−1.418e−03
Fig. 5. Contour plots of displacement component along the collagen molecule axis, u
y
in (mm): (a) collagen only; (b) mineralized fibril, interfaces perfectly
bonded, (c) mineralized fibril, no cross-links; (d) mineralized fibril, enzymatic cross-links only; (e) mineralized fibril, enzymatic cross-links and low level of
non-enzymatic cross-links, N ¼ 1; and (f) mineralized fibril, high level of non-enzymatic cross links, N ¼ 10. Cross-links are indicated as ‘‘
K
’’.
curves for models with one additional non-enzymatic
cross-link. The addition of a non-enzymatic cross-link
significantly outside the interface region dominated by
interface sliding does not alters the model response. If,
however, the non-enzymatic cross-link is located such that it
blocks sliding, see
Fig. 5
e, the material response changes.
Then, a reduced nonlinearity and a higher modulus are
found. While the ultimate stress changes only little, the
failure response is brittle as evident from the pronounced
snap back. The strain to ultimate stress is much reduced
and as a consequence the toughness—defined as the area
under the stress–strain curve until the ultimate stress—is
significantly lowered.
Fig. 7
depicts the influence of cross-link density on the
mechanical response. Non-enzymatic cross-links were
placed randomly along the collagen–collagen interfaces,
with N the number of cross-links per 1/2 unit cell. An
increase in N increases the initial modulus (which remains
however below that of the model with perfectly bonded
interfaces), reduces the nonlinearity, and the strain to reach
the ultimate stress. Failure occurs due to mineral fracture
and subsequent debonding along the mineral collagen
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