PAMM · Proc. Appl. Math. Mech. 9, 141 – 142 (2009) / DOI 10.1002/pamm.200910045

A viscoelastic-diffusion model for cartilage replacement material

Marcus Stoffel1,∗, Dieter Weichert1, and Ralf Müller-Rath2

1 Institut für Allgemeine Mechanik, RWTH Aachen2 Klinik für Orthopädie und Unfallchirurgie, RWTH Aachen

In order to predict deformations and internal stresses of articular cartilage replacement material, two viscoelastic diffusionmodels are proposed in the present study. Also, the remodeling effect of the material seeded with human cells is verifiedexperimentally.

c© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

In this investigation cellular condensed collagen gel is used to develop a soft tissue that may in future replace human articularcartilage. Our goal is to propose a constitutive model in order to predict stresses and strains of the acellular basic material. Forthis reason, a material model is developed, accounting for viscoelasticity and diffusion. In the latter application as replacementtissue especially the damping property of the investigated material is of great importance. Therefore, two diffusion modelsare presented, aiming a most precise prediction of the relaxation behaviour. A material parameter identification process bymeans of tension tests is applied and the simulated results are validated by compression experiments. A second issue of thisstudy is the verification of the remodeling effect in the cellular replacement material. For this purpose, a bioreactor has beendeveloped which is used for cyclic stimulations of specimens seeded with chondrocytes. Biological evaluations about theamount of pronounced cell activities are carried out by means of histological sections.

2 Constitutive model for acellular material

The deformation state is expressed in the Eulerian description for finite strains. Here, we decompose the time derivative of theCauchy stress tensor in the following three parts:

σ̇ij = Cijklε̇kl + C̃ijkl (ε̇) + σ̇dij . (1)

The first term on the right hand side of Eq. 1 covers the pure elastic material behaviour by means of Hooke’s law with a fourth-order stiffness tensor connected with the strain tensor in rate form. The second term accounts for the viscoelastic behaviour ofthe material represented by a stiffness tensor depending on the strain rate tensor.In the present study, we focuse on the third term standing for the diffusion behaviour of the soft tissue. In order to obtain amost realistic prediction of the stress rate tensor σ̇ij two different types of evolution equations are proposed. The first one isabbreviated with VED1 model in its form

σ̇ij = −Dσij (2)

with a diffusion parameter D to be defined later. This equation leads in the one dimensional case to an exponential function

σ11 = Ae−Dt (3)

with time t in the exponent.The second type of evolution equation for the diffusion behaviour is the VED2 model in the form

σ̇ij =12D2 σij

ln

[σij

sup(√

σijσij

)] (4)

leading in the one dimensional case to the expression

σ11 = Ae−D√

t (5)

∗ Corresponding author E-mail: stoffel@iam.rwth-aachen.de, Phone: +49 241 80 94589, Fax: +49 241 80 92231

c© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

142 Short Communications 2: Biomechanics

given by Betten [1] and extended in the present study for three dimensional analysis.Due to the propagation of fluid through the solid material of the soft tissue the amount of liquid fraction and, hence, the damp-ing property can vary during a deformation. For this reason, a deformation dependent diffusion coefficient D is introducedas

D (εv) = D0 − D1|εv| (6)

with D0 and D1 as diffusion constants. Eq. 6 is used for compression states, only, accounting for the loss of liquid. Forpositive values of the volume strain εv , the parameter D is kept constant.Further more, it is considered in this investigation that the Young’smodulus can vary through the thickness of the material due to fabri-cation influences. In the core a higher liquid concentration is presentthan in the outer zones which is taken into account in the constitutivemodel by defining a gradient material. This part of the modelling aswell as the finite element approximation are described in detail in [2].The material parameters are identified by uni-axial tension tests atdifferent strain rates and with relaxation states [2]. In order to val-idate the proposed material model compression tests with cylindicalspecimens are carried out and the measurements are compared to sim-ulation results shown in Fig. 1. The measured force, recorded by aload cell, are presented together with two simulated force evolutionsusing VED1 and VED2 models. Both types of diffusion models leadto a good prediction of the measurement. However, the simulationusing the VED2 model predicts the measured force nearly identical.

Time / s0 2 4 6 8 10 12

For

ce /

N

-15

-10

-5

0

Dis

plac

emen

t / m

m

-1.5

-1.0

-0.5

Measured forceCalculated force with VED1Calculated force with VED2Measured displacement

Fig. 1 Compression test and simulations

3 Experiments with cellular material

Based on this mechanical modelling of the acel-lular basis material, the remodeling effect can beverified in a next step by means of a bioreactor.In Fig. 2 the experimental set-up is shown pro-ducing cyclic stimulations on a cellular specimenby an unbalanced mass. During the experimenta nutrient medium and a gas exchange are pro-vided around the specimen. The force acting onthe sample is recorded by a load cell and the num-ber of cycles are registered by a forked light bar-rier. All experiments must be prepared under ster-ile conditions and are carried out in an incubatorat constant temperature of 37oC. The biologicalevaluation is performed by means of histologicalsections as shown in Fig. 4, where a small part ofthe cross section area of a compression specimenis shown after 400000 cycles.

Unbalanced mass

Load cell

Cellular specimen

EngineForked light barrier

Fig. 2 Bioreactor

100 µm

Developed protein

Fig. 3 Histological section

Due to the mechanical stimulation additional proteins have been produced which did not occur in unstimulated referencesamples. This experimental result can now be intensified by using higher cell concentrations and higher number of loadingcycles.

4 Discussion

The present study showed that the deformations and internal forces of condensed collagen samples could be predicted bythe applied VED models. Moreover, it was verified that a remodeling occurs in cellular specimens which were stimulatedmechanically. This gives the opportunity to extend this investigation by including the change of the internal structure in theconstitutive modelling.

References

[1] J. Betten, Creep Mechanics, Springer-Verlag, 2005.[2] M. Stoffel, D. Weichert, R. Müller-Rath, Modeling of articular cartilage replacement materials, Arch. Mech, 61, 1, 1-19, 2009.

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