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Two-layered quasi-3D finite element model of the oesophagus
Medical Engineering & Physics 26 (2004) 535–543 www.elsevier.com/locate/medengphy
Two-layered quasi-3D finite element model of the oesophagus Donghua Liao a,b, Jingbo Zhao a, Yanhua Fan a,c, Hans Gregersen a,d, a
Center of Excellence in Visceral Biomechanics and Pain, Aalborg Hospital and Institute of Health Technology, Aalborg University, Aalborg, Denmark b Biomedical Engineering Center, Beijing University of Technology, Beijing, China c China–Japan Friendship Hospital, Beijing, China d National Center of Ultrasound in Gastroenterology, Haukeland University Hospital, Bergen, Norway Received 29 August 2003; received in revised form 4 March 2004; accepted 20 April 2004
Abstract Analysis of oesophageal mechanoreceptor-dependent responses requires knowledge about the distribution of stresses and strains in the layers of the organ. A two-layered and a one-layered quasi-3D finite element model of the rat oesophagus were used for simulation. An exponential pseudo-strain energy density function was used as the constitutive equation in each model. Stress and strain distributions at the distension pressures 0.25 and 1.0 kPa were studied. The stress and strain distributions depended on the wall geometry. In the one-layered model, the stress ranged from 0.24 to 0.38 kPa at a pressure of 0.25 kPa and from 0.67 to 2.57 kPa at a pressure of 1.0 kPa. The stress in the two-layered model at the pressure of 0.25 and 1.0 kPa varied from 0.52 to 0.64 kPa and from 1.38 to 3.84 kPa. In the two-layered model, the stress was discontinuous at the interface between the muscle layer and the mucosa–submucosa layer. The maximum stress jump was 1.67 kPa at the pressure of 1.0 kPa. The present study provides a numerical simulation tool for characterising the mechanical behaviour of a multi-layered, complex geometry organ. # 2004 IPEM. Published by Elsevier Ltd. All rights reserved. Keywords: Layered oesophagus; Constitutive equations; Stress; Strain; Finite element analysis
1. Introduction The oesophagus is a multi-layered tube. The submucosa–mucosa layer differs from the muscle layer in terms of structure, zero-stress state and material constants [1,2]. However, most studies of oesophageal motility and mechanical function consider the oesophagus as a homogeneous (one-layered) structure. Oesophageal motility disorders are often associated with structural remodelling of the oesophagus. The oesophageal remodelling in achalasia [3] and systemic sclerosis [4] are well known examples. In studies of the remodelling in other multi-layered tissues, the stress distributions throughout the wall play a role for functions of mechanosensitive receptors in the wall [5–7]. Therefore, data on the material properties, i.e. the stress–strain distributions, in the different layers of the Corresponding author. The Research Administration, Aalborg Hospital, Hobrovej 42A, DK-9100 Aalborg, Denmark. Tel.: +4599322533; fax: +45-98154008. E-mail address: [email protected] (H. Gregersen).
oesophagus will aid the understanding of the relationship between the stress, remodelling and mechanosensory function. Recently, the stress and the strain of the layered oesophagus at the zero-stress state [1,2,8,9] and the pressurised state [9,10] were analysed by assuming the oesophageal geometry as an axi-symmetric cylinder. However, it is well known that the oesophagus uses folding of the mucosal layer to advantage its normal function [11]. Thus, the initial oesophageal geometry is complex and cannot be simplified as a cylindrical model. Hence, the theoretical methods used for analysis of mechanical behaviour in cardiovascular tissue are of limited use in the oesophagus [12–15]. The passive mechanical behaviour of the layered oesophagus based on real geometry is yet unknown. In this study, a computer-based finite element model was generated from images of snap-frozen rats’ oesophagus. This model, when coupled with the material constants of each layer, allows simulation of deformations, and stress–strain distributions of the
1350-4533/$ - see front matter # 2004 IPEM. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.medengphy.2004.04.009
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layered oesophageal wall during homeostatic loading. A quasi-3D model is introduced as the first step to estimate the relationships between the layered oesophageal morphology and the stress–strain distribution at the in vivo length with physiological distension pressures of 0.25 and 1.0 kPa.
2. Materials and methods 2.1. Animal experiments Seven Wistar rats weighing 300 g were used in this study. Two rats were used for generating the finite element geometric model and five rats were used for obtaining the material constants. Approval of the protocol was obtained from the Danish Committee for Animal Experimentation. The experiment for the mechanical constant determination was described in detail by Gregersen [11] and Liao et al. [9]. The following describes the geometrical model reconstruction. The animals were anaesthetised with sodium pentobarbital (50 mg kg1 i.p.). Following laparatomy, the
calcium antagonist papaverine (60 mg kg1) was injected into the lower thoracic aorta through a cannula (22 G/25 mm) in order to abolish contractile activity in the oesophagus. After obtaining smooth muscle relaxation, the oesophagus was dissected free from adjacent tissue and its in situ length was measured. The oesophagus was then cut at the proximal and distal ends, and it was immediately placed in physiological saline solution. Cannulas were inserted into both ends of the oesophageal lumen. The oesophageal lumen was perfused with 1% Evans Blue solution for about 180 s in a small organ bath which contains 1% Evans Blue solution and then flushed for about 10 s with physiological saline solution. The oesophageal segment was fixed in a steel frame and then immersed into liquid nitrogen in the horizontal position to minimise gravity effects. Using a scalpel, transverse sections of the tissue were cut in the frozen state at different locations. The crosssections were videotaped immediately by a CCD camera (SXC-151P, Sony, Japan) attached to a stereomicroscope (Zeiss, Germany) with multiple external light sources. Fig. 1(a) shows a cross-sectional image of an oesophageal section.
Fig. 1. Generation of finite element meshes: (a) a rat oesophageal cross-section image obtained from frozen slices, (b) an initial cylinder surface approximating the inner surface, (c) the data clouds measured from the oesophageal inner edge and (d) a 48 nodes 16 elements two-layered 3-D oesophageal mesh generated using the Continuity programme.
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2.2. Geometry measurements SigmaScan Pro 4.0 (Jandel Scientific, CA, USA) was used to measure the co-ordinates of the inner edge, outer edge and thickness of the muscle layer from the images. The muscle layer thickness measured histologiv cally at every 20 in the circumference was 0:264 0:012 mm (mean SD). Hence, the muscle layer thickness was assumed uniform in this study, and the mean muscle layer thickness was calculated at four different quadrants around the oesophagus [1,2,16]. A unit length of a 3-D oesophageal model was built as a first approach and the cross-sections were assumed uniform in the longitudinal direction. Hence, with the coordinates information of one slice, the data clouds of the entire model could be obtained. 2.3. Numerical analysis As a first approximation, the residual strain and the residual stress were not taken into account, the no-load state was therefore considered as the stress-free state in this study. 2.3.1. Mesh generation A higher order element (eight nodes, tricubic Hermite interpolation 3-D element) was selected due to the inner irregular boundary. The degree of freedom of each node in one direction is 8, including three first derivative terms, three second-order cross-derivative terms and a triple cross-derivative term [17–19]. The oesophageal finite element mesh was first created by two cylindrical surfaces using four nodes bicubic Hermite interpolation 2-D element with approximate size of the inner and outer surfaces (Fig. 1(b)). The inner and the outer cylinder surfaces were fitted to the inner wall and the outer wall data clouds using the Continuity software (UC San Diego, USA) (Fig. 1(c)). Hence, a one-layered oesophageal model with 32 nodes and eight elements and a two-layered model with 48 nodes and 16 elements were obtained by linking the nodes of the inner and outer surface meshes together in radial direction (Fig. 1(d)). 2.3.2. Mechanical properties The stress–strain curve obtained from both the submucosal–mucosal layer and the muscle layer showed an exponential, large deformation pattern. Consequently, the muscle layer, the submucosal–mucosal layer and the intact oesophageal wall were modelled by a pseudostrain density function of the exponential type [7,20,21]: q0 w ¼ ðC=2Þ½expðQÞ
ð1Þ
2 2 where Q ¼ a1 Ehh þ a2 Ezz þ a3 Err2 þ 2a4 Ehh Ezz þ 2a5 Err Ezz þ 2a6 Ehh Err . All Eij (i;j ¼ h;r;z) terms are Green
537
strains relative to the zero-stress state, h, r, z are the circumferential, radial and axial directions, respectively. The parameter C expresses a scale to all stress components, and the parameters ai are the material constants, which affect non-linearity and anisotropy of the material. ai (i ¼ 1; 2; 3) are related to the stress– strain relationship in the circumferential, longitudinal and radial direction, respectively [22]. The Cauchy stress, r, calculated from the partial derivatives of Eq. (1), can for the three directions be expressed as: rh ¼ Cð1 þ 2Eh Þða1 Eh þ a4 Ez þ a6 Er ÞeQ rz ¼ Cð1 þ 2Ez Þða4 Eh þ a2 Ez þ a5 Er ÞeQ rr ¼ Cð1 þ 2Er Þða6 Eh þ a5 Ez þ a3 Er ÞeQ
ð2Þ
According to equilibrium equation in cylindrical coordinates, we have drr rr rh þ ¼0 dr r
ð3Þ
with the boundary conditions rr jr¼ri ¼ Pi ;
rr jr¼ro ¼ Po ¼ 0
ð4Þ
where ri, ro and Pi, Po are the radii and the pressure at the inner (mucosal) surface and outer (muscle) surface, respectively. The internal pressure and the axial force acting on the sample can be calculated from Eqs. (1)– (6) by numerical integration: ð ri Pi ¼ C½ð1 þ 2Er Þða6 Eh þ a5 Ez þ a3 Er Þ ð1 ro
þ 2Eh Þða1 Eh þ a4 Ez þ a6 Er ÞexpðQÞ
dr r
ð5Þ
and F¼
pr2i Pi
ð ro þ 2p C½ð1 þ 2Ez Þða4 Eh þ a2 Ez ri
þ a5 Er ÞexpðQÞr dr
ð6Þ
Each oesophageal layer as well as the intact oesophageal wall was modelled as a cylinder when unloaded and pressurised, with an open sector as the stress-free configuration. Hence, the material constants of the muscle layer, mucosa–submucosa layer and the intact layer could be estimated by matching pressure–radius relationship of Eq. (5) to the biaxial experimental data of the different layers using the non-linear least-square method [7,11,20,21,23]. In order to decrease the influence of the initial value on the curve fitting, all curves were fitted 4–5 times with different initial values. The coefficients C and a1 were stable and relatively independent of the initial value. Hence, the coefficients C and a1 were fixed at the stable values and the remaining coefficients were obtained by curve fitting using Eq. (5) again with known C and a1 to the experimental data.
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The material constants in Table 1 are based on our previous study on the layered oesophagus [9] and were used in present numerical models. The coefficients of the intact wall were used as the material parameters for the one-layered model. The coefficients of the muscle and submucosal–mucosal layer were used for the twolayered model. 2.3.3. Boundary conditions A longitudinal stretch ratio of 1.25 was applied in the model to approximate the stretch ratio in vivo [1]. Distension pressures of 0.25 and 1.0 kPa were applied as loaded conditions and the displacements in the longitudinal direction were constrained at one end of the oesophagus. 2.3.4. Finite element solution method [18,19] The numerical analysis was done on an SGI workstation using Continuity software (UC San Diego, USA). To achieve the convergence, the FE equation was solved using the Newton–Raphson iteration to each load step. 2.3.5. Error analysis and solution convergence [18,19] Numerical solutions were characterised by the total strain energy of the body WT. The percent error in total strain energy was computed from: PWT ¼
WT WT 100% WT
ð7Þ
WT was the total strain energy estimated from a highly refined model in the absence of exact analytic solutions. 2.4. Statistics The data were normal-distributed and accordingly expressed as mean SE. Parametric statistics in terms of ANOVA and Student’s t-test were used. Differences were considered significant when P < 0:05.
3. Results The stress and strain distributions in the non-linear anisotropy layered oesophageal model were calculated. Solution convergence was tested in the one-layered model for passive inflation (0.25 kPa) with 4, 8 and 16 circumferential elements. PWT was decreased from 18.43% with four elements to 6.07% with eight elements. Considering the delicate balance between accurate result and the computational expenses, the mesh with eight elements for the one-layered model and 16 elements for the two-layered model were chosen for the numerical analysis. 3.1. Stress–strain distributions The luminal pressure distended the oesophageal wall and the irregular mucosa distended gradually with the pressure. The undeformed and the deformed luminal cross-sectional domain and the circumferential strain distribution at a luminal pressure of 0.25 and 1.0 kPa for the one-layered and the two-layered wall are shown in Figs. 2a–d. The corresponding circumferential stresses for the one-layered and the two-layered models are illustrated in Figs. 2e–h. The wall deformation after the distension pressure of 0.25 and 1.0 kPa are given in Figs. 2a and c for the one-layer model and Figs. 2b and d for the two-layer model by comparing the undeformed (dashed lines) and the deformed geometry data. Figs. 2a–d show that the strain distributions in both the one-layered model and the two-layered model were continuous and that negative strains existed near the mucosal buckles. The stress differed between the one-layer model and the twolayered model. The stress was discontinuous at the interface between the submucosa–mucosa layer and the muscle layer in the two-layered model. The maximum stress jumps at the pressure of 0.25 and 1.0 kPa were 0.44 and 1.67 kPa. Four radial cross lines A–A, B–B, C–C and D–D were selected at positions where the stress and strain varied significantly (Figs. 2g and h) for illustration of the stress–strain distribution throughout the wall. The
Table 1 Coefficients of the constitutive equation (n ¼ 5)
a1
a2
a3
a4
a5
a6
C (kPa)
Intact oesophagus Muscle layer Submucosal– mucosal layer
1:35 0:20
1:82 0:46
0:58 0:54
0:19 0:16
0:28 0:11
0:48 0:20
3:45 0:38
1:16 0:32 6:10 0:84
1:85 0:57 0:93 0:23
0:92 0:41 0:62 0:38
0:31 0:10 0:25 0:11
0:24 0:07 1:53 0:58
0:23 0:11 1:18 0:22
2:42 0:32 1:76 0:25
Notes: The data are mean SE; n, sample size. One way ANOVA was used for comparing the mechanical constants among the intact oesophageal, the muscle layer and the mucosal–submucosal layer model, P < 0:01. Student’s t-test was used to compare a1 and a2 obtained from the different models, P ¼ 0:002.
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Fig. 2. The oesophageal undeformed sections (domain rounded by the dashed lines), the deformed section and circumferential stress and strain distributions. The strain distribution in the one-layered model (a, c) and in the two-layered model (b, d) with pressure of 0.25 kPa (a, b) and 1.0 kPa (c, d). The stress distribution in the one-layered model (e, g) and in the two-layered model (f, h) with pressure of 0.25 kPa (e, f) and 1.0 kPa (g, h). The radial lines A–A, B–B, C–C and D–D in (g) and (h) indicate four locations where the stress and strain varies significantly.
circumferential strain distributions throughout the onelayered model and the two-layered model at the radial cross lines A–A, B–B, C–C and D–D with the luminal pressure of 0.25 and 1.0 kPa are shown in Fig. 3. The corresponding circumferential stress distributions are
illustrated in Fig. 4. The figures show that the strains and the stresses were not evenly distributed throughout the wall. The strain distributions in the one-layered and the two-layered model had a similar pattern. The strains in the two models were almost identical at the
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Fig. 3. The circumferential strain (Ehh) distributions throughout the two-layered model and the one-layered model at different positions. 0 and 1 in the abscissas indicate the inner surface and the outer surface. A–A, B–B, C–C and D–D are the radial lines illustrated in Figs. 2g and 2h. t, _, the two-layered model with distension pressure of 0.25 and 1.0 kPa; l, , the one-layered model with distension pressures of 0.25 and 1.0 kPa.
pressure of 0.25 kPa. At the pressure of 1.0 kPa, the strain of the one-layered model was higher than that of the two-layered model. The highest stresses were found near the inner surface in both models. The stresses in the submucosal–mucosal layer were significantly higher than the muscle layer stresses. The stress gradient across the mucosal layer was also higher than that in the muscle layer. For instance, the stress gradient throughout the line D–D of Fig. 2h at the pressure of 0.25 and 1.0 kPa were 0.95 and 6.78 kPa mm1 in the submucosal–mucosal layer and 0.10 and 0.42 kPa mm1 in the muscle layer.
4. Discussion The present study provides the first layered oesophageal finite element model. Two pressurised anisotropic oesophageal quasi-3D models were developed. The stress–strain distributions in both the one-layered and the two-layered model were not uniform due to the irregular geometry and the anisotropic material properties. The stress distribution in the two-layered models differed from that in the one-layered model due to the differences in structure and material properties in the layered model. The circumferential stress in the submucosal–mucosal layer was higher than that in the
muscle layer. The quasi-3D model can be further developed for generating a real 3-D oesophageal model reconstructed from the clinical medical images. This model can also be extended to simulate the mechanical behaviour of other layered organs such as blood vessels and the small intestine. Distension was used for mechanical stimulation in numerous previous animal [24–26] or human [27,28] visceral sensory studies. The circumferential tension (stress), and strain during distension were considered important parameters for the mechanosensitive receptors. The circumferential tension and stress in previous studies were calculated in terms of Laplace’s law where the oesophagus were simplified to a straight thin walled cylinder and the tensions or stresses across the wall were assumed homogeneous [16,27,28]. However, both the one-layered and the two-layered model in this study showed that the circumferential stress and strain throughout the oesophageal wall at the pressurised state were not uniform. Electrophysiological studies indicate that afferent nerve fibres located in the mucosal layer and the intraganglionic laminar ending (IGLE) located in the muscle layer function as mechanoreceptors [24,26]. The neurons respond in an intensity-dependent manner to oesophageal distension [24]. Hence, determination of the stress or tension and strain locally is important for understanding the
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Fig. 4. The circumferential stress (rhh) distributions in the two-layered and the one-layered model at different positions. 0 and 1 in the abscissa indicate the inner surface and the outer surface. A–A, B–B, C–C and D–D are the lines shown in Figs. 2g and h. t, _, the two-layered model with distension pressure of 0.25 and 1.0 kPa; l, , the one-layered model with distension pressures of 0.25 and 1.0 kPa.
mechanical environment at receptor sites. According to Laplace’s law, the stress or tension located at mucosal afferent fibres is expected to be identical to that at the muscle afferent fibres. It means that the mechanical stimulation intensity is uniform throughout the wall. However, that assumption is not true when the 3-D oesophageal geometry as well as the anisotropic mechanical properties are considered. The stress distribution varies as functions of the geometry, external load as well as the material structures. The stress in the mucosal layer was higher than that in the muscle layer in both the two-layered model and the one-layered model. The peak stress for the one-layered model and the twolayered model at the pressure of 1.0 kPa were 2.6 and 3.9 kPa, i.e. the peak stress in the one-layered model has an error of about 30% compared to the twolayered model. Hence, the material structures in the layers must be taken into account for the stress analysis. In this study, the 3-D model was established based on only one section of the rat oesophagus. The assumption of the uniform muscle layer thickness has been widely used in the previous cylindrical oesophageal model [1,2,16]. However, the muscle layer thickness should be measured point by point around the oesophageal rather than by the mean value measured from some specific locations. The thickness variation will influence the pattern of the stress distribution
[29]. The real 3-D non-uniform thickness oesophageal numerical model will be established using the ultrasonography technology in our future study. The luminal pressure in the human oesophagus at the resting state is less than 1.0 kPa [28] and in vitro ultrasonic experiments showed that the rabbit oesophageal folds entirely disappeared with the luminal pressure of 0.25 kPa (unpublished data). Hence, only the distension pressure of 0.25 and 1.0 kPa was selected in this study for performing the oesophagus mechanical behaviour under physiological pressure conditions. The degree of correspondence between the numerical simulation and the experimental data [13] with the distension pressure of 0.25 kPa was compared between the maximum outer radius and the measured outer radius. The relative error (e ¼ ðjRexperiment Rnumerical j= Rexperiment Þ) is 11.2% and 10.8% for the one-layered and the two-layered model. The present numerical results and the previous theoretical study [9] shows large differences but were also based on different premises. For example, the minimum circumferential stretch ratio of the intact oesophagus with the distension pressure of 0.25 kPa was approximately 1.38 in the theoretical analysis, whereas in the present numerical analysis, the datum was approximately 1.03. The primary reason for such differences is that the irregular inner boundary was used in the numerical study whereas an axi-symmetric
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cylinder model was used in the theoretical analysis. The present numerical study considers the irregular boundary in terms of the buckling of the submucosa layer but the buckling mechanical behaviour is not taken into consideration. In the future, the buckling behaviour of the mucosa and submucosa will be taken into account and a three-layered oesophagus model developed where the submucosal and mucosal layer will have their own material properties. This will likely reduce any error between computed and measured data. The no-load state was defined as the stress-free state in this study. However, previous studies have shown that the no-load state is not a true stress-free state and that residual stresses and strains exist. The residual stress and residual strain analysis for the buckling configuration such as the oesophagus need further study and thus the residual stress and residual strain were neglected in this study. The current model took the passive behaviour of the oesophagus into account while the muscle contraction was neglected. Because the function of the whole digestive system including the oesophagus depend on both contractile and non-contractile structures presenting active and passive mechanical properties [26], the passive studies furnish a base upon which the active state can be better understood according to the well known Hill’s model. The oesophageal model within muscle contraction needs further study. The present study devises a method to estimate the local stress and strain distribution across the oesophageal wall. This study does not provide direct evidence about whether stress or strain are the stimulus for the afferent response but the numerical method can be employed in the future study on this issue. In conclusion, the circumferential stress and strain variation was due to the irregular oesophageal geometry as well as non-homogeneous and anisotropic wall properties. The oesophagus must therefore be analysed based on its real geometry and the oesophageal wall must be modelled as at least a two-layered system, each layer with its own material constants. The numerical layered oesophageal model may prove useful for studying biomechanical remodelling and to reveal the relationship between the mechanical behaviour and the mechanoreceptor response.
Acknowledgements Karen Elise Jensen’s Foundation and the Danish Technical Research Council are thanked for financial support.
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Two-layered quasi-3D finite element model of the oesophagus Donghua Liao a,b, Jingbo Zhao a, Yanhua Fan a,c, Hans Gregersen a,d, a
Center of Excellence in Visceral Biomechanics and Pain, Aalborg Hospital and Institute of Health Technology, Aalborg University, Aalborg, Denmark b Biomedical Engineering Center, Beijing University of Technology, Beijing, China c China–Japan Friendship Hospital, Beijing, China d National Center of Ultrasound in Gastroenterology, Haukeland University Hospital, Bergen, Norway Received 29 August 2003; received in revised form 4 March 2004; accepted 20 April 2004
Abstract Analysis of oesophageal mechanoreceptor-dependent responses requires knowledge about the distribution of stresses and strains in the layers of the organ. A two-layered and a one-layered quasi-3D finite element model of the rat oesophagus were used for simulation. An exponential pseudo-strain energy density function was used as the constitutive equation in each model. Stress and strain distributions at the distension pressures 0.25 and 1.0 kPa were studied. The stress and strain distributions depended on the wall geometry. In the one-layered model, the stress ranged from 0.24 to 0.38 kPa at a pressure of 0.25 kPa and from 0.67 to 2.57 kPa at a pressure of 1.0 kPa. The stress in the two-layered model at the pressure of 0.25 and 1.0 kPa varied from 0.52 to 0.64 kPa and from 1.38 to 3.84 kPa. In the two-layered model, the stress was discontinuous at the interface between the muscle layer and the mucosa–submucosa layer. The maximum stress jump was 1.67 kPa at the pressure of 1.0 kPa. The present study provides a numerical simulation tool for characterising the mechanical behaviour of a multi-layered, complex geometry organ. # 2004 IPEM. Published by Elsevier Ltd. All rights reserved. Keywords: Layered oesophagus; Constitutive equations; Stress; Strain; Finite element analysis
1. Introduction The oesophagus is a multi-layered tube. The submucosa–mucosa layer differs from the muscle layer in terms of structure, zero-stress state and material constants [1,2]. However, most studies of oesophageal motility and mechanical function consider the oesophagus as a homogeneous (one-layered) structure. Oesophageal motility disorders are often associated with structural remodelling of the oesophagus. The oesophageal remodelling in achalasia [3] and systemic sclerosis [4] are well known examples. In studies of the remodelling in other multi-layered tissues, the stress distributions throughout the wall play a role for functions of mechanosensitive receptors in the wall [5–7]. Therefore, data on the material properties, i.e. the stress–strain distributions, in the different layers of the Corresponding author. The Research Administration, Aalborg Hospital, Hobrovej 42A, DK-9100 Aalborg, Denmark. Tel.: +4599322533; fax: +45-98154008. E-mail address: [email protected] (H. Gregersen).
oesophagus will aid the understanding of the relationship between the stress, remodelling and mechanosensory function. Recently, the stress and the strain of the layered oesophagus at the zero-stress state [1,2,8,9] and the pressurised state [9,10] were analysed by assuming the oesophageal geometry as an axi-symmetric cylinder. However, it is well known that the oesophagus uses folding of the mucosal layer to advantage its normal function [11]. Thus, the initial oesophageal geometry is complex and cannot be simplified as a cylindrical model. Hence, the theoretical methods used for analysis of mechanical behaviour in cardiovascular tissue are of limited use in the oesophagus [12–15]. The passive mechanical behaviour of the layered oesophagus based on real geometry is yet unknown. In this study, a computer-based finite element model was generated from images of snap-frozen rats’ oesophagus. This model, when coupled with the material constants of each layer, allows simulation of deformations, and stress–strain distributions of the
1350-4533/$ - see front matter # 2004 IPEM. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.medengphy.2004.04.009
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layered oesophageal wall during homeostatic loading. A quasi-3D model is introduced as the first step to estimate the relationships between the layered oesophageal morphology and the stress–strain distribution at the in vivo length with physiological distension pressures of 0.25 and 1.0 kPa.
2. Materials and methods 2.1. Animal experiments Seven Wistar rats weighing 300 g were used in this study. Two rats were used for generating the finite element geometric model and five rats were used for obtaining the material constants. Approval of the protocol was obtained from the Danish Committee for Animal Experimentation. The experiment for the mechanical constant determination was described in detail by Gregersen [11] and Liao et al. [9]. The following describes the geometrical model reconstruction. The animals were anaesthetised with sodium pentobarbital (50 mg kg1 i.p.). Following laparatomy, the
calcium antagonist papaverine (60 mg kg1) was injected into the lower thoracic aorta through a cannula (22 G/25 mm) in order to abolish contractile activity in the oesophagus. After obtaining smooth muscle relaxation, the oesophagus was dissected free from adjacent tissue and its in situ length was measured. The oesophagus was then cut at the proximal and distal ends, and it was immediately placed in physiological saline solution. Cannulas were inserted into both ends of the oesophageal lumen. The oesophageal lumen was perfused with 1% Evans Blue solution for about 180 s in a small organ bath which contains 1% Evans Blue solution and then flushed for about 10 s with physiological saline solution. The oesophageal segment was fixed in a steel frame and then immersed into liquid nitrogen in the horizontal position to minimise gravity effects. Using a scalpel, transverse sections of the tissue were cut in the frozen state at different locations. The crosssections were videotaped immediately by a CCD camera (SXC-151P, Sony, Japan) attached to a stereomicroscope (Zeiss, Germany) with multiple external light sources. Fig. 1(a) shows a cross-sectional image of an oesophageal section.
Fig. 1. Generation of finite element meshes: (a) a rat oesophageal cross-section image obtained from frozen slices, (b) an initial cylinder surface approximating the inner surface, (c) the data clouds measured from the oesophageal inner edge and (d) a 48 nodes 16 elements two-layered 3-D oesophageal mesh generated using the Continuity programme.
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2.2. Geometry measurements SigmaScan Pro 4.0 (Jandel Scientific, CA, USA) was used to measure the co-ordinates of the inner edge, outer edge and thickness of the muscle layer from the images. The muscle layer thickness measured histologiv cally at every 20 in the circumference was 0:264 0:012 mm (mean SD). Hence, the muscle layer thickness was assumed uniform in this study, and the mean muscle layer thickness was calculated at four different quadrants around the oesophagus [1,2,16]. A unit length of a 3-D oesophageal model was built as a first approach and the cross-sections were assumed uniform in the longitudinal direction. Hence, with the coordinates information of one slice, the data clouds of the entire model could be obtained. 2.3. Numerical analysis As a first approximation, the residual strain and the residual stress were not taken into account, the no-load state was therefore considered as the stress-free state in this study. 2.3.1. Mesh generation A higher order element (eight nodes, tricubic Hermite interpolation 3-D element) was selected due to the inner irregular boundary. The degree of freedom of each node in one direction is 8, including three first derivative terms, three second-order cross-derivative terms and a triple cross-derivative term [17–19]. The oesophageal finite element mesh was first created by two cylindrical surfaces using four nodes bicubic Hermite interpolation 2-D element with approximate size of the inner and outer surfaces (Fig. 1(b)). The inner and the outer cylinder surfaces were fitted to the inner wall and the outer wall data clouds using the Continuity software (UC San Diego, USA) (Fig. 1(c)). Hence, a one-layered oesophageal model with 32 nodes and eight elements and a two-layered model with 48 nodes and 16 elements were obtained by linking the nodes of the inner and outer surface meshes together in radial direction (Fig. 1(d)). 2.3.2. Mechanical properties The stress–strain curve obtained from both the submucosal–mucosal layer and the muscle layer showed an exponential, large deformation pattern. Consequently, the muscle layer, the submucosal–mucosal layer and the intact oesophageal wall were modelled by a pseudostrain density function of the exponential type [7,20,21]: q0 w ¼ ðC=2Þ½expðQÞ
ð1Þ
2 2 where Q ¼ a1 Ehh þ a2 Ezz þ a3 Err2 þ 2a4 Ehh Ezz þ 2a5 Err Ezz þ 2a6 Ehh Err . All Eij (i;j ¼ h;r;z) terms are Green
537
strains relative to the zero-stress state, h, r, z are the circumferential, radial and axial directions, respectively. The parameter C expresses a scale to all stress components, and the parameters ai are the material constants, which affect non-linearity and anisotropy of the material. ai (i ¼ 1; 2; 3) are related to the stress– strain relationship in the circumferential, longitudinal and radial direction, respectively [22]. The Cauchy stress, r, calculated from the partial derivatives of Eq. (1), can for the three directions be expressed as: rh ¼ Cð1 þ 2Eh Þða1 Eh þ a4 Ez þ a6 Er ÞeQ rz ¼ Cð1 þ 2Ez Þða4 Eh þ a2 Ez þ a5 Er ÞeQ rr ¼ Cð1 þ 2Er Þða6 Eh þ a5 Ez þ a3 Er ÞeQ
ð2Þ
According to equilibrium equation in cylindrical coordinates, we have drr rr rh þ ¼0 dr r
ð3Þ
with the boundary conditions rr jr¼ri ¼ Pi ;
rr jr¼ro ¼ Po ¼ 0
ð4Þ
where ri, ro and Pi, Po are the radii and the pressure at the inner (mucosal) surface and outer (muscle) surface, respectively. The internal pressure and the axial force acting on the sample can be calculated from Eqs. (1)– (6) by numerical integration: ð ri Pi ¼ C½ð1 þ 2Er Þða6 Eh þ a5 Ez þ a3 Er Þ ð1 ro
þ 2Eh Þða1 Eh þ a4 Ez þ a6 Er ÞexpðQÞ
dr r
ð5Þ
and F¼
pr2i Pi
ð ro þ 2p C½ð1 þ 2Ez Þða4 Eh þ a2 Ez ri
þ a5 Er ÞexpðQÞr dr
ð6Þ
Each oesophageal layer as well as the intact oesophageal wall was modelled as a cylinder when unloaded and pressurised, with an open sector as the stress-free configuration. Hence, the material constants of the muscle layer, mucosa–submucosa layer and the intact layer could be estimated by matching pressure–radius relationship of Eq. (5) to the biaxial experimental data of the different layers using the non-linear least-square method [7,11,20,21,23]. In order to decrease the influence of the initial value on the curve fitting, all curves were fitted 4–5 times with different initial values. The coefficients C and a1 were stable and relatively independent of the initial value. Hence, the coefficients C and a1 were fixed at the stable values and the remaining coefficients were obtained by curve fitting using Eq. (5) again with known C and a1 to the experimental data.
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The material constants in Table 1 are based on our previous study on the layered oesophagus [9] and were used in present numerical models. The coefficients of the intact wall were used as the material parameters for the one-layered model. The coefficients of the muscle and submucosal–mucosal layer were used for the twolayered model. 2.3.3. Boundary conditions A longitudinal stretch ratio of 1.25 was applied in the model to approximate the stretch ratio in vivo [1]. Distension pressures of 0.25 and 1.0 kPa were applied as loaded conditions and the displacements in the longitudinal direction were constrained at one end of the oesophagus. 2.3.4. Finite element solution method [18,19] The numerical analysis was done on an SGI workstation using Continuity software (UC San Diego, USA). To achieve the convergence, the FE equation was solved using the Newton–Raphson iteration to each load step. 2.3.5. Error analysis and solution convergence [18,19] Numerical solutions were characterised by the total strain energy of the body WT. The percent error in total strain energy was computed from: PWT ¼
WT WT 100% WT
ð7Þ
WT was the total strain energy estimated from a highly refined model in the absence of exact analytic solutions. 2.4. Statistics The data were normal-distributed and accordingly expressed as mean SE. Parametric statistics in terms of ANOVA and Student’s t-test were used. Differences were considered significant when P < 0:05.
3. Results The stress and strain distributions in the non-linear anisotropy layered oesophageal model were calculated. Solution convergence was tested in the one-layered model for passive inflation (0.25 kPa) with 4, 8 and 16 circumferential elements. PWT was decreased from 18.43% with four elements to 6.07% with eight elements. Considering the delicate balance between accurate result and the computational expenses, the mesh with eight elements for the one-layered model and 16 elements for the two-layered model were chosen for the numerical analysis. 3.1. Stress–strain distributions The luminal pressure distended the oesophageal wall and the irregular mucosa distended gradually with the pressure. The undeformed and the deformed luminal cross-sectional domain and the circumferential strain distribution at a luminal pressure of 0.25 and 1.0 kPa for the one-layered and the two-layered wall are shown in Figs. 2a–d. The corresponding circumferential stresses for the one-layered and the two-layered models are illustrated in Figs. 2e–h. The wall deformation after the distension pressure of 0.25 and 1.0 kPa are given in Figs. 2a and c for the one-layer model and Figs. 2b and d for the two-layer model by comparing the undeformed (dashed lines) and the deformed geometry data. Figs. 2a–d show that the strain distributions in both the one-layered model and the two-layered model were continuous and that negative strains existed near the mucosal buckles. The stress differed between the one-layer model and the twolayered model. The stress was discontinuous at the interface between the submucosa–mucosa layer and the muscle layer in the two-layered model. The maximum stress jumps at the pressure of 0.25 and 1.0 kPa were 0.44 and 1.67 kPa. Four radial cross lines A–A, B–B, C–C and D–D were selected at positions where the stress and strain varied significantly (Figs. 2g and h) for illustration of the stress–strain distribution throughout the wall. The
Table 1 Coefficients of the constitutive equation (n ¼ 5)
a1
a2
a3
a4
a5
a6
C (kPa)
Intact oesophagus Muscle layer Submucosal– mucosal layer
1:35 0:20
1:82 0:46
0:58 0:54
0:19 0:16
0:28 0:11
0:48 0:20
3:45 0:38
1:16 0:32 6:10 0:84
1:85 0:57 0:93 0:23
0:92 0:41 0:62 0:38
0:31 0:10 0:25 0:11
0:24 0:07 1:53 0:58
0:23 0:11 1:18 0:22
2:42 0:32 1:76 0:25
Notes: The data are mean SE; n, sample size. One way ANOVA was used for comparing the mechanical constants among the intact oesophageal, the muscle layer and the mucosal–submucosal layer model, P < 0:01. Student’s t-test was used to compare a1 and a2 obtained from the different models, P ¼ 0:002.
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Fig. 2. The oesophageal undeformed sections (domain rounded by the dashed lines), the deformed section and circumferential stress and strain distributions. The strain distribution in the one-layered model (a, c) and in the two-layered model (b, d) with pressure of 0.25 kPa (a, b) and 1.0 kPa (c, d). The stress distribution in the one-layered model (e, g) and in the two-layered model (f, h) with pressure of 0.25 kPa (e, f) and 1.0 kPa (g, h). The radial lines A–A, B–B, C–C and D–D in (g) and (h) indicate four locations where the stress and strain varies significantly.
circumferential strain distributions throughout the onelayered model and the two-layered model at the radial cross lines A–A, B–B, C–C and D–D with the luminal pressure of 0.25 and 1.0 kPa are shown in Fig. 3. The corresponding circumferential stress distributions are
illustrated in Fig. 4. The figures show that the strains and the stresses were not evenly distributed throughout the wall. The strain distributions in the one-layered and the two-layered model had a similar pattern. The strains in the two models were almost identical at the
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Fig. 3. The circumferential strain (Ehh) distributions throughout the two-layered model and the one-layered model at different positions. 0 and 1 in the abscissas indicate the inner surface and the outer surface. A–A, B–B, C–C and D–D are the radial lines illustrated in Figs. 2g and 2h. t, _, the two-layered model with distension pressure of 0.25 and 1.0 kPa; l, , the one-layered model with distension pressures of 0.25 and 1.0 kPa.
pressure of 0.25 kPa. At the pressure of 1.0 kPa, the strain of the one-layered model was higher than that of the two-layered model. The highest stresses were found near the inner surface in both models. The stresses in the submucosal–mucosal layer were significantly higher than the muscle layer stresses. The stress gradient across the mucosal layer was also higher than that in the muscle layer. For instance, the stress gradient throughout the line D–D of Fig. 2h at the pressure of 0.25 and 1.0 kPa were 0.95 and 6.78 kPa mm1 in the submucosal–mucosal layer and 0.10 and 0.42 kPa mm1 in the muscle layer.
4. Discussion The present study provides the first layered oesophageal finite element model. Two pressurised anisotropic oesophageal quasi-3D models were developed. The stress–strain distributions in both the one-layered and the two-layered model were not uniform due to the irregular geometry and the anisotropic material properties. The stress distribution in the two-layered models differed from that in the one-layered model due to the differences in structure and material properties in the layered model. The circumferential stress in the submucosal–mucosal layer was higher than that in the
muscle layer. The quasi-3D model can be further developed for generating a real 3-D oesophageal model reconstructed from the clinical medical images. This model can also be extended to simulate the mechanical behaviour of other layered organs such as blood vessels and the small intestine. Distension was used for mechanical stimulation in numerous previous animal [24–26] or human [27,28] visceral sensory studies. The circumferential tension (stress), and strain during distension were considered important parameters for the mechanosensitive receptors. The circumferential tension and stress in previous studies were calculated in terms of Laplace’s law where the oesophagus were simplified to a straight thin walled cylinder and the tensions or stresses across the wall were assumed homogeneous [16,27,28]. However, both the one-layered and the two-layered model in this study showed that the circumferential stress and strain throughout the oesophageal wall at the pressurised state were not uniform. Electrophysiological studies indicate that afferent nerve fibres located in the mucosal layer and the intraganglionic laminar ending (IGLE) located in the muscle layer function as mechanoreceptors [24,26]. The neurons respond in an intensity-dependent manner to oesophageal distension [24]. Hence, determination of the stress or tension and strain locally is important for understanding the
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Fig. 4. The circumferential stress (rhh) distributions in the two-layered and the one-layered model at different positions. 0 and 1 in the abscissa indicate the inner surface and the outer surface. A–A, B–B, C–C and D–D are the lines shown in Figs. 2g and h. t, _, the two-layered model with distension pressure of 0.25 and 1.0 kPa; l, , the one-layered model with distension pressures of 0.25 and 1.0 kPa.
mechanical environment at receptor sites. According to Laplace’s law, the stress or tension located at mucosal afferent fibres is expected to be identical to that at the muscle afferent fibres. It means that the mechanical stimulation intensity is uniform throughout the wall. However, that assumption is not true when the 3-D oesophageal geometry as well as the anisotropic mechanical properties are considered. The stress distribution varies as functions of the geometry, external load as well as the material structures. The stress in the mucosal layer was higher than that in the muscle layer in both the two-layered model and the one-layered model. The peak stress for the one-layered model and the twolayered model at the pressure of 1.0 kPa were 2.6 and 3.9 kPa, i.e. the peak stress in the one-layered model has an error of about 30% compared to the twolayered model. Hence, the material structures in the layers must be taken into account for the stress analysis. In this study, the 3-D model was established based on only one section of the rat oesophagus. The assumption of the uniform muscle layer thickness has been widely used in the previous cylindrical oesophageal model [1,2,16]. However, the muscle layer thickness should be measured point by point around the oesophageal rather than by the mean value measured from some specific locations. The thickness variation will influence the pattern of the stress distribution
[29]. The real 3-D non-uniform thickness oesophageal numerical model will be established using the ultrasonography technology in our future study. The luminal pressure in the human oesophagus at the resting state is less than 1.0 kPa [28] and in vitro ultrasonic experiments showed that the rabbit oesophageal folds entirely disappeared with the luminal pressure of 0.25 kPa (unpublished data). Hence, only the distension pressure of 0.25 and 1.0 kPa was selected in this study for performing the oesophagus mechanical behaviour under physiological pressure conditions. The degree of correspondence between the numerical simulation and the experimental data [13] with the distension pressure of 0.25 kPa was compared between the maximum outer radius and the measured outer radius. The relative error (e ¼ ðjRexperiment Rnumerical j= Rexperiment Þ) is 11.2% and 10.8% for the one-layered and the two-layered model. The present numerical results and the previous theoretical study [9] shows large differences but were also based on different premises. For example, the minimum circumferential stretch ratio of the intact oesophagus with the distension pressure of 0.25 kPa was approximately 1.38 in the theoretical analysis, whereas in the present numerical analysis, the datum was approximately 1.03. The primary reason for such differences is that the irregular inner boundary was used in the numerical study whereas an axi-symmetric
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cylinder model was used in the theoretical analysis. The present numerical study considers the irregular boundary in terms of the buckling of the submucosa layer but the buckling mechanical behaviour is not taken into consideration. In the future, the buckling behaviour of the mucosa and submucosa will be taken into account and a three-layered oesophagus model developed where the submucosal and mucosal layer will have their own material properties. This will likely reduce any error between computed and measured data. The no-load state was defined as the stress-free state in this study. However, previous studies have shown that the no-load state is not a true stress-free state and that residual stresses and strains exist. The residual stress and residual strain analysis for the buckling configuration such as the oesophagus need further study and thus the residual stress and residual strain were neglected in this study. The current model took the passive behaviour of the oesophagus into account while the muscle contraction was neglected. Because the function of the whole digestive system including the oesophagus depend on both contractile and non-contractile structures presenting active and passive mechanical properties [26], the passive studies furnish a base upon which the active state can be better understood according to the well known Hill’s model. The oesophageal model within muscle contraction needs further study. The present study devises a method to estimate the local stress and strain distribution across the oesophageal wall. This study does not provide direct evidence about whether stress or strain are the stimulus for the afferent response but the numerical method can be employed in the future study on this issue. In conclusion, the circumferential stress and strain variation was due to the irregular oesophageal geometry as well as non-homogeneous and anisotropic wall properties. The oesophagus must therefore be analysed based on its real geometry and the oesophageal wall must be modelled as at least a two-layered system, each layer with its own material constants. The numerical layered oesophageal model may prove useful for studying biomechanical remodelling and to reveal the relationship between the mechanical behaviour and the mechanoreceptor response.
Acknowledgements Karen Elise Jensen’s Foundation and the Danish Technical Research Council are thanked for financial support.
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