Finite element modeling of human skin using an isotropic, nonlinear elastic constitutive model

Abstract

The collagen network in skin is largely responsible for the nonlinear mechanical stress-strain response of skin. We hypothesize that the force-stretch response of collagen is governed by the entropics of long-chain molecules. We show that a constitutive model derived from the statistical mechanics of long-chain molecules, corresponding to the fibrous collagen network in skin, captures the mechanical response of skin. A connection between the physiologically meaningful parameters of network molecular chain density and free length of collagen fibers and the constitutively significant parameters of initial modulus and limiting stretch is thus established. The relevant constitutive law is shown to have predictive capabilities related to skin histology by replicating in vivo and in vitro experimental results. From finite element simulations, this modeling approach predicts that the collagen network in hypertrophic scars is more dense and the constituent collagen fibers have shorter free lengths than in healthy skin. Additionally, the model is shown to predict that as rat skin ages, collagen network density increases and fiber free length decreases. The importance of knowledge of the in situ stress state for analyzing skin response and validating constitutive laws is also demonstrated.

Introduction

Human skin is a heterogeneous material composed of collagen and elastin fibers in a proteoglycan matrix. Acting together, these components are responsible for the mechanical behavior of skin as manifested in stress–strain curves. These curves are characterized by a low-stiffness region at small strains followed by a dramatic increase in stiffness as the strain becomes large (Payne, 1991). It is generally accepted that during the initial, low-stiffness region, the collagen fibers align themselves parallel to the maximum stretch direction while either the elastin (Clark et al., 1996; Sanders et al., 1995; Daly, 1982) and/or the proteoglycan matrix (Pereira et al., 1991) provides resistance to deformation. Once the collagen fibers are sufficiently aligned, further extension of the skin requires extension of the collagen fibers, resulting in a significant increase in skin stiffness.

The most popular instrument for determining the mechanical properties of skin in vivo has been the extensometer, characterized by two tabs glued to the skin and driven apart from each other. The extensometer has been used both for studying the directional properties of skin (Ferguson and Barbenel, 1981; Alexander and Cook, 1977; Stark, 1977) as well as for clinical evaluation of healthy and diseased skin (Clark 1996, Clark 1987; Leung et al., 1984; Burlin et al., 1977). Various researchers have used in vitro tests to determine properties of healthy (Oxlund et al., 1988; Lanir and Fung, 1974) and burned (Dunn et al., 1985) skin.

Attempts to model the mechanical behavior of skin have focused on specific behavioral aspects of the tissue, such as viscoelasticity (Barbenel and Evans, 1977; Christensen et al., 1977) or nonlinear elasticity (Alexander and Cook, 1977). Quasistatic viscoelastic effects are only appreciable at nonphysiologic stress levels however (Daly, 1982; Wilkes et al., 1973). Alexander and Cook (1977) used a phenomenological nonlinear strain energy function to model skin, but their results did not speak to the histologic basis for the observed behavior. Others have focused on developing microstructurally based models to explain specific traits of the bulk tissue response directly attributable to one component of the tissue, such as the collagen fibers (Haut and Little, 1972). However, this model is applicable to single collagen fibers, and the constitutive law for individual fibers has not yet been incorporated into a network formulation to model bulk tissue behavior.

We hypothesize that since the mechanical behavior of skin at large stretches is dominated by the macromolecular collagen network, a constitutive model based on the entropy change upon stretching of long-chain molecules using physiologically meaningful parameters to represent the collagen network in skin accurately models the elastic behavior of skin. The recent success of statistical mechanics models to describe stretching of DNA (Marko and Siggia, 1995) motivates this hypothesis. This modeling approach is fully compliant with the rubrics of continuum mechanics and results in a specific form for the strain energy density. Consequently, by incorporating this constitutive law into a finite element simulation, results from various experiments can be compared, and changes in a model parameter associated with a skin property such as collagen density predict a change in skin histology between two samples. Additionally, we hypothesize that anisotropic behavior of skin is modeled using an initially isotropic constitutive law that develops anisotropy in the presence of an anisotropic stress state.