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This is to certify that the
dissertation entitled
Experimental and Analytical Deve10pment of

A Poroelastic Finite Element Model for Tendon

presented by

Theresa Staton Atkinson

has been accepted towards fulﬁllment
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Ph . D . . Mechanics
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EXPERIMENTAL AND ANALYTICAL DEVELOPMENT OF
A POROELASTIC FINITE ELEMENT MODEL FOR TENDON

By

Theresa Staton Atkinson

A Dissertation

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Materials Science and Mechanics

1998

ABSTRACT

EXPERINIENTAL AND ANALYTICAL DEVELOPMENT OF A POROELASTIC
FINTTE ELEMENT MODEL FOR TENDON

By

Theresa Staton Atkinson

Mechanical models which allow the mechanical response of tendon to be predicted
and quantiﬁed are important in the development and assessment of orthopaedic
reconstruction techniques. In the ﬁrst study an autogenous patellar tendon ACL
reconstruction was performed in a goat model in order to gain ﬁrst hand insight into the
assessment of reconstruction techniques. Extensive tendon and fat pad proliferation were
observed along with signiﬁcant reductions in the biomechanical properties of the host
tendon. An existing mechanical model was used to obtain a description of the tensile
response of the tissue. While these data helped explain some of the clinical complications
documented in the reconstructed joint, they did not describe the role of the ﬂuid within the
healing tissue. Experimental evidence suggests that the tensile behavior of tendon is a
function of the collagen structure of the tissue and the tissue hydration. The models
currently available do not offer a means by which the hydration effects might be explicitly
explored. In order to study potential inﬂuences of water content on tendon tensile
response a ﬁnite element model of a subfascicle (a microstructural element of tendon) was

constructed in the second study. The collagen ﬁber morphology reﬂected in the model

interacted with the interﬁbrillar matrix to produce behaviors similar to those seen in
tendon and ligament during tensile, cyclic, and relaxation experiments conducted by
others. Although this model exhibited mechanical responses which were similar to those
observed in whole tendon and ligament, it was preliminary in nature and as such contained
some undesirable compromiSes. In the third study a more detailed description of the
subfascicular microstructure was incorporated into the model. This model was shown to
exhibit reasonable relaxation and tensile responses as well as a realistic, positive pressure
proﬁle throughout the subfasicle. In the fourth study experiments were performed to
support the development of the subfascicle model and its extension to whole tendon. The
experimental data suggested that small portions of tendon exhibit a higher tensile modulus,
a slower rate of relaxation and a lower amount of relaxation in comparison to larger
specimens from the same location in the same tendon. In the ﬁfth study the subfascicle
model was able to match subfascicle relaxation and constant strain rate tensile responses
as described in the previous experimental study. In addition, a fascicle model, consisting of
two subfascicles surrounded by epitenon, was created to investigate potential interactions
between subfascicles and the connective tissue membrane. This analysis suggested that the
presence of connective tissues in tendon may play an important role in deﬁning the whole
tendon relaxation response. In the ﬁnal study the subfascicle model was utilized in the
development of a recruitment model tendon. This work suggested that subfascicle
organization within a tendon specimen also plays a role in the development of the
relaxation response. These studies highlight the importance of the collagen microstructure

in the development of the time varying responses of tendon.

To my husband Pat whose insight, advice, and
loving support has comforted and inspired me
during the course of this research and to my parents
who encouraged me to persevere.

ACKNOWLEDGMENTS

I would like acknowledge Dr. Roger Haut for his guidance and support
throughout the course of this project. Dr. Haut inspired me to face the challenges my
research presented and made it possible for me to present and publish my research. I
would also like to acknowledge Dr. Nicholas Altiero for his contributions to my research
and for his assistance in preparing my work for publication. I also express my gratitude to
Dr. Amoczky and Dr. Beck for their participation on my committee and for making
themselves available to me for consultation throughout my years of graduate education. I
thank my co-authors, without whom I would not have been able to complete my studies:
Pat Atkinson, Benjamin Ewers, and Vince Mendenhall. Finally I also wish to thank all
those in the lab who have worked with me on a daily basis and who have made my
graduate experience so memorable: Bill Newben'y, Rich Banglmaier, Dana Dvorchek-

Driksna, Cliff Beckett, and Jane Walsh.

TABLE OF CONTENTS

LIST OF TABLES ......................................................................................................... vi
LIST OF FIGURES ............. , .......................................................................................... vii
INTRODUCTION .......................................................................................................... 1
CHAPTER 1

Patellar Tendon And Inﬁapatellar Fat Pad Healing

After Harvest Of An ACL Graft ..................................................................................... 10
(to be published July 1998 in Journal of Surgical Research)

Abstract ......................................................................................................................... 1 1
Introduction .................................................................................................................. 12
Materials and Methods .................................................................................................. 13
Results .......................................................................................................................... 17
Discussion ..................................................................................................................... 19
CHAPTER 2

A Poroelastic Model That Predicts Some Phenomenological Responses

of Ligaments and Tendons ............................................................................................. 30
(published in November 1997 in Journal of Biomechanical Engineering)

Abstract ......................................................................................................................... 31
Introduction .................................................................................................................. 31
Methods ........................................................................................................... -. ............ 34
Results .......................................................................................................................... 39
Discussion ..................................................................................................................... 41
CHAPTER 3

A Microstructural Poroelastic Model For Patellar Tendon ............................................. 63
(published in June 1997 ASME Summer Bioengineering Conference Proceedings)
Abstract ......................................................................................................................... 64
Introduction .................................................................................................................. 64
Methods ........................................................................................................................ 65
Results .......................................................................................................................... 66
Discussion ..................................................................................................................... 67
CHAPTER 4

The Tensile and Stress Relaxation Responses of Human Patellar Tendon

Varies with Specimen Cross Sectional Area ................................................................... 71
(submitted for consideration in Journal of Biomechanics)

Abstract ......................................................................................................................... 72

Introduction .................................................................................................................. 73

Methods ........................................................................................................................ 75
’ - Results .......................................................................................................................... 79

Discussion ..................................................................................................................... 80

CHAPTER 5

Extension of a Microstructural Model for a Subfascicle Toward a Description

of Whole Tendon ................. . .......................................................................................... 97

(intended for submission to Journal of Biomechanics)

Abstract ......................................................................................................................... 98

Introduction .................................................................................................................. 99

Methods ...................................................................................................................... 102

Results ........................................................................................................................ 107

Discussion ................................................................................................................... 109

CHAPTER 6

A Subfascicle Recruitment Model for Tendon .............................................................. 131

Abstract ....................................................................................................................... 132

Introduction ................................................................................................................ 132

Methods ...................................................................................................................... 133

Results ........................................................................................................................ 137

Discussion ................................................................................................................... 139

CHAPTER 7

Conclusions and Recommendations for Future Work ................................................... 156

Appendix A

Porous Elastic Material Model ..................................................................................... 165

Appendix B

Subfascicle model Abaqus code ................................................................................... 169

Appendix C

Fascicle model Abaqus code ........................................................................................ 177

LIST OF TABLES

CHAPTER 1
Table 1: Biomechanical and biochemical data for goat patellar tendons
and infrapatellar fat pads following harvest of a patellar tendon graft

to reconstruct the ACL (mean i one standard deviation) ................................................ 25
CHAPTER 2
Table 1: Material Coefﬁcients ........................................................................................ 51
CHAPTER 4
Table 1: Description of Specimens ................................................................................. 88
CHAPTER 5
Table 1: Material Coefﬁcients for Subfascicle FEM ...................................................... 1 19
Table 2: Fascicle FEM Model Relaxation Response to 2% Strain ................................. 119
CHAPTER 7
Table 1: Comparison of Relaxation and Creep ............................................................ 163

LIST OF FIGURES

CHAPTER 1

Figure 1 ............................... . .......................................................................................... 27
Figure 2 ......................................................................................................................... 28
Figure 3 ......................................................................................................................... 29
CHAPTER 2

Figure 1 ......................................................................................................................... 54
Figure 2a ....................................................................................................................... 55
Figure 2b ....................................................................................................................... 56
Figure 2c ....................................................................................................................... 57
Figure 3 ......................................................................................................................... 5 8
Figure 4a ....................................................................................................................... 59
Figure 4b ....................................................................................................................... 60
Figure 5a ....................................................................................................................... 61
Figure 5c ....................................................................................................................... 62
CHAPTER 3

Figure 1 ......................................................................................................................... 69
Figure 2 ......................................................................................................................... 70
Figure 3 ......................................................................................................................... 70
Figure 4 ......................................................................................................................... 70
CHAPTER 4

Figure 1 ......................................................................................................................... 91
Figure 2 ......................................................................................................................... 92
Figure 3 ......................................................................................................................... 93
Figure 4 ......................................................................................................................... 94
Figure 5 ......................................................................................................................... 94
Figure 6 ......................................................................................................................... 95
Figure 7 ......................................................................................................................... 95
Figure 8 ......................................................................................................................... 96
CHAPTER 5

Figure 1 ....................................................................................................................... 122
Figure 2 ....................................................................................................................... 123
Figure 3 ....................................................................................................................... 124
Figure 4 ....................................................................................................................... 125
Figure 5 ....................................................................................................................... 126
Figure 6 ....................................................................................................................... 127

Figure 7a ............................................................................... 128

Figure 7b ..................................................................................................................... 129
Figure 8 ....................................................................................................................... 130
CHAPTER 6

Figure 1 ....................................................................................................................... 147
Figure 2 ....................................................................................................................... 147
Figure 3 ....................................................................................................................... 148
Figm'e 4 ....................................................................................................................... 149
Figure 5 ....................................................................................................................... 150
Figure 6 ....................................................................................................................... 150
Figure 7 ...................................................... ' ................................................................. 151
Figure 8 ....................................................................................................................... 151
Figure 9 ....................................................................................................................... 152
Figure 10 ..................................................................................................................... 152
Figure 11 ..................................................................................................................... 153
Figure 12 ..................................................................................................................... 153
Figure 13 ..................................................................................................................... 154
Figure 14 ..................................................................................................................... 154
Figure 15 ..................................................................................................................... 155
CHAPTER 7

Figure 1 ....................................................................................................................... 164
Figure 2 ....................................................................................................................... 164

INTRODUCTION

Most currently available tendon and ligament models consider the tissue to be
entirely composed of collagen (Belkoff and Haut, 1992, Hurschler et al., 1997, Kwan and
Woo, 1989, Stouffer et al., 1985,) and utilize a distribution function to describe the
recruitment of collagen and capture the inﬂuence of the collagen structure on the tensile
response (Belkoff and Haut, 1992, Hurschler et a1., 1997, Kwan and Woo, 1989).
However, the collagen in tendon has been described as being arranged in subfascicles and
fascicles (collections of subfascicles) which are enclosed by connective tissue sheaths
(Y ahia and Drouin, 1988). Danylchuk (1978) describes the collagen fasciculi as the
portion of tendon responsible for its tensile su‘ength. Kastelic (1980) modeled the collagen
fascicle and suggested that the nonlinear tensile response of tendon might arise from
collagen recruitment within the fascicle.

More recently, experimental studies have suggested that other tissue components,
such as water (which comprises 60-70% of the weight of tendons and ligaments), might
play a signiﬁcant role in the tensile behavior of ligaments and tendons. For example,
experiments suggest that human patellar tendons exhibit higher tensile modulus and
ultimate strength when tested in a bath environment versus in air (Haut and Powlison,
1990). Similarly, the tendon is stiffer when tested in a hydrating solution versus a
dehydrating solution (Haut, et. a1 1995). Furthermore, both studies suggest that the time

dependent responses of human tendon are inﬂuenced by water. For example, in relaxation

tests, specimens tested in a saline bath exhibit a larger and more rapid load relaxation
versus those tested in air (Haut and Powlison, 1990). Similarly, specimens tested in a
hydrating bath have greater load relaxation than those tested in a dehydrating bath (Haut
et al., 1995).

In ligaments, increasing the water content causes the tissue to respond with greater
cyclic load relaxation relative to ligaments with lower water content (Chimich et al.,
1992). Data from cyclic tests also demonstrate monotonic decreases in rat tail tendon
diameter from cycle to cycle (Lanir et al., 198 8). Similar behavior has been documented in
the rat medial collateral ligament (MCL) (Thielke et al., 1995). This deformation suggests
darsiﬁcation of the collagenous structure and exudation of ﬂuid, and is consistent with
reports that glycosaminoglycans (GAGs) and water are exuded during tensile strain tests
(Lanir et al.,l988; Hanniﬁn and Amoczky, 1994).

Effects associated with extracellular water have been observed experimentally, but
few analytical models address the mechanism of its action. Chen et a1. (1993) utilized a
ﬁnite element analysis (FEA) to study permeability effects in tendon and ligament. The
model includes a ﬂuid phase moving around regular cylinders (meant to represent collagen
ﬁbers). Chen and Vanderby (1994) proposed a directionally sensitive permeability for
tendons. The inﬂuence of water has also recently been incorporated in a model of the
rabbit MCL (Wilson et al., 1994). Wilson’s FEA model utilizes a continuum matrix of
' poroelastic material and spring elements attached at nodes to include the elastic stiffness
of collagen. The model predicts that pressure within the MCL is negative during tensile

deformation, implying that ﬂuid ﬂows into the structure. This result, however, is contrary

to current experimental data indicating positive internal pressures (Chen et al., 1995) in
rabbit patellar tendon during tensile stretch and ﬂuid exudation in the canine ﬂexor tendon,
rabbit MCL, and rat tail tendon under tensile strain (Hannifm and Amoczky, 1994; Thielke
et al., 1995; Lanir et al., 1988). Finally, most recently Chen et a1. (1997) proposed a
poroelastic cylinder model for tendon. In this model, however, it was necessary to assume
an unreasonable Poisson’s ratio of 0.65 in order to achieve the desired positive pressures
upon extension.

These contrary results suggest that continuum characterizations of the tendon may
not reﬂect mechanisms important to predicting the role of water in tensile load response.
Lanir et a1. (1988) describe the tendon on a microscopic scale as consisting of an
arrangement of collagen ﬁbrils embedded in a hydrophilic gel. While there is controversy
in the literature concerning the hierarchical levels of collagen, it is generally accepted that
ﬁbrils are combined to form sub-fasciculi, fasciculi, and ﬁnally the whole structure
(Danylchuk et. a1, 1978;Viidik 1990; Yahia and Drouin 1988). It has been proposed that
the morphology of collagen ﬁbrils can be complex, ranging from being helically wound in
the sub-fascicle (termed fascicle by Yahia and Drouin, 1988) in the patellar tendon to a
both helical and planar waveforms in the anterior cruciate ligament (ACL). In the sub-
fasciculi containing helically wound ﬁbrils, the peripheral ﬁbrils appear as undulated
helices, while the central ones are normal helices (Y ahia and Drouin, 1989). It has been
previously suggested that the collagen in tendon and ligament interacts in such a way that
it compresses the gelatinous matrix and creates a sensitivity to strain rate in the tissue

(Lanir, 1978).

The purpose of this disseration was to build a microstructurally accurate model for
a patellar tendon which could be used to study how the ﬂuid present in tendon contributes
to the tissue’s mechanical responses.

The experimental study described in the ﬁrst chapter of this dissertation was
performed to gain insight into tendon healing following harvest of a patellar tendon graft.
Changes in the tendon and the infrapatellar fat pad were examined in a goat model
following reconstruction of the ACL, using an autogenic patellar tendon graft This animal
model approximates the clinical scenario for ACL reconstruction and the anthropometry
of the human knee. Extensive proliferation was observed in the patellar tendon during the
healing process. This proliferation appeared to have a negative effect on the fat pad which
became ﬁbrotic. A mechanical model for tendon formulated by Belkoff and Haut (1992),
was ﬁt to the tensile test data to quantify changes in the tensile response. This model
suggested that the collagen structure in the tendon degraded during the healing process,
becoming more disorganized This work highlighted a need for models to help advance
our tmderstanding of the structure-function relationship in tendon such that a greater
understanding of the healing process might be gained and healing might be improved

In the second chapter a microstuctural ﬁnite element model for tendon was
developed. In this development we hypothesized that longitudinal deformation of the
more helically oriented ﬁbrils in the subfascicles of patellar tendon might compress the
matrix to generate hydraulic pressures and movement of unbound water during tensile
stretch. By this mechanism the load might be shared between collagen and the

surrounding matrix to inﬂuence the tensile response of the entire ligamentous or

tendonous structure. We assumed that the subfascicle may be considered a characteristic
structure of ligament and tendon, which therefore should exhibit behaviors observed in the
gross structure. The ﬁnite element model exhibited stress relaxation and strain rate
sensitivity which were dependent on the level of hydration, and were qualitatively similar
to those reported by other researchers. There were, however, several aspects of this
model which were undesirable: a sealed boundary condition was applied to the model
periphery, the “matrix” portion of the model alone reﬂected pressures in the subfascicle,
and the tensile response of the model was linear. In order to address the limitations of
this early model, a new, more microstructually accurate model was developed in Chapter
3. This model exhibited a continuous positive pressure proﬁle under tensile load and a
nonlinear tensile response.

In Chapter 4 a series of mechanical tests were performed in which quarters of
human patellar tendons were sequentially sectioned. The tests were designed to describe
both collagen recruitment, via a constant strain rate test, and the time dependent stress
relaxation response of specimens harvested from a common location within the same
donor. The objectives of these tests were to determine whether the responses of small
portions of tendons, which were composed of several subfascicles, were qualitatively
similar to those of whole tendon and to identify a relationship between mechanical
behaviors of the small and large portions of tendon within the same subject. The
experimental studies suggested that the constant deformation, stress relaxation response of
human patellar tendon decreases as a frmction of the specimen cross sectional area. These

experiments documented a 50% reduction in the rate and the amount of relaxation as the

specimen cross sectional area decreases from 20 to 1 m2. The larger specimens which
exhibited an increased relaxation response also contained a higher percentage of ﬂuid than
the smaller specimens. This additional ﬂuid may have contributed to the increased
relaxation in the larger specimens. .

In Chapter 5 a quantitative subfascicle model was developed based on the
experimental data collected in Chapter 4. Using this model potential explanations for the
variation in the tissue’s relaxation responses were investigated. The model’s relaxation
response was observed to increase when the water content of the model was increased,
consistent with the experimental data. In addition the subfascicle model was extended to
form a fascicle model consisting of two fascicles surrounded by an epitenon (a connective
tissue membrane). The transversely oriented ﬁbers within the epitenon caused the
subfascicles to be pressed together under a tensile load, causing an increased relaxation
response. This study suggests that both the tissue ﬂuid content and the presense of
oriented connective tissues surrounding the collagen fascicles induce the large relaxation
response characteristic of whole human patellar tendon.

In the sixth chapter an analytic model was created to extend the response
characteristics of the subfascicle into a model which was capable of describing tendon
specimens of arbitrary cross section. The model was ﬁt to experimental data described in
Chapter 4 using the Marquardt’s nonlinear least squares method The ﬁtted parameters
suggest that the tissue hydration and subfascicle organization are separate inﬂuences on
the mechanical response. The ﬁtted models were also utilized to simulate the tissue’s

creep response. The creep and relaxation responses were nearly equivalent for the model

as posed. When a more aggressive relationship between strain and the rate of relaxation
was assumed the model predicted more relaxation than creep, consistent with Thornton et
al’s (1997) experimental observations in the rabbit medial collateral ligament. Further
study is required to develop an experimental basis for the strain/relaxation rate function.

In the ﬁnal chapter concluding remarks are made relative to the beneﬁts and
drawbacks of the modeling approach applied in these studies. Potential extensions of the
research are proposed.

Appendix A contains background information documenting the porous elastic
material model. This material model was utilized to describe the matrix in the subfascicle
model described in Chapter 2. This material was selected because it stiffens under
compression and was therefore thought to be useful in producing the tensile stiffening
response characteristic of tendon. As this model is not commonly used to describe
biologic material, it was therefore eliminated in favor of a linear poroelastic material in
subsequent chapters.

Appendix B contains the Abaqus code for the subfascicle model.

Appendix C contains the Abaqus code for the fascicle model.

REFERENCES

Belkoff, SM. and Haut, RC. (1992) Microstructurally based model analysis of y -
irradiated tendon allografts. J. 0thop. Res. 10, 461-464.

Chen, C., and Vanderby, R., 1997 , A poroelastic model of streaming potential and
interstitial ﬂuid ﬂow in ligament and tendon, In Proceedings of the 1997 Bioengineering
Conference. Sun River, Oregon, pp. 185-186.

Chen, C., McCabe, R., and Vanderby, R. Jr. (1995) Two electrokinetic phenomena
in rabbit patellar tendon: pressure and voltage. In Proceedings of the 1995 Bioengineering
Conference. Beaver Creek, Colorado.

Chen, C. T. and Vanderby, R. (1994) 3-D ﬁnite element analysis to investigate
anisotropic permeability for interstitial ﬂuid ﬂow in ligaments and tendons. In Trans. of the
40th Annual Meeting of Orthop. Res. Soc. New Orleans, LA.

Chen, C. T., Vanderby, R., Graf, B. K., and Malkus, D. S. (1993) Interstitial ﬂuid
ﬂow in ligaments and tendons: effects of ﬁbril spacing and ﬂuid properties. In Proceedings
of the 1993 Bioengineering Conference, Breckenridge, Colorado.

Chimich, D. D., Shrive, N. 6., Frank, C. B., Marchuk, L., and Bray, R. C. (1992)
Water content alters viscoelastic behaviour of the normal adolescent rabbit medial
collateral ligament. J. Biomech. 25, 831-837.

Danylchuk, K.D., Finlay, J .B. and Krcek, JP. (1978) Microstructural organization
of human and bovine cruciate ligaments. Clinical Orthopaedics and Related Research.
131, 294-298.

Hannaﬁn, J .A. and Amoczky, SP. (1994) Effect of cyclic and static tensile loading
on the water content and solute diffusion in canine ﬂexor tendons: an in-vitro study. J.
Orthop. Res. 12, 350-356.

Haut, TL, and Haut, RC. (1997) The state of tissue hydration determines the
strain-rate-sensitive stiffness of human patellar tendon. J. Biomech. 30, 79-82.

Haut, RC. and Powlison, AC. (1990) The effects of test environment and cyclic
stretching on the failure properties of human patella tendons. J. Orthop. Res. 8, 532-540.

Hurschler, C., Loitz-Ramage, B., and Vanderby, R., (1997) A structurally based
stress-stretch relationship for tendon and ligament. J. of Biomech. Eng. 119, 392-399.

Kastelic, J ., Palley, 1., and Baer, E. (1980) A structural mechanical model for
tendon crimping. J. Biomech. 13, 887-893.

Kwan, M.K., and Woo, S. L-Y. (1989) A structural model to describe the
nonlinear stress-strain behavior for parallel-ﬁbered collagenous tissues. J. Biomech. Eng.
111, 361-363.

Lanir, Y., 1978, “Structure-strength relations in mammalian tendon,” Biophysical
J., Vol. 24, pp. 541-554.

Lanir, Y., Saland, E. L, and Foux, A., 1988, “Physico-chemical and
microstuctural changes in collagen ﬁber bundles following stretch in-vitro,” Biorheology
J., Vol. 25(4), pp. 591-604.

Stouffer, DC, Butler, DL, and Hosny, D. (1985) The relationship between crimp
pattern and mechanical response of human patellar tendon-bone units. J. of Biomech. Eng.
107, 158-165.

Thielke, R.J., Vanderby, R. Jr., and Grood, ES. (1995) Volumetric changes in
ligaments under tension. In Proceedings of the 1995 Bioengineering Conference.
Breckenridge, Colorado.

Thornton, G.A., Oliynyk, A., Frank, CB, and Shrive, N.G., 1997, Ligament creep
cannot be predicted from stress relaxation at low stress: A biomechanical study of the
rabbit medial collateral ligament, J. Orthop. Res., 15: 652-656.

Viidik, A. 1990, “Structure and function of normal and healing tendons and
ligaments” in Biomechanics of Diarthrodial Joints Vol 1, (Edited by Mow, V.C.,
Ratcliffe, A., and Woo, S. L-Y.), p. 3-12, Springer-Verlag, N.Y.

Wilson, A. N., Frank, C. B., Shrive, N. G. (1994) The behaviour of water in the
rabbit medial collateral ligament. In Second World Congress of Biomechanics, (Edited by
Blankevoort, L. and Kooloos, J. G. M.), pp. 226b. Amsterdam, The Netherlands.

Yahia, L. H., and Drouin, G., 1989, “Microscopical investigation of canine
anterior cruciate ligament and patellar tendon: collagen fascicle morphology and
architecture,” J. Orthop. Res., Vol. 7, pp. 243-251.

Yahia, L. H., and Drouin, G., 1988, “Collagen structure in human anterior cruciate
ligament and patellar tendon,” J. Mat. Sci, Vol. 23, pp. 3750-3755.

Chapter 1:

Patellar Tendon And Infrapatellar Fat Pad Healing After Harvest
Of An ACL Graﬁ

 

Theresa S. Atkinson, Patrick J. Atkinson, Mendenhall, H.Vincent, and Roger C. Haut

10

ABSTRACT

Clinical studies have documented proliferation of the host patellar tendon and
ﬁbrosis extending into adj acent tissues after reconstruction of the injured anterior cruciate
ligament (ACL) using the central one third of the patellar tendon (PT) as the graft. Such
generalized arthroﬁbrosis has been implicated in knee locking and as possible source of
anterior knee pain. However, it is not clinically feasible to measure changes in tendon
morphology and mechanical properties and degeneration of peripheral tissues over time
following graft harvest. In a rabbit experimental model proliferative changes in the tendon
and the infrapatellar fat pad have been documented following harvest of a central third
tendon graft without ACL reconstruction. Studies in larger animals have shown signiﬁcant
reductions in the strength and stiffness of the healing patellar tendon, but without
assessment of the peripheral tissue response. In the current study an ACL reconstruction
was performed in a goat model using an autogenous patellar tendon graft. Extensive
tendon and fat pad proliferation were observed along with signiﬁcant reductions in the
biomechanical properties of the host tendon. Signiﬁcant fat pad ﬁbrosis was documented
using biochemical methods. The current data conﬁrm that harvest of an autogenous PT
graft for reconstruction of the ACL results in signiﬁcant changes in the PT and adjacent
tissues. These data may help explain some of the clinical complications documented in the

reconstructed joint.

11

INTRODUCTION

Reconstruction of the injured ACL using a full thickness central one third patellar
tendon graft is currently standard practice (Aﬂietti et al., 1994), due to the graft’s superior
strength and the ability to anchor the graft via patellar and tibial bone blocks. However,
signiﬁcant reductions in the strength and stiffness of the healing patellar tendon have been
observed in experimental studies (Burks et al., 1990, Linder et al., 1994, Kamps et al.,
1994). In addition, clinical (Berg, 1992) and experimental (Atkinson et al., 1996) studies
have documented a doubling of the cross sectional area and signiﬁcant changes in the
length of the healing tendon following harvest of the graft. Increased stress on the host
tendon has been implicated in initiating this remodeling process. Other structures in the
knee joint have also shown poor responses following patellar tendon graft harvest.
Clinical studies have documented proliferation, or ﬁbrosis, extending into tissues adjacent
to the patellar tendon, such as the infrapatellar fat pad (Rosenburg et al., 1992, Olglivie-
Harris and Giddens, 1994) and the joint capsule (Richmond and Assal, 1991, Cosgarea et
al., 1994). Such proliferation may lead to loss of extension and contribute to post-surgical
knee pain (Richmond and Assal, 1991). Advanced fat pad ﬁbrosis (Hoffa’s syndrome) is a
clinically recognized source of anterior knee pain (Magi et al., 1991). Clinical studies have
suggested that the removal of ﬁbrotic fat pads can help alleviate pain (France et al.,
Kempf, 1980, Krebs and Parker, 1994). Our rabbit experimental model, in which a patellar
tendon graft is harvested without ACL reconstruction, shows proliferation in the patellar
tendon and ﬁbrosis in the fat pad at six and twelve weeks following resection of the

central third (Atkinson et al., 1996). At 52 weeks the patellar tendon shows

12

biomechanical and histological evidence of a return to normal tissue, and the fat pad
shows decreased ﬁbrosis. In our rabbit model, no bone blocks are excised during resection
of the central third patellar tendon, and fat pad disruption is minimal. In the current study
changes in the tendon and fat pad following reconstruction of the ACL, using an autogenic
patellar tendon graft, were evaluated in a goat model. This model more closely
approximates the clinical scenario for ACL reconstruction and the anthropometry of the
human knee than the previous rabbit model. We hypothesized that this model would also
exhibit proliferation in the patellar tendon, signiﬁcant reduction in the biomechanical
properties of the tendon, and ﬁbrosis of the fat pad similar to our observations following
resection of the central third of the patellar tendon in the rabbit model.
MATERIALS AND METHODS

The right limbs of eighteen female goats (>40kg, >4yrs, mixed breed) were
subjected to patellar tendon graft harvest and ACL reconstruction: six animals were
sacriﬁced at each time point of time zero, 12 weeks and 52 weeks. The left limb served as
an unoperated control for each animal. Time zero animals were sacriﬁced following
surgery, while 12 and 52 week animals were allowed unrestricted mobilization on a farm.
The procedures used in this study conformed to the Quig for the Care _a_nd Use of
W, National Research Council, Revised 1996.
(Motive Procedure

The patellar tendon (PT) was exposed via a lateral skin incision and clearing of the
overlying subcutaneous tissue. A partial thickness PT graft, approximately 1 mm thick and

8 mm wide was harvested. Longitudinal incisions were made at the medial and lateral

l3

margins of the tendon and a 1/3 thickness transverse incision was made in the tendon.
Bone blocks, the width of the autograft, were then harvested at the patella and tibia Two
1 mm diameter holes were drilled through these blocks, and non-absorbable #2 suture
material was passed through the holes to aid in passage and ﬁxation of the graft. The
patella was then medially subluxed and the joint was held in full ﬂexion. The fat pad and
synovium surrounding the ACL were dissected longitudinally to provide visibility of the
ACL insertions. The ACL was incised at the tibial and femoral insertions and removed A
tibial tunnel was constructed by drilling from the tibia, medial to the patellar tendon
insertion, to a point slightly anterior and medial of the. normal ACL insertion. A
posteriorly placed femoral tunnel was then drilled through the joint to the anterior origin
of the lateral head of the gastrocnemius. A cancellous bone screw and spiked washer
were placed in the femur. Isometry of the tunnel placement was checked at ﬁrll extension
and ﬂexion using marked cotton tape. If necessary, the femoral tunnel was widened to
insure isometric placement. The graft was then passed through the bone tunnels. The
non-absorbable suture material was wrapped around the unthreaded end of the bone
screw, and the screw was tightened Small Kirschner wires were used to additionally ﬁx
the bone block. The graft was then placed under manual tension and the joint was cycled
20 times. The non-absorbable sutures on the tibial graft end were wrapped around a tibial
bone screw and washer and a small force was applied to tension the graft. The tibial
screw was tightened, and the tibial bone block additionally ﬁxated with small Kirschner

wires. The ACL graft was then observed through the entire range of motion. If

14

impingement was noted, a femoral notchplasty was performed. The joint capsule,
subcutaneous tissue, and skin were then closed using non-absorbable sutures.
Biomeghm'cal Analysis:

Following sacriﬁce, the fat pad was resected and immediately immersed in 10%
buffered formalin. Patella-patellar tendon-tibia complexes were isolated from unoperated
and operative sides at 12 and 52 weeks. The patellar tendon cross sectional area (CSA)
was taken as the average of measurements taken at three longitudinal points using an area
micrometer (Butler et al., 1983). At time zero the boundaries of the operated tendon were
difﬁcult to determine, thus these were harvested leaving some peripheral, non-load bearing
tissues intact. The CSAs of these time zero tissues were determined following mechanical
testing using photographs of intact tendon cross sections (NIH Image 1.6). Patellar tendon
length was deﬁned as the distance ﬁom the insertion at the distal patella to the insertion
into the tibia and was measured by one individual with vernier calipers (Burks et al., 1990,
Linder et al., 1994, Kamps et al., 1994). The patella and tibia were potted in a specially
designed stainless steel box and tube, respectively. A stainless steel pin was carefully
installed through a hole in the side of the tube to extend through both cortices of the mid-
tibia to provide additional ﬁxation. Mechanical experiments were conducted in a 0.15 M
phosphate buffered saline (PBS) bath (pH 7.2) maintained at 37°C. The potted tibia was
mormted on a vertical plate with an angle of 20 degrees between the long axis of the
patella and the tibial shaft in the sagittal plane. To eliminate lateral loading, the vertical
plate containing the tibia was secured to a X-Y table base and a universal joint connected

the patella to the crosshead. The complex was preconditioned at 3% strain for 20 cycles at

15

1 Hz. It was immediately extended to failure in a materials testing machine (Instron model
1331) at a nominal strain rate of 100 %/s. An established analytic model (Belkoff and
Haut, 1992) was used to obtain a numerical description of the load-displacement data.
This model contains 2 parameters to describe the “toe” or initial nonlinear region of
response (u and 0’) and a stiffness (k) to characterize the linear region. Based on the
original development of the model (Belkoff and Haut, 1992), the parameter tr describes
the elongation required to straighten 50% of tendon ﬁbers and o is the standard deviation
about that mean. The stiffness parameter (k) was determined from the load/displacement
response and the u and 0' values were obtained using an iterative curve ﬁtting program.
The ultimate (failure) load was determined ﬁom the load/displacement response. A tissue
modulus was estimated by the product of stiffness and original tendon length divided by
the CSA. These data provided an estimate of the substance tensile modulus and yield a
measure of the quality of the tissue.

The fat pads of both limbs from two animals per time point were histologically
processed, using standard methods, and stained with H & E. A portion of every test and
unoperated fat pad was also assayed for its collagen content. Brieﬂy, the wet fat pad
tissue was minced and six 2.0-2.5 mg aliquots were weighed for each limb. The aliquots
were freeze dried with liquid nitrogen, pulverized, then processed according to established
methods to determine the content of hydroxyproline (Stegemann, 1958). The collagen
content was then obtained from the hydroxyproline content (7.46 ug collagen/pg

hydroxyproline).

l6

A randomized ANOVA was used to compare the fat pad hydroxyproline, CSA,
length, ultimate load, stifﬁress and model parameters (11 and 0') of the operated tendons at
different time points (p50.05, SigmaStat, Jandel Corp., San Rafael CA) and where
signiﬁcant differences were identiﬁed Student-Newman-Keuls (SNK) post-hoe testing was
applied Paired t-tests were used to compare the operative and contralateral unoperated
data (p50.05). Correlations (p50.05) between fat pad hydroxyproline and patellar tendon
cross sectional area, length, and stiffness were also obtained to investigate whether
changes in tendon size or properties were related to changes in fat pad collagen content.
RESULTS

One animal from the 12 week group and one from the 52 week group were not
tested due to technical problems. The CSA of the operated tendon signiﬁcantly increased
ﬁ'om time zero to 12 and from 12 to 52 weeks (Table 1). The normalized (operated
limb/contralateral limb) average CSA of the PT at time zero was 54%, while at 12 and 52
weeks the ratio was 237% and 184%, respectively. The tendon length decreased
signiﬁcantly ﬁ'om time zero to 12 weeks, and did not return to the unoperated length by
5 2 weeks. The ultimate load supported by the bone-tendon-bone complex increased
signiﬁcantly between time zero (22% of the unoperated limb load) and 12 weeks (54% of
the unoperated limb load), and did not return to the unoperated level by 52 weeks. Four
operated bone-tendon-bone preparations and 1 unoperated preparation failed due to
avulsion at the distal pole of the patella and the remaining failed in the tendon
midsubstance. The stiffness of the operated preparation increased signiﬁcantly between

time zero and 12 weeks (41% versus 69% of unoperated), and between 12 and 52 weeks

17

(69% versus 87%). The time zero tissue exhibited a “toe” response which was similar to
that of the unoperated tissue, but with a lower stiffness (Figure 1). In contrast, at later
times the operated tissue exhibited a more pronounced “toe region” region than the
unoperated tissue (as evidenced by generally larger u and O' values for the operated
tendon). Histologic sections of the tendons indicated that the defect ﬁlled with collagen
which appeared disorganized at 12 weeks and begans to show signs of order at 52 weeks
(Figure 2). The estimated tensile modulus at time zero was 72% of the unoperated value.
This parameter decreased signiﬁcantly between time zero and 12 weeks (to 26% of
unoperated). No signiﬁcant change in the estimated modulus was detected between 12
and 52 weeks.

At time zero the tendon and fat pad morphology was not different from the
unoperated tissues. Normal fat pads generally showed small amounts of collagen
throughout the adipose tissue (Figure 3a). Extensive collagen inﬁltration was observed in
fat pads at both 12 and 52 weeks (Figure 3b,c). Though not quantiﬁed in this study,
neovascularization was also observed in fat pads from operated joints at these time points.
The hydroxyproline content of the fat pad signiﬁcantly increased between time zero and
12 weeks to 648% of the unoperated fat pad (Table 1). At 52 weeks the hydroxyproline
content remained signiﬁcantly higher than the unoperated level, however it dropped
slightly ﬁom the 12 week level to 617% of the unoperated level. Increases in
hydroxyproline content of the fat pad were correlated with increasing cross sectional area,

and with decreasing length of the patellar tendon.

l8

DISCUSSION

This study was designed to investigate the effects of ACL reconstruction in a goat
model using an autogenous PT graft on the mechanical and histological properties of the
host patellar tendon. We hypothesized that extensive proliferation of the host tendon,
signiﬁcant reductions in the biomechanical properties of the tendon, and remodeling of the
fat pad would follow surgery. Furthermore we suggested these changes would be similar
to those observed in our rabbit model, where the ACL was not reconstructed.

Our results support these hypotheses, demonstrating signiﬁcant changes to the
tendon and fat pad in the goat model following ACL reconstruction using an autogenous,
ipsilateral patellar tendon graft. The unoperated patellar tendon modulus was similar to
that reported in Ng et al. (1995) (319 MPa vs. 302.5 MPa) but both of these were much
higher than that reported by Jackson et a1. (1993) (147.3 MPa). The ultimate load in the
unoperated tendon was much higher than that documented in earlier studies (4334 N vs.
516.7 N (Ng et al., 1995) and 2714 N (Jackson et al., 1993), possibly due to the larger
cross sectional area of the tendons (41.9 mm2 vs. 13.7 mm2 (Ng et al., 1995) and 28.3
mm2 (Jackson et al., 1993)). This difference in size was most likely the result of different
sized animals utilized in the studies (>40 kg in the current study vs. 33.2 kg average (Ng
et al., 1995) and >25 kg (Jackson et al., 1993)). The operated tendon tissue was
hypertrophic and the biomechanical properties were signiﬁcantly reduced compared to
unoperated limbs at all time points. The increase in the operated tendon cross section,
decrease in tendon length, and improvement in tendon stiffness over time observed in the

current study were also observed by Jackson et a1. (1993) between 6 weeks and 6 months.

19

These changes were also similar to those observed in our rabbit model: at 12 weeks CSA
237% (goat) of unoperated as compared to 201% (rabbit), stiffness 69% compared to
75%, 0' and p values higher than unoperated, and at 52 weeks some improvement toward
unoperated levels. The ultimate load of the host tendon remained much below that found
in the rabbit model (at 12 weeks 52% of unoperated, as compared to 80% in the rabbit
model). This might be a result of the relatively large graft harvested in the current study
(at time zero the CSA was 54% of the unoperated side, versus 67% in the rabbit model).
Signiﬁcant remodeling of the fat pad was evidenced by the replacement of adipose tissue
by collagen, and development of new blood vessels. These changes were similar to those
observed in our rabbit model (Atkinson et al., 1996). They were more signiﬁcant than
those noted in Muneta et al’s ( 1993) rabbit model, where the ACL was reconstructed with
an achilles tendon autograft and the animal was allowed limited motion. This difference
may be due to increased compression applied to the fat pad in the current study resulting
from the increased cross sectional area of the healing patellar tendon. This is supported by
our ﬁnding that increased fat pad hydroxyproline content was correlated with tendon
CSA. On the other hand, the genesis of ﬁbrosis in the fat pad may be associated with
biochemical changes not considered in the current study. The inﬂux of enzymes
associated with the healing patellar tendon, may encourage remodeling in the adjacent fat
pad due to their shared vasculature (Linder et al., 1994, Paulos et al., 1983, Kohn et al.,
1995)

A limitation of the current study was that the harvest procedure differed from that

in the previous rabbit model: a partial thickness graft versus a central third in the earlier

20

studies. In both procedures, however, the character of the response appeared the same.
This suggested that the tendon response may be initiated when stresses in the host are
increased due to the removal of load bearing tissue. This etiology was previously
suggested by observations in our rabbit model (Atkinson et al., 1996). Another limitation
was that a visual technique was utilized to measure the time zero tendon cross sections.
This likely caused the tendon area to be overestimated in the test case, where the cut
surface likely allowed excessive ﬂuid to be imbibed and expand the tendon, thereby
reducing the time zero modulus. A further limitation of the study was that, while we
associated anterior knee pain with ﬁbrosis of the fat pad, we did not actually assess pain.
In future studies it may be possible to assess pain by observation of the animal’s stance
forces (Bray et al., 1992) or gait. Finally, both the patellar tendon and fat pad were
incised, thus making it difﬁcult to determine whether the changes in the fat pad were
related to the healing of the patellar tendon or to incision of the fat pad itself. The
previous rabbit model, however, had minimal disruption of the fat pad and signiﬁcant fat
pad ﬁbrosis was still observed. Taken together these data seem to suggest that the healing
tendon may have a large inﬂuence on fat pad healing. Further studies are required to
examine fat pad healing independent of tendon healing.

In conclusion, the current study suggests that signiﬁcant remodeling of both
tendon and fat pad take place following harvest of autogenous patellar tendon grafts.
These changes were similar to those noted in our previous rabbit model. Increased stress
on the patellar tendon due to harvest of load bearing tissue for ACL reconstruction may

trigger the tendon proliferation and result in ﬁbrosis of the fat pad. These changes may

21

help explain, in part, post-surgical complications such as infrapatellar contracture
syndrome and anterior knee pain.
ACKNOWLEDGMENT:

This investigation was supported, in part, by DePuy Incorporated, Warsaw,
Indiana. The authors wish to gratefully acknowledge Mrs. Jane Walsh for her preparation
of the histology slides, William Newberry and Benjamin Ewers for their assistance in the
mechanical testing, and Mrs. Dana Dvorchek-Driksna for performing the biochemical

assays on fat pads.

22

REFERENCES:

Aglietti, P., Buzzi, R., Zaccherotti, G., and DeBiase, P. Patellar Tendon Versus
Doubled Semitendinosus and Gracilis Tendons for Anterior Cruciate Ligament
Reconstruction. The American Journal of Sports Medicine 22(2):211, 1994.

Atkinson,lP., DeCamp, C., Kamps, B., Hespenheide, B., Zukosky, D., and Haut,
R. Healing response of the patellar tendon after removal of its central third and
implantation of an augmentation device. Trans Orthop Res Soc 43:751, 1996.

Belkoff, S. M. and Haut, R. C. Microstructurally based model analysis of gamma-
irradiated tendon allografts. Journal of Orthopaedic Research 10:461, 1992.

Berg, E. Intrinsic Healing of a patellar tendon Donor Site Defect after Anterior
Cruciate Ligament reconstruction. Clinical Orthopaedics 2782160, 1992.

Bray, R., Shrive, N., Frank, C., and Chimich, D. The early effects of joint
immobilization on medial collateral ligament healing in an acl-deﬁcient knee: a gross
anatomic and biomechanical investigation in the adult rabbit model. Journal of
Orthopaedic Research 10:157, 1992.

Burks, R., Haut, R., and Lancaster, R. Biomechanical and Histological
Observations of the Dog Patellar Tendon After Removal of its Central One-third. The
American Journal of Sports Medicine 18: 146, 1990.

Butler, D., Hulse, D., Kay, M., Grood, E., Shires, P., D'ambrosia, R., and Shoji,
H. Biomechanics of Cranial Cruciate Ligament Reconstruction in dog. Veterinary Surgery
12(3):113, 1983.

Cosgarea, A., DeHaven, K., and Lovelock, J. The surgical treatment of
arthroﬁbrosis of the knee. Amer J Sports Med 22(2):]84, 1994.

France, E., Paulos, L., Abbott, P., and Roberts, P. Failure Characteristics of the
medial Collateral Ligament of the Knee: Effects of high strain rate. Aviation Space and
Environmental Medicine 582488, 1987.

Jackson, D. W., Grood, E. S., Goldstein, J. D., Rosen, M. A., Kurzweil, P. R.,
Cummings, J. E, Simon, T. M. A comparison of patellar tendon autograft and allograft
used for anterior cruciate ligament reonstruction in the goat model. Amer J Sports Med
21(2):]76, 1993.

Kamps, B. S., Linder, L. H., DeCamp, C. E., and Haut, R. C. The inﬂuence of

immobilization versus exercise on scar formation in the rabbit patellar tendon after
excision of the central third. American Journal of Sports Medicine 22:803, 1994.

23

Kempf, F. Treatment of mensicus defects with reference to our operative method:
subtotal menisectomy, resection of hoffa's corpus adiposum, inner drainage. Arch Orthop
T raumat Surg 96:95, 1980.

Krebs, V. and Parker, R. Arthroscopic resection of an extraossifying chondroma of
the inﬁ‘apatellar fat pad: end stage hoffa's disease? Arthrosc: J Arthrosc Rel Surg
10(3):301, 1994.

Kohn, D., Deiler, S., and Rudert, M. Arterial blood supply of the infrapatellar fat
pad: anatomy and clinical consequences. Arch Orthop Trauma Surg 114:72, 1995.

Linder, L., Sukin, D., Burks, R., and Haut, R. Biomechanical and histologic
properties of the canine patellar tendon after removal of the medial third. The American
Journal of Sports Medicine 22(1): 136, 1994.

Muneta, T., Yamamoto, H., Takakuda, K., Sakai, H., and Furuya, K. Effects of
Postopertative Immobilization on the Reconstructed Anteior Cruciate Ligament: An
Experimental Study in Rabbits. AmerJ Sports Med 21(2):305, 1993.

Ng, 6., Cakes, B. W., Deacon, O. W., McLean, I. D., and Lampard, D.
Biomechanics of Patellar Tendon Autograft for Reconstruction of the Anterior Cruciate
Ligament in the Goat: Three Year Study. The Journal of Bone and Joint Surgery 13:602,
1995.

Magi, M., Branca, A., Bucca, C., and Langerame, V. Hoffa disease. Ital J Orthop
Traumatol l7(2):211, 1991.

Paulos, L., Butler, D., Noyes, F., and Grood, E. Inna-articular Cruciate
Reconstruction. Clinical Orthopaedics and Related Research 172:78, 1983.

Oglivie-Harris, D. and Giddens, J. Hoffa's disease: Arthroscopic resection of the
infrapatellar fat pad. Arthrosc: J Arthrosc Rel Surg 10(2): 184, 1994.

Richmond, J. and Assal, M. Arthroscopic management of arthroﬁbrosis of the
knee, including infrapatellar contracture syndrome. Arthrosc: J Arthrosc Rel Surg
7(2):l44, 1991.

Rosenburg, T. D., Franklin, J. L., Baldwin, G. N., and Nelson, K. A. Extensor
Mechanism Function After Patellar Tendon Graft Harvest for Anterior Cruciate Ligament
Reconstruction. The American Journal of Sports Medicine 20(5):519, 1992.

Stegemann, H. Mikro vestimmurg von hydroxyproline mit chloramin-tund p-
dimethylamino benzaldehyd. Hoppe-Seylers Z. Physiol Chem 41, 1958.

24

ABLE

Table l: Biomechanical and biochemical data for goat patellar tendons and infrapatellar fat
pads following harvest of a patellar tendon graft to reconstruct the ACL (mean i one

standard deviation).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Measured Parameter Unoperated 0 weeks 12 weeks 52 weeks
(n=1 6) (n=6) (n=5) (n=5)
Tend“ 0°53 41-9ﬁ3-9" 22.8:5.3""’-" 99.3:12. 1“" 77.0:t14.8“'d
Sectional Area Qnmz)
Tendon Length (mm) 408:2.4 4151,24” 36.4ﬂ.3"" 30.6i4,0a.b.d
Ultimate Load (N) 4334:8314 946i244arbrc 23541-6754." 24543316244!
“mile Sﬁfﬁm 319i45" 130:25""’" 221:26“'"-" 276:29""’-"
/mm)
model parameter 11 0683:480 0.292:0. 142 1.730:0.929" 1193:0435"
(M)
model parameter a 0502:0337 0194:0124 1090:0598" 0906:0455"
(my
Tensile Modulus (MPa) 313ﬂ2 255:96’” 80i12“'d 1 15:34 "1""
Fat Pad Collagen 22-3i25-6 23.1:21.7”'" 144.6:80.6"'" 137.6:69.2"-"
dag/mg wet tissue)

 

a - signiﬁcantly different than unoperated, paired t-test (p<0.05).
b - signiﬁcantly different ﬁom 12 weeks, ANOVA (p<0.05).
c - signiﬁcantly different ﬁ'om 52 weeks, ANOVA (p<0.05).

d - signiﬁcantly different from time 0, ANOVA (p<0.05).

25

 

LEGEND

Figure 1: Composite load-elongation curves for the unoperated and three operated
groups constructed using the average values of the parameters 11 (elongation required to
straighten 50% of tendon ﬁbers), 0' (the standard deviation about that mean) and the linear

stiffness (K) as determined by the non-linear curve ﬁtting program

Figure 2: Tendon coronal sections (H&E, 100X) A) Typical unoperated tendon, B) 12
and C) 52 week operated patellar tendons. Note a minimal distribution of mature
ﬁbroblasts in the unoperated tissue versus the marked hypercellularity and cell immaturity
in the operated tissues at 12 and 52 weeks. Also note poor collagen alignment at 12 weeks
but improved collagen alignment and crimping at 52 weeks.

Figure3: Fat pad cross sections (H&E, 40X). A) Time zero operated and unoperated fat
pads, B) 12 week operated fat pads showing extensive ﬁbrosis and neovascularization
(arrow) and

C) 52 weeks showing reduced ﬁbrosis.

26

 

 

 

 

 

  
 
 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1400
1200
—control / l1
1000 ,_, —"'"tirne zero I,___
_ — 12 meks 1"
2 .. .. / "’
TD, 800 .. 52 Weeks 1 ——,¢ / L
2 / / r
/
400
200 4|.
0 T r l
0 1 2 3 4 5 6

elongation (mm)

 

 

 

Figure 1

27

 

Figure 2

28

 

Figure 3

Footnotes:
1 Sponsored in part by DePuy, Incorporated of Warsaw Indiana.

29

Chapter 2:

A Poroelastic Model That Predicts Some Phenomenological
Responses of Ligaments and Tendons

 

Theresa S. Atkinson, Roger C. Haut, and Nicholas J. Altiero

30

ABSTRACT:

Experimental evidence suggests that the tensile behavior of tendons and ligaments
is in part a function of tissue hydration. The models currently available do not offer a
means by which the hydration effects might be explicitly explored To study these effects
a ﬁnite element model of a collagen sub-fascicle, a substructure of tendon and ligament,
was formulated The model was microstructurally-based and simulated oriented collagen
ﬁbrils with elastic-orthotropic continuum elements. Poroelastic elements were used to
model the interﬁbrillar matrix. The collagen ﬁber morphology reﬂected in the model
interacted with the interﬁbrillar matrix to produce behaviors similar to those seen in
tendon and ligament during tensile, cyclic, and relaxation experiments conducted by
others. Various states of hydration and permeability were parametrically investigated,
demonstrating their inﬂuence on the tensile response of the model.

INTRODUCTION:

Tendons and ligaments provide stability and control for the motion of j oints. Thus,
their function plays an important role in musculoskeletal biomechanics. Assessment of the
mechanical properties of these tissues is important in development of various surgical
techniques for joint reconstruction. In these assessments, the role of collagen ﬁbers in the
tissue has been highlighted and correlated to various disease states. Models which refer to
various collagen morphologies have been used to characterize the results of these studies
(Belkoff and Haut, 1991; Lanir, 1978; Kwan and Woo, 1989).

More recently, experimental studies have suggested that other tissue components,

such as water (which comprises 60-70% of the weight of tendons and ligaments), might

31

play a signiﬁcant role in the tensile behavior of ligaments and tendons. For example,
experiments suggest that human patellar tendons exhibit higher tensile modulus and
ultimate strength when tested in a bath environment versus in air (Haut and Powlison,
1990). Similarly, the tendon is stiffer when tested in a hydrating solution versus a
dehydrating solution (Haut, et. al 1995). Furthermore, both studies suggest that the
viscoelastic properties of human tendon are inﬂuenced by water. For example, in
relaxation tests, specimens tested in a saline bath exhibit a larger and more rapid load
relaxation versus those tested in air (Haut and Powlison, 1990). Similarly, specimens
tested in a hydrating bath have greater load relaxation than those tested in a dehydrating
bath (Haut et al., 1995).

In ligaments, increasing the water content causes the tissue to respond with greater
cyclic load relaxation relative to ligaments with lower water content (Chimich et al.,
1992). Data from cyclic tests also demonstrate monotonic decreases in rat tail tendon
diameter ﬁom cycle to cycle (Lanir et al., 198 8). Similar behavior has been documented in
the rat medial collateral ligament (MCL) (Thielke et al., 1995). This deformation suggests
densiﬁcation of the collagenous structure and exudation of ﬂuid, and is consistent with
reports that glycosaminoglycans (GAGs) and water are exuded during tensile strain tests
(Lanir et al.,l988; Hanniﬁn and Amoczky, 1994).

Effects associated with extracellular water have been observed experimentally, but
few analytical models address the mechanism of its action. Chen et al. (1993) utilized a
ﬁnite element analysis (FEA) to study permeability effects in tendon and ligament. The

model includes a ﬂuid phase moving around regular cylinders (meant to represent collagen

32

ﬁbers). Chen and Vanderby (1994) proposed a directionally sensitive permeability for
tendons. The inﬂuence of water has also recently been incorporated in a model of the
rabbit MCL (Wilson et al., 1994). Wilson’s FEA model utilizes a continuum matrix of
poroelastic material and spring elements attached at nodes to include the elastic stiffness
of collagen. The model predicts that pressure within the MCL is negative during tensile
deformation, implying that ﬂuid ﬂows into the structure. This result, however, is contrary
to current experimental data indicating positive internal pressures (Chen et al., 1995) in
rabbit patellar tendon during tensile stretch and ﬂuid exudation in the canine ﬂexor tendon,
rabbit MCL, and rat tail tendon under tensile strain (Hanniﬁn and Amoczky, 1994; Thielke
et al., 1995; Lanir et al., 1988).

These contrary results suggest that continuum characterizations alone of the
tendon may not reﬂect mechanisms important to predicting the role of water in tensile load
response. Lanir et a1. (1988) describe the tendon on a microscopic scale as consisting of
an arrangement of collagen ﬁbrils embedded in a hydrophilic gel. While there is
controversy in the literature concerning the hierarchical levels of collagen, it is generally
accepted that ﬁbrils are combined to form sub-fasciculi, fasciculi, and ﬁnally the whole
structure (Danylchuk et. a1, 1978;Viidik 1990; Yahia and Drouin 1988). It has been
proposed that the morphology of collagen ﬁbrils can be complex, ranging ﬁom being
helically wound about a sub-fascicle (termed fascicle by Yahia and Drouin, 1988) axis in
the patellar tendon to a both helical and planar waveforms in the anterior cruciate ligament
(ACL). In the sub-fasciculi containing helically wound ﬁbrils, the peripheral ﬁbrils appear

as undulated helices, while the central ones are normal helices (Yahia and Drouin, 198 9).

33

It has been previously suggested that the collagen in tendon and ligament interacts in such
a way that it compresses the gelatinous matrix and creates a sensitivity to strain rate in the
tissue (Lanir, 1978). We hypothesize that longitudinal deformation of the more helically
oriented ﬁbrils in the sub-fascicular structure of patellar tendons and ACLs might
compress the matrix to generate hydraulic pressures and movement of unbound water
during tensile stretch. By this mechanism the load might be shared between collagen and
the surrounding matrix to inﬂuence the tensile response of the entire ligamentous or
tendonous structure. Danylchuk (1978) states that the connective tissue sheaths between
fascicles and sub-fascicles, contain collagen which is oriented perpendicular to the long
axis of the structure and thus provides a binding function rather than signiﬁcant tensile
strength. We therefore assumed that the sub-fascicle may be considered a characteristic
structure of ligament and tendon which therefore should exhibit behaviors observed in the
gross structure

METHODS:

The ﬁbril structure and matrix interaction were idealized for the purpose of
building a simple model. This simple model consists of an outer ring composed entirely
of helically oriented ﬁbrils, and an internal region comprised of a water-based matrix (Fig.
1). These regions interact to support tensile loads. Further simpliﬁcations of this
conceptual model were made in order to provide the basis for a ﬁnite element model. The
sub-fascicle’s cross section was assumed to be circular and of constant diameter
throughout its length. Thus, it was only necessary to model a representative section ﬁom

the center of a centrally located sub-fascicle.

34

A ﬁnite element model, consisting of 84, 20 noded 3-dimensional continuum
elements, was deve10ped using a commercial code (ABAQUS, Hibbitt, Karlsson, and
Sorensen, Inc. Pawtucket RI.) The cubic elements consisted of nodes at the midpoint of
each edge possessing three degrees of freedom for displacement, and nodes at the corners
possessing an additional degree of freedom for pressure. The model’s radius, 37.5 urn,
was an average of those scaled from micrographs of the human patellar tendon (Y ahia and
Drouin 1988). The length of the model was chosen to be 10.0um to maintain an
approximately 1 to 1 relationship between the lengths of the element sides in a majority of
the elements and to help avoid numerical ill-conditioning. The ﬁbrous outer ring was
modeled as an orthotropic poroelastic material, where the selection of orthotropic material
properties was intended to produce deformations consistent with helically oriented
collagen ﬁbrils. The ﬁbril alignment, measured as a 62° inclination from vertical on one
SEM of a tendon (Y ahia and Drouin, 1988, Figure 8), was attained by setting the 2
direction of the orthotropic outer ring at this inclination. For simplicity, the ﬁbril
morphology was assumed to be purely helical, at a constant angle of orientation through
the sub—fascicle’s thickness.

Little information is available documenting material properties of collagen sub-
fascicles. One study suggests that rabbit patellar tendon fascicles have a modulus of
approximately 225MPa (Yamamoto et al., 1995). Rat tail tendon (RTT) has been
described as being composed of 1-3 collagen fascicles (Kastelic et. al, 1978) with a tensile
modulus of approximately 6OOMPa (Haut, 1983). In this study the tensile modulus in the

ﬁbril direction was assumed to be 6OOMPa (Table 1). Unfortunately, there is no

35

information on the transverse or shear moduli of collagen sub-fascicles or fascicles. We
assumed that the ﬁbrous portion of the model was weak transverse to the ﬁbril direction.
Thus, the transverse moduli were assumed to be one order of magnitude less than the ﬁbril
direction modulus, and the shear moduli less than the transverse. Poisson’s ratios
consistent with an orthotropic material were then selected.

The cenﬁal core was modeled as nonlinear poroelastic. The poroelastic material
model, originally developed to describe soil behavior, is similar in formulation to the
biphasic model currently applied to cartilage (Suh and Spilker, 1994; Mow et al., 1984).
The poroelastic matrix of tendon and ligaments may be similar to cartilage in that it is a
ﬁber embedded permeable structure which contains water, proteoglycans and other
constituents (Mow and Hayes, 1991; Thielke et al., 1995).

The nonlinear poroelastic material stiffens upon compression. This is accomplished

through moderation of the shear modulus in the following way:

G=[3(1—2v)(1+e,)](P + P. )exp(8,, )

2(1 + v)( V°'

 

where eo is the initial voids ratio (volume ﬁaction of ﬂuid/volume ﬁaction of solid), P is
the internal pressure, P, is the elastic ultimate strength, 8"”; = In J" is the elastic portion of
volume change, v is Poisson’s ratio, and 1c is the log bulk modulus (relating the logarithm
of pressure to the dilatation) (Abaqus Theory Manual, Zienkiewicz and Naylor, 1972).
Thus, G increases with compaction and pressure. Assuming that the volume fraction of
water in tendon is similar to the weight ﬁaction (70% of the wet weight is water), an

initial voids ratio of 2.33 (=.7/.3) was used for the nonlinear poroelastic portion of the

36

model (Haut, 1993; Mow and Hayes 1991), however, eo=l .0 (50%) was also
investigated. Based on the assumption that ﬂuid in tendon is similar to that in other
tissues, the log bulk modulus of the nonlinear poroelastic core was obtained from data for
ﬂuid in the human annulus ﬁbrosis (Best et al., 1984). Poisson’s ratio was assumed to be
0.49. The tensile modulus of the material was then derived such that the initial shear
modulus of the inner core approximately matched the shear moduli of the outer
poroelastic ring.

Fluid ﬂow was assumed to obey Darcy’s law, i.e.:

v = - k dp/dr

where v is the ﬂuid velocity, k is the permeability, and p is the pressure. This law is
applicable to low ﬂow rate problems and is used in biphasic models of cartilage (Mow and
Hayes, 1991). The inﬂuence of permeability on ﬂow rate can be complex and is known to
be a nonlinear function of deformation in cartilage (Mow et al. 1984). Since k is unknown
for tendon, it was assumed constant. In the past, similar assumptions have been utilized in
cartilage models (Suh and Spilker, 1994). The permeability of cartilage has been reported
from 1.45x10'” (bovine) to 2.17x10'”m’/Ns (human patellar groove) (Mow and Hayes,
1991). Permeabilities in the range 1.0x10‘12 to 1.45x10‘27m’ms were investigated in the
current study.

The orthotropic poroelastic material of the outer ring was assumed to be capable
of holding water with permeabilities similar to those of the core portion of the model.
Initial voids ratios of 1.0 and 2.33, similar to those used in the core were assigned to this

portion, but values as low as 0.01 were also examined. All the materials in the model

37

were assumed to be fully saturated and the outer ring was assumed to be perfectly
attached to the inner core.
Widens:

All loading was prescribed through displacement control. The bottom of the model
was constrained to planar motion with 4 nodes separated by 90° constrained to radial
motion. The top of the model was also constrained to planar motion (r and 0 ﬁee). A
uniform displacement was applied across the top of the model. The top and bottom were
assumed to be sealed in order to represent conditions in a long thin fascicle at the center
of a tendon. The outer boundary of the model was also sealed. This sealed boundary
condition was dictated by the use of continuum elements in the ring portion of the model
(rather than modeling discrete collagen ﬁbers), which would experience a positive
dilatation under tensile loading and thus tend to draw in water if a perfectly draining
boundary condition were applied. This drawing in of water would compete with the
movement of water ﬁ'om the core part of the model and thus obscuring effects the model
was designed to study. The sealed boundary condition forced ﬂuid to ﬂow according to
gradients developed as result of the applied stress and not as a function of a drainage
pressure applied as a boundary condition.

Relaxation experiments, similar to those performed by Haut et al. (1995), were
simulated with the model with an initial, suddenly applied, 3% strain followed by a 1805
period of constant deformation. Constant strain-rate tensile tests, also similar to those in
Haut et al. (1995), were simulated using constant strain rates of 0.5%/s and 50%/s.

Cyclic extension tests, similar to those performed by Chen (1995), Chimich et al. (1992),

38

and Hanniﬁn and Amoczky (1994), were simulated using a 0.5 cycle per second 4% strain
triangular ramp.

Parametric studies were also performed during the simulated relaxation and
constant strain rate tests in which moduli, permeability, and voids ratio were varied in both
the ring and core portions of the model.

Newton’s method with backward time integration was utilized to solve the coupled
ﬂow and deformation equations simultaneously. Since pilot studies indicated that shear
deformations exceeded 10%, nonlinear deformation terms were included in the analysis.
Due to the unsymmetric nature of the coupled ﬂow equations, the unsymmetric matrix
solver wasused in all solutions.

RESULTS
Constant Deformation Relgag'on:

In the simulated relaxation experiments the model indicated an initially high
reaction force which slowly relaxed to a lower, steady state level (Fig. 2 (a)). The
relaxation response was highly sensitive to permeability and water content. Increasing
permeability caused the sub-fascicle to relax to the steady state response faster. Ifk was
less than 1x10'19m4/Ns, the relaxation phenomenon was lost in the time frame of the
current study (0—180s). If it was greater than 1x10“, the initial peak in the force was lost.
Increased initial wate' content, reﬂected by an increase in e,, tended to increase the peak
force, steady state force and the amount of relaxation. Higher percentages of relaxation
resulted when the water content and voids ratio were the same throughout the sub-

fascicle, rather than varied between the ﬁbrous ring and the poroelastic core.

39

The internal pressures predicted during the constant deformation experiments also
exhibited a dependence on water content and permeability. During this test the internal
pressure within the sub-fascicle was initially high, but decreased to a lower steady state
value (Fig. 2 (b)). Pressures central to the sub-fascicle were high, and decreased radially
(Fig. 2 (c)). The core portion of the model always experienced a negative dilatation. At
steady state the pressures in the model were either positive, slightly negative or close to
zero depending on the material properties selected. Combinations of properties which
produced large negative pressures in the ﬁbrous ring portion of the model resulted in
negative steady state pressures. Increasing eo increased the initial internal pressure. ,
Increasing k increased the rate of pressure decay. Increased ultimate tensile strength of the
poroelastic core, Pt, tended to decrease the peak pressure in the middle of the fascicle. The
pressure at each location was constant through the model’s length, and the initial pressure
resulting in the ﬁbrous ring was always negative.

The deformation of the ring elements was such that the ﬁber axis became more
aligned with increasing axial load causing the element to twist helically (Fig. 3). In
addition, the radial deformation during these relaxation tests indicated that the sub-
fascicle’s radius decreased slightly as a function of time.

During cyclic loading the sub-fascicle’s internal pressures followed the applied
loading (Fig. 4 (a)). The peak internal pressures were positive, and the pressures when the
displacement returned to zero were slightly negative. The peak pressures and axial force

varied as a function of the extension, decreasing with increasing numbers of cycles. During

40

the extension, the pressures at the center of the model were higher than those at the edge
(Fig- 4 (b))-
jljensile Deformation at Vﬂ'ed Rates;

During simulated constant strain rate experiments the sub-fascicle stiffness was
dependent on initial water content, as deﬁned by e. As the initial water content was
increased, the model’s stiffness increased (Fig. 5 (a)). The sub-fascicle’s stiffness was also
dependent on the values chosen for k. Increasing k caused the stifﬁress to decrease for
1x10'12 < k < 1x10'15m4/Ns. At low permeability, there was no difference in stifﬁress due
to changes in e... Increasing the various elastic moduli caused the model’s stiffness to
increase accordingly.

The model’s stiffness was dependent on strain rate if k>1x10'”m’/Ns (Fig. 5 (b)).
Faster rates of extension produced a stiffer response. Increasing P. , or the transverse
elastic moduli in the ﬁbrous ring portion, or decreasing the permeability in any part of the
model, decreased the sensitivity of stiffness to strain rate.

The force/deformation response of the model was linear for small strains.
DISCUSSION:

Many of the model’s predicted relaxation behaviors were consistent with
experimentally-observed tendon and ligament data. The peak force, rate of relaxation and
relaxed force exhibited by the model depended on co, the initial ratio of water to substrate.
When the amount of water in the sub-fascicle decreased, the initial force, steady state
force, and the rate of relaxation decreased. This caused the reduced relaxation modulus

(slope of the normalized force, logarithm time curve) to decrease. This behavior is

41

consistent with Haut et al.’s (1995) observations and exhibits the same time varying
character as that noted in Lanir et al. (1988) and Chimich et al.’s (1992) work. The peak
force and rate of relaxation shown in the model also depended on k. Parametric studies
demonstrated that the ability of the elastic part of the model to take on water was
primarily governed by the permeability parameter. In the current model the core portion
exhibited dilatation and water ﬂow behaviors hypothesized to exist in the matrix of a sub-
fascicle. The ring portion loaded the matrix in a twisting manner, which we hypothesize is
similar to the effect produced by collagen ﬁbrils within the sub-fascicle. In this model the
ring also acted as a sink for water. Water moved out of the nonlinear poroelastic core
material into the ring portion of the model similar to how we hypothesize it would move
out of the real sub-fascicle, and ultimately out of the entire structure. Water motion out of
tissues has been observed experimentally (Chimich et al., 1992; Lanir et al., 1988; Hanniﬁn
and Amoczky, 1994). The water motion phenomena in the model was readily seen in the
simulated relaxation, where water initially trapped in the compressed poroelastic core of
the model, diffused out into the ﬁbrous ring portion thus softening the poroelastic portion
and decreasing the overall tensile stress.

The model’s pressure proﬁle was also similar to that in a recently reported
experiment which indicated that the internal pressure developed in a rabbit patellar tendon
during cyclic loading is positive and follows the loading proﬁle (Chen et al., 1995). The
model also predicted a concurrent decrease in the diameter of the sub-fascicle during the
test. This phenomenon was consistent with experimental observations in rat tail tendon

(Lanir et al., 1988) and in rat MCL (Thielke et al. 1995). It has been suggested that this

42

decreasing cross sectional area indicates compaction and water motion out of the
interﬁbrillar matrix (Hannifm and Amoczky, 1994; Lanir et al., 1988; Thielke et al., 1995).
These phenomena are consistent with observations made in the current parametric studies
using the poroelastic model.

During cyclic extension, the model’s internal pressures were positive on extension,
negative on the retmn to zero displacement, and the peak pressures and forces decreased
with increasing number of cycles. This behavior was similar to Chen et al.’s (1995)
experimental pressure measurements in the rabbit patellar tendon. This pressure pattern
was also consistent with water motion radially out of the poroelastic portion (into the
elastic portion) during extension and back in when the displacement returned to zero.
These predictions suggest that ﬂuids tend to leave the sub-fascicle during cyclic loading,
consistent with recent experimental ﬁndings (Hanniﬁn and Amoczky, 1994; Lanir et al.,
1988). The predicted relaxation during cyclic loading was similar to that observed in
ligaments (Chimich et al., 1992).

In Chen et al.’s (1995) experiment, the magnitude of the measured pressure was
signiﬁcantly less than the tensile stress, suggesting a lack of load sharing between the
ﬁbers and matrix. In the current study, cyclic loading tests showed the peak predicted
pressures in the matrix portion of the sub-fascicle were similar to the average stress
(average force/area). Thus, the current model suggested there was load sharing. It is
possible that Chen’s pressure probe provided a leakage path for water. On the other hand,
the model describes a very small unit of the tendon and predicted pressures might not

reﬂect gross pressures measured in the tendon. This difference might also be attributed to

43

possible differences in the water content of rabbit patellar tendon and human tendon. As
the pressures predicted in the model also depend on the voids ratio and permeability,
additional parameter studies performed under cyclic loading may provide more insight
into these experimentally observed behaviors.

In simulated constant rate of deformation tensile tests (10%/s) the sub-fascicle’s
stiffness increased when the voids ratio (water ﬁaction) increased Similar behavior has
been noted in tendons when the test environment was used to alter the water content
(Haut and Powlison, 1990, Haut et al., 1995). This behavior resulted in the model when
the voids ratio was increased in either both the elastic and poroelastic parts or just in the
poroelastic part. Variation of the initial amount of water in the elastic part does not in
itself change the stifﬁress of the elastic portion. However, anything that alters the ability
of the elastic portion to accept water ﬁom the poroelastic portion effected the stiffness of
the poroelastic portion and thus the stiffness of the total model. The sub—fascicle stiffness
also exhibited a dependence on permeability. Increasing the permeability tended to make
the model more compliant. Although no direct evidence exists documenting the
permeability of tendons, previously Haut and Powlison attributed the increased
compliance of y—irradiated tendons to increased permeability. At lower permeability,
changes in the voids ratio did not effect the sub-fascicle stifﬁress. This suggests that the
water acts as a stiffening agent only when the permeability of the tendon is sufﬁciently
high. The force-deformation curve provided by the model was linear, unlike the force-
deformation curves for a gross tendon or groups of fascicles (Butler, et. a1 1986). This

discrepancy is attributed to the simpliﬁed ﬁbril morphology incorporated in the model.

Factors such as variation of ﬁbril orientation through the sub-fascicle thickness, the
undulation of ﬁbrils, sub-fascicle recruitment and sub-fascicle interactions were not
examined in the current study, but should be addressed in future models.

When the rate of deformation was changed from 10%/s to other rates of extension
the model exhibited rate sensitivity. During fast extension (50%/s) the model predicted a
stiffer response than that druing slow extension (.5%/s), similar to Haut et al. (1995).
This behavior was highly dependent on the tissue permeability, with no rate dependence
exhibited at k<1x10’”m‘/Ns. This suggests that the water motion in the tissue might
cause rate sensitivity when the permeability of the tendon is sufﬁciently high.

Deformation in the model’s outa' ring was consistent with collagen ﬁbrils re-
orienting into a tighter, more axially aligned helix. This deformation produced a wringing
effect in the sub-fascicle, compressing the hydrophilic gel portion to generate positive
pressure. This particular deformation may then be very important in producing many of
the observed mechanical behaviors particular to tendons and ligaments.

Although the model exhibits behaviors consistent with many experimentally
observed behaviors, several problems remain which must be addressed in its further
development. The distribution and morphology of collagen ﬁbrils in the sub-fascicle have
been simulated by the use of a ﬁbrous ring attached to the outside of the core, matrix
material. As the ring material is somewhat compressible, it experienced a positive
dilatation under tensile load leading to negative negative pressures in the ring. Yet, on the
other hand, the model exhibited a wringing out of the core and thus water motion out of

the sub-fascicle. We hypothesize that water ﬂow effects exhibited by the core of our

45

model will occur throughout a real sub-fascicle as real ﬁbrils experience rigid body motion
motion as they align along the loading axis. Improvements such as distributing elements
throughout the matrix to represent ﬁbrils, might be a more realistic microstructural model
of the sub-fascicle. Another limitation of the current model is that the thin connective
tissue sheaths which separate sub-fascicles have not been included in the model.
However, we hypothesize that these thin sheaths do not play an active role in the tensile
response of tendon and ligament, rather they act to bind sub-fascicles and fascicles into the
gross structure. The model’s tensile response was compared qualitatively to that of
whole tendon or ligament, however interactions between subfascicles and extra-fascicular
materials must be evaluated experimentally before a whole tendon model can be
developed Strategies to utilize the model to quantitatively. describe whole tendon and
ligament behavior remain to be explored. Appropriate material properties must also be
obtained The bulk modulus (1c) of the interstitial ﬂuid, the fascicle’s elastic modulus, the
angle of ﬁbril orientation and other geometric properties need to be measured. Other
material properties, such as the transverse elastic moduli and the shear moduli might be
assumed, or could be derived based on SEM studies of ﬁbril reorientation during tensile
loading with the use of this analytical model. The model also suggests that parameters
such as the tissue permeability (k), and voids ratio (e) are extremely important. These
parameters might be measured using standard permeability tests and measurements of
water content.

The model demonstrates that when structural aspects, i.e. collagen morphology

and orientaﬁon, and aspects of how collagen interacts with the hydrophilic base matrix

46

material are taken into account, a reasonable mechanical response results. The sub-fascicle
modeled here is similar in form to those found in both the patellar tendon and ACL. Other
fascicle morphologies also exist in the ACL and require further study, but the approach
taken in modeling this structure may serve as a basis for such a model. We suggest that
future improvements in the understanding of ligament and tendon mechanics will
necessarily involve the study of microstructure and the measurement of parameters not
previously considered Thus, we propose that analytical models‘which incorporate tissue

water content and microstructural information will be important tools in future research.

47

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50

Table 1 Material Coefﬁcients

 

 

 

 

 

Elastic Moduli Value Poisson’s Ratio Value Shear Moduli Value
E1 30MPa v12 .01 Gl .5 MPa
E2 6OOMPa v23 .25 G2 .5 MPa
E3 30MPa v13 .25 G3 .5 MPa

 

 

 

 

 

 

51

 

Figure legends:
Fig. l Conceptualization of the geometry of a fascicle used as the basis of the ﬁnite

element model.

Fig. 2 (a) The effects of voids ratio and permeability on stress relaxation in a constant
deformation (3% strain) test. The forces were normalized by the peak force in the eo=2.33,
k=1 .45x10‘19m’st load case. For each load case, k and co were constant throughout the
model. The initial normalized force in the load cases e°=l , k=1.45x10‘19 m’/Ns and
5.45x10'l8 m‘/Ns was 0.95. The material constants for the outer ring are given in Table 1.
The material constants in the poroelastic matrix were v=0.45, 1c=0.047, and P,=3.4x105

Pa.

Fig. 2 (b) The effects of voids ratio and permeability on fascicle internal pressure during

stress relaxation a constant deformation (3% strain) test. The pressures were normalized

by the peak for in the eo=2.33, k=l .45x10'19m’/Ns load case.

Fig. 2 (c) The variation of pressure in the radial direction during the eo=1.0 and

k=l .45x10'19 m’/Ns load case.
Fig. 3 Deformation of the model at 3% strain (magniﬁed x 10) during a constant

deformation test. The elements shear to the left, bringing the ﬁber axis (initially oriented

at 62° to the right) more directly in line with the tensile load.

52

Fig. 4 (a) Fascicle internal pressures during cyclic deformation. Pressures were normalized
by the peak positive pressure. Orthotropic material properties are given in Table 1,
e°=0.01 and k=1.45x10'l9m’/Ns. Poroelastic properties were eo=1.0,

k=1.45x10‘“m"/Ns, v=0.49, P.=3.4x10‘ Pa, and tt=0.047.

Fig. 4 (b) Fascicle pressures as a function of radial distance at the peak of the lst cycle.

Fig. 5 (a) The effect of voids ratio and permeability during constant strain rate tensile
deformation (10%/s). Forces were normalized by the force at 4.8% strain in the eo=2.33,
k=1 .45x10‘19m4/Ns load case. The lines for the load cases eo=l, k=1.45x10’19 m’/Ns and
eo=2.33, k=1 .45x10'19 m4/Ns are on top of one another. The material constants in the ring
portion of the model are given in Table l. The material constants in the poroelastic matrix

were v=0.45, lc=0.047, and P.=3.4x10‘ Pa.

Fig. 5 (b) The effect of strain rate on the tensile response of the fascicle. For both tests
e°=2.33 and k=1 .45x10'17m’/Ns. The material constants in the ring portion of the model
are given in Table 1. The material constants in the poroelastic matrix were v=0.45,

tt=0.047, and P.=3.4x10‘ Pa.

53

 

 

 

 

 

 

 

54

 

normalized force

 

 

 

 

----- e =1 .0,k=1 .45e-19
- — - e =1 .0,ké.45e-18
- - - e =2.33,k=1.45e-19
— - e =2.33,k=5.45e-18

 

 

 

L'ul-"u In. H ”anon—'I- — "

 

 

0.7-
0.65 : : t i l l 4
0204oeom10012014015°1°°
time,s
Fig.2(a)

55

normalized pressure

 

 

 

 

 

 

 

 

 

‘ l
I
0.3 l:
.1 ----- e =1 .0,k=1 .45e-19
0.6 ‘l' i“. - - - e =1 .0,k=5.45e-18
'. - - - e =233,k=1.45e-19
0,4 ‘1 -“ — -e d.33,k=5.45e-18
0.2 . \‘ “ ‘
\- - _ _‘.‘ _ ._ - _ ._ ::;.;'_"_' 2 L':.'.’.’.:.'_' °_: '_'.'.'.a'.&1.s t.- a-..
o : a : : r 4. r : :
h — _ _ — — — —.--—...—"—' '—?'—.'&'J.'-I.H HH—-—
-o.2
o 20 too 120 140 160 180

so , so
time,s
Fig.2(b)

56

 

    

 

0.4 matrix —-1

normalized pressure

 

 

 

0.3 ..
0.2 0
0.1 ..
0 t a t I . t
0 5 10 15 20 25 30
radial position, micrometers
Fig. 2 (c)

57

30

 

15
no. 4 (a)

time! S

10

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.8 “

-O.2

. iﬂi u
4. 2 o
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q
6.
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59

 

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matrix—1

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normalized pressure

I

 

I.
‘-
d
d
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~1o 15 20 25 30 85

 

radial position, micrometers
Ho- 40:)

60

 

 

 

       
    

 

 

9 8 7
1 1 1
has
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.2 drammm
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7. 6. 5
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023 page

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% strain

Fig. 5 (a)

61

normalized force

 

0.9 4
0.84
0.7 .
0.6 -
0.5 -
0.4 -
0.3 «
0.2 -

0.1‘

 

    

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% strain
Fig. 5 (b)

62

 

 

 

— 50%/s
---s%m

 

 

 

Chapter 3:

A Microstructural Poroelastic Model For Patellar Tendon

 

Theresa Atkinson, Roger Haut, and Nicholas Altiero

63

ABSTRACT:

In order to study potential inﬂuences of water content on tendon tensile response a
ﬁnite elemmt model of a subfascicle (a microstructural element of tendon) was
constructed. Although this model exhibited mechanical responses which were similar to
those observed in whole tendon and ligament, it was preliminary in nature and as such
contained some undesirable compromises. These compromises related to boundary
conditions imposed as a result of the simpliﬁcation of the subfascicular geometry. In the
current study a more detailed description of the subfascicular microstructure was
encorporated in a model. This model was shown to exhibit reasonable relaxation and
tensile responses as well as a realistic pressure proﬁle throughout the subfasicle.
INTRODUCTION:

Predicting and measuring the mechanical response of tendon is important in the
development and assessment of various orthopaedic reconstruction techniques (which
frequently utilize these tissues as graft materials). Recent experimental studies (Haut et
al., 1995, Haut and Powlison, 1990) indicate that water, which comprises approximately
60-70% of the total weight of the tendon, might play a signiﬁcant role in dictating the
tensile response of ligaments and tendon. Several studies performed by other researchers
(Hanniﬁn and Amoczky, 1994, Lanir et al, 1995, Thielke et al, 1995) suggest that ﬂuids
are exuded from these tissues and internal pressures are positive under tensile loads.
Predicting how variations in water content might inﬂuence tendon mechanical response is
desirable given the variation in hydrating methods and solutions utilized by investigators in

the evaluation of tendon and ligament response. The effect of hydration on tensile

behavior of tendon was recently investigated with a ﬁnite element model of a patellar
tendon subfascicle (Atkinson et al., 1997). The model exhibited stress relaxation and strain
rate sensitivity which were dependent on the level of hydration, and were qualitatively
similar to those reported by other researchers. There were, however, several aspects of
this model which were rmdesirable: a sealed boundary condition was applied to the model
periphery, the “matrix” portion of the model alone reﬂected pressures in the subfascicle,
and the tensile response of the model was linear. In order to address the limitations of
this early model, a new, more microstructually accurate model has been developed
METHODS:

The original model was based on Yahia and Drouin’s (1988) description of the
microstucture of tendon and ligament wherein he identiﬁes a helical arrangement of
collagen in the sub-fascicular structures present in PT and the ACL. Jozsa et a1 (1991) also
describes a ﬁbre (subfascicle) composed of helically oriented collagen ﬁbres as the basic
unit of tendon found in various human tendons, i.e.: Achilles, quadriceps, and extensor
pollicis longus. We idealized the helically oriented collagen ﬁbers as rings of ﬁbers
separated by matrix (Figure 1).

In the original model the collagen ﬁber effects were contained in a single ring of ﬁbers
located on the periphery of the model. In the current model the ﬁber effect was modeled
with a distribution of concentric rings throughout the model.

In order to incorporate both solid and ﬂuid effects into the mechanical response of the
model, poroelastic material was utilized, which has the same form as the linear biphasic

model when linear elastic behavior is assumed for the solid portion. An orthotropic

65

poroelastic material was utilized to simulate oriented ﬁbers within 8 ﬁbrous rings, where
the E2 direction represented the ﬁber modulus. Material properties were selected to
achieve a material which was strongest in the ﬁber direction, weaker in the orthogonal
directions and weakest in shear (E1=E3= 300MPa, E2= 6OOMPa, Gua=3MPa) in order to
achieve a twisting effect consistent with tightening of helically wound ﬁbers. Poisson’s
ratios for the orthotropic materials were selected to be consistent with the elastic moduli.
The ﬁber orientation angle within each ring varied from 5° from vertical at the innermost
ring to 25° at the outermost. The matrix material was assumed to be isotropic poroelastic
(E=2MPa, v=0.1).

As the fascicle is a reasonably long continuous structure, a 3 dimensional slice ﬁ'om the
middle was modeled. The top and bottom surfaces of the model were sealed to represent
conditions in a long ﬁber. The outer boundary of the model was assumed to be perfectly
draining. The bottom plane of the model was constrained to in plane motions with 4,
nodes 90° apart, additionally constrained to radial motion. The top plane was assumed to
deform uniformly in the z direction with r and 0 free.

The model was utilized to simulate a relaxation experiment with 3% strain induced in
lms and tensile tests at strain rates of .1%/s and 10%/s.

RESULTS:

Positive pressures throughout the model indicated a negative dilatation during tensile
stretch (Fig. 2). The model’s relaxation resulted from water exudation (Fig. 3). Water
moved radially ﬁom the center to the periphery. The nonlinear tensile response resulted

ﬁom reorientation of the simulated ﬁbers with the load (Fig. 4).

66

DISCUSSION:

Incorporation of the microstructural aspects of tendon structure, such as ﬁber
organization, in poroelastic structures can lead to models which exhibit realistic tensile
behaviors. These new mathematical models for parallel ﬁbered connective tissues may
prove especially useful in understanding the roles of ﬁber organization and tissue hydration
during various states of healing. The new models will, however, require further validation

studies.

67

REFERENCES:

Atkinson, T.S., Haut, RC. and Altiero, NJ. (1997) A poroelastic model that
predicts some phenomenological responses of ligaments and tendons. J. Biomech. Eng.
119, 400-405.

Hannaﬁn, J. A. and Amoczky, S. P., 1994, “Effect of cyclic and static tensile
loading on the water content and solute diffusion in canine ﬂexor tendons: an in-vitro
study,”J. Orthop. Res., Vol. 12, pp. 350-356.

Haut, T.L., and Haut, RC. (1997) The state of tissue hydration determines the
str'ain-rate-sensitive stifﬁress of human patellar tendon. J. Biomech. 30, 79-82.

Haut, R. C. and Powlison, A. C., 1990, “The effects of test environment and cyclic
stretching on the failure properties of human patella tendons,” J. Orthop. Res., Vol. 8, pp.
532-540.

Jozsa, L., Kannus, P., Balin, J.B., Reffy, A., 1991, Three-Dimensional
Ultrastructure of Human T endons, Acta Anat, Vol. 142, pp. 306-312, 1991.

Lanir, Y., 1978, “Structure-strength relations in mammalian tendon,” Biophysical
J., Vol. 24, pp. 541-554.

Lanir, Y., Saland, E. L., and Foux, A., 1988, “Physico-chemical and
microstructural changes in collagen ﬁber bundles following stretch in-vitro,” Biorheology
J., Vol. 25(4), pp. 591-604.

Thielke, R.J., Vanderby, R. Jr., and Grood, ES. (1995) Volumetric changes in
ligaments under tension. In Proceedings of the 1995 Bioengineering Conference.

Breckenridge, Colorado.

Yahia, L. H., and Drouin, G. (1988) Collagen structure in human anterior cruciate
ligament and patellar tendon. J. Mat. Sci. 23, 3750-3755.

68

 

 

 

 

 

 

collagen / ’
ﬁbers

 

ﬁber
orientation

 

Figure 1

69

 

 

 

 

 
    
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
  

  
   

 

 

 

 

 

 

 

1 1 1.2
F/Fo
”’0 Normalized
0.8 ~ .
_ Relaxation
05" Pressure
. . . 0.4 .. 4»
Dlstrlbutlon
O i T O l
0 0.5 rim 1 0 o 5 mo 1
Figure2 Figure3
1 . . :
Normalized Tensile c
A 0.8 or Response
4".
g 0.6 ——
r;
g 0.4 —— o
E 0.2 _ .10/0/3
- - - -10°/o/s
O i l
0% 2% 4% 6% 8%
% strain

 

 

 

Figure 4

70

 

Chapter 4:

The Tensile and Stress Relaxation Responses of Human Patellar
Tendon Varies with Specimen Cross Sectional Area

 

Theresa S. Atkinson, Benjamin Ewers and Roger C. Haut

71

ABSTRACT:

In order to provide insight into the mechanical response of the collagen fascicle
structures in tendon, a series of constant strain rate and constant deformation, stress
relaxation mechanical tests were performed on human patellar tendon specimens of
various sizes. These data described the stress relaxation and constant strain rate tensile
responses as a function of cross sectional area and water content. The experimental data
suggested that small portions of tendon exhibit a higher tensile modulus, a slower rate of
relaxation and a lower amount of relaxation in comparison to larger specimens ﬁ'om the
same location in the same tendon. The variation of the mechanical response with respect
to specimen cross sectional area was nonlinear. These data suggest that the structural
level present in a specimen has a strong inﬂuence on its tensile and stress relaxation

responses.

72

INTRODUCTION:

Most currently available tendon and ligament models consider the tissue to be
entirely composed of collagen (Belkoff and Haut, 1992, Hurschler et al., 1997, Kwan and
Woo, 1989, Stouffer et al., 1985,) and utilize a distribution function to describe the
recruitment of collagen and capture the inﬂuence of the collagen structure on the tensile
response (Belkoff and Haut, 1992, Hurschler et al., 1997, Kwan and Woo, 1989).
However, the collagen in tendon has been described as being arranged in subfascicles and
fascicles (collections of subfascicles) which are arclosed by a connective tissue sheaths
(Y ahia and Drouin, 1988). Danylchuk (1978) describes the collagen fasciculi as that
portion of tendon responsible for its tensile strength. Kastelic (1980) modeled the collagen
fascicle and suggested that the nonlinear tensile response of tendon might arise from
collagen recruitment within the fascicle.

More recently, the ﬂuid present in these tissues has been shown to contribute to
their mechanical responses (Chen et al., 1995, Chen et al., 1994, Chen et al., 1993,
Chimich et a1, 1992, Haut and Haut, 1997, Haut and Powlison, 1990, Thielke et al, 1995).
Recent efforts have been made to improve the characterization of these tissues to take the
inﬂuence of ﬂuid into account (Atkinson et al., 1997, Chen and Vanderby, 1997, Wilson
et al., 1994). Atkinson et al. (1997) modeled the collagen subfascicle. This model
achieved stress relaxation and strain rate sensitivity via interaction between the
subfascicle’s collagen ﬁber structure and an internal hydrated matrix. The model response

was qualitatively similar to that reported for whole tendon and ligament.

73

Previous experimental studies have suggested that the tensile load/ deformation
response of small portions of tendon is qualitatively similar to that of whole tendon (Butler
et al., 1986, Stouffer et al., 1985). These experiments, however, do not describe how the
properties of small portions of tendon relate to those of larger segments in the same
subject, nor do they document time dependent phenomena such as stress relaxation
behavior. Currently, the mechanical behavior of the microstructural elements of tendon,
i.e. fascicles and subfascicles, is not well deﬁned and it is unclear how the mechanical
response of whole tendon might be attributed to the collective responses of these
structures.

In the current study a series of mechanical tests were performed in which quarters
of human patellar tendons were sequentially sectioned. The tests were designed to
describe both collagen recruitment, via a constant strain rate test, and the time dependent
stress relaxation response of specimens harvested ﬁ'om a common location within the
same donor. The objectives of these tests were to determine whether the responses of
small portions of tendons, which were composed of several subfascicles, were qualitatively
similar to those of whole tendon and to identify a relationship between mechanical
behaviors of the small and large portions of tendon within the same subject. We
hypothesized that the small portions of tendon would behave qualitatively like whole
tendon, exhibiting a nonlinear stress strain response and stress relaxation. We also
hypothesized that the whole tendon response might be described by linear superposition of

the responses of the smallest pieces of tendon.

74

METHODS:

All testing was performed using four pairs of human cadaver knees obtained from
the Michigan Tissue Bank (Table 1). Specimens were maintained at -20° C until the day
prior to testing, when they were thawed in room temperature 0.1 M phosphate buffered
saline (PBS), pH 7.2, in preparation for dissection. All soft tissue structures, excluding the
patellar tendon, were removed from the patellar and tibial bone blocks. The paratenon and
fat pad were removed from the tendon surface. The patellar tendons were separated into
quarters with bone blocks maintained at each end (Figure 1). Six specimens were selected
from the quarters: one ﬁ'om each cadaver and an additional specimen from cadavers 2 and
3 (Table 1). The remaining tissue was returned to -20°C storage. The specimens were
inspected under a dissecting microscope and damaged portions were removed. At all
times the specimens were kept moist with a spray of 0.1M PBS. Specimens were potted
in grips using room temperature curing epoxy. The cross sectional area of each specimen
was measured at 3 locations using a constant pressure area micrometer (Butler et al.,
1983). The length of each piece was measured from bone to bone at 3 locations. The
specimens were then stored overnight at -4 C. The following day the specimens were
equilibrated at least 60 minutes at room temperature in distilled water. This bath was
selected to increase the tendon’s hydration and thereby enhance the tissue’s hydration
dependent response (Haut and Haut, 1997). The tissues were mounted for tests in a servo-
hydraulic test machine (Instron model 1331) in a vertical orientation with the patellar bone

block attached to a 100 lb load cell. The specimen was positioned such that it was axially

75

aligned between the patellar and tibial bone blocks. The specimen was immersed in a 37°
C distilled water bath and allowed to equilibrate, while slack, for 5 minutes. A small
preload was applied (2N) and the specimen alignment was visually veriﬁed A constant
strain (2%) relaxation experiment was then conducted. The peak strain was achieved at a
cross head displacement rateof 123 mm/s (the maximum displacement rate of the
equipment) and was held constant for 180s while force data was gathered at 15 Hz.
Immediately following relaxation, the specimen was returned to slack for 2s, then
subjected to a subfailure tensile test (peak strain of 5%) at a grip-to-grip strain rate of
l%/s (sample rate 100 Hz).

Following the tensile test the specimen was returned to slack, the bath was drained
and a portion of the specimen was carefully resected (approximately 1/4 of the cross
section). This resected portion was gently patted with gauze to remove excess ﬂuid, then
weighed on a digital scale (Sartorius, model R160D) to determine its wet weight. The
cross sectional area of this portion was determined using the constant pressure area
micrometer. The remaining, intact piece of tendon was removed from the test ﬁxture and
allowed to re-equilibrate in a distilled water bath for at least 30 minutes. Aﬁer re-
equilibration the specimen was again mounted in the test ﬁxture and retested following the
protocol described above. This process was repeated (except that for very small
specimens the preload was 0.2N) until it was not physically possible to harvest a smaller
portion of the tendon.

Following the mechanical testing each piece of each specimen was dried in an oven

and the dry weight determined The total wet weight of the specimen at each level of

76

dissection was then determined by adding the wet weights of the pieces that made up the
specimen. The total dry weight was similarly obtained using the dry weights. The percent
of water at each level of dissection was assumed to be 1- total dry/wet weight.

As it is possible that the serial sectioning protocol used in the ﬁrst series of tests
may have induced some artifact due to repeated testing, a second series of experiments
was performed to verify the trends observed in the ﬁrst series. The specimens in the
second series were those which had been previously set aside and returned to the freezer
during preparation for the ﬁrst series of tests (Table 1). One quarter sized specimen was
selected from each cadaver to serve as the “large” sized specimen. Two small specimens
were harvested to help insure that at least one of the specimens would be able to complete
the test protocol. As it has been previously reported that the mechanical response of
human patellar tendon varies spatially across the structure (Chun et al., 1989), the “small”
specimens were harvested from locations consistent with those of the contralateral “large”
specimen (Figure 1). The specimens’ cross sectional areas and lengths were determined
as described in the ﬁrst series of experiments. The specimens were potted and mounted as
described earlier. In these tests the experimental protocol developed in the ﬁrst series of
tests was repeated, however, at the conclusion of the tensile test the whole specimen was
transected The specimen’s wet weight and cross sectional area were determined. Each
specimen was then dried in an oven and reweighed to determine the dry weight. The
percentage of ﬂuid in the tissue at the time of test was assumed to be equal to 1- dry/wet

weight.

77

The data from the ﬁrst series of tests were analyzed to determine the stiffness in
the linear range (N/mm cross head displacement) during the constant strain rate tests. The
end of the toe region of the tensile response (the start of the linear region) was identiﬁed
as the point where the difference between the line describing the linear range and the
experimental curve exceeded 5%. A paired t-test was used to compare the length of the
“toe” region in the large and smallest specimens. A tensile modulus was calculated by
multiplying the stiffness by the specimen length and dividing by the cross sectional area.
The rate of relaxation (N/lns) and amount of relaxation ((l-ForceJForceimtial) were
obtained ﬁ‘om the normalized relaxation response. The tensile modulus, rate of relaxation,
and amount of relaxation were plotted versus cross sectional area for each specimen, and
appropriate trend lines were ﬁt to the data. An F-test was used to determine whether
there were signiﬁcant differences between the slopes of the ﬁtted lines. Where there were
no signiﬁcant differences, the data were combined and a general trend identiﬁed. Paired t-
tests where also used to identify signiﬁcant differences between the responses of the small
and large specimens in the second series of tests (one of the two small specimens was
selected at random and paired with it’s contralateral large specimen) to conﬁrm that the
differences observed in the ﬁrst series of tests were present when a specimen was tested
only once. The correlation of tensile modulus, rate of relaxation and amount of relaxation
with water content and the correlation of water content with cross sectional area was

obtained within each specimen. Statistical signiﬁcance in all tests was set at p<0.05.

78

RESULTS:

In the ﬁrst series the initial specimen cross sectional areas ranged from 3.2 to 16.2
m2, and the ﬁnal, dissected specimens ranged in size ﬁom 4x10‘2 to 0.2 m2. In the
second series of tests the cross sectional areas of the larger portions of tendon ranged
from 14.5 to 21.7 mmz, while those of the small portions ranged from 0.1 to 2.6 m2. The
tensile response of a small portion of tendon (cross sectional area < 1 m2) was nonlinear
with a toe region persisting to between 0.3 to 1.4% strain (Figure 2), while that of larger
specimens exhibited a signiﬁcantly longer toe region to approximately 1.4 - 2.8 % strain.
The tensile modulus was found to increase nonlinearly with decreasing cross sectional area
in each series of experiments. A natural logarithm, linear relation ﬁt these data well
(Figure 3) . There was no signiﬁcant difference between the slopes of the ﬁtted curves for
specimens ﬁom cadavers 1,2,4 and 3 (11 location), therefore these data were combined and
a general trend line was plotted. The medial-central (1c) specimen from cadaver 3
exhibited a modulus-area trend with a signiﬁcantly steeper slope (Figure 4). The general
effect of specimen cross sectional area on modulus was consistent between all specimens.
The small specimens ﬁom the second series of tests also exhibited tensile moduli which
were signiﬁcantly larger than the moduli of the matched contralateral larger specimens.
These data followed the general trend demonstrated in the ﬁrst series of experiments (data
points on Figure 4).

The rate of relaxation increased nonlinearly with increasing cross sectional area,
and the rate-area relationship was also described well by a natural logarithm, linear

function (Figure 3). There was no signiﬁcant difference between the slopes of the rate-area

79

functions for all specimens, therefore a general trend line was computed using all
specimens (Figure 5). The small specimens in the second series of tests relaxed at a rate
which was signiﬁcantly slower than that of the larger specimens. The trend was
comparable to that observed in the ﬁrst series of tests (points on Figure 5).

The amount of relaxation nonlinearly increased with increasing cross sectional
area, and a natural logarithm, linear function ﬁt the data (Figure 3). There was a
signiﬁcant difference in the slope of the relaxation/area frmction for specimens 2 (1c) and 3
(11), as compared to the other specimens. Two trend lines describing the amount of
relaxation/area relationship were therefore generated to describe these groups, but the
cross sectional area effect was consistent between specimens (Figure 6). The large
specimens from the second series of tests also relaxed signiﬁcantly more than the smaller
specimens (data points on Figure 6).

A strong overall linear relationship was identiﬁed between the rate of relaxation
and the amount of relaxation for all specimens (Figure 7). There was also a signiﬁcant
positive correlation between the percentage of water present and the rate and amount of
relaxation a specimen exhibited for specimens 1,2(1c),3(lc), and 4 (Figure 8). The
percentage of water in the larger pieces was generally larger than that of the smaller,
however there was only a signiﬁcant correlation in specimen 3(ml).

DISCUSSION:

Previously, the mechanical strength of tendon (Danylchuk et al., 1978) and the

“toe” region of the tensile response (Atkinson et al., 1997, Kastelic et. al., 1980) have

been attributed to the collagen fasciculi within the tissue. There have been, however, few

80

mechanical studies that attempt to isolate these structures to evaluate the tensile response
of human tendon fasciculi and no documentation of their time varying response. The
intent of the current study was to document the mechanical response of tendon as the
specimen was sequentially sectioned into a specimen which might be described as a small
group of collagen subfascicles. The experiments demonstrated that small specimens
exhibited a slightly nonlinear tensile response, rmlike that of larger specimens where the
“toe” region persisted to greater strains. The tensile modulus of the specimens increased
nonlinearly as the specimen cross section decreased. In constant strain, stress relaxation
tests the smaller specimens relaxed at a slower rate than the larger specimens, and they did
not relax as much as the larger specimens. These ﬁndings suggest that there may be
structural inﬂuences in both the tensile and relaxation, or time dependent, responses of
tendon beyond those found in a small group of subfascicles. There was also signiﬁcant
overall correlation between water content and the rate and amount of relaxation, with the
Specimens with higher water content relaxing faster and relaxing more. These data
suggested that tissue hydration also played a role in the production of the time dependent
response of the tissue.

The tensile modulus obtained for large specimens in the current study compared
favorably to those previously reported in Haut and Haut (1997). In that study halves of
human patellar tendon exhibited an average tensile modulus of 203 MPa (average cross
sectional area of 52 m2) compared to an average of 200 MPa for similarly sized
specimens in the current study (as obtained using the trend function ﬁt to the data). The

modulus of small specimens exceeded that of large specimens in the current study.

81

Previous studies have also suggested that the tensile modulus of a small piece of tendon is
greater than that of whole tendon (Butler et al., 1987, Stoeffer et al., 1985). Butler et al.
(1986) suggests that the modulus of a small specimen of tendon may be more descriptive
of the collagen in tendon as the “fascicle initial length and cross section can be more
accurately determined” and “ﬁber bundles...tend to be more parallel than in whole tissues”.
The increased modulus could also have been due to a decrease in the amount of areolar
connective tissue in smaller samples of the tendon (Danylchuk et al, 1978, Yahia and
Drouin, 1988). Danylchuk et al. (1978) previously suggested that “a precise deﬁnition of
the tensile strength of ligament ought to take into account the relative contributions of
collagen fasciculi and connective sheaths...”. In the current study the toe region of the
tensile response of the small specimens was minimal. This ﬁnding suggested that some
mechanism other than collagen recruitment within the fascicle or subfascicle, potentially
whole subfascicle or fascicle recruitment, contributed to the creation of the “toe” region in
a large specimen.

The rate of relaxation for the large specimens in the current study approached that
observed in Haut and Haut (1997) (0.143 N/ln(s) for an area of 52 mm"). Also in the Haut
and Haut (1997) study, tissues tested in a dehydrating solution (sucrose) exhibited a
slower rate of relaxation than those tested in distilled water. A similar trend was observed
in the current study where the rate of relaxation decreased with decreasing ﬂuid content.
Previously Stouffer et al. (1985) reported that no relaxation was observed in tendon
specimens ranging in size ﬁom 0.33-0.72 mm2 during tensile testing, where the specimens

were “slowly elongat ” then held at a ﬁxed position for several minutes to facilitate

82

strain measurement in a 0.9 M, 37°C saline bath . In that study it was the authors intent
to describe the specimens steady state tensile response and the slow rate of extension may
have allowed ﬂuids inside of the specimens to escape during deformation thus losing the
relaxation response. On the other hand, the distilled water bath selected in the current
study likely magniﬁed the relaxation response, as intended

In the current experiments the large specimens, which relaxed faster and more than
smaller specimens, also tended to contain more ﬂuid This ﬂuid might have caused the
tissue to swell and therefore exhibit a higher permeability and more relaxation. The
inﬂuence of ﬁee ﬂuid might also explain the strong linear relationship identiﬁed between
the rate of relaxation and the amount of relaxation. On the other hand, the faster rate of
relaxation in the large pieces might also be attributed to structural inﬂuences, as the larger
pieces which exhibited faster relaxations also exhibited longer toe regions in their tensile
response. This longer toe suggests that there might have been more “squeezing” effect in
these tissues as the collagen structures reorient to align with the load As the specimen
cross sectional area and the water content of the tissue were not signiﬁcantly correlated in
most cases, it may be that both structure and ﬂuid content contribute to the creation of the
relaxation response.

The cmrent study was limited in that it was not possible to document the
microstructure present in the tested specimens, as the specimens were dehydrated for
determination of water content. The dehydrated portions, however, were examined under
a light microscope and axially aligned structures were identiﬁed. The study was also

limited in that it was not possible to measure the amount of ﬂuid in the specimens prior to

83

each test. As ﬂuids are known to move out of these tissues when stretched (Hannafm and
Amoczky, 1994, Lanir et al., 198 8), the “pre-test” ﬂuid content was likely higher than
that measured in the current study. The study was also limited in that distilled water was
used instead of a physiologic saline solution. In the future these response data need to be
studied in physiological baths. In addition to these limitations, the study was limited in
that the number of cadaver tendons included in the study, 4, was small in number.
However, the trends in mechanical responses observed in the current study were
consistent between cadavers and in specimens drawn from various sites within the
tendons, suggesting that the observed variation of mechanical response was general.

In conclusion, the experiments suggested that while small portions of tendon
behave qualitatively similar to large portions, there were signiﬁcant quantitative effects of
specimen size on the stress relaxation and constant strain rate, tensile responses of the
tendon. These experimental data indicated that the whole tendon response could not be
predicted by linear superposition of the responses of small portions of the tendon, thus
disproving our initial hypothesis. These data also suggest that tissue ﬂuid and structural
organization of collagenous structures larger than the subfascicle likely play major roles in

the response of a whole tendon.

84

REFERENCES:

Atkinson, T.S., Haut, RC. and Altiero, NJ. (1997) A poroelastic model that
predicts some phenomenological responses of ligaments and tendons. J. Biomech. Eng.
119, 400-405.

Atkinson, T.S., Haut, RC. and Altiero, NJ. (1996) A microstructural poroelastic
model for patellar tendon. In Proceedings of the 1997 Bioengineering Conference,
Sunriver, Oregon.

Belkoff, SM. and Haut, RC. (1992) Microstructurally based model analysis of y -
irradiated tendon allografts. J. Othop. Res. 10, 461-464.

Butler, D.L., Noyes, F.R., Walz, K.A., and Gibbons, M.J. (1987) Biomechanics of
human knee ligament allograft treatment. In Trans. of the 33rd Annual Meeting of the
Orthop. Res. Soc. San Francisco, CA.

Butler, D.L., Kay, M.D., and Stouffer, DC. (1986) Comparison of material
properties in fascicle-bone units from human patellar tendon and knee ligaments. J.
Biomech. 19, 425-432.

Butler, D., Hulse, D., Kay, M., Grood, E., Shires, P., D'ambrosia, R., and Shoji,
H. (1983) Biomechanics of Cranial Cruciate Ligament Reconstruction in dog. Veterinary
Surgery. 12, 113.

Chen, CT, and Vanderby, R. (1997) A poroelastic model of streaming potential
and interstitial ﬂuid ﬂow in ligament and tendon. Advances in Bioengineering, 36, 185-
186.

Chen, C., McCabe, R., and Vanderby, R. Jr. (1995) Two electrokinetic phenomena
in rabbit patellar tendon: pressure and voltage. In Proceedings of the 1995 Bioengineering
Conference. Beaver Creek, Colorado.

Chen, C. T. and Vanderby, R. (1994) 3-D ﬁnite element analysis to investigate
anisotropic permeability for interstitial ﬂuid ﬂow in ligaments and tendons. In Trans. of the
40th Annual Meeting of Orthop. Res. Soc. New Orleans, LA.

Chen, C. T., Vanderby, R., Graf, B. K., and Malkus, D. S. (1993) Interstitial ﬂuid
ﬂow in ligaments and tendons: effects of ﬁbril spacing and ﬂuid properties. In Proceedings
of the 1993 Bioengineering Conference, Breckenridge, Colorado.

Chimich, D. D., Shrive, N. G., Frank, C. B., Marchuk, L., and Bray, R. C. (1992)
Water content alters viscoelastic behaviour of the normal adolescent rabbit medial
collateral ligament. J. Biomech. 25, 831-837.

85

Chun, K.J., Butler, D.L, Bukovec, D.B., Gibbons, M.J., and Stouffer, DC. (1989)
Spacial variation in material properties of fascicle-bone units from human patellar tendon.
In Trans. of the 35th Annual Meeting of the Orthop. Res. Soc. Las Vegas, Nevada.

Danylchuk, K.D., Finlay, J .B. and Krcek, JP. (1978) Microstructural organization
of human and bovine cruciate ligaments. Clinical Orthopaedics and Related Research.
131, 294-298.

Hannaﬁn, J.A. and Amoczky, SP. (1994) Effect of cyclic and static tensile loading
on the water content and solute diffusion in canine ﬂexor tendons: an in-vitro study. J.
Orthop. Res. 12, 350-356.

Haut, T.L., and Haut, RC. (1997) The state of tissue hydration determines the
strain-rate-sensitive stifﬁress of human patellar tendon. J. Biomech. 30, 79-82.

Haut, RC. and Powlison, AC. (1990) The effects of test environment and cyclic
stretching on the failure properties of human patella tendons. J. Orthop. Res. 8, 53 2-540.

Hurschler, C., Loitz-Ramage, B., and Vanderby, R., (1997) A structurally based
stress-stretch relationship for tendon and ligament. J. of Biomech. Eng. 119, 392-399.

Kastelic, J ., Palley, 1., and Baer, E. (1980) A structural mechanical model for
tendon crimping. J. Biomech. 13, 887-893.

Kwan, M.K., and Woo, S. L-Y. (1989) A structural model to describe the
nonlinear stress-strain behavior for parallel-ﬁbered collagenous tissues. J. Biomech. Eng.
111, 361-363.

Lanir, Y., Saland, E. L., and Foux, A. (1988) Physico-chemical and
microstructural changes in collagen ﬁber bundles following stretch in-vitro. Biorheology J.
25, 591-604.

Stouffer, DC, Butler, DL, and Hosny, D. (1985) The relationship between crimp
pattern and mechanical response of human patellar tendon-bone units. J. of Biomech. Eng.
107, 158-165.

Thielke, R.J., Vanderby, R. Jr., and Grood, ES. (1995) Volumetric changes in
ligaments under tension. In Proceedings of the 1995 Bioengineering Conference.
Breckenridge, Colorado.

Wilson, A. N., Frank, C. B., Shrive, N. G. (1994) The behaviour of water in the

rabbit medial collateral ligammt. In Second World Congress of Biomechanics, (Edited by
Blankevoort, L. and Kooloos, J. G. M.), pp. 226b. Amsterdam, The Netherlands.

86

Yahia, L. H., and Drouin, G. (1988) Collagen structure in human anterior cruciate
ligament and patellar tendon. J. Mat. Sci. 23, 3750-3755.

87

TABLE 1: Description of Specimens

 

 

 

 

 

 

 

 

 

 

 

cadaver age sex cause of series 1 series 2
number death harvest location, harvest location,
left/right knee, left/right knee,
initial cross cross sectional
sectional area area (m2)
(m2)
1 52 M M.C. 1c, right, 3.2 me, right, 8.7
Infarction me, light, 2.3
mc, rightLl .3
2 17 M motor 11, right, 15.4 1c, left, 2.7
vehicle lc,right, 9.8 lc, light, 0.8
accident 1c, right, 1.2
3 52 F ventricular 1c, right, 6.1 11, left, 18.1
ﬁbrillation mc, right, 16.2 11, right, 0.6
11, right, 1.2
4 19 F motor 11, left, 5.4 mm, right, 17.7
vehicle mm, left, 0.2
accident mm, left, 0.1

 

88

 

Figure Legends:

Figure 1: The specimens were harvested to provide approximately quarter sized pieces of
tendon for the ﬁrst series of experiments (which were subsequently sectioned during the
testing) and one quarter sized piece and two smaller pieces for the second series of tests.
The small pieces in the second series of tests were harvested from a location consistent
with that of the contralateral quarter sized piece, to help minimize the inﬂuence of spatial
variation across the tendon. The harvest locations were denoted as: ll=most lateral,

lc=lateral-central, mc=medial-central, and mm=most medial.

Figure 2: The constant strain rate, tensile responses of smaller specimens exhibited a

shorter toe region than that of larger specimens (specimen 3, lateral-central).

Figure 3: The data obtained by sequentially sectioning each specimen was plotted versus

the specimen cross sectional area and a natural-logarithm, linear relation ﬁt the data well.

Figure 4: A natural logarithm, linear relationship existed between specimen cross sectional
area and tensile modulus for specimens in series 1. The data from series 2, denoted by the

points (0), follow the trends observed in series 1.
Figure 5: The natural logarithm, linear relationship describing the variation in the rate of

relaxation (as obtained from the normalized relaxation response) with specimen cross

sectional area ﬁ'om experimental series 1 and data points from specimen tested in series 2.

89

Figure 6: The natural logarithm, linear relationship describing the variation in the amount
of relaxation with specimen cross sectional area ﬁom experimental series 1 and data points

ﬁom specimen tested in series 2.

Figure 7: The relationship between the amount of relaxation and the rate of relaxation

exhibited by a specimen was linear.

Figure 8: There was a signiﬁcant correlation between the percentage of ﬂuid present in the
specimen and the rate and amount of relaxation in 4 of the 6 specimen tested. The plot
describes a typical relationship between the percentage of ﬂuid and the rate and amount of

relaxation exhibited (specimen 2, lateral-central).

90

Right Tendon

Contralateral Left
Tendon

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
  

 

I
if

Specimens for Specimen for
Series 2 Series 1

 

Figm'e 1

91

 

180
160
140
120
100
80
60
4O
20
0

force (N)

 

0% 1 % 2% 3% 4% 5%
large specimen Strain

 

 

 

 

 

\
\

 

\

 

 

force (N)
I;

 

A

 

 

 

 

 

 

 

 

 

0.5
o .
0% 1 % 2% 3% 4% 5%
small specimen strain
Figure 2

92

 

 

 

% relaxation

 

 

 

 

 

 

 

 

 

 

 

 

60
50 O ‘ a»
40 [I
30 . /
20 74y = 9.2298Ln(x) + 29.494 ——
10 R2 = 0.8582

0

O 2 4 6 3

cross sectional area (sq mm)

 

rate of relaxation
0.07 b
0.06 + .

0.05 71/

0.04 . /'
0.03 74—3, = 0.0112Ln(x) + 0.0388 7

0.02 R 2 = 0.8298
0.01
0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 2 4 6 8
cross sectional area (sq mm)

 

 

 

Modulus (MPa)
900
800
700
600
500
400

300 y = -89.674Ln(x) + 587.45

200 R 2 = 0.8663
100

0

0 2 4 6

cross sectional area (sq mm)

 

 

 

93

Figure 3

 

i

1800

9

tensile modulus (MPa)

 

 

 

 

 

 

 

 

 

 

specimens 1,2,3(li), and 4

 

- — -1 standard deviation

 

 

specimen 3 (lo)

 

 

 

 

----- 1 standard deviation

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5 10 1 5 20 25
cross sectional area (square mm)

 

Figure 4

 

0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01

rate of normalized relaxation (N/Lns)

 

 

”

specimens 1.2.3.4
— — - 1 standard deviation

5 10 15 20 25
cross sectional area (square mm)

 

Figure 5

94

 

 

 

percent relaxed

 

 

o'l’
‘0-
.-

 

I
n
.0
.0
'

 

 

 

 

o0-
......
.
p"

 

 

 

 

 

 

 

 

ﬂ—i
'“d'
"

 

 

 

- — -1 standard deviation

 

 

 

 

specimens 1,2(mc),3(lc).and 4

specimens 2(Ic) and 3(ll)

 

 

----- 1 standard deviation

I I

10 15 20

cross sectional area (square mm)

25

 

Figure 6

 

90
80
70
60
50
40
30
20

% relaxed [(Fi-Ff)/Fi]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

,10
IC

 

 

10-

 

 

,i‘i'b
,rnc
I
l

 

ODDDIO
ymwmmd

q

 

 

 

I
I
itted line

 

 

 

 

 

 

 

 

 

 

 

0.02

0.04 0.06 0.08 0.1
rate of relaxation (F/Fi/int)

0.12

 

Figure 7

95

 

0.14

 

 

percent relaxed
and

 

rate of relaxation * 1000

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

140 r +
130 _ 0 percent relaxed I
I rate of relaxation /

120 - /
110 /
100 y,‘

90 477/- j'.‘ .

80 / . 74——

70 ‘

O
60 L
50
20% 25% 30% 35% 40%

percentage of fluid in sample

 

Figure 8

96

 

Chapter 5:

Extension of a Microstructural Model for a Subfascicle Toward a
Description of Whole Tendon

 

Theresa S. Atkinson, Roger C. Haut and Nicholas J. Altiero

97

ABSTRACT:

A ﬁnite element model of a collagen subfascicle (a microstructural element of
tendon) was previously constructed to describe the role of solid structure and ﬂuid motion
in the mechanical response of tendon. In the current study, this model was able to
quantitatively match subfascicle relaxation and constant strain rate tensile responses as
described in a previous experimental study. The subfasicle model was extended to form a
simple fascicle model consisting of two subfascicles surrounded by epitenon. In previous
experiments, a difference in the rate and amount of relaxation was observed between
tendon specimens with small cross sectional areas (<1 m2) and large specimens (20
m2), with large specimens relaxing faster and more than small. The fascicle and
subfascicle models were utilized to explore whether increased subfascicle hydration and/or
the presence of an epitenon layer surrormding fascicles in the large specimen could help
explain this variation. The subfascicle model suggested that increased ﬂuid in the
subfascicle might explain a portion of the increased relaxation response exhibited by larger
specimens. The fascicle model suggested that transversely oriented ﬁbers in the
connective tissues surrounding the fascicles can produce subfascicle interaction, which will
increase the rate and amount of relaxation. The analysis suggested that the presence of
connective tissue in tendon may play an important role in deﬁning the relaxation response

of a whole tendon.

98

INTRODUCTION:

Predicting and measuring the mechanical response of tendon is important in the
development and assessment of various orthopaedic reconstruction techniques which
frequently utilize these tissues as graft materials. Lanir (1979) hypothesized that the
collagen structure has an important effect on the tissue’s function. Many models assume
that a tendon or ligament’s mechanical response can be predicted if the geometric
organization and mechanical properties of the collagen are known (Belkoff and Haut,
1992, Hurschler et al, 1997, Kastelic et al, 1980, Kwan and Woo, 1989, Stouffer et al.,
1985). Tissue hydration has also been shown to contribute to the mechanical response of
the tissue (Chen et al., 1995, Chen et al., 1994, Chen et al., 1993, Chimich et al, 1992,
Haut and Haut, 1997, Haut and Powlison, 1990, Thielke et al, 1995). However, at this
time, relatively little is known about how the collagen microstructure and ﬂuid motion in
tendon contribute to the observed relaxation and tensile responses.

The microstructure of tendon has been described as being composed of collagen
fascicles which in turn are composed of subfascicles (Danylchuk et al, 1978, Yahia and
Drouin, 1988). The subfascicles are the smallest repeating structural element of the tissue
and their structure in the patellar tendon has been documented by Yahia and Drouin
(1989). Atkinson et al. (1997a) suggested that the collagen subfascicle might be the
flmdamental structural unit within the tendon. They devised a ﬁnite element method
(FEM) model of the structure, based on descriptions by Yahia and Drouin (1988), where a
band of collagen was helically oriented about a central core of matrix. This model

suggested that the helical orientation causes the collagen to compress the interﬁbrillar

99

matrix causing ﬂuid motion and relaxation. The model’s mechanical response, however,
was compared qualitatively to that of whole tendon or ligament since the response of a
subfascicle is largely unknown.

More recently, an experimental study suggests that very small pieces (cross-
sectional areas) of tendon, which may be composed of less than 10 subfascicles, exhibit
relaxation and tensile responses which are qualitatively similar to that of whole tendon
(Atkinson et al., 1998). This study, however, demonstrated signiﬁcant quantitative
differences in the rate and amount of relaxation between tendon specimens with small and
large cross sectional areas (Figure 1). The experiments documented a 50% reduction in
the rate and the amount of relaxation when the specimen cross sectional area decreased
from 20 to 1 m2. In the study the larger specimens, which exhibited the increased
relaxation response, also contained a higher percentage of ﬂuid than the smaller
specimens. The experimental data for the small specimens suggested that the subfascicle
model produced an appropriate mechanical response. However, the variation of the
relaxation response with specimen size suggested that a linear superposition of
independent subfascicles could not be used to predict the whole tendon response. Linear
superposition would result in a rate and amount of relaxation in whole tendon equivalent
to those of the subfascicle.

Two potential explanations for the observed increase in relaxation with increasing
specimen cross section readily present themselves. It is possible that the additional ﬂuid
present in the large specimens may signify more ﬂuid within the specimen’s constituent

subfascicles. This increased ﬂuid may have contributed to the observed increased

100

relaxation response. It is also possible that connective tissue structures present between
subfascicles and fascicles, likely present in greater proportion in the large specimens, may
play a role in increasing the rate and amount of relaxation. In ligament, fascicles are
covered by a connective tissue sheath termed epitenon (Danylchuk et al, 1978, Clark and
Sidles, 1990, Jozsa et al 1991). Thin projections ﬁ'om the epitenon, termed endotenon,
subdivide the fascicles into subfascicles (Danylchuk et al., 1978). The endotenon and
epitenon are described as areolar connective tissues carrying small blood vessels,
lymphatics and nerves (Elliott, 1965). These areolar tissues bind the fasciculi into
functionally independent units (Danylchuk et al., 1978, Chowdhury et al, 1991), allowing
them to move with respect to each other (Y ahia et a1, 1994, Clark and Sidles, 1990).
Danylchuk et al. (1978) describes the collagen ﬁbers within the epitenon as having the
same diameter as those in the fasciculi. They are, however, “randomly situated in a coiled
manner along the long axis of the fasciculi” (Danylchuk et al, 1978). In an extensive study
of human tendon microstructure Jozsa et a1 (1991) describes the collagen ﬁbrils in the
epitenon as “crossed over each other and the longitudinal axis of the tendon at various
angles forming an irregular network.” Some of the ﬁbers of the epitenon are fused with
those of the fascicle (Jozsa et al, 1991, Yahia and Drouin, 1988). The epitenon may
provide a binding function and thereby play a role in the relaxation response in tendon.
The current study was performed to explore whether increased ﬂuid in the
subfascicle or the presence of connective tissue structures might have produced the
increasing rate and amount of relaxation observed in the large specimens in the previous

experimental study. In order to explore these possibilities, a quantitative subfascicle

101

model was developed using previously obtained experimental data (Atkinson et al., 1998).
The ﬂuid content in this model was then increased to determine whether this might
increase the rate and amount of relaxation. The model was also utilized in the
construction of a fascicle model, consisting of two subfascicles surrounded by epitenon.
Fiber orientations within the epitenon were visualized with scanning electron microscopy
(SEM). These orientations were simulated to explore whether the presence of an epitenon
layer might enhance the relaxation response.

METHODS:

The subfascicle ﬁnite element model utilized in the current study was a
modiﬁcation of the previously described subfascicle model (Atkinson et al, 1997a,b).
Brieﬂy, the model represents Yahia and Drouin’s (1989) description of a subfascicle in
patellar tendon (wherein collagen ﬁbers are wrapped in a helix about the subfascicle’s
axis) using a representative 3-D section of a cylindrical subfascicle with a 50 um radius
(Figure 2). The helically oriented collagen ﬁbers were wrapped around the periphery of
the model, and the matrix within the subfascicle was collected in the center. The top and
bottom surfaces of the model were sealed (as the subfascicle is a long and thin structure)
and the outer boundaries were assumed to be perfectly draining. The bottom plane of the
model was constrained to in plane motions with 4 nodes, 90° apart, additionally
constrained to radial motion. The top plane was assumed to deform uniformly in the z
direction with r and 0 ﬁne. The matrix portion, in the center of the model, was assumed
to be a linear isotropic poroelastic material. An orthotropic poroelastic material simulated

the helically oriented ﬁbers within the ﬁbrous rings, where the E2 direction represented the

102

ﬁber modulus. The properties of the orthotropic material were selected to achieve a nearly
incompressible material, which was weak in shear. These properties allowed the ﬁber
portion to helically twist in a nearly rigid body fashion. The ﬁber direction of the
orthotropic outer ring was a 20° declination ﬁom vertical, the approximate ﬁber
orientation scaled from SEM images presented by Yahia and Drouin (1988) and the crimp
angle exhibited in young rat tail tendon (Kastelic et al, 1978). Fluid ﬂow was assumed to
obey Darcy’s law and the permeability was assumed to be constant.

This modiﬁed subfacicle model was ﬁt to relaxation data for a small specimen
taken from the previous experimental study (cadaver 2, lateral-central harvest location,
Atkinson et a1, 1998). This specimen exhibited a stiffness and rate of relaxation which
were near the average for similarly sized specimens. It was assumed that this experimental
data contained the response of 7 subfascicles (the number of subfasicles, each with a
radius of 50 pm, that ﬁt into the specimen cross sectional area). As the fasciculi provide
the majority of the tensile strength in tendon (Danylchuk et al., 1978, Yahia and Drouin,
198 8), the contribution of other connective tissue structure to the strength or stiffness of
the specimen was neglected. Each subfascicle was assumed to carry 1/7 th of the load.

As the endotenon is a relatively thin projection of the epitenon, its inﬂuence was neglected
and each of the seven subfascicles were assumed to exhibit equal, normalized, relaxation
responses. The potential inﬂuence of any epitenon present in the specimen was also
neglected at this point, as the inﬂuence of this structure was to be evaluated later in the
study. The relaxation test was simulated by axially deforming the subfascicle model to 2%

strain in 1 ms, while leaving the top of the model free to undergo radial and angular

103

deformations, then holding this strain for 180 s. Fluids were allowed to drain freely from
the lateral bormdaries of the model at all times. Constant strain rate tensile tests were
modeled at a deformation rate of l %/s, while the boundary of the model was assumed to
drain ﬁ'eely.

An iterative ﬁtting process was employed to obtain a quantitative match between
the experimental relaxation response and the response of the subfascicle FEM model. This
process involved maintaining a constant relationship between the tensile moduli in the
“ﬁber” part of the model (Ez/EI=E2/E3=2, so that the transverse moduli were less than the
ﬁber modulus, E). The Poisson’s ratios were held constant, and the shear moduli were
assumed to be equal and much smaller than the tensile moduli. The elastic modulus of the
matrix was assumed to be equal to the shear moduli of the “ﬁbers,” as the matrix was
assumed to be much softer than the collagen ﬁber portion of the model. This also
decreased the number of independent parameters included in the model. The Poisson’s
ratio of the matrix was assumed to be 0.2, as experimental studies of ligament suggest that
these tissues might be highly compressible (Thielke, et al., 1995). The iterative ﬁtting
process utilized a linear interpolation scheme, where the ﬁber and matrix moduli were
sequentially varied to match the initial and ﬁnal forces in the relaxation experiment. The
model permeability was then varied to ﬁt the time varying character of the response. The
tensile response of the model was compared to the experimental response to verify the
model’s ﬁt.

Once a ﬁtted model was obtained, it was used to examine whether increased

subfascicle hydration might explain the increased relaxation observed in the large

104

specimens of the previous experimental study (Atkinson et al., 1998). As there does not
appear to be an accepted practice for simulating increased water content in tendon, the

increased ﬂuid was simulated using three independent methods. In the ﬁrst approach the
permeability (k) of the subfascicles in the large, more hydrated specimen was determined

using the relationship suggested by Argoubi and Shirazi-Adl (1996):

k-k 4H1") 2ex 1+e -1 uationl
° e,(1+e) p 1+e0 eq

where eo was the voids ratio (V guidN .01“) and kowas the permeability ﬁ‘om the “ﬁtted”

 

 

subfascicle model, and e was the voids ratio of a subfascicle ﬁ'om a large specimen. M is a
constant used to ﬁt experimental data In the current study, M was assumed to be 1 as
larger values yield unrealistic increases in the pameability. The voids ratio in the “ﬁtted”
subfascicle model was 1.857, based on the amount of ﬂuid (65% ﬂuid) in the small
specimen (Atkinson et al 1998). The subfascicles in large specimens from the previous
study contained approximately 75% ﬂuid, which translated into a voids ratio of 3. Using
these values in equation 1 suggested that the subfascicles in large specimens would have a
permeability twice that of the small specimens. In the second and third approaches,
changes in water content were simulated by varying the steady-state material properties of
the matrix portion of the model. In an isotropic poroelastic or biphasic material the
Poisson’s ratio is a measure of the ﬂuid efﬂux through the tissue, with small ratios
associated with greater efﬂux (Mow et a1, 1991), and thus more movable ﬂuid present in
the tissue. Therefore in the second approach Poisson’s ratio of the matrix was set to 0.0

in the model to simulate increased movable ﬂuid in the large specimens. It has also been

105

suggested that the aggregate modulus of a biphasic material might be a function of the
ﬂuid content of the tissue. McFarland et al. (1986) observed a decreased tensile modulus
concurrent with an increase in the ﬂuid content in patellar tendon sections used as grafts.
Thus, the steady state modulus of the tendon may decrease when the water content
increases. In the third approach an increase of ﬂuid within the subfascicle was simulated
by decreasing the modulus of the matrix in the subfascicle model by 10%.

Although increased ﬂuid in the subfascicle may increase the relaxation response, it
is also possible that structural elements outside of the subfascicle, such as the epitenon,
may play a role. In order to examine whether an epitenon layer surrounding fascicles
enhances the relaxation response, a simple fascicle model was constructed from two
“ﬁtted” subfascicle models surrounded by an thin epitenon layer (Figure 3). The epitenon
was assumed to be perfectly attached to the subfascicles, based on Jozsa et al (1991) and
Yahia and Drouin’s (1988) observations of binding ﬁbers between the structures. The
thickness of the epitenon layer (1/ 12 of the fascicle mm'or axis) was taken from average
thicknesses measured in coronal transmission electron micrographs of human anterior
cruciate ligament (Hart et al, 1990 ). The ﬁber orientations in the epitenon were
visualized for two human patellar tendons, where the tissues were prepared using
Danylchuk et al’s (1978) methods. Brieﬂy, the tissues were ﬁxed for 9 days (Histochoice)
then transversely sectioned with a razor. The sections were digested in hyaluronidase
(15000 units/ 150ml sodium acetate buffer, pH 5.4, 0.1 M, 37 °C) for 12 h, dehydrated in
increasing concentrations of ethanol, then critical point dried, mounted on aluminum stubs,

sputter coated and examined with a Jeol JSM 6400V scanning electron microscope.

106

Danylchuk et al. (1978) suggested that only the orientation of the collagen ﬁbers in the

epitenon distinguished it from the fasciculi. The epitenon was therefore simulated using
the collagen ﬁber material model. The orientation of the ﬁbers was varied based on our

micrographs. The relaxation response of the fascicle was compared to that of two non-

connected subfascicles.

RESULTS:

In simulated relaxation the subfascicle FEM model exhibited a peak force, rate of
relaxation and relaxed force which were qualitatively similar to that deﬁned by 1/7th of the
response of a small specimen from the previous experimartal study (Figure 4). The
material properties required to obtain agreement between the subfascicle model and this
experimental relaxation response (Table 1) resulted in a subfascicle with a total effective
modulus of 0.85 GPa. The pressure in the model was positive at all times with a peak
internal pressure in the center of the model of 6.5 MPa or 0.8% of the effective modulus.
Although the model coefﬁcients were obtained by ﬁtting the model to a small specimen’s
relaxation response, the model’s tensile response was similar to the specimen’s tensile
response, except the model’s response was linear while that from the experiments was
slightly nonlinear (Figure 5).

In relaxation simulations, when the subfascicle’s water content was assumed to
increase ﬁom 65% to 75%, the permeability, by equation 1, increased to 10.2e-19 m’le
or two times the “ﬁtted model” permeability. This produced a rapid rate of relaxation,
such that steady-state was achieved approximately 10s faster than with the original

permeability (Figure 6, arrows). The peak and steady state forces in the model were

107

equivalent to those in the 65% water content case. When an increase in ﬂuid content was
simulated by decreasing the Poisson’s ratio of the matrix from 0.2 to 0.0, the model
exhibited negligible change in the rate and amount of relaxation. However, when the
modulus was decreased 10%, the rate and amount of relaxation increased approximately
12%.

The SEM visualization of the patellar tendon revealed epitenon surrounding the
fascicles in the patellar tendon (Figure 7a). The SEM micrographs suggested a
disorganized, but relatively transverse, ﬁber direction in the epitenon (Figure 7b). As the
ﬁber direction in the epitenon appeared to vary somewhat from location to location in the
micrographs, ﬁber orientations from 0° (horizontal, transverse to the length of the tendon)
to -60° from horizontal were investigated in the fascicle model. In simulated relaxation
tests with the fascicle model the pressure in each subfascicle was positive and continuous
for all epitenon ﬁber directions (Figure 8). The fascicle model indicated that the rate and
amount of relaxation increased as the direction of the collagen in the epitenon became
more transversely oriented (Table 2). With more transverse orientations of the collagen
the epitenon tended to push the subfascicles together under a tensile load, increasing the
amormt of twisting in each subfascicle, resulting in higher internal pressures in the model.
At the -5° ﬁber orientation the amount of relaxation and the rate of relaxation of the
structure increased by 39% and 18%, respectively, over that of two independent fascicles.
In cases where the rate and amount of relaxation were increased, the pressure in the center
of the subfascicles was also increased For the -5° case the pressure increased 13% over

that without an epitenon. In comparison, with an isotropic epitenon layer the internal

108

pressure and the amount of relaxation were 39% and 66% less than a model with no
epitenon.
DISCUSSION:

The subfascicle FEM model exhibited behaviors that were qualitatively similar to
those of a representative small specimen from a previous experimental study. The
collagen ﬁber modulus suggested by the model, 6.7 GPa, was somewhat higher than that
previously reported for pure collagen, 0.6 - 2 GPa (Haut, 1983; Lanir, 1979). The
effective modulus of the whole structure (0.85 GPa) was, however, within this range. In
the subfascicle model the collagen ﬁbers accormt for approximately one quarter of the
subfascicle’s cross section. This amormt was based on the assumption that the
interﬁbrillar matrix accounted for a substantial portion of the subfascicle’s cross sectional
area This assumption was based on the relatively high volume of water in tendon and
ligament as indicated by the ratio of wet to dry weight (Chimich, et al., 1992). The
previously reported collagen moduli suggest that collagen may account for a somewhat
greater portion of the subfascicle’s cross section than was assumed in the model. The
linearity of the subfascicle’s tensile response was attributed to the relatively crude collagen
ﬁber representation utilized in the current study. In a subfascicle the collagen ﬁbes are
distributed throughout the cross section and Yahia and Drouin (1989) suggests that the
peripheral ﬁbers are undulated and helically oriented, while those at the center form a
simple helix. In a previous study, a more sophisticated subfascicle model was developed
which incorporated distributed collagen ﬁbers which were inclined at various helical angles

(Atkinson et al, 1997b). This model exhibited a nonlinear tensile response. However, this

109

model was not utilized in the current study, as the radial distribution of collagen ﬁber
orientations in the subfascicles is unknown, and the model was not well suited to the
iterative ﬁtting process used in the study due to the solution time required. In the future it
may be possible to obtain a description of the orientation of collagen ﬁbers within the
subfascicle using SEM.

An important limitation of ﬁtting the subfascicle model to the experimental data
was that the experimental specimen likely contained some connective tissues as well as
subfascicles. This endotenon may have contributed to the observed relaxation response
which was attributed to the subfascicle. In the small specimens utilized in the previous
experimental study (Atkinson et al, 1998), however, the epitenon was likely transected or
removed. The relaxation response observed was therefore more likely closer to that of a
single subfascicle than that observed in large specimens where more fascicles were likely
surrounded with an intact epitenon.

In the current study it was suggested that the relaxation response was attributed to
ﬂuid motion out of the subfascicle, resulting in a local redistribution of ﬂuid This concept
is consistent with Atkinson et al’s (1998) experimental data which suggests that relaxation
does not result from gross ﬂuid motion out of the specimen as decreased rates of
relaxation were observed in specimens with shorter ﬂow paths (smaller cross sections),
rather than faster rates. It is also consistent with Hanniﬁn and Amoczky’s (1994) ﬁndings
that large molecules are not taken up by tendon under cyclic load, suggesting that large
scale ﬂuid transfer between the tendon and its surrounding does not occur. The

permeability of the subfascicle obtained in the currmt study, 10'19 m‘/Ns, was below the

110

10'15 m‘/Ns range obtained in conﬁned compression testing of 3.175 mm diameter
coupons ﬁom rabbit ﬂexor tendon (Malaviya, et al. 1995). Conﬁned compression yields a
gross effective permeability in the direction of compression. However, as the patellar
tendon is generally loaded in tension, this measure may not be reasonable. In the fascicle
model it was suggested that the relaxation response of whole tendon may result from ﬂuid
motion out of subfascicles into adjoining connective tissues. In this case the subfascicle
permeability, suggested here, would more accurately describe the state of ﬂuid ﬂow in the
tissue. The current study also suggested that there might be local regions of high pressure
within the subfascicles in the tendon. However the pressure between fascicles, in the
connective tissues, might be lower. A low pressure between fascicles would be consistent
with low pressures (.001 MPa) measured within rabbit patellar tendon under cyclic load
(Chen et al, 1995).

In the previous experimental study (Atkinson et al, 1998), large specimens
contained a greater percentage of ﬂuid and exhibit faster rates of relaxation and a larger
amount of relaxation than smaller specimens. This trend was also observed in Haut and
Haut’s ( 1997) study where specimens tested in a distilled water relaxed faster than those
tested in a dehydrating sucrose solution. There does not appear to be consensus in the
literature on how to accommodate variation in water content into a biphasic analysis. In
the current study three methods were utilized to reproduce the inﬂuence of variation in
tissue hydration. Variations in permeability and Poisson’s ratio could not reproduce the
variation in response observed in large specimens. Decreasing the elastic modulus of the

matrix caused a moderate increase in the relaxation response of the subfascicle FEM,

lll

suggesting that a portion of the difference between small and large tendon specimens
might be due to variation in water content. However, the current study would suggest that
variations in the water content of the subfascicles can only account for a portion of the
observed variation of the relaxation response.

The fascicle model suggested that transversely oriented ﬁbers in the epitenon
increase the subfascicle deformations thereby increasing the pressurization of the matrix,
resulting in an increase in the rate and amount of relaxation. This model described a
single, simple subfascicle, but it suggests that a transversely oriented epitenon might cause
the relaxation response to continue to increase as greater numbers of subfascicles and
fascicles are grouped together to represent the larger specimens. The epitenon has been
described as a continuous network of connective tissues throughout the tissue (J ozsa et
al., 1991, Yahia and Drouin, 1989) composed of transversely oriented ﬁbers (Danylchuk
et al., 1978). The possible existence of a continuous transversely oriented network may be
further supported by experimental studies documenting a 10 MPa transverse modulus for
human medial collateral ligament (Quapp and Weiss, 1997). The potential inﬂuence of the
epitenon in a large section of tendon therefore appeared to be more signiﬁcant than the
inﬂuence of increased ﬂuid, and may help explain the marked increase in relaxation of
larger specimens observed in the previous experiments.

There were several limitations in the current study. The collagen ﬁber angle in the
model was assumed to be 20° declined from vertical. It was not possible to verify this
assumption as the specimens tested in the experimental study were dehydrated for

determination of water content. Previors SEM studies, however, suggest that this

112

i
orientation is reasonable (Yahia and Drouin, 197 8, Kastelic et al, 1978). Another

limitation was that a rather simpliﬁed model was ﬁt to the experimental data, however this
model provided a reasonable approximation to the subfascicle tensile response and yielded
a positive pressure proﬁle. It was also impossible to determine the morphology of the
connective tissues as the specimens in the previous experimental study were destroyed
during the determination of water content. SEM, however, evidence suggests that the
ﬁbers in these tissues are not longitudinally aligned in the tendon (Y ahia and Drouin, 1988
Danylchuk et al, 1978, Jozsa et al, 1991). A further limitation was that the fascicles were
assumed to be aligned longitudinally in the tendon. In the anterior cruciate ligament (ACL)
the arrangement of the fascicles and subfascicles has been described as 3-dimensional
where these units exhibit an undulating course and are arranged in various directions and
are interwoven (Strocchi, et al 1992, Elliott, 1965). Danylchuk indicates that the ratio of
the area occupied by the connective tissue sheaths to the area occupied by the collagen
fasciculi varies along the longitudinal course of the ACL, further suggesting a structure in
which the fasciculi are not perfectly parallel. If a similar structure exists in the patellar
tendon, it might also play a role in the production of the relaxation response. In future
studies it will be important to document the microstructure of tendon specimens so that
models may be created which represent the speciﬁc microstructural elements present.

In conclusion, the current study suggested that connective tissues surrounding the
collagen fasciculi might exhibit a binding function that theoretically could increase the
pressurization of collagen subfascicles to generate more and faster relaxation in fascicles

versus individual subfascicles. This effect is postulated to become even more pronounced

113

as many fascicles are grouped together to form a whole tendon. The study further
suggested that increases in the degree of tissue hydration can lead to increased relaxation

responses in ligaments and tendons.

114

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Clark, J .M., and Sidles, J.A., 1990, “The interrelation of ﬁber bundles in anterior
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Zaffagrini, S., and Rugger, A., 1992, “The human anterior cruciate ligament: histological
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118

Table 1: Material Coefﬁcients for Subfascicle FEM

 

 

 

 

 

 

 

 

 

 

 

 

 

Moduli Value
ﬁber: E1, E3 3350 MPa
ﬁber: E2 6700 MPa
ﬁber: v23 0.00
ﬁber: v12,v13 0.49
ﬁber: G1,G2,G3 105 MPa
ﬁber: k 5.1e—l9m4/Ns
matrisz 105 MPa
matrix: v 0.20
matrix: k 5.1e—19m4/Ns

 

Table 2: Fascicle FEM Model Relaxation Response to 2% Strain

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

collagen internal rate of amount of
ﬁber angle in pressure in relaxation relaxation
epitenon subfascicle
degrees MPa N/ln(s) (1 - T/I~"’)%
0 10.6 .0170 16.5
-5 7.0 .0167 21.6
-10 7.6 .0165 20.7
-20 7.9 .0164 17.3
-30 7.3 .0160 16.3
-40 6.0 .0143 14.5
-50 5.2 .0122 12.8
-60 4.4 .0102 10.8
isotropic 3.8 .0045 5.3
no epitenon 6.5 .0141 15.5

 

119

Figures:
Figure 1: Normalized force (HO/Fm) data from Atkinson et al. (1998) showing a
reduced rate and amount of relaxation exhibited by a subfascicular bundle with a cross

sectional area of 0.04 mm2 vs. a larger specimen with a cross section of 16.2 mmz.

Figure 2: The subfascicle structure was idealized with a layer of helically oriented

collagen and a central region of matrix.

Figure 3: A simple fasCicle model was created where two subfascicles were joined

together and bound by an oriented epitenon layer.

Figure 4: The relaxation response of the subfascicle model compared favorably to
experimental data for a subfascicle (as deﬁned by 1/7 th of the response of a specimen

which was likely composed of 7 subfascicles) obtained in a previous study.

Figure 5: The tensile response of the subfascicle model was similar to experimental data

obtained in a previous study.

Figure 8: The increased water content simulated in the subfascicle model by: increasing

the model permeability, decreasing the Poisson’s ratio of the matrix, and by varying the

elastic modulus of the matrix portion of the model. The arrows on the plot describe the

120

time at which a steady state condition was achieved in the ﬁtted model and in the ﬁtted
model with the permeability increased to twice the initial permeability. Decreasing the
elastic modulus of the matrix resulted in a faster rate of relaxation and more relaxation,

consistent with the trend observed in the previous experimental study.

Figure 7: SEM images of human patellar tendon in coronal section: (a) epitenon layers

surrounding fasciculi (X60), (b) transverse collagen ﬁbers in epitenon sheath (X13,000).

Figure 8: A representative pressure proﬁle in a subfascicle contained in a fascicle where

the epitenon ﬁbers were oriented at -20 degrees.

121

 

1 00%
90%
80%
70%
60%
50%
40%
30%
20%
1 0%

0%

Normalized Force (F/Fi)

 

50

100
time (s)

150

 

200

 

Figm'e 1

122

 

 

 

 

 

 

 

 

 

 

Figure 2

123

collagen
ﬁbers in
subfascicles

freely
draining

Figure 3

matrix

A— ——-

@@

 

 

124

 

collagen
ﬁbers in
epitenon

 

 

0.140

 

0.120

 

0.100

 

0.080

 

 

 

 

foroe (N)

0.060 .
—— experiment

— model

 

 

 

0.040

 

0.020

 

 

 

 

 

 

 

 

 

 

 

0.000
0 20 40 60 80 100120140160180

time (s)

 

 

 

Figure 4

125

 

 

0.4

 

0.35

 

0.3

 

0.25

 

 

0.2

Force (N)

 

 

 

 

0.15

 

/ —experiment

0-1 / —model

0.05 /
o /

0% 1% 2% 3% 4% 5%
strain

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5

126

 

normalized force (F/Fo)

 

 

—fitted model (v=0.2, E=105 MPa, ko=5.1e-19m4/Ns)

 

-—'k=2k0

 

 

----- v=o.o
—---E=95MPa

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.82 steady statel steady statel
tor k=2ko tor titted
0.8
20 40 60
time (s)
Figure 6

127

 

 

 

Figure 7a

128

 

Figure 7 b

129

pressure = peak positive
internal pressure

  
   

 

 

 

 

 

pressure = O
(freely

draining) epitenon layer

Figure 8

130

Chapter 6:

A Subfascicle Recruitment Model for Tendon

 

Theresa S. Atkinson

131

ABSTRACT:

The tensile strength of tendon and ligament has been attributed to their collagen
fasciculi. These fasciculi are composed of subfascicles. Previous ﬁnite element modeling
of subfascicles suggested that these structures may be responsible for the time varying
response exhibited by tendon. An analytic model for tendon, wherein subfascicles are
sequentially recruited during tensile extension, was formulated in the current study.
Relationships daived ﬁ'om the subfascicle ﬁnite element model described in Chapter 5 and
from the experimental studies described in Chapter 4 were utilized to deﬁne the rate of
subfascicle relaxation within the model. This model was able to ﬁt experimental data
obtained in a previous study. This ﬁt suggested tissue hydration and subfascicular
organization inﬂuence the relaxation response. The ﬁtted model was also used to simulate
creep. The creep simulation indicated that the experimentally observed difference in creep
and relaxation (% creep < % relax) was possible if the rate of relaxation rapidly decreased
with increasing strain.

INTRODUCTION:

Analytic models for tendon and ligament have been developed to describe collagen
ﬁber recruitment during elongation. These models, however, are incapable of predicting
the time varying response. Quasi-linear viscoelastic theory has been successfully utilized
to characterize the viscous response of the tissue (Kwan et al., 1993, Huang et al., 1997),
however this model does not provide a characterization of the collagen structure.
Recently new models for tendon and ligament have been proposed which utilize

poroelastic descriptions of the tissue, thereby including the ﬂuid portion of the tissue

132

(Chen and Vanderby 1997, Atkinson et al, 1997). Chen and Vanderby’s model treats the
tendon as a continuum, neglecting structural inﬂuences. Atkinson et al’s model of a
collagen subfascicle (1997) provides a description of collagen ﬁbers interacting with a
hydrated matrix to produce the time varying response characteristic of tendon. While this
model provides a useful interpretive tool, an extension of this model to describe whole
tendon remains largely unattainable due to the computational complexity of such a model.

In the current study an analytic model for tendon was developed based on Belkoff
and Haut’s (1992) ﬁber recruitment model and on Atkinson et al’s (1997) subfascicle
ﬁnite element model. In this way the phenomenological response of the subfascicle was
extended to describe whole tendon. In this model subfascicles are sequentially tensioned to
produce the mechanical response of portions of tendon of arbitrary size. The model was
ﬁt to experimental data obtained in the study described in Chapter 4. The errors
associated with the model’s ﬁt to the data were then assessed in order to examine the
appropriateness of the model.

METHODS:

The analytic model was based on an earlier model (Belkoff and Haut, 1992). In the
current model it was assumed that the responses of tendon, or portions thereof, to
elongation loads may be described by sequentially tensioned subfascicles. When the
subfascicle ﬁnite element model was tensioned, water contained in the matrix was
pressurized by the lateral compression of twisted collagen ﬁbers. This caused ﬂuid to
move out of the subfascicle and the force generated by the subfascicle decreased. A

similar force response was observed for tendon specimens with small cross sections tested

133

in the experimental study described in Chapter 4 (Figure 1). During a constant strain,
stress relaxation test the normalized force, time response can be described by the function:
mom...“ = - ocln(t/to) + B,

where t is the time following subfascicle deformation, to is 1 second, or is the dimensionless
rate of relaxation, and the amount of relaxation (percentage) was characterized by B.

In simulated stress relaxation in the subfascicle FEM, the rate of relaxation was
inﬂuenced by the peak strain level. This sensitivity was a result of the ﬁber geometry
incorporated in the model, i.e. as the subfascicle ﬁnite element model was stretched,
collagen ﬁbers rotated and aligned with the applied load. The ﬁbers therefore offered less
lateral compression to the matrix. This reduced lateral compression resulted in a matrix
which was less pressurized at the peak deformation and, as the ﬂuid ﬂow rate in the model
is directly related to ﬂuid pressure by Darcy’s law, the relaxation rate decreased. A
ﬁmctional relationship between subfascicle strain and the rate of relaxation was derived by
ﬁtting a function through normalized relaxation rates obtained using the subfascicle model
described in Chapter 2 to simulate relaxation at strain levels ranging from 1-12%:
or’ = orc(66.3ez + 1.72 +99) 8/L s 12%
or’ = 0 5/L > 12%,
where or is the rate of relaxation for small strains (e<l%), or’ is the modiﬁed rate of
relaxation, and e is strain.

In the previous experimental study (Chapter 4) the amount a specimen relaxed (B)
was found to be directly related to its rate of relaxation (Figure 2, reprinted ﬁom Chapter

4). This experimentally derived relationship was included in the analytic model:

134

B = 7.12a’ + .9903.
where e is the subfascicle strain and L is the initial specimen length.

In the analytic model each subfascicle was assumed to begin to relax as soon as it
was recruited, thus the force generated by the subfascicle was described by the function:
F = k -[0t’ln(t/to) + B] 08 = k 0[0t’1n(t/to) + 7.12 or’ +9903] 08,
where F [unitsz N] was the force generated by the subfascicle, k [N/mm] was the
subfascicle stiffness and 8 [mm] was its elongation.

A Gaussian function was assumed to describe the recruitment of subfascicles
during an extension 8:
m5) = exp[(5-u)2/20'2]/[O'~/ﬁ].

The tr parameter described the amount of deformation required to tension 50% of the
subfascicles in the tissue. Sigma (0') was the standard deviation of subfascicle recruitment
lengths about that mean. This recruitment function multiplied by the increment of
displacement yields the percentage of the total number of subfascicles present in the
specimen which are recruited at any given extension. This percentage, multiplied by the
total number of subfasicles, yields the total number of subfascicles tensioned for a given
displacement. As the total number of subfascicles present in a sample of tendon may not
be known (although it might be possible to estimate), the number of subfascicles and the
subfascicle stiffness were combined Given the total number of subfascicles in a specimen,
the model would yield the average subfascicle stiffness.

The complete analytic model was therefore:

135

F(5,t) = W i i (a. - 5.)(6,.. — 6.)R(8,-.1)T(8. — 8,4.- — o),

i=50 j=50

where t: - tj is the time following the tensioning of a recruited group of subfascicles
deﬁned by R(55)(5,-+1 - 5,), 8, - 5,- is the elongation of the group, 50 is the initial deformation
(generally zero), 8 is the current deformation, and F is the total force in the specimen at
any time and displacement.

The modiﬁed sensitivity coefﬁcimts (BOdF/dB, where B = k, 1.1, 0’, and oz) for the
model parameters were plotted for both tensile extension and stress relaxation (Figure 3).
These plots indicated that the or and k terms were related (they exhibited similarly shaped
sensitivity curves) during tensile extension, therefore the rate term (or) could not be
distinguished from the stiffness term (k) using the tensile response. However, during
relaxation the parameters were distinguishable. Therefore the parameters ll, 0', and k
were obtained by ﬁtting tensile experimental data using a nonlinear curve ﬁtting procedure
(Marquardt’s method, Press, et al, 1988). The ﬁtted u, o, and k values were then held
constant and the relaxation response was simulated If the relaxation response was too fast
or too slow, a new rate of relaxation was assumed. This new rate was then utilized in a
subsequent ﬁtting of the u, 0', and k values using the tensile test data. The iterative ﬁtting
process continued until the parameters converged and a ﬁt to both tensile and relaxation
responses was obtained.

The analytic model was ﬁt to data obtained in previous testing of tendon as
described in Chapter 4: the largest and smallest dissections of patellar tendons from the

ﬁrst series of tests and all specimens from the second series of test. The ﬁtted models

136

were used to obtain a simulated creep response for each specimen. The creep response
was compared to the relaxation response.

Several further numerical exercises were performed in order to assess the ability of
the model to reproduce the experimentally observed tensile and relaxation responses. The
differences between the expaimental data and the model (the residuals) were plotted for
both relaxation and tensile tests so that any trends in the residuals might be identiﬁed
Trends in the residuals signify systematic errors in the model. A sequential parameter
estimation scheme was also utilized to obtain the model parameters for one specimen.
Here the model was initially ﬁt using the ﬁrst 490 data points, then 491 data points were
considered and the parameter values obtained from the previous iteration were used as the
initial “guess”. This process continued until 500 data points were considered The
variation in each parameter was plotted versus the number of data points considered.
Trends in this plot also indicate systematic inadequacies in the model.

RESULTS:

The analytic model was able to ﬁt the tensile and relaxation responses exhibited by
both small and large specimen (Figure 4). The ﬁt was best in cases where the peak force
in the relaxation experiment was similar to or less than that for the same deformation in
the tensile tests. The specimen stiffness obtained from the model increased linearly with
specimen cross sectional area (Figure 5). The rate of subfascicle relaxation increased
nonlinearly with specimen cross sectional area (Figure 6). The deformation required to
recruit 50% of the subfascicles (u) of the smallest portions of tendon were much smaller

than those of largest specimen, but there was no signiﬁcant difference for the small and

137

large groups. There was a signiﬁcant positive correlation between the model parameter p
and the rate of relaxation (Figure 7). There was not a signiﬁcant correlation between 1.1
and the percentage of ﬂuid present in the specimen (the percentage of ﬂuid was obtained
in the experimental study described in Chapter 4).

What the creep was simulated with a ﬁtted model the response was similar to the
relaxation response (Figure 8). A reduced creep response, consistent with Thornton et
al’s (1997) observation, was obtained when a more rapid rate of change of the rate of
relaxation (or) with respect to strain was assumed (a linear relationship: or’= or“(1-20* a,
note 8 <5% for all simulations) (Figure 9).

The residual plots for all of the tensile tests exhibited a characteristic sinusoidal
shape (Figure 10). The percent error in the ﬁt was initially large, when the forces were
small, but diminished with time and deformation (Figure 11). The signal error due to
electrical noise was measured during a test where no specimen was in place. This error
was similar in magnitude to the residuals from the model, however, there was no apparent
pattern to this signal (Figure 12). The residual plots for all of the relaxation tests also
exhibited a characteristic shape (Figure 13), and again the percent error in the ﬁt
diminished with time (Figure 14).

The parameters identiﬁed in the sequential estimation scheme tended to increase
with increasing numbers of observations (Figure 15). This change was most signiﬁcant for
k and n which increased by 8% with the addition of 10 observations and was least

signiﬁcant for 0t (not plotted) which did not change.

138

DISCUSSION:

In the analytic model, the average subfascicle rate of relaxation was greater in
specimens of larger cross section, similar to the gross rate of relaxation observed
experimentally. This implies that each constituent subfascicle in a large specimen tended to
relax at a faster rate than in a small specimen. A similar trend was observed when
Atkinson et al. (1998) compared the response of a fascicle ﬁnite element model to that of
a single subfascicle. The fascicle model suggested that the presence of connective tissue
sheaths surrounding the fascicles in the large specimen might produce enhanced relaxation.
As the specimen cross sectional area increases the amount of connective tissue present in
the specimen increases (Figure 4, Chapter 5). This increased presence of connective tissue
may explain the nonlinear increase in the relaxation response as specimen cross sectional
area increased. The analytic model suggested that the increased rate was correlated with
increased tissue disorganization as described by the u term in the model. An increased
level of disorganization and structural variation might increase interaction between
subfascicles and thereby increase the rate and amount of relaxation. As rt and the
percentage of ﬂuid in the specimens were not correlated, it appears that both water
content and the organization of subfascicles may increase the rate of relaxation.

The model also suggested that the stiffness of the specimen increased
approximately linearly with increasing cross sectional area. The stifﬁress provided by the
model is a steady state property, as the time varying character of the response is accounted
for in or. This linear trend suggests that the steady state modulus of tendon is nearly

constant, not dependent on the cross section. This appears reasonable as this modulus

139

may be attributed to the collagen ﬁbers within the specimen. The cross sectional areas of
the specimens were measured using a constant pressure area micrometer which displaces
ﬂuid ﬁom the specimen. The cross sections measured might therefore be described as the
“solid” or steady state cross sections, predominantly composed of collagen.

When the ﬁtted model was used to examine the creep response, the creep and
relaxationresponses were similar. However when the rate (or) was assumed to decrease
rapidly with increasing strain, the model predicted less creep than relaxation This is
essentially the mechanism proposed by Thornton et al. (1997), however the analytic model
provides a tool with which the strain sensitivity of the time varying responses of the tissue
might be described. Further experimental studies are required to investigate the inﬂuence
of strain on the relaxation response, however pilot studies suggest a decreased relaxation
response as the strain level increases.

The residuals plotted for the tensile simulation exhibit a sinusoidal nature (Figure
10). This error might be due to excitation of the natural frequency of the testing system.
During this experiment the testing machine had to accelerate very quickly to achieve the
constant rate of deformation desired in the test. Accelerations in the range of 80,000 m/s2
were calculated using the experimental displacement data and sample rate. This sudden
acceleration likely excited the natural ﬁequency of the testing system causing a resonance
to be superposed on the signal. A similar pattern of residuals was observed when Belkoff
and Haut’s (1992) model was ﬁt to tensile test data for a rabbit patellar tendon tested at a
similar displacement rate. This suggests that this error was an undesirable product of the

test ﬁxture and was not related to the modiﬁcations made to the model in the current

140

study. As the current model was able to describe the character of the tensile response and
as the largest errors were close in magnitude to the signal error from the load cell, these
errors likely do not present a serious problem for most applications.

When the residuals were examined in the stress relaxation simulation a large
degree of error was again noted in the early portion of the response. This is consistent
with the use of the natural logarithm - linear function to simulate the time rate of decay of
the force. This function tends to positive inﬁnity as time approaches zero. The function
can therefore introduce signiﬁcant error in the early response of the model. In previous
studies (Fung et al., 1972) a series of exponentials was used to describe this time change.
These functions are well behaved, equal to 1 at zero and tending to zero for longer times.
It was demonstrated, however, that the coefﬁcients in these series were not unique. In the
future other relaxation functions should be evaluated to address this problem. However,
the ability of the current function to ﬁt the remainder of the data and it’s relative simplicity
have caused others to recommend it’s use (Kwan et al., 1983).

In the sequential estimation scheme the stifﬁress and u parameters were found to
increase with increasing numbers of observations, suggesting an inadequacy in the model.
It is possible that the stiffening function used in the model (the equation used to decrease
the rate as the deformation increases) was not sufﬁciently steep. The model thus requires
a higher stiffness as more long term data is considered in order to maintain the linear
force-elongation response at later time points. The need for a steeper function was also
suggested by the creep simulations which showed creep equal to relaxation with the

current model, contrary to experimental data (Thornton, et al., 1997). When a more

141

aggressive relaxation rate/ strain relationship was employed the creep response was
surpressed and the model predicted less creep, consistent with Thornton et al.’s (1997)
data. Further experimental studies are required to determine the appropriate peak strain-
relaxation rate relationship.

While the present model was able to ﬁt the experimental tensile and relaxation
responses, several limitations exist within its development. The most important limitation
was that the average rate of subfascicle relaxation was noted to vary with specimen cross
section, but no explicit relationship was incorporated into the model. The model and
Atkinson et al’s previous experimental study (1998) suggest that the rate of relaxation
increases with specimen cross section according to a natural logarithmic relationship.
While the physical basis for this relationship is currently unknown, it appears reasonable to
suggest that small increases in a specimen cross section at the “fascicular” level present a
more signiﬁcant increase in the complexity of the potential subfascicle interactions than
similar increases at the “whole tendon” level. Further ultrastructural visualization and
structural modeling may provide a more detailed understanding of this observed relation.
In addition the model incorporates a rate of relaxation/amount of relaxation relationship,
derived from experimental studies, which was attributed to the tissue hydration. Further
experimental studies are required to conﬁrm the source of this relationship and to further
explore the inﬂuence of hydration.

In conclusion, the model was able to ﬁt the available experimental data and may be

used to simulate the creep response. Additional studies are required to develop explicit

142

 

relationships between trends observed in the model and the subfascicle morphology of the

tested specimen.

143

REFERENCES:

Atkinson, T.S., Ewers, B. and Haut, R.C., 1998, The Tensile and Stress
Relaxation Responses of Human Patellar Tendon Varies with Specimen Cross Sectional
Area, submitted to J. Biomechanics, April 1998.

Atkinson, T.S., Haut, RC. and Altiero, NJ. (1997) A poroelastic model that
predicts some phenomenological responses of ligaments and tendons. J. Biomech. Eng.
119, 400-405.

Atkinson, T.S., Haut, RC. and Altiero, NJ. (1996) A microstructural poroelastic
model for patellar tendon. In Proceedings of the 1997 Bioengineering Conference,
Sunriver, Oregon.

Belkoff, S. M. and Haut, R.C., 1992, “Microstructurally based model analysis of y
- irradiated tendon allografts”, J. Othop. Res.,Vol. 10, pp. 461-464.

Chen, C., and Vanderby, R., 1997, A poroelastic model of streaming potential and
interstitial ﬂuid ﬂow in ligament and tendon, In Proceedings of the 1997 Bioengineering
Conference. Sun River, Oregon, pp. 185-186.

Danylchuk, K.D., Finlay, J .B. and Krcek, J.P., 1978, “Microstructural organization
of human and bovine cruciate ligaments”, Clinical Orthopaedics and Related Research,
No. 131, pp. 294-298.

Elliott, DH, 1965, Structure and function of mammalian tendons, Biol. Rev., Vol.
40, pp. 394-421.

Fung, Y.C., Perrone, N, and Anliker, M., editors, 1972, Biomechanics: Its
Foundations and Objectives, Prentice-Hall, Inc.

Huang, C., Wang, V.M., Cohen, N.P., Bucchieri, J.S., Pawluk, R.J., Pollock,
R.G., and Mow, V.C., 1997, Nonlinear viscoelastic properties of the inferior glenohumeral
ligament, In Proceedings of the 1997 Bioengineering Conference, Sunriver, Oregon.

Kwan, M.K., Lin, T.H—C., and Woo, S. L-Y, 1993, “On the viscoelastic properties
of the anteromedial bundle of the anterior cruciate ligament,” J. Biomechanics, Vol.
26(4), pp. 447-452.

Press, W.H., Flannery, B.P., Tevolsky, S.A., and Vetterling, W.T., 1988,

“Modeling of Data” in Numerical Recipes: The Art of Scientific Computing, Cambridge
Unversity Press, N.Y., pp. 498-525.

144

Strocchi, R., DePasquale, V., Gubellini, P., Facchini, A., Marcacci, M., Buda, R.,
Zaffagrini, S., and Rugger, A., 1992, The human anterior cruciate ligament: histological
and ultrastructural observations, J. Anat, Vol 180, pp. 515-519.

Thornton, G.A., Oliynyk, A., Frank, CB, and Shrive, N.G., 1997, Ligament creep

cannot be predicted ﬁ'om stress relaxation at low stress: A biomechanical study of the
rabbit medial collateral ligament, J. Orthop. Res., 15: 652-656.

145

FIGURES LEGENDS:

Figure 1: Stress relaxation of a specimen tested in the study described in Chapter 4. The
time rate of change of the force is ﬁtted well by a natural logarithm function.

Figure 2: The rate of relaxation and the percent relaxation of the specimen tested in
Chapter 4 were linearly related.

Figure 3: Modiﬁed sensitivity coefﬁcients for the model parameters. The similarity
between the or and k curves for extension indicates that these parameters have a similar
inﬂuence on the tensile response. The relaxation response was therefore used to obtain or.

Figure 4: Representative examples of the model’s ability to ﬁt experimentally obtained
tensile and relaxation responses in specimens of both large and small cross sectional areas.

Figure 5: The stiffness derived from the model increased linearly with cross sectional area.

Figme 6: The average subfascicle rate of relaxation decreased with decreasing specimen
cross sectional area.

Figure 7: The average subfascicle rate of relaxation increased with increasing u,
suggesting more disorganization (a longer toe region) led to faster rates of relaxation.

Figure 8: The model predicted slightly less creep than relaxation for a typical tendon
specimen.

Figure 9: The model predicted signiﬁcantly less creep than relaxation when the rate of
relaxation was assumed to decay rapidly with increasing strain.

Figure 10: The error in the model’s ﬁt to the tensile data exhibited a sinusoidal shape

Figure 11: The error constituted a large percentage of the measurement initially, but at
longer times and larger elongations it constituted a small percentage.

Figure 12: Signal noise ﬁ'om the load cell used in the experiments during a tensile test
when no specimen was in place.

Figure 13: The error in the model’s ﬁt to the relaxation data decayed with time.

Figm'e 14: The error was a large percentage of the measurement initially, then a smaller
percentage later.

Figure 15: The estimated parameter values increased with increasing numbers of
observations.

146

 

 

cm C39 relaxation at 2% strain

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1
A 0.95 \
8 g 0.9
5 g 0.85 -
8 .3 0.8 e
g 3 0.75
g a 0.7 ~— .4
5 g o 65 ,_ y = -0.0513Ln(x) + 0.8498 __
c V ' R2 - o 9779
E; 0.6 --—— - - —1
0.55 x=t/ls;[0.007-SS]
0.5 l l
O 1 2 3 4 5
normalized time (t (s) /1 s)
Figmel
90 d
80 .
70 ﬁ/
5 a
E 60
u. D
A O
“a 50 /
B D
E 40 o 1,lc
E I 2,Ic ‘
\o 30 Cl 2,1110
° A 3,mc
20 A 3," ‘—
' tiar- e
10 T e In _—
0
0.02 0.04 0.06 0.08 0.1 0.12 0.14
rate of relaxation (F/Fi/Int)
Figure2

147

 

 

 

Modiﬁed Sensitivities inTension

 

 

 

 

sensitivity ﬁdF/tﬁ)

 

0 0.5 1 1 .5 2 2.5
displacement at 1%Is

 

 

Modiﬁed Sensitivities in Relaration

 

 

 

E

u. k
'D

a —m’
2‘ .
g ----- rate
0

0)

 

 

 

O 25 50 75 100 125 150
time (s)

 

 

Figure 3

148

 

 

force (N)

 

—- exp large
---- model large
— - exp small
-- -- model small

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time (s)
25
20 I
A 15 —exp. large
3 -— model large
g - - exp small
LE 10 / - -- model small
5 /
0 41/ ‘—
o 0.5 1 1-5 2 2'5
elongation (mm)

 

149

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

k (model derived)
350
300
250 ' y = 12.525x + 13.827
R2 = 0.7315
200
150 I
o / o
100 7
. //
50 P/
O O
O
O 5 1O 15
area (sq. mm)
Figure 5
0.08
0.07 ’
0 O
0.06
2 /
g 0.05 y = 0.0057Ln(x) + 0.0516
1: 2 _
g 0. 04 ‘ R _ 0.7579
E
.3 0.03 l»
0.02
0.01
0
0 5 10 15
cross sectional area (square mm)

 

 

 

 

Figure 6

150

 

 

 

 

 

rate
i\‘(
v Q

4»

 

y = 0.0209x + 0.0459
n2 = 0.4485

 

 

 

 

,ca

 

 

 

 

 

 

CD

 

1.5

 

Figure 7

 

1.8
1.6
1.4

1.2

0.8

0.6
now

"0.4

0.2

 

—-—relaxatlon

 

50 100 150
tlme (s)

200

 

Figure 8

151

 

 

 

 

1 .2
creep

 

 

 

P
00
’—

 

relaxation

 

 

Normalized elongation
Normalized Force
9 o
4:- O)

 

.0
N

 

 

 

 

 

 

O

50 1 00 1 50 200
time (s)

O

 

 

 

Figure 9

 

error (F(model) - F(exp)) in fit to cll3941

 

0.1

 

 

 

0.05

 

 

 

 

 

 

-0.05 i

 

 

-0.1

 

 

 

 

 

 

 

-0.15
0 0.5 1 1.5 2 2.5

elongation (mm)

 

 

 

Figure 10

152

 

% error in fit to CLL3941

 

20°/o
0°/0
-20°/o
-40°/o
'60°/o ‘
'80°/o
-1 00°/o

,... -120°/o .
0 0.5 1 1.5 2 2.5

elongation (mm)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Barre“

 

0.8
0.6
0.4
0.2

0.2
0.4
-0.6
0.3

 

26.5 27 27.5 28 28.5 29 29.5 30

 

 

 

Figure 12

153

 

error in fit to cll3941

 

0.5

 

0.4

 

0.3

 

0.2

 

0.1

 

 

-0.1

 

-0.2

 

 

 

 

 

-0. 3

 

50 100 150 200

time (s)

 

Figure 13

 

40%

20%

0°/o

-20%
-40%
-60%
-80%

"' ’ '1 00%

 

°/o error in fit to cll3941

 

 

 

 

 

 

 

 

 

 

 

50 100 1 50 200
time (s)

 

Figure 14

154

 

 

 

percentage change of parameters
in sequential regression

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10%
8°/o ‘ g r.
6% W i a k
4°/o J . mu
2°/ I T sigma
O
A A A A
0% I a A .

 

490 492 494 496 498 500
# data points in sequential regression

 

 

 

Figure 15

155

Chapter 7:

Conclusions and Recommendations for Future Work

 

Theresa S. Atkinson

156

The studies described in this dissertation represent a continuing effort to develop a
structurally realistic model for tendon. Earlier efforts by other researchers focused on
structural modeling of the collagen portion of the tissue without regard for the ﬂuid
portion. While these models were capable of reproducing the characteristic nonlinear
tensile response of the tissue, they could not be used to quantify the inﬂuence of ﬂuid in
the tissue. The models described in this dissertation were based upon the concept that ﬂuid
motion in the tissue arises when the collagen ﬁbers in the tissue to compress the ﬂuid ﬁlled
interstitial matrix. These models demonstrate that ﬂuid motion at the subfascicular and
fascicular level is likely important in the production of the observed time varying
mechanical responses of tendon.

While the ﬁnite element models in Chapters 3 and 5 provided a reasonable
description of a single fascicle or a simple fascicle, they remain far from a model of a
whole tendon. At this time the computational power required to create a whole tendon
model composed of hundreds of fascicle models poses a signiﬁcant challenge. The ﬁnite
element models described in this dissertation do, however, provide a tool which can be
used to explore the mechanical responses associated with these microstructural elements.
This type of information was used as the basis for the subfascicle recruitment model for
tendon described in Chapter 6. Similar to previously posed collagen recruitment models,
the analytic model provided a phenomenologic description of tendon and ligament. While
the analytic model has the advantage of being relatively easy to apply to a variety of
tissues, it requires validation and further development to provide a more explicit

expression of the collagen/interstitial ﬂuid interaction.

157

Although the studies contained in this dissertation shed light on the possible
mechanical interactions of the ﬂuid and solid portions of tendon, many issues remain to be
explored One issue raised in the course of this research was the potential importance of
the connective tissues surrounding the fascicles in the production of the relaxation
response. In order to investigate whether the continuous network of connective tissue in
tendon (the epitenon) enhances the relaxation response, a set of pilot experiments were
performed The medial and lateral halves of a single human patellar tendon (78 year old
male, congestive heart failure) were tested in distilled water. The test protocol was similar
to that used in the experiments described in Chapter 4. Brieﬂy, specimens were held at a
constant strain of 2% for 1805 in a relaxation test. Immediately following the relaxation
test a tensile test was performed at a strain rate of 1% strain/second Following the initial
test, several slits were made along the length of the specimens such that each specimen
was divided into eighths which remained connected to the bone blocks at each end. These
slits in the tendon were intended to distrn'b the connective tissue network surrounding the
subfascicles and thereby decrease the rate and amount of relaxation produced. The
specimens were re-equilibrated in distilled water for 1 hour, then retested

This protocol resulted in a reduced rate and amount of relaxation in the sectioned
tendon in comparison with its intact response. The amount of relaxation was reduced by
14% in one case and 13% in the other. The rates of relaxation were reduced by 11% and
13%.

These results support the hypothesis that the system of connective tissue in tendon

plays a role in the production of the relaxation response. However, further testing is

158

required. In these tests the pieces of the subdivided tendon were relatively large compared
to the smallest specimen tested in Chapter 4. Further subdividing might produce larger
differences in the relaxation response. The structure of the tested specimen should also be
veriﬁed using microscopic or scanning electronic microscopic techniques.

Another area where further investigation is warranted is in the simulation of the
creep response of the tissue. Thornton et al. (1997) found that the rabbit medial collateral
ligament creeped less than it relaxed. They suggested that the mechanisms of creep and
relaxation differ, with ﬁber reorientation limiting the creep response. In Chapter 6 of this
dissertation the analytic model was used to examine creep. This work suggested that the
tissue would creep less than it relaxed if the rate of relaxation decreased rapidly with
increasing strain. In other words, if the decay of the stiffness of the tissue was limited as
the strain increased, the creep would also be limited. A similar trend was identiﬁed in the
subfascicle ﬁnite element model in a series of simulated experiments.

The simulated experiments were performed as follows. A relaxation test was
simulated with a peak strain of 4% imposed The peak force from this simulation was then
imposed on the model in a separate simulated creep test.

Experiments with the model indicated that the subfascicle creeped more than it
relaxed for smaller ﬁber orientation angles (more transverse orientations) (Table 1). At
larger ﬁber angles (more axially aligned) the subfascicle creeped less than it relaxed. At a
75 degree ﬁber orientation, there was 75 % more relaxation than creep. In this case, and in
each case where the model creeped less than it relaxed, the rate of change of the rate of

relaxation with strain was greater than that in cases with more creep (Figure 1). In the

159

cases with reduced creep, the rate of relaxation was slower than for cases where there was
more creep (Figure 1). In order to examine whether the reduced creep response was a
result of the relatively low amounts of relaxation noted in the 73-80 degree cases, another
simulation was performed. The matrix modulus was decreased (for the 75 degree ﬁber
angle case) to achieve an increased amount of relaxation. This increased relaxation did
not result in increased creep (Table 1), suggesting that the creep response was more
sensitive to the ﬁber orientation than to the relaxation response. The internal pressure in
the model, which governs the rate of relaxation, was also lower in cases where less creep
was noted (Figure 2). The pressure increased more slowly with increasing strain in cases
with less creep.

The simulated experiments demonstrated that when the helical wrapping of the
model’s ﬁbers about the matrix was reduced, the time variation of the axial force in the
model was also reduced This mechanism was similar to that suggested by Thornton et
al.’s (1998) who found that the crimping of collagen was reduced following creep. This
reduced crimp results in a more axially aligned collagen ﬁbers which likely have limited
interaction with the ﬂuid ﬁlled matrix. The time variation of the force generated by the
collagen was therefore reduced. In the future the creep and relaxation responses for
human tendon should be examined. It will be important to examine the responses of
specimens with both small and large cross sectional areas to determine whether the creep
response originates in the subfascicle, or whether it varies with specimen cross sectional
area similar to the relaxation response. It would also be valuable to examine the collagen

ﬁber organization in specimens at the conclusion of relaxation and creep tests. This

160

information might shed further light on the role of the collagen morphology in the
production of mechanical responses.

In addition to the efforts outlined above, further experimentation is required to
expand the basis for the analytic model described in Chapter 6 and provide for its
validation. Experimental development of a functional relationship between the rate of
relaxation and the water content of the tissue is required This work might involve
measuring the permeability of the tendon via a conﬁned compression creep test. Future
validation of the analytic model might involve prediction of the creep response of a
specimen using a model ﬁt to its relaxation and tensile responses.

In conclusion, it is clear that further ultrastructural study and modeling of human
and animal tendon microstructure is required. The studies contained in this dissertation
provide a step in this process, but future studies will enhance our understanding of these
tissues. As indicated throughout this ﬁnal chapter, future research efforts should continue
to focus the roles of ﬂuid and collagen structure in the production of the mechanical
responses of tendon and ligament. These efforts promise to continue to provide insights
which will assist in the developmmt of techniques and devices to maintain the optimal
function of tendon and ligament.

References:

Thornton, G.A., Oliynyk, A., Frank, CB, and Shrive, N.G., 1997, Ligament creep
cannot be predicted from stress relaxation at low stress: A biomechanical study of the
rabbit medial collateral ligament, J. Orthop. Res., 15: 652-656.

Th0rnton,GM, Sutherland, CA, Barclay, LD, Leask, GP, Marchuk, LL, Frank,

CB, Shrive, NG, 1998,Creep is resisted by ﬁbre recruitement in ligament: Evidence ﬁom
altered crimp patterns, Trans. 44th An. Meeting Orthop. Res. Soc., pp. 45.

161

Figure Legends:

Figure l: The rate of relaxation decreased with increasing strain in the subfascicle model
(with material properties as deﬁned in Chapter 5 and with ﬁber angles of 70 and 75
degrees). Note that slower rates of relaxation occur with the 75 degree ﬁber angle. Also
note that the rate of relaxation decreases more rapidly with increasing strain in the 75

degree case as compared to the 70 degree case.

Figure 2: The peak internal pressure in the subfascicle model (at the start of the creep test)
increased with increasing peak strain (the point where the creep load was fully applied).
The internal pressure in the 75 degree case was lower than that in the 70 degree case. The
rate of change of the peak internal pressure with increasing strain in the 75 degree case

was less than in the 70 degree case.

162

TABLES:

TABLE 1: Comparison of Relaxation and Creep

 

 

 

 

 

 

 

 

A—G % relax % creep
80 degrees 1.4% , 1.2%
75 degrees 6.5% 1.6%
75 degrees, reduced Em = .8E......,,,-x 7.5% 1.6%
73 degrees 9.7% 9.3%
70 degrees 13.4% 15.0%
60 degrees 24.1% 37.8%

 

 

 

163

 

 

 

0.01 8

 

0.016

 

 

~
~
§~

 

 

0.014

y = -0.0007x + 0.0179

 

0.012 -\

 

 

F12: 0.9735

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

0.01 \T\ I

0.006 1
I 75 degees y = -0.001x + 0.0121
0004 ‘ R2- 0 9996 —‘
— Linear (75 degees) ' '
0.002 1
0‘ ---Linear(70degees)
0 1 2 3 4 5 6
Figure l
O 70 R2: 0.9999
12 ‘ I 75 ’7
I

10 — Linear (75) 4’ r"

3* ---Linear(70) 1’

6 y = 2.6675x + 0.1004

4 4, R2: 09994

’ r
(//
2 5
O ..
0 1 2 3 4

 

 

 

 

peak strain (percent)

 

Figure 2

164

 

 

APPENDIX A

Porous Elastic Material Behavior

165

The porous elastic material model was utilized in the original version of the
subfasicle ﬁnite element model. This model was created to describe the behavior of
certain clays which are highly compressible.

(1) d6 = -Kd(1np)

e = voids ratio

volume holes (ﬂuid)/volume solid

p = ﬂuid pressure
(2) e = volumetric strain = 81 + 62 + £3 = -(e0 -e)/(l+eo)
(3) d6 = -(K/(1+eo))d(1np) ‘
integrate (3) 1c
(4) 6 = -K(1nP)/(1+eo) e l_\:L
(5) e = AVNo = (v.v.,)/v0 = V/vo - 1 = 1’1 P
(6) -1c(lnp)/(1+eo)= J - 1
(7) (1n(1/p))K/(1+eo) = J - 1
(8) Up = (po + pt)/(p + p) note: p(, is the initial stress,

so that when p=0 you
get some stress, i.e.: lnpo - ln0

1n(po +91) - 111031)

p is the tensile strength such that

as p goes to -p. (-p because you are pulling)
VNo = no ,

The shear modulus of the materials is derived:
(9) de = -K d(lnp) = «K damn/6M

(10) p = 'Ome-n
(11) d(lnp) = dp/p = 'dOm/OM

(12) dO'M/dE = = = tangent bulk modulus

166

(13)

(14)

(15)

(17)

(18)

recall de = -(l(/(l + eo))d dd".,,.,..,,/o......m combine the derivative of (4)
above with equation (1 l)
K = om(1+ eo)/- K

recall: K = E/(3(l-21))) G=E/(2(1+u))
K = mow/(30-20))

G2(1+u)/(3(1-20))= o....(1+ eo)/- x
G = 3om(1+ e°)(1-Zu)/(-2(l+1>)ic)
recall: om = p

G = 3(l+eo)(l-21))p/(2(I+‘D)K)

In Abaqus p=p+p. as explained above and an additional term exp(e°.,o.) is added on, 8°...“ =
an°

therefore:

G = 3( 1-21211 1+e9) (p+p,)l
2(1+u)1<

This was utilized in the ﬁrst model because it provided a non-linear stiffening like tendon.
The nonlinear poroelastic material stiffens upon compression. This is accomplished
through moderation of the shear modulus in the following way:

 

31-2v l I
[1

167

where e0 is the initial voids ratio (volume fraction of ﬂuid/volume fraction of solid), P is
the internal pressure, P. is the elastic ultimate strength, adv... = 1n Je1 is the elastic portion of
volume change, v is Poisson’s ratio, and K is the log bulk modulus (relating the logarithm
of pressure to the dilatation) (Abaqus Theory Manual, Zienkiewicz and Naylor, 1972).
Thus, G increases with compaction and pressure. Assuming that the volume fraction of
water in tendon is similar to the weight ﬁaction (70% of the wet weight is water), an
initial voids ratio of 2.33 (=.7/.3) was used for the nonlinear poroelastic portion of the
model (Haut, 1993; Mow and Hayes 1991), however, eo=l .0 (50%) was also
investigated. Based on the assumption that ﬂuid in tendon is similar to that in other
tissues, the log bulk modulus of the nonlinear poroelastic core was obtained from data for
ﬂuid in the human annulus ﬁbrosis (Best et al., 1984). Poisson’s ratio was assumed to be
0.49. The tensile modulus of the material was then derived such that the initial shear
modulus of the inner core approximately matched the shear moduli of the outer
poroelastic ring.

References:

Best, B. A., Setton, L. A., Guilak, A., Ratcliffe, A., Weidenbaum, M. and Mow,
V. C., 1989, “Permeability and compressive stifﬁress of annulus ﬁbrosus: variation with
site and composition,” Trans. Orthop. Res. Soc., Vol. 14, p. 354.

Haut, R. C., 1993, ‘The mechanical and viscoelastic properties of the anterior
cruciate ligament and of ACL fascicles,” The Anterior Cruciate Ligament: Current and
Future Concepts, (Edited by Jackson, D. W., et al.), Raven Press, Ltd, New York.

Mow, V. C. and Hayes, W. C., 1991, Basic Orthopaedic Biomechanics, Raven
Press, Ltd, New York, New York, pp. 143- 243.

Zienkiewicz, DC. and Naylor, D.J., 1972, “The adaptation of critical state solid

mechanics theory for use in ﬁnite elements”, Stress-Strain Behavior of Soils, Parry,
R.H.G. ed, Foulis and C0, , pp. 537-543.

168

APPENDIX B

Subfascicle model Abaqus code

169

*heading,unsymm
“single tall fascicle
*node

9,-10e—6,-10e-6
81,10e-6,-10e-6
153,10e-6,10e-6
225,-10e-6,10e-6
288,-10e-6,-7.5e-6

*ngen

9,81,9

153,225,9

81,153,9

225,288,9

‘node

289,-7.5e-6,-7.5e-6
295,7.5e—6,-7.5e-6

33 l,-7.5e-6,7.5e-6
337,7.5e—6,7.5e-6

*ngen

289,295,l

331,337,]

*nset,nset=cl , generate
289,295,]

*nset,nset=c2, generate

33 1,337,1

*nﬁll

cl,02,6,7

‘nset, nset=center,generate
9,288,9

*node
l,-42.4264e-6,-42.4264e-6
109,60e-6,0

181,0,60e-6
280,-49.88818e-6,-33.33421e-6
‘ngenJine=c

1,109,9,313
109,181,9,3l3

18 1 ,280,9,313

*nset, nset=outer,generate
l,280,9

*nﬁll,bias=.95
outer,center,8 , 1

*nset, nset=bottom,generate
1,337

**

170

*node
2409,-10e-6,-10e-6,75e-6
2481,10e-6,-10e-6,75e-6
2553,10e-6,10e-6,75e-6
2625 ,- 1 0e-6, 1 0e-6,75e-6
2688,- l 0e—6,-7.5e-6,75e-6
*ngen

2409,248 1,9

2553,2625,9

248 1,2553,9

2625,2688,9

*node
2689,-7.5e-6,-7.5e-6,75e-6
2695,7.5e-6,-7.5e-6,75 e-6
2731,-7.5e-6,7.5e-6,75e—6
2737,7.5e6,7.5e-6,75e-6
’"ngen

2689,2695,l

2731,2737,1
*nset,nset=clt,generate
2689,2695,1
*nset,nset=c2t,g enerate
273 1,2737,l

*nﬁll

clt,c2t,6,7

‘nset, nset=centert,generate
2409,2688,9

*node
2401,-42.4264e-6,-42.4264e-6,75e-6
2509,60e6,0,75e-6

25 8 1 ,0,60e-6,75e-6
2680,-49.88818e-6,-33.33421e-6,75e-6
*ngen,line=c

2401 ,2509,9,2713
2509,2581,9,2713
2581,2680,9,2713

*nset, nset=outert,generate
2401,2680,9

*nﬁll,bias=.95
outert,centert,8 , 1
*nset,nset=top, generate
2401,2737,1

*nﬁll

bottom,top,6,400

**

171

*element,type=C3d20RP
1,297,299,313,311,1097,1099,1113,1111,298,306,312,304,
1098,] 106,1 1 12,1 104,697,699,7l3,711
2,299,301,315,313,1099,1101,1115,1113,300,308,3l4,306,
1100,] 108,1 1 14,1 106,699,701,715,713
3,313,315,329,327,1113,1115,1129,1127,314,322,328,320,
1114,] 122,1 128,1 120,713,715,729,727
4,311,313,327,325,1111,1113,1127,1125,312,320,326,318,
1112,] 120,1 126,1 1 18,71 1,713,727,725
5,9,27,297,279,809,827,1097,1079,l8,290,296,288,
818,1090,1096,1088,409,427,697,679
6,27,45,299,297,827,845,1099,1097,36,292,298,290,
836,1092,1098,1090,427,445,699,697
7,45,63,301,299,845,863,1 101,1099,54,294,300,292,
854,1094,1100,1092,445,463,701,699
8,63,81,99,301,863,881,899,] 101,72,90,302,294,
872,890,]102,1094,463,481,499,701
9,279,297,311,261,1079,1097,1111,1061,296,304,310,270,
1096,1104,1110,1070,679,697,711,661
10,301,99,117,315,1101,899,917,1115,302,108,3l6,308,
1102,908,1116,1108,701,499,517,715
11,261,311,325,243,1061,1111,1125,1043,3]O,318,324,252,
1110,1118,1124,1052,661,711,725,643

12,315,] 17,135,329,l 1 15,917,935,1 129,316,126,330,322,
1116,926,1130,1122,715,517,535,729
13,243,325,207,225,1043,l125,1007,1025,324,332,2l6,234,
1124,1132,1016,1034,643,725,607,625
14,325,327,189,207,1125,1127,989,1007,326,334,198,332,
1126,1134,998,1132,725,727,589,607
15,327,329,171,189,l127,1129,971,989,328,336,180,334,
1128,1136,980,1134,727,729,57l,589
16,329,135,153,171,1129,935,953,971,330,l44,162,336,
1130,944,962,1136,729,535,553,571
*element,type=c3d20rp
17,7,25,27,9,807,825,827,809,16,26,18,8,816,826,818,808,
407,425,427,409
32,277,7,9,279,1077,807,809,1079,286,8,288,278,
1086,808,1088,1078,677,407,409,679
33,5,23,25,7,805,823,825,807,14,24,16,6,
814,824,816,806,405,423,425,407
48,275,5,7,277,1075,805,807,1077,284,6,286,276,
1084,806,1086,1076,675,405,407,677
49,3,21,23,5,803,821,823,805,12,22,14,4,812,822,814,804,
403 ,421 ,423,405
64,273,3,5,275,1073,803,805,1075,282,4,284,274,

172

1082,804,1084,1074,673,403,405,675
65,1,19,21,3,801,819,821,803,10,20,12,2,810,820,812,802,
401,419,421 ,403

80,271,] ,3,273,1 O7] ,801 ,803,1073,280,2,282,272,
1080,802,1082,1072,671,401,403,673

*elgen

17,15,18,1,1,l,1,3,800,80
33,15,18,1,1,1,],3,800,80
49,15,18,1,1,1,1,3,800,80
65,15,18,1,1,1,1,3,800,80
*elgen

1,1,1, 1,1,1,13,,800,80
2,1,1,1,1,1,1,3,800,80
3,1,1,1, 1 1,,1,3,800,80
4,1,1,,1,1 1,1,3,800,80
5,1,1 ,1,1,1, 1,3,800,80
6,1,1 ,1,1,1,1,3,800,80
7,1,1,1,1,1,1,3,800,80

8, 1 ,1,1,1 ,1,],3,800,80
9,1,1,1,1,1,1,3,800,80
10,1,1,1,1,1,1,3,800,80
11,1,1,1,1,1,1, 3, 800, 80
12,1,,1,1,1,1,1,3 800, 80
13,1,1,1,1,l,1,3, 800, 80
14,1,1,1,1,1,l,3, 800, 80
15,1,1,1, 1,1,1,3, 800, 80
16,1,1, 1,1,1,,1,3 800, 80
32,1,1,1,1 1,,1,3,800,80
48,1,1, 1,1,1,1,3,800,80

64,1,1,l,1,1,1,3,800,80
80,1,1,1,1,1,1,3,800,80
*elset,elset=matrix,generate
1,48

65,80

81,128

145,160

161,208

225,240
*elset,elset=ﬁbers,generate
49,64

129,144

209,224

*nset, nset=drain,generate
1,271,18

801,1071,18

173

1601,1871,18
2401,2671,18
*nset,nset=top2, generate
2403,2409
2689,2737
2412,241 8
2422,2427
2430,2436
2439,2445
2448,2454
2457,2463
2466,2472
2475 ,248 1
2484,2490
2493,2499
2502,2508

25 1 1,25 17
2520,2526
2529,2535
2538,2544
2547,2553
2556,2562
2565,2571
2574,2580
2583,2589
2592,2598
2601 ,2607

26 10,26 1 6

26 19,2625
2628,2634
2637,2643
2646,2652
265 5,266 1
2664,2670
2673,2679
2682,2688
*nset,nset=rside,generate
73, 145 ,9
473,545,9
873,945 ,9
1273, 1345,9
1673, 1 745,9
2073,2 145,9
2473,2545,9
*orientation,name=orientl ,deﬁnition=nodes,system=cylindrical

174

313,713

1,70
*nset,nset=alln,generate
1,337

401,737

801,1137

1201,1537

1601,1937

2001,2337

2401,2737

*initial conditions, type=saturation
alln,1

‘initial conditions,type=ratio
alln,2.3

**

“oriented collagen properties (ring)

it

*solid section,elset=ﬁbers,material=collagen,orientation=orient1
*material,name=collagen

*elastic,tprengineering constants
3350,6700,3350,.49,0,.49,105,105,

105

‘permeabilityspeciﬁc weight=1

5.1e-13

**

"matrix core properties:

#*

*solid section,elset=manix,materialanater
I‘material,namea2vater
*elastic,type=isotropic

105,.2

*permeability,speciﬁc weight=1
5.1e-13

*boundary

bottom,3

253,2

37,1

109,2

18 1,1

drain,8
*restart,write,ﬁ'equency=1
*step,nlgeom,inc=200
*soils,consolidation,utol=5 e5
1e-6,.001
*boundary,type=displacement

175

top,3,3, 1 .5e-6

*endstep

*step,inc=200
*soils,consolidation,utol=5e5,end=ss
.001,180,.06,100,.001

*endstep

**
**

"note ** = a comment line

"in order to run a creep simulation
"comment out the above load case
"and uncomment out the below load
"case (i.e. add ** before the lines
“above starting at *step and ending at
** *endstep, then delete ** from the
"lines below)

#*
**

***step,nlgeom,inc=200,amplitude=ramp
***soils,consolidation,utol=5e5
**1e-10,0.001

****enter the peak force from the relaxation
****test divided by 257 (the number of nodes)
****as the load here (i.e. top,3,load)
"*cload

**top,3,0.01033e-7

***endstep

***step,inc=200
***soils,consolidation,utol=5e5,end=ss
**1e-20,l80,.06,100,.001

***endstep

176

APPENDIX C

Fascicle model Abaqus code

177

*heading,unsymm

“tall 2 fascicle model
*node

9,-10e-6,-10e-6
81,10e-6,-10e-6
153,10e-6,10e-6
225,-10e-6,10e-6
288,-10e-6,-7.5e-6

*ngen

9,81,9

153,225,9

8 1 , 153,9

225,288,9

*node
289,-7.5e-6,-7.5e-6
295,7.5e6,-7.5e-6

33 l ,-7.5e-6,7.5e-6
337,7.5e-6,7.5e-6

*ngen

289,295,]

331,337,1

*nset,nset=c1 , generate
289,295,]

I"nset,nset=c2, generate
331,337,]

*nﬁll

cl ,c2,6,7

*nset, nset=center,generate
9,288,9

*node

1,-42.4264e-6,-42 .4264e6
109,60e-6,0

181,0,60e-6
280,-49.88818e-6,-33.33421e-6
*ngen,line=c

1,109,9,313
109,181,9,313
181,280,9,313
*nset,nset=outer, generate
1,280,9

*nﬁll,bias=.75
outer,center,8 , 1
*nset,nset=bottom,generate
1,337

**

178

*node
2409,-10e-6,-10e-6,75e-6
2481,10e-6,-10e-6,75e-6
2553,10e-6,10e-6,75e—6
2625,-10e—6,10e-6,75e-6
2688,-10e-6,-7.5e-6,75e-6
*ngen

2409,248 1,9

2553,2625,9

248 1 ,255 3,9

2625 ,2688,9

*node
2689,-7.5e—6,-7.5e-6,75e-6
2695 ,7.5e-6,-7.5 e-6,75 e-6
2731,-7.5e-6,7.5e-6,75e-6
2737,7.5e6,7.5e-6,75e—6
‘ngen

2689,2695,1

2731,2737,1

*nset,nset=c l t,generate
2689,2695 , l
‘nset,nset=c2t,generate
273 l ,2737,1

*nﬁll

cl t,02t,6,7

*nset, nset=centert,generate
2409,2688,9

*node
2401,-42.4264e-6,-42.4264e-6,75e-6
2509,60e6,0,75e-6

25 8 1 ,0,60e-6,75e-6
2680,-49.88818e-6,-33.33421e-6,75e-6
‘ngen,line=c
2401,2509,9,2713
2509,2581,9,2713

258 1 ,2680,9,2713

*nset, nset=outert, generate
2401,2680,9

*nﬁll,bias=.75
outert,centert,8 , 1
*nset,nset=t0p,generate
2401,2737,1

*nﬁll

bottom,top,6,400

it

179

*element,type=C3d20RP
1,297,299,313,311,1097,1099,1113,1111,298,306,312,304,
1098,]106,1112,1104,697,699,713,711
2,299,301,315,313,1099,1101,1115,1113,300,308,3l4,306,
1100,1108,1114,1106,699,701,715,713
3,313,315,329,327,1113,1115,1129,1127,314,322,328,320,
1114,] 122,1 128,1 120,713,715,729,727
4,311,313,327,325,1111,1113,1127,1125,312,320,326,318,
1 112,1120,1126,1118,711,713,727,725
5,9,27,297,279,809,827,1097,1079,18,290,296,288,
818,1090,1096,1088,409,427,697,679
6,27,45,299,297,827,845,1099,1097,36,292,298,290,
836,1092,1098,1090,427,445,699,697
7,45,63,30],299,845,863,1 101,1099,54,294,300,292,
854,1094,1100,1092,445,463,701,699
8,63,81,99,301,863,881,899,1101,72,90,302,294,
872,890,] 102,1094,463,481,499,701
9,279,297,3]1,261,1079,1097,1111,1061,296,304,310,270,
1096,1104,1110,1070,679,697,711,661
10,301,99,117,315,1101,899,917,1115,302,108,316,308,
1102,908,1116,1108,701,499,517,715
11,261,311,325,243,1061,1111,1125,1043,310,318,324,252,
1110,]118,1124,1052,661,711,725,643

12,3 15,1 17,135,329,1 1 15,917,935,l 129,3 16, 126,330,322,

1 116,926,1130,1122,715,517,535,729
13,243,325,207,225,]043,1125,1007,1025,324,332,216,234,
1124,1132,1016,1034,643,725,607,625
14,325,327,189,207,1125,1127,989,1007,326,334,198,332,
1126,1134,998,1132,725,727,589,607
15,327,329,171,189,1 127,1129,971,989,328,336,180,334,
1128,1136,980,1134,727,729,571,589
16,329,135,153,171,1129,935,953,971,330,144,162,336,
1130,944,962,1 136,729,535,553,571
*element,type=c3d20rp
17,7,25,27,9,807,825,827,809,16,26,18,8,816,826,818,808,
407,425,427,409
32,277,7,9,279,1077,807,809,1079,286,8,288,278,
1086,808,1088,1078,677,407,409,679
33,5,23,25,7,805,823,825,807,14,24,16,6,
814,824,8l6,806,405,423,425,407
48,275,5,7,277,1075,805,807,1077,284,6,286,276,
1084,806,1086,1076,675,405,407,677
49,3,21,23,5,803,821,823,805,12,22,14,4,812,822,814,804,
403,421 ,423 ,405
64,273,3,5,275,1073,803,805,1075,282,4,284,274,

180

1082,804,1084,1074,673,403,405,675
80,271,1,3,273,1071,801,803,1073,280,2,282,272,
1080,802,1082,1072,671,401,403,673
66,19,37,39,21,819,837,839,82l,28,38,30,20,828,838,830,820,
419,437,439,421
65,1,19,21,3,801,819,821,803,10,20,12,2,810,820,812,802,
401,419,421,403
69,73,91,93,75,873,891,893,875,82,92,84,74,882,892,884,874,
473,491,493,475
75,201,183,181,199,100],983,981,999,192,182,l90,200,992,982,990,1000,
601,583,581,599

*elgen

17,15,18,1,1,1,1,3,800,80

33,15,18,1,1,1,1,3,800,80

49,15,18,1,1,l,l,3,800,80

69,4,18,1,1,1,1,3,800,80

75,5,18,1,1,1,1,3,800,80

 

*elgen

1,1,1,1,1,1,1,3, 800, 80
2,1, 1,1,,1,1,1,3 800,80
3,1,1,1,1,,1,13,,80080
4,1,], 1,], 1,1,3, 800, 80
5,1,1,1,1 ,,1,1,3 800, 80
6, 1 ,1 ,1,1,l,1,3, 800, 80
7,1 ,1 ,,1,1,1,1,3 800, 80
8,1,1,1,1 ,1 ,1,3, 800, 80
9,1,1,1, 1 ,1,1,3, 800, 80
10,1,1, 1,,1,1,1,3 800, 80
11,1,1, 1,1,1,1, 3, 800, 80
12,1,1, 1 ,,1,1,1,3 800, 80
13,1,1, 1 ,1,,1,13, 800, 80
14,1,1,1 ,1,1,1,3, 800, 80
15,1,1,1,,1,1 1,,3 800,80
16,1,1,1,1,1,1,3,800,80
32,1,1, 1,1,1 ,1,3,800,80
48,1,1,1,1,1,1,3,800,80
64,1,l,1,1,1 1,,3,800,80
65,1,1,1,1,1,1,3,800,80
66,1,1, 1,,1,1 1,3,800,80
80,1,1, 1,1,1 ,1,3,800,80
‘node

2809,120e-6,-10e-6
2881,140e-6,-10e-6
2953,140e-6,10e-6
3025,120e-6,10e-6

181

3088,120e-6,-7.5e-6
*ngen

2809,2881,9

2953,3025,9

288 l ,2953,9

3025,3088,9

*node
3089,122.5e—6,-7.5e—6
3095,137.5e—6,-7.5e-6
3131,122.5e-6,7.5e-6
3137,137.5e-6,7.5e-6
*ngen

3089,3095,1

3131,3137,1
1"nset,nset=clr, generate
3089,3095,1
*nset,nset=c2r, generate
3131,3137,1

*nﬁll

clr,c2r,6,7
I‘nset,miet=centrr,generate
2809,3088,9

*node
2801,87.5736e6,-42.4264e-6
2909,190e-6,0
2981,130e-6,60e-6
3080,8011182e—6,-33.33421e-6
‘ngen,line=c
2801,2909,9,3113
2909,2981,9,3113
2981,3080,9,3113
"nset,nset=outerr, generate
2801,3080,9
*nﬁll,bias=.75
outerr,centrr,8 , 1
*nsetnset=bottomr,generate
2801,3137

**

*node
5209,120e-6,-10e-6,75e-6
5281,140e-6,-10e-6,75e-6
5353,140e-6,10e-6,75e-6
5425,120e-6,10e-6,75e-6
5488,120e-6,-7.5e-6,75e-6
*ngen

182

5209,5281,9

5353,5425,9

5281,5353,9

5425,5488,9

*node
5489,122.5e-6,-7.5e-6,75e-6
5495,137.5e-6,-7.5e-6,75e-6
5531,122.5e6,7.5e—6,75e-6
5537,137.5e-6,7.5e-6,75e-6
*ngen

5489,5495,1

5531,5537,1
‘nsetnset=c1tr,generate
5489,5495,1
*nsetnset=c2tr,generate
5531,5537,1

*nﬁll

c1tr,c2tr,6,7
*nseLnschentertrgenerate
5209,5488,9

*node
5201,87.5736e—6,-42.4264e-6,75e-6
5309,190e-6,0,75e-6
5381,130e-6,60e-6,75e-6
5480,8011 182e-6,-33.33421e-6,75e-6
*ngen,line=c
5201,5309,9,5513
5309,5381,9,5513
5381,5480,9,5513
*nset,nset=outertr, generate
5201,5480,9

*nﬁll,bias=.75
outert,centertr,8 , l
*nset,nset=topr,generate
5201,5537 ,1

*nﬁll

bottomr,topr,6,400

**

*element,type=C3d20RP
241,3097,3099,3113,3111,3897,3899,3913,391l,3098,3106,3112,3104,
3898,3906,3912,3904,3497,3499,3513,3511
242,3099,3101,3115,3113,3899,3901,3915,3913,3100,3108,3114,3106,
3900,3908,3914,3906,3499,3501,3515,3513
243,3113,3115,3129,3127,3913,3915,3929,3927,3114,3122,3128,3120,
3914,3922,3928,3920,3513,3515,3529,3527

183

244,3111,3113,3127,3125,3911,3913,3927,3925,3112,3120,3126,3118,
3912,3920,3926,3918,35]1,3513,3527,3525
245,2809,2827,3097,3079,3609,3627,3897,3879,2818,3090,3096,3088,
3618,3890,3896,3888,3209,3227,3497,3479
246,2827,2845,3099,3097,3627,3645,3899,3897,2836,3092,3098,3090,
3636,3892,3898,3890,3227,3245,3499,3497
247,2845,2863,3101,3099,3645,3663,3901,3899,2854,3094,3100,3092,
3654,3894,3900,3892,3245,3263,3501,3499
248,2863,2881,2899,3101,3663,3681,3699,3901,2872,2890,3102,3094,
3672,3690,3902,3894,3263,3281,3299,3501
249,3079,3097,3111,3061,3879,3897,3911,3861,3096,3104,3110,3070,
3896,3904,39 10,3870,3479,3497,351 1,3461
250,3101,2899,2917,3115,3901,3699,3717,3915,3102,2908,3116,3108,
3902,3708,3916,3908,3501,3299,3317,3515
251,3061,3111,3125,3043,3861,3911,3925,3843,3110,3118,3124,3052,
3910,3918,3924,3852,3461,3511,3525,3443
252,3115,2917,2935,3129,3915,37]7,3735,3929,3116,2926,3130,3122,
3916,3726,3930,3922,3515,3317,3335,3529
253,3043,3125,3007,3025,3843,3925,3807,3825,3124,3132,3016,3034,
3924,3932,3816,3834,3443,3525,3407,3425
254,3125,3127,2989,3007,3925,3927,3789,3807,3126,3134,2998,3132,
3926,3934,3798,3932,3525,3527,3389,3407
255,3127,3129,2971,2989,3927,3929,3771,3789,3128,3136,2980,3134,
3928,3936,3780,3934,3527,3529,3371,3389
256,3129,2935,2953,2971,3929,3735,3753,3771,3130,2944,2962,3136,
3930,3744,3762,3936,3529,3335,3353,3371

*element,type=c3d20rp
257,2807,2825,2827,2809,3607,3625,3627,3609,2816,2826,2818,2808,
3616,3626,3618,3608,3207,3225,3227,3209
272,3077,2807,2809,3079,3877,3607,3609,3879,3086,2808,3088,3078,
3886,3608,3888,3878,3477,3207,3209,3479
273,2805,2823,2825,2807,3605,3623,3625,3607,2814,2824,2816,2806,
3614,3624,3616,3606,3205,3223,3225,3207
288,3075,2805,2807,3077,3875,3605,3607,3877,3084,2806,3086,3076,
3884,3606,3886,3876,3475,3205,3207,3477
289,2803,282],2823,2805,3603,3621,3623,3605,2812,2822,2814,2804,
3612,3622,3614,3604,3203,3221,3223,3205
304,3073,2803,2805,3075,3873,3603,3605,3875,3082,2804,3084,3074,
3882,3604,3884,3874,3473,3203,3205,3475
320,3071,2801,2803,3073,3871,3601,3603,3873,3080,2802,3082,3072,
3880,3602,3882,3872,3471,3201,3203,3473
307,2837,2855,2857,2839,3637,3655,3657,3639,2846,2856,2848,2838,
3646,3656,3648,3638,3237,3255,3257,3239
317,3037,3019,3017,3035,3837,3819,3817,3835,3028,3018,3026,3036,
3828,3818,3826,3836,3437,3419,3417,3435

184

*elgen
257,15,18,1,l,1,1,3,800,80
273,15,18,1,1,1,1,3,800,80
289,15,18,1,1,1,l,3,800,80
307,8,18,1,1,1,l,3,800,80
317,3,18,1,1,1,1,3,800,80

*elgen

241,1,1, 1,1 ,1,1,3,800,80
242,1,1,1,1,1,1,3,800,80
243,1,1,1,1,1,1,3,800,80
244,1,1,1 ,1 ,1,1,3,800,80
245,1,l,1,1,1,1,3,800,80
246,1,1,1,1,1,1,3,800,80
247,1,1, l, 1,1,1,3,800,80
248 ,,1,1 l,1,1,1,3,800,80
249,1,1,1,1,1,1,3,800,80
250,1,1, 1,1, 1,1,3,800,80
251,1,1, 1, 1 ,1,1,3,800,80
252,1,1,1, 1 ,1,1,3,800,80
253,1,1,1,1,1,1,3,800,80
254,1,1,1,1,1,1,3,800,80
255,1,1,1,1,1,1,3,800,80
256,1,1,1,1,1,1,3,800,80
272,1,1,1,1,1,1,3,800,80
288,1,1,1, 1,1,1,3,800,80
304,1,1,1,1,1,1,3,800,80
320 ,,1,1 1,1,1,1,3,800,80
*node

5601,62.5e6,-22.96e-6
5602,65e-6,-22.96e-6
5603,67.5e6,-22.96e-6
5613,62.5e6,22.96e—6
5614,65e—6,22.96e-6
5615,67.5e-6,22.96e-6
*nseunset=j oinl , generate
5601,5603

‘nseunseFjoinZ, generate
5613,5615

*nﬁll

join],ioin2,4,3

I"node
6201,62.5e—6,-22.96e-6,75e-6
6202,65e6,-22.96e-6,75e-6
6203,67.5e—6,-22.96e-6,75e-6
6213,62.5e6,22.96e-6,75e-6

185

6214,65e-6,22.96e-6,75e-6
6215,67 .5e-6,22.96e-6,75e-6
*nset,nset=j oinlt,generate
6201,6203
*nset,nset=j oin2t,generate
6213,6215
*nﬁll
joinltj oin2t,4,3
*nset,nset=extrab, generate
5601,5615
*nset,nset=extrat,generate
6201,6215
*nﬁll
extrab,extrat,6, 100
*element,type=c3d20rp
48 l,91,5602,5608,109,891,5802,5808,909,5601,5605,5607,100,
5801,5805,5807,900,49 1 ,5702,5708,509
483,109,5608,5614,127,909,5808,5814,927,5607,5611,5613,118,
5807,58]1,5813,918,509,5708,5714,527
482,5602,3071 ,3053,5608,5802,3871,3853,5808,5603,3062,5609,5605,
5803,3862,5809,5805,5702,347 1 ,3453,5 708
484,5608,3053,3035,5614,5808,3853,3835,5814,5609,3044,5615,561l,
5809,3844,5815,5811,5708,3453,3435,5714
485,891,5802,5808,909,169l,6002,6008,1709,5801,5805,5807,900,
6001,6005,6007,1700,1291,5902,5908,1309
487,909,5808,5814,927,l709,6008,6014,1727,5807,58l1,5813,918,
6007,60]1,6013,]718,1309,5908,5914,1327
486,5802,387 1 ,3853,5808,6002,467 1 ,4653,6008,5803,3862,5809,5 805,
6003 ,4662,6009,6005,5 902,427] ,425 3,5 908
488,5808,3853,3835,5814,6008,4653,4635,6014,5809,3844,5815,5811,
6009,4644,6015,6011,5908,4253,4235,5914
489,1691,6002,6008,1709,2491,6202,6208,2509,6001,6005,6007,1700,
6201,6205,6207,2500,2091,6102,6108,2109
490,6002,467],4653,6008,6202,5471,5453,6208,6003,4662,6009,6005,
6203,5462,6209,6205,6102,5071,5053,6108
491,1709,6008,6014,1727,2509,6208,6214,2527,6007,601 1,6013, 1718,
6207,62]1,6213,2518,2109,6108,6114,2127
492,6008,4653,4635,6014,6208,5453,5435,6214,6009,4644,6015,601 1,
6209,5444,6215,6211,6108,5053,5035,6114
*node
6302,20e-6,-54.62e-6
6301,20e-6,-56.735e-6
6300,20e-6,-58.85e-6
6326,] 10e-6,-54.62e-6

6325,] 10e-6,-56.735e-6

186

6324,110e-6,-58.85e-6
6902,20e-6,-54.62e-6,75e-6
6901,20e-6,-56.735e-6,75e-6
6900,20e—6,-58.85e-6,75e—6
6926,] 10e-6,-54.62e-6,75e-6
6925,]10e-6,-56.735e-6,75e-6
6924,110e-6,-58.85e-6,75e-6
*nset,nset=1eftl,generate
6300,6302
*nset,nset=rightl,generate
6324,6326

*nﬁll

leftl,rightl,8,3
‘nset,nset=baseb,generate
6300,6326
*nsetnset=leftlt,generate
6900,6902
‘nsetmset=rightlt,generate
6924,6926

'nﬁll

leftlt,rightlt,8,3
*nset,nset=basebt,generate
6900,6926

*nﬁll

baseb,basebt,6, 1 00

*node

6329,20e-6,58.85e-6
6328,20e6,56.735e-6
6327,20e6,54.62e-6
6353,110e-6,58.85e-6
6352,110e-6,56.735e-6
6351,] 10e-6,54.62e-6
6929,20e6,58.85e-6,75e-6
6928,20e6,56.735e-6,75e-6
6927,20e6,54.62e—6,75e-6
6953,110e-6,58.85e-6,75e-6
6952,110e-6,56.735e—6,75e-6
6951,] 10e-6,54.62e-6,75e-6
*nset,nse1=leﬁ12,generate
6327,6329
*nset,nset=right12,generate
635 1,6353

*nﬁll

left12,right12,8,3
*nset,nset=baseb2,generate

187

6327,6353

*nset,nseFleftlt2, generate

6927,6929

*nset,nset=rightlt2,generate

6951,6953

*nﬁll

leftlt2,rightlt2,8,3

‘nset,nset=basebt2,generate

6927,6953

*nﬁll

baseb2,basebt2,6,100

*elementtype=c3d20rp
600,37,6300,6302,39,837,6500,6502,839,46,6301,48,38,846,6501,848,838,
437 ,6400,6402,439

606,837,6500,6502,839, 1637,6700,6702,1639,846,6501,848,838,]646,6701,1648, 1638,
1237,6600,6602,1239
612,1637,6700,6702,1639,2437,6900,6902,2439,1646,6701,1648,1638,
2446,6901 ,2448,2438,2037,6800,6802,2039

601 ,6300,6306,6308,6302,6500,6506,6508,6502,6303,6307,6305,6301 ,
6503,6507,6505,650 1 ,6400,6406,6408,6402
605,6324,2837,2839,6326,6524,3637,3639,6526,2828,2838,2830,6325,
3628,3638,3630,6525,6424,3237,3239,6426
611,6524,3637,3639,6526,6724,4437,4439,6726,3628,3638,3630,6525,
4428,4438,4430,6725,6624,4037,4039,6626
617,6724,4437,4439,6726,6924,5237,5239,6926,4428,4438,4430,6725,
5228,5238,5230,6925,6824,4837,4839,6826
618,183,6327,6329,181,983,6527,6529,981,174,6328,172,182,974,6528,972,982,
583,6427,6429,581

624,983,6527,6529,98 l, 1783,6727,6729,178 1,974,6528,972,982,1774,6728,1772, 1782,
1383,6627,6629, 1 381
630,1783,6727,6729,1781,2583,6927,6929,2581,1774,6728,1772,1782,
2574,6928,2572,2582,2183,6827,6829,2181
619,6327,6333,6335,6329,6527,6533,6535,6529,6330,6334,6332,6328,
6530,6534,6532,6528,6427,6433,6435,6429

623,6351,2983,298 1,6353,6551,3783,378 1,6553,2992,2982,2990,6352,
3792,3782,3790,6552,645],3383,3381,6453
629,6551,3783,3781,6553,6751,4583,4581,6753,3792,3782,3790,6552,
4592,4582,4590,6752,6651,4183,4181,6653
635,6751,4583,4581,6753,6951,5383,5381,6953,4592,4582,4590,6752,
5392,5382,5390,6952,6851,4983,4981,6853

*elgen

601,4,6,1,1,1,1,3,200,6

1619,4,6,1,1,1,1,3,200,6

*elset,elset=matrix,generate

1,48

188

69,72

8 1,128

149,152

161,208

229,232

241,288

3 17,320

321,368

397,400

401,448

477,480
*elset,elset=center,generate
481,492
*elset,e1set=epil,generate
65,66

75,80

225,226

235,240

145,146

155,160

*elset,els et=epir, generate
307,3 14

387,394

467,474
*elset,elset=epib,generate
600,617
*elset,elset=epit,generate
618,635

*elset,els et=ﬁbers, generate
49,64

129,144

209,224
*elset,elset=ﬁbersr,generate
289,304

369,384

449,464
*nset,nset=drain,generate
6300,6324,3

6500,6524,3

6700,6724,3

6900,6924,3

6329,6353,3

6529,6553,3

6729,6753,3

6929,6953,3

189

18 1,271,] 8

1,37, 1 8

98 1,1071, 1 8

801,837,] 8

1781,1871,18

1601,1637,18

2581,2671,18

2401 ,2437, 18

2837,2981,18

3637,3781,18

4437,458 1,1 8

5237,5381,18

‘nset,nset=topl, generate

2689,2737,1

2409,2688,9

2408,2687,9

2407,2686,9

2406,2685,9

2405 ,2684,9

2404,2683,9

2403,2682,9

*nset,nset=topright,generate

5489,5537,1

5209,5488,9

$208,548 7,9

5207,5486,9

5206,5485,9

5205,5484,9

5204,5483,9

5203,5482,9

*orientation,name=orientl ,deﬁnition=nodes,system=cylindrical
3 13,713

1,70
*orientation,name=orientr,deﬁnition=nodes,system=cylindrical
31 13,3913

1,70
*0rienta1ion,name=orimtel,deﬁnition=nodes,system=cylindrical
3 13,713

1,-10
*orientati0n,name=orienter,deﬁnition=nodes,system=cylindrical
31 13,3913

1,-10
*orientation,name=orienteb,deﬁnition=nodes,system=cylindrical
5608,6208

1 ,-10

190

*nset,nset=alln,generate
1,337

401,737

801,] 137

1201 , 1537

1601,] 937

2001,2337

2401,2737

2801,3137

3201,3537

3601,3937

4001,4337

4401,4737

4801,5137

5201,5537

5601,5615

5701,5715

5801,5815

5901,5915

6001,6015

6101,61 15

6201,6215

6300,6353

6400,6453

6500,6553

6600,6653

6700,6753

6800,6853

6900,6953
*nset,nset=botjoin,generate
5601,5615
*nset,nset=botside,generate
6300,6353
*nset,nset=topall,generate
2401,2737

5201,5537

6201,6215

6900,6953
*nsetmstholdgenerate
253,260

109,] 16

3053,3060

2909,2916
*elset,elset=leftpress,generate
165,166

191

161,161
169,169
164,164
171,171
173,174
187,192
177,178
203,208
193,194
219,224
209,210
235,240
225,226
155,160
145,146
75,80
65,66
1,1

5,6

9,9
11,11
4,4
13,14
27,32
17,18
43,48
33,34
59,64
43,50
75,80
65,66
139,144
129,130
81,81
85,86
89,89
91,91
84,84
93,94
107,112
97,98
123,128
113,114

*initial conditions, type=saturation

alln,l

' “‘5. M
' 4

‘4‘

*initial conditions,type=ratio
alln,2.3
it

“material properties for the ﬁbers
**

*solid section,elset=ﬁbers,material=collagen,orientation=orient1
*material,name=collagen

*elastic,type=engineering constants
3350,6700,3350,.49,0,.49,105,105,

105

*permeability,speciﬁc weight=1

5 .1 e- 1 3

‘solid section,elset=ﬁbersr,material=coll,orientation=orientr
*materialmame=coll

*elastic,type=engineering constants
3350,6700,3350,.49,0,.49,105,105,

105

‘permeabilityspeciﬁc weight=1

5. 1 e- 1 3

3*

"material properties for the epitenon layer

3*

*solid section,elset=epil,material=sheathl,orientation=orientel
*material,name=sheathl

*elastic,type=engineering constants
3350,6700,3350,.49,0,.49,105,105,

105

**elastic,type=isotropic

"105,.2

*permeability,speciﬁc weight=1

5.] e13

*solid section,elset=epir,material=sheathr,orientation=orienter
*material,name=sheathr

*elastic,type=engineering constants
3350,6700,3350,.49,0,.49,105,105,

105

**elastic,type=isotropic

"105,2

*permeability,speciﬁc weight=1

5.1e-13

*solid section,elsWepibmaterial=sheathb,orientation=orienteb
*material,name=sheathb

*elastic,type=engineering constants
3350,6700,3350,.49,0,.49,105,105,

105

193

 

**elastic,type=isotropic

**105,.2

*permeability,speciﬁc weight=1

5.1e-13

*solid section,elset=epit,material=sheatht,orientation=orienteb
*material,name=sheatht
*elastic,type=engineering constants
3350,6700,3350,.49,0,0.49,105,105,

105

**elastic,type=isotropic

"105,.2

*permeability,speciﬁc weight=1

5. 1 e-13

**

"material properties of the material between
"the fascicles:

**

*solid section,elset=center,material=hard
*material,name=hard
*elastic,type=isotropic

105,.2

*permeability,speciﬁc weight=1

5.1 e- l 3

**

“material props. for the "cores"
**

*solid section,elset=mat1ix,material=water
*mateﬁalmamewater
*elastic,type=isotropic

105,.2

*pemeability,speciﬁc weight=1
5.1 e-13

*bormdary

bottom,3

bottomr,3

botjoin,3

botside,3

5608,2

5608,]

5 61 1,1

56 14, 1

5605,]

5602,]

5607,2

hold,2

194

 

drain,8
*restart,write,frequency= 1
*step,nlgeom,inc=200
*soils,consolidation,utol=5e5
1 e6, .00 1
*boundary,type=displacement
topall,3,3,1.5e6

*endstep

*step,inc=200
*soils,consolidation,utol=5e5,end=ss
.001,180,.06,100,.001
*endstep

195

"111111111111117