Abstract—A multiaxial testing rig has been designed to investigate mechanical properties of soft tissue membranes. This approach has the advantage over biaxial loading in that it can be used to investigate soft tissue membranes with complex structural architecture. A finite element model of tissue mechanics has been used to analyze the experimental data in order to evaluate the stress-strain relationship, and a forward solve algorithm developed to estimate material parameters values for a given constitutive law. The multiaxial testing rig and the finite element analysis have been used to evaluate the constitutive properties of in-vivo human skin.

Keywords—Finite element modeling, in-vivo human skin, multiaxial experiments.

I. INTRODUCTION

Biaxial loading experiments are commonly used to investigate two-dimensional mechanical behavior of soft tissue membranes [1-5]. This technique is considered superior to other methods, such as indentation and uniaxial testing, as biaxial loading can investigate the anisotropic properties observed in most types of soft tissue. The anisotropy of soft tissue is linked to its microstructure and normally aligns with the predominant fiber direction [6]. This anisotropy does, however, limit the appropriateness of biaxial testing, as different tissues often lack simple uniform fiber orientations. Biaxial testing can only be applied to samples with homogenous fiber directions, which are aligned with the experimental loading axes. In order to investigate more complicated tissue architectures, a testing rig utilizing multiaxial loading of membranes has been developed. This approach has the advantage that samples can be loaded in multiple directions simultaneously, and removes the need to remount the sample in order to test along multiple axes. The multiaxial test does, however, present a complicated mechanical problem that cannot in general be analyzed analytically. A finite element analysis, with a mathematical representation of the geometry, has been coupled with the multiaxial testing protocol in order to analyze the experimental data. This analysis is used to evaluate the strain fields for inhomogeneous samples and to evaluate material parameter values for the constitutive stress-strain relationship.

II. METHODOLOGY

The multiaxial testing rig, Fig 1, is equipped with a circular array of sixteen displacement actuators (Physike Instruments), each with a travel range of 50 mm and a resolution of 0.1 µm. This arrangement allows deformations to be imposed along eight separate axes of the sample simultaneously. Each actuator is equipped with a custom build 2-D force transducer, with a range of 2 N and a resolution of 1 mN. These transducers measure the force vectors associated with each of the sixteen attachment points of the tissue sample. The geometry of the sample, and its deformations, are recorded using a high resolution CCD camera (Atmel). The actuator control, data acquisition from the force transducers, and the image acquisition from the camera are all performed using an integrated software system developed with LabView (National Instruments).This software supports actuator control in either position or force feedback modes, allowing deformation states to be specified in terms of stress or strain.

Fig 1: The multiaxial testing rig.

Investigating Stress-Strain Properties of in-vivo Human Skin using Multiaxial Loading Experiments and Finite Element Modeling.

Y. A. Kvistedal, P. M. F. NielsenBioengineering Institute, University of Auckland, Auckland, New Zealand

The deformations in the tissue are traced between different deformation states by performing a two-dimensional cross-correlation on subsets of the images [7]. This technique traces the displacements of small regions, typically 64x64 pixels, with an accuracy of up to one twentieth of a pixel. The accuracy of the cross-correlation relies on high-contrast high-frequency information within the images, which is provided by staining the tissue sample with fine-grain carbon powder before each experiment. Data from experiments are combined in a finite element model, to analyze the sample deformations. Fig 2 illustrates the geometry represented by a circular mesh of 192 bi-linear elements. The displacements of the nodal positions of a deformed state are used to illustrate the deformation of the geometry upon applying loading at the sixteen attachment nodes. An addition 800 internal data points are traced using the cross-correlation technique. The displacements of these are used for geometrical fitting when using the experimental data to estimate the material parameters of constitutive laws. Strain fields for both of the principal strain components can be calculated from the deformations of the finite element mesh. This is particularly useful when performing experiments on inhomogeneous samples, such as scar tissue, as local internal variations in stiffness can be quantitatively analyzed [8]. Inhomogeneities in the strain field have previously been reported for homogenous tissue under biaxial loading [5]. This effect is illustrated in Fig 3 where a large increase in strain is observed around the sixteen attachment points. The implications of these

inhomogeneities are that there will be errors in the constitutive stress-strain relationship if position vs. force data from the boundary of biaxial experiments are used to calculate the material parameters. This approach relies on the assumptions of homogenous strain fields. A technique has been developed to estimate material parameters of constitutive laws using a forward solve algorithm. A typical experiment has multiple loading cycles, where each consists of up to fifteen deformation states with associated boundary forces. With an initial estimate of the material parameters, the constitutive model initially solves the forward problem for each deformed state of a loading cycle, by applying the recorded boundary forces to the undeformed geometry. The algorithm then proceeds by applying small changes to each of the material parameters, and for each change solving the forward problem for all of the deformed states. For each solution, geometric residuals are evaluated between the simulated deformations and the experimentally determined displacements at each of the data point. As comparisons are made between the displacements at each deformed state of a loading cycle, a total of ~10,000 residuals are used for the optimization. Once the geometric residuals are obtained, a non-linear, least square, optimization algorithm evaluates new estimates of the material parameters in the constitutive law. The complete forward solve procedure is repeated until the parameter estimates satisfy a specified optimizer criterion, which indicates that an optimal solution is found.

Fig 3: A strain field showing the first principal strain component, on a gray scale from 0.4 to 0.8, in in-vivo human skin subject to mutiaxial loading. Inhomogeneities in the strain field are observed around the sixteen attachment nodes.

Fig 2: The finite element model of the test sample containing all data froma single deformed state of an experiment on in-vivo human skin. The lightgrey mesh represents the undeformed geometry. The deformationsoccurring when the sample is subject to loading, indicated by the forcevector arrows, are represented by the black mesh. The dots indicate the datapoints used to evaluate geometrical residuals during estimations of theconstitutive stress-strain relationship.

III. RESULTS

Experiments have successfully been performed on inhomogeneous rubber membranes, sheep diaphragm and in-vivo human skin. Fig 4 shows stress-strain curves generated using an exponential strain energy constitutive law [9]. The material parameters used were estimated from experiments on human skin for three individuals. Differences in the mechanical behavior of human skin have previously been reported to be dependent on body location and subject age [3, 4]. The experiments in Fig 4 were performed at identical body locations of age and gender matched subject, illustrating that differences in mechanical behavior of skin also exists between similar individuals.

IV. DISCUSSION

The combination of multiaxial loading experiments and finite element analysis is shown to be successful for investigating the mechanical properties of homogenous and inhomogenous soft tissue membranes. Estimates of material parameter values for a constitutive law describing the stress-strain properties of human skin have been obtained from multiple experiments. The variations between the estimated parameter values indicate that differences of mechanical properties exist between similar individuals. In order to accurately describe the mechanical behavior of most types of soft tissue, a constitutive model has to account for the anisotropic and non-linear stress-strain properties. There are many constitutive laws presented in the literature that are capable of modeling these properties in different soft tissue types [9-11]. Most soft tissues does, however, also exhibit strong viscoelastic behavior showing

stress relaxation and creep, and a constitutive law will not be complete without incorporating these time dependent effects. The multiaxial testing rig has successfully obtained viscoelastic measurements. However, only quasi-static deformation data has currently been modeled, due to a lack of constitutive laws capable of modeling this behavior. A limitation of the current multiaxial testing rig is that it can only obtain data in 2-D. The finite element analysis has thus correspondingly modeled the tissue as a 2-D membrane using plain stress theory. In order to improve the model accuracy further, more detailed information of the tissue microstructure in the third dimension is required. We are currently investigating combining multiaxial loading with optical coherence tomography, as a method for determining this information.

ACKNOWLEDGMENT

The authors would like to thank Duane Malcolm, Poul Charette, Sharif Malak, Rob Kirton, Martyn Nash, Kevin Augenstein, Peter Hunter, and David Budgett for help and contributions to the development and testing of the multiaxial rig and the finite element analysis techniques.

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Fig 4: The stress-strain relationship generated from a constitutive modelusing material parameter values estimated from in-vivo experiments onhuman skin, of three individuals.

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