An in silico biomechanical analysis of the stent–esophagus interaction

Abstract

Despite all technological innovations in esophageal stent design over the past 20 years, the association between the stent design’s mechanical behavior and its effect on the clinical outcome has not yet been thoroughly explored. A parametric numerical model of a commercially available esophageal bioresorbable polymeric braided wire stent is set up, accounting for stent design aspects such as braiding angle, strut material, wire thickness, degradation and friction between the wires comprising a predictive tool on the device’s mechanical behavior. Combining this tool with complex multilayered numerical models of the pathological in vivo stressed, actively contracting and buckling esophagus could provide clinicians and engineers with a patient-specific window into the mechanical aspects of stent-based esophageal intervention. This study integrates device and soft tissue mechanics in one computational framework to potentially aid in the understanding of the occurrence of specific symptoms and complications after stent placement.

Appendices

Appendix 1: Esophageal histology

The esophageal wall is composed of four well-defined layers: the mucosa, the submucosa, the muscularis externa and the adventitia (Fig. 8) (Gregersen 2003; Gregersen et al. 2000). The mucosal layer is build up from connective tissue and can be subdivided in the epithelium, the lamina propria and the muscularis mucosa. Contrary to the mucosa, where very fine collagen fibrils are loosely and randomly organized, collagen fibrils in the submucosa are organized in thick aligned fibers running helically down the esophagus (Natali et al. 2009). Due to this difference in structure, the submucosa plays a stronger mechanical role in offering resistance to the deformation of the esophageal wall than the mucosa does. The submucosa is loosely interconnected (Sommer et al. 2013) with the muscularis externa which is mainly composed of smooth and striated muscle cells (the transition from striated to smooth muscle is spread throughout the middle third of the esophagus). This muscle layer can be subdivided in two layers, the inner one having muscle cells arranged in the circumferential and the outer one arranged in the axial direction. The muscle layer is surrounded by a layer of soft connective tissue, the adventitia, which supports the flexibility of the esophagus and its integration in the thorax (Gregersen 2003). In this study, the esophagus was modeled as a five-layered system consisting of the mucosa (M), the collagen-rich submucosa (SM), a virtual interfacial layer (IF) and the inner and the outer muscle layer of the muscularis externa (ME). The interfacial layer was introduced to incorporate the loose interconnection between the submucosa and the muscularis externa into our models.

Fig. 8

Image credit: biophoto associates/science photo library

Light micrograph of a cross section through the human esophagus depicting the mucosa, submucosa, the loose connective tissue in between the submucosa and the muscle layers (modeled as the interfacial layer in the finite element models), the circular and the longitudinal muscle layers.

Appendix 2: Constitutive modeling

The esophagus exhibits, as most other biologic soft tissues, a large deformability, together with nonlinear, pseudo-elastic and anisotropic material properties due to its aforementioned layer-specific composition (Fan et al. 2004; Yang et al. 2006a). Constitutive models are capable of characterizing this mechanical behavior through a functional relation between stresses and strains (Famaey et al. 2013). Following Fung et al., anisotropic hyperelastic material properties can be adequately represented constitutively by means of a strain energy function (SEF) (Fung 1993). The general material model applied in this study describes active esophageal tissue. Active contraction of the smooth (and striated) muscle cells is the most important feature of the peristaltic movement of the esophagus. In this model, the muscle cells were modeled to be oriented with their longest axes along the contractile direction and evenly distributed along the depth and height of the muscle tissue (Murtada et al. 2010). The contractile unit consists of two thin filaments (actin) and one thick filament (myosin) in a side-polar structure with cross-bridges attaching these filaments. These units are modeled to be arranged in series, separated by dense bodies forming long contractile fibers inside the muscle tissue (see Fig. 2, middle). In addition, they are assumed to contract uniformly and acting as a single unit. The three element Hill model has been developed to handle this muscle behavior in more detail (Hill 1970). The model considers the tissue as a composite of a ’contractile element’ connected with a ‘series elastic element’ to describe the active contraction of the muscle, and a ’parallel element’ describing the connective tissue (Gregersen et al. 2000). These contractile elements behave similarly in smooth muscle and striated muscle. The proposed constitutive model was additively decomposed in three constituents: an isotropic matrix material constituent, an anisotropic constituent attributed to the dispersion of collagen fibers in the tissue and an anisotropic smooth cell constituent. Due to the esophagus’s composition and our choice for a five-layered model, the three constituents will never contribute simultaneously in a specific layer. Several research groups have modeled the matrix and fiber constituent through phenomenological (Sokolis 2010; Stavropoulou et al. 2009) or microstructurally motivated (Sommer et al. 2013; Yang et al. 2006c) constitutive descriptors. In this study, we chose to model the passive deformational response of the intact esophagus based upon the pioneering work of Gasser et al. in arterial tissue mechanics (Gasser et al. 2006). The third, active contraction, constituent was motivated by Murtada et al.’s model of mechanical smooth-muscle activation (Murtada et al. 2010). Famaey et al. (2013) similarly combined Gasser’s passive and Murtada’s active material model in a cardiovascular setting, studying the stiffness degradation and identifying the critical loading regimes during arterial clamping.

1.1 A.2.1 Kinematics

To set forth our material model, we first must introduce some key kinematic quantities. For soft tissues undergoing large deformation, the deformation is typically quantified through the deformation gradient ${\varvec{F}}$, that is the gradient of the deformation map $\varvec{\chi }$ with respect to the undeformed position ${\varvec{X}}$.

$$\begin{aligned} {\varvec{F}}=\frac{\partial {\varvec{x}}}{\partial {\varvec{X}}}=\frac{\partial \varvec{\chi } \left( {\varvec{X}} \right) }{\partial {\varvec{X}}} \end{aligned}$$

The local volume ratio of the material, which is close to 1 for nearly incompressible materials, is defined by the Jacobian J

$$\begin{aligned} J=\hbox {det}\left( {\varvec{F}} \right) \end{aligned}$$

The deformation gradient ${\varvec{F}}$ can be decomposed multiplicatively into a spherical (dilatational) part $J^{1/3}{\varvec{I}}$ and a unimodular (distortional) part $\bar{{\varvec{F}}}$, so that $\det \left( {\bar{{\varvec{F}}}} \right) =1$.

The deformation of the material will be defined in terms of the invariants of the right and left Cauchy–Green tensors, denoted C and b, respectively, and their isochoric counterparts, denoted $\bar{{\varvec{C}}}$ and $\bar{{\varvec{b}}}$, respectively, associated with $\bar{{\varvec{F}}}$.

With $\lambda {_r}$, $\lambda {_\theta }$ and $\lambda {_z}$ being the principal distortional stretches in the radial, circumferential and axial direction, the invariants describing the deformation of the isotropic matrix material can be expressed as follows:

$$\begin{aligned} \bar{I}_{1}= & {} \lambda {_r} ^{2}+\lambda _\theta ^{2}+\lambda _z ^{2} \\ \bar{I}_{2}= & {} \frac{1}{2}\left[ \bar{I}_{1}^{2}-\left( {\lambda _r^{4}+\lambda _\theta ^{4}+\lambda _z ^{4}} \right) \right] \\ \bar{I}_{3}= & {} \lambda _r ^{2}\lambda _\theta ^{2}\lambda _z ^{2} \end{aligned}$$

while the stretches along the fiber or muscle cells direction are described as follows:

$$\begin{aligned} \bar{I}_{4} ^\mathrm{fib}= & {} \lambda _\theta ^{2}\mathrm{cos}^{2}\alpha ^\mathrm{fib1}+\lambda _z ^{2}\mathrm{sin}^{2}\alpha ^\mathrm{fib1} \\ \bar{I}_6 ^\mathrm{{fib}}= & {} \lambda _\theta ^{2}\mathrm{cos}^{2}\alpha ^\mathrm{fib2}+\lambda _z ^{2}\mathrm{sin}^{2}\alpha ^\mathrm{fib2} \\ \bar{I} ^\mathrm{smc}= & {} \lambda _\theta ^{2}\mathrm{cos}^{2}\alpha ^\mathrm{smc}+\lambda _z ^{2}\mathrm{sin}^{2}\alpha ^\mathrm{smc} \end{aligned}$$

with $\alpha ^\mathrm{fib1}, \alpha ^\mathrm{fib2}, \alpha ^\mathrm{smc}$ the angle between the first fiber, the second fiber, the smooth/striated muscle cell, respectively, and the circumferential direction.

To account for the dispersion of collagen fibers along their averaged direction, Gasser et al. (2006) proposed the use of the following pseudo-invariants

$$\begin{aligned} \begin{array}{l} \bar{I}_{4} ^\mathrm{fib{\prime }}=\kappa \bar{I}_{1} +\left( {1-3\kappa } \right) \bar{I}_{4} ^\mathrm{{fib}} \\ \bar{I}_{6} ^\mathrm{fib{\prime }}=\kappa \bar{I}_{1} +\left( {1-3\kappa } \right) \bar{I}_{6} ^\mathrm{{fib}} \\ \end{array} \end{aligned}$$

in which $\kappa$ is a structure parameter representing the fiber distribution in an integral sense and describes its ‘degree of anisotropy’ ($\kappa =0$ corresponds to an anisotropic non-dispersed fiber state, while $\kappa =1/3$ corresponds to the totally dispersed fiber, and thus isotropic, state). This parameter should be determined from histological data.

1.2 A.2.2 Strain energy function

As the esophageal tissue can be considered nearly incompressible due to the high liquid content of the ground substance (Gregersen et al. 2000; Natali et al. 2009), we additively decomposed the SEF $\Psi$ in a volumetric $\Psi ^{vol}$ and a deviatoric $\Psi ^\mathrm{dev}$ part.

$$\begin{aligned} \Psi =\Psi ^\mathrm{vol}+\Psi ^\mathrm{dev} \end{aligned}$$

The deviatoric part was additively decomposed in the isotropic contribution of the matrix material $\Psi ^\mathrm{mat}$, the anisotropic contribution of two families of collagen fibers $\Psi ^\mathrm{fib1}$ and $\Psi ^\mathrm{fib2}$, and the contribution of the smooth/striated muscle cells $\Psi ^\mathrm{smc}$.

$$\begin{aligned} \Psi ^\mathrm{dev}=\Psi ^\mathrm{mat}+\Psi ^\mathrm{fib1}+\Psi ^\mathrm{fib2}+\Psi ^\mathrm{smc} \end{aligned}$$

1.2.1 A.2.2.1 Volumetric bulk material

The volumetric free energy $\Psi ^\mathrm{vol}$ was characterized by the hyperelastic constitutive Arruda–Boyce model, which can be expressed as:

$$\begin{aligned} \Psi ^\mathrm{{vol}}=\frac{K_0 }{2}\left( {\frac{J^{2}-1}{2}-\hbox {ln}\left( J \right) } \right) \end{aligned}$$

with $K_{0}$ being the bulk modulus. To ensure near-incompressibility, the bulk modulus had to be set high enough to ensure the ratio of the bulk modulus to the initial shear modulus $\frac{K_{0}}{\mu _{0}}$ is large enough. In this study, this ratio was chosen to be equal to 20.

1.2.2 A.2.2.1 Non-collagenous ground matrix

The extracellular matrix is characterized by means of an incompressible neo-Hookean model, i.e.,

$$\begin{aligned} \Psi ^\mathrm{{mat}}=\frac{1}{2}c\left( {\bar{I}_{1}-3 }\right) \end{aligned}$$

where $\bar{I}_{1}$ denotes the previously defined first invariant of the Cauchy–Green tensor, and c the neo-Hookean parameter, which characterizes the matrix’s stiffness through its initial shear modulus ($c=\mu _{0})$. The neo-Hookean material model is a special case of the polynomial hyperelastic material model (also called generalized Rivlin model).

1.2.3 A.2.2.3 Collagen fibers

The collagen fibers have no strength under compression and can thus only contribute to the SEF when under tension. Following Gasser et al. (2006), we particularized their transversely isotropic free energy function for the $i\mathrm{th}$ family of collagen fibers as follows:

$$\begin{aligned} \Psi ^\mathrm{fib_i }=\frac{k_1 }{2k_2 }\left[ {\mathrm{exp}\left( {k_2 \left[ {\bar{I}_i ^\mathrm{fib^{\prime }}-1} \right] ^{2}-1} \right) } \right] \end{aligned}$$

in which $k_2 >0$ is a dimensionless parameter and $k_1 >0$ is a stress-like parameter to be determined from mechanical tests of the tissue.

1.2.4 A.2.2.4 Smooth and striated muscle cells

The muscle cells form an integral part of the matrix constituent, both in their passive and active state. Murtada et al. developed a mechanochemical model describing muscle contraction and relaxation (Murtada et al. 2010). This model is based on the sliding filament theory first described by Huxley (1957). The theory states how the muscle contraction is the result of the relative sliding between thin actin and thick myosin filaments, which is accomplished by configurational changes in the cross-bridges between these filaments. These configurational cross-bridges changes are modeled through a four-state model in which cycling between these states is governed by chemically catalyzed (de)phosphorylation of the myosin heads. The rate parameters of transition between these states may differ significantly depending on the method of activation and the muscle type. For esophageal muscle tissue, no isometric and/or isotonic contraction experiments are available (ex infra- Sect. A.2.5), so we simplified the model by assuming all the myosin heads to be in their force generating state for this model. This lead to the following strain energy component:

$$\begin{aligned} {\Psi ^\mathrm{smc}}=\frac{1}{2}\mu _\mathrm{smc} \left[ {\sqrt{\bar{I}^\mathrm{{smc}}}+u_\mathrm{{rs}} -1 }\right] ^{2} \end{aligned}$$

where $\mu _\mathrm{{smc}}$ characterized the stiffness of the actin-myosin filament apparatus and $u_\mathrm{{rs}}$ stands for the averaged normalized relative sliding between both filaments. Due to the strong hyperelasticity of the esophageal muscle tissue (ex infra), a quadratic degree did not suffice to phenomenologically describe the tissue. A quartic degree proved to be more appropriate (ex infra):

$$\begin{aligned} {\Psi ^\mathrm{{smc}}}=\frac{1}{2}\mu _\mathrm{{smc}} \left[ {\sqrt{\bar{I} ^\mathrm{{smc}}}+u_\mathrm{{rs}} -1} \right] ^{4} \end{aligned}$$

1.2.5 A.2.2.5 Layer-specific equations and material properties

The full three-constituent material model summarized to:

$$\begin{aligned} \Psi =\Psi ^\mathrm{{vol}}+\Psi ^\mathrm{{dev}}=\Psi ^\mathrm{{vol}}+\Psi ^\mathrm{{mat}}+\Psi ^\mathrm{fib1}+\Psi ^\mathrm{fib2}+\Psi ^\mathrm{{smc}} \end{aligned}$$

For the mucosa (M) with its loosely and randomly thin collagen fibrils, the full material model was simplified to isotropic hyperelastic behavior. The same behavior was used for the interfacial layer (IF) between the submucosa and the muscle tissue.

$$\begin{aligned} \Psi ^{\textit{M}}=\Psi ^\mathrm{{IF}}=\Psi ^\mathrm{{vol}}+\Psi ^\mathrm{{mat}} \end{aligned}$$

The submucosa with its thick aligned fibers running helically down the esophagus was modeled as follows:

$$\begin{aligned} \Psi ^\mathrm{{SM}}=\Psi ^\mathrm{{vol}}+\Psi ^\mathrm{{mat}}+\Psi ^\mathrm{fib1}+\Psi ^\mathrm{fib2} \end{aligned}$$

Both muscle layers consist fully out of smooth/striated muscle cells and no cumulative collagen fibers are present, which lead to:

$$\begin{aligned} \Psi ^\mathrm{{ME}}=\Psi ^\mathrm{{vol}}+\Psi ^\mathrm{{mat}}+\Psi ^\mathrm{{smc}} \end{aligned}$$

Table 3 Fitted constitutive parameters

Fig. 9

Comparison of averaged experimental results on ovine tissue samples (extracted from Sommer et al. (2013)) and the constitutive material model fit

1.3 A.2.3 Evaluation of the constitutive parameters

The material parameters introduced by the outlined constitutive model need to be calibrated based on specific experimental setups. As we did not dispose of any esophageal tissue, we had to rely on literature to pinpoint the quantitative values of these parameters. Several studies have focused on characterizing the passive material properties of each layer of the esophageal wall (Fan et al. 2004; Liao et al. 2003; Natali et al. 2009; Sokolis 2010, 2013; Sommer et al. 2013; Stavropoulou et al. 2009; Yang et al. 2006b, c). Unfortunately, these articles are not conclusive on layer-specific mechanical behavior. While Liao et al. (2003), Fan et al. (2004) and Yang et al. (2006a) concluded that the muscle tissue is stiffer than the (sub)mucosal tissue, the opposite was concluded in Stavropoulou et al. (2009), whereas Yang et al. (2006b) and Sommer et al. (2013) stated that the mucosa–submucosa is initially more compliant and becomes stiffer than the muscle layer at increasing pressures. These discrepancies can be caused by tissue samples originating from different species (where different ratios in layer thicknesses might be seen), differences in chemical tissue preparation to ensure passive relaxed muscle behavior and the experimental protocol using only uniaxial or multiaxial data. Regarding the latter, only a fraction of these studies (Liao et al. 2003; Sokolis 2010, 2013; Stavropoulou et al. 2009) focused on the characterization of the multiaxial passive behavior of esophageal tissue, mostly based on extension–inflation tests. Sommer et al. (2013) investigated the heterogeneity of the multilayered composite esophageal structure by performing (layer-specific) uniaxial tensile, biaxial tensile and extension–inflation tests on ovine esophagi. We chose to calibrate the constitutive model based on this study for two reasons. First, Sommer et al.’s experimental results constitute the most complete multiaxial data set available up to date. Second, the mucosa–submucosa was found to be initially more compliant and become stiffer at increasing pressures, similar to the results described in the only available study on porcine data (Yang et al. 2006b), which is often considered as the best mechanical surrogate for human tissue. The constitutive parameters for each layer of our model were calibrated in a nonlinear least squares fitting procedure in which the constitutive biaxial response of the model was compared to the ex vivo results. As Sommer et al. studied the combined mucosa–submucosa layer and the full double-layered muscle layer, some assumptions had to be made while fitting the constitutive parameters to their averaged experimental results. Focusing on the mucosa and submucosa, the esophageal histology (“Appendix 1”) allowed us to assume a mechanically similar non-collagenous ground matrix for both these layers. All stiffening of the tissue based on the recruitment of the anisotropic distributed embedded wavy collagen fibrils was assumed to be attributed to the submucosa’s collagen content only. For the virtual interfacial layer introduced to incorporate the loose connective tissue connection between the submucosa and the muscularis externa, we assumed isotropic hyperelastic behavior one order of magnitude softer than the mucosal layer. Both the circumferential and longitudinal muscle layers were assumed to mechanically behave identically except for their smooth muscle cell orientations. To fit the material parameters to the experimentally measured passive strength of the full (two-layered) muscularis externa slab, we assumed the muscularis externa to behave as a single layer with their smooth muscle cells oriented per Sommer et al.’s fitted ‘collagen fiber’ orientations, leading to an angle of $55.3^{\circ }$ with respect to the circumferential direction. The fitted constitutive material parameters can be found in Table 3, and the fit is depicted in Fig. 9.

As can be seen in Fig. 9, a second-degree muscle cell invariant dependency of the strain energy function proved inadequate to describe the strong hyperelastic behavior of the muscle cells. The aforementioned quartic muscle cell invariant dependency strain energy potential model resulted in a better fit.

1.4 A.2.4 FEA implementation

The outlined constitutive model was implemented in the Abaqus user subroutines UMAT and VUMAT, a family of implicit (Abaqus-Standard) and explicit (Abaqus-Explicit) subroutines, respectively, designed to define a material’s mechanical behavior, in which the stress state in the local material orientations gets updated at the end of the increment for which it is called. In the implicit subroutine, the material Jacobian matrix $\frac{\partial \sigma }{\partial \epsilon }$ also needs to be fed back to the FEA solver at the end of each increment (Simulia Corp 2014).

1.5 A.2.5 Limitations of the constitutive model

Although the used dataset (Sommer et al. 2013) for the constitutive material fit comprises, to the authors’ knowledge, the most complete multiaxial study on esophageal tissue, the fitted mechanical parameters can only fully reflect the correct passive behavior of the material following an appropriate design of the experiments. The active response of the esophageal muscle layers in situ should be quantified in ex vivo or in vivo isometric and/or isotonic contraction experiments. Due to a lack of these types of experiments on esophageal smooth/striated muscle cell tissue in literature, the four-state model describing the kinetics of the actin-myosin power stroke (Murtada et al. 2010) and the viscous evolution law of the average normalized relative sliding between both filaments $\dot{u}_\mathrm{rs.}$ could not be quantified. For that reason, future work on the experimental characterization of active esophageal muscle tissue (through ex vivo or in vivo isometric and/or isotonic contraction experiments) should be applauded. Murtada et al. (2016) recently proposed a newer mathematical model in which the intrinsically coupled active and passive wall contributions by smooth muscle cells to the total Cauchy stress in the wall was described as the sum of the passive and active stresses, in which the active stress contribution is no longer part of the strain energy function, which could be very interesting if fitted to esophageal muscle tissue. As the smooth muscle cell filaments act like active dashpots which contract proportionally to the resulting force acting upon them (three element Hill model (Fung 1970)), the active tension of smooth muscle cells typically increases until it reaches a local maximum point. The ability of the muscle to develop active tension depends on the initial stretch state of the muscle, e.g., (Gregersen et al. 2008; Murtada et al. 2016). This mechanical phenomenon is not included into the model due to the simplifications (no viscous evolution law for $\dot{u}_\mathrm{rs.})$ to the smooth muscle cell strain energy potential contributions (which had to be taken due to unavailable experimental data). Consequently, the applied constitutive model might overestimate the contractile force of the esophageal muscle tissue at higher stretches, which is schematically illustrated in Fig. 10.

Fig. 10

Schematic illustration of a stress–strain diagram for esophageal muscle tissue. The outlined and calibrated constitutive model will overestimate the esophageal muscle’s total force compared to the physiological total force

Additionally, as this study sets forth to develop patient-specific esophagus models, we have to note that Gregersen et al. (2008) have recently shown that the human esophageal properties and function deteriorate with age (the wall becomes stiffer and the maximal active esophageal muscle tension deteriorated after the age of 40 years). The proposed constitutive material parameter fit was not tuned and/or scaled to the considered patient’s esophageal function or age, but based on the passive multiaxial study of ovine tissue by Sommer et al. (2013).