Computational Models for the Mechanical Investigation of Stomach Tissues and Structure

Abstract

Bariatric surgery is performed on obese people aiming at reducing the capacity of the stomach and/or the absorbing capability of the gastrointestinal tract. A more reliable and effective approach to bariatric surgery may integrate different expertise, in the areas of surgery, physiology and biomechanics, availing of a strong cooperation between clinicians and engineers. This work aimed at developing a computational model of the stomach, as a computational tool for the physio-mechanical investigation of stomach functionality and the planning of bariatric procedures. In this sense, coupled experimental and numerical activities were developed. Experimental investigations on pig and piglet stomachs aimed at providing information about stomach geometrical configuration and structural behavior. The computational model was defined starting from the analysis of data from histo-morphometric investigations and mechanical tests. A fiber-reinforced visco-hyperelastic constitutive model was developed to interpret the mechanical response of stomach tissues; constitutive parameters were identified considering mechanical tests at both tissue and structure levels. Computational analyses were performed to investigate the pressure–volume behavior of the stomach. The developed model satisfactorily interpreted results from experimental activities, suggesting its reliability. Furthermore, the model was exploited to investigate stress and strain fields within gastric tissues, as the stimuli for mechanoreceptors that interact with the central nervous system leading to the feeling of satiety. In this respect, the developed computational model may be employed to evaluate the influence of bariatric intervention on the stimulation of mechanoreceptors, and the following meal induced satiety.

Appendix A

Constitutive parameters were identified by the inverse analysis of experimental activities. Aiming at investigating the complex distribution of stomach tissues, experimentations performed by Zhao et al.56 were analysed. Tensile tests were developed on tissue specimens harvested from pig stomachs, considering the influence of location, as fundus, corpus and antrum, wall layer, as connective layer and muscular layers, and direction, as circumferential and longitudinal ones. Specimens loading occurred at low strain rate, leading to almost equilibrium response. Mathematical procedures led to the following model formulation that allowed interpreting results from tensile tests:

$${\mathbf{P}}^{\infty } \left( {\mathbf{C}} \right) = \gamma^{\infty } {\mathbf{P}}^{0} \left( {\mathbf{C}} \right) = 2\gamma^{\infty } {\mathbf{F}}\frac{{\partial W^{0} \left( {\mathbf{C}} \right)}}{{\partial {\mathbf{C}}}} = 2\gamma^{\infty } {\mathbf{F}}\frac{{\partial W_{m}^{0} \left( {\mathbf{C}} \right)}}{{\partial {\mathbf{C}}}} + 2\gamma^{\infty } {\mathbf{F}}\frac{{\partial W_{fL}^{0} \left( {I_{4} } \right)}}{{\partial {\mathbf{C}}}} + 2\gamma^{\infty } {\mathbf{F}}\frac{{\partial W_{fC}^{0} \left( {I_{6} } \right)}}{{\partial {\mathbf{C}}}} = {\mathbf{P}}_{m}^{\infty } \left( {\mathbf{C}} \right) + {\mathbf{P}}_{fL}^{\infty } \left( {I_{4} } \right) + {\mathbf{P}}_{fC}^{\infty } \left( {I_{6} } \right)$$

(A.1)

$${\mathbf{P}}_{m}^{\infty } \left( {\mathbf{C}} \right) = - \gamma^{\infty } pF^{ - T} + C_{1}^{\infty } \exp \left[ {\alpha_{1} \left( {I_{1} - 3} \right)} \right]\left( {2{\mathbf{F}} - {2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}I_{1} {\mathbf{F}}^{ - T} } \right)$$

(A.2)

$${\mathbf{P}}_{fL}^{\infty } = 2\left( {{{C_{4}^{\infty } } \mathord{\left/ {\vphantom {{C_{4}^{\infty } } {\alpha_{4} }}} \right. \kern-0pt} {\alpha_{4} }}} \right)\left\{ {\exp \left[ {\alpha_{4} \left( {I_{4} - 1} \right)} \right] - 1} \right\}{\mathbf{F}}\left( {{\mathbf{a}}_{0} \otimes {\mathbf{a}}_{0} } \right)$$

(A.3)

$${\mathbf{P}}_{fC}^{\infty } = 2\left( {{{C_{6}^{\infty } } \mathord{\left/ {\vphantom {{C_{6}^{\infty } } {\alpha_{6} }}} \right. \kern-0pt} {\alpha_{6} }}} \right)\left\{ {\exp \left[ {\alpha_{6} \left( {I_{6} - 1} \right)} \right] - 1} \right\}{\mathbf{F}}\left( {{\mathbf{b}}_{0} \otimes {\mathbf{b}}_{0} } \right)$$

(A.4)

where $\gamma^{\infty } = 1 - \sum\limits_{i = 1}^{n} {\gamma^{i} }$ is a parameter that specifies the equilibrium relative stiffness of the tissue, while parameters $C_{r}^{\infty } = \gamma^{\infty } C_{r}$ (r = 1,4,6) specify equilibrium initial stiffness terms. The specific formulation of the deformation gradient F was derived by considering the orientation of the specific tensile test along longitudinal or circumferential direction, the boundary conditions, as null values of normal stress components along lateral directions, and the incompressibility constraint.10

The comparison between model results and experimental data was performed by a cost function, which specified a measure of relative error, rather than an absolute one to provide a better approximation for both low and high result values:

$$\Upomega \left( {\varvec{\upomega}} \right) = \frac{1}{u}\sqrt {\sum\limits_{j = 1}^{u} {\left[ {2 - \frac{{P_{j}^{\exp } \left( {\lambda_{j}^{\exp } } \right)}}{{P_{j}^{\bmod } \left( {{\varvec{\upomega}},\lambda_{j}^{\exp } } \right)}} - \frac{{P_{j}^{\bmod } \left( {{\varvec{\upomega}},\lambda_{j}^{\exp } } \right)}}{{P_{j}^{\exp } \left( {\lambda_{j}^{\exp } } \right)}}} \right]}^{2} }$$

(A.5)

where ω is the set of constitutive parameters, u the number of experimental data, $\lambda_{j}^{\exp }$ the jth experimentally imposed input data (in terms of strain), $P_{j}^{\exp }$ the jth experimentally measured output data (in terms of stress), $P_{j}^{\bmod }$ the model output data evaluated by assuming constitutive parameters ω and input condition $\lambda_{j}^{\exp }$. With regard to each stomach region and each wall layer, the cost function was evaluated considering results from tensile tests developed along both longitudinal and circumferential directions. Optimization techniques were adopted to minimize the cost function,7 leading to the constitutive parameters $C_{r}^{\infty } ,\alpha_{r} \;(r = 1, 4, 6)$ for connective layer and muscular layers from fundus, corpus and antrum. Data from tensile tests along circumferential and longitudinal directions only were at disposal. Aiming at the almost univocal identification of constitutive parameters,7 all the gastric tissues were assumed to have the same mechanical contribution from the isotropic ground matrix. It follows that the same values of C₁ and α₁ parameters were assumed for connective layer and muscular layers from fundus, corpus and antrum.

Viscous parameters τⁱ and γⁱ were identified by analyzing relaxation data from structural tests developed on piglet stomachs. The following model formulation was assumed to interpret exponential decay of the normalized pressure P^norm with time8,39:

$$P^{norm} \left( t \right) = \gamma^{\infty } + \sum\limits_{i = 1}^{n} {\gamma^{i} \exp \left[ { - \frac{t}{{\tau^{i} }}} \right]}$$

(A.6)

Again, parameters identification was performed by minimizing the discrepancy between experimental and model results. Two viscous branches were assumed to contemporarily minimize the number of parameters and correctly interpret the trend of experimental data.12 The same viscous parameters were assumed for all the stomach tissues. Subsequently, the instantaneous initial stiffness terms $C_{r} = C_{r}^{\infty } /\gamma^{\infty } (r = 1,4,6)$ were calculated.

The developed computational model of the stomach was exploited to evaluate and enhance its capability in interpreting the stomach mechanical behavior. Inflation tests were analyzed to compare computational results and data from experimental tests. Directly assuming the identified constitutive parameters led to imprecise results. In detail, the pressure–volume curves from computational and experimental analyses showed similar stiffness values (as the curve slope in the quasi-linear region), but at different inflation conditions. The situation is typical in the field of soft tissue mechanics, because of unsuitable post-processing of data from mechanical tests at tissue level. In detail, the tensile response of soft biological tissues shows an initial toe region and a subsequent quasi-linear tract. Experimental results are usually post-processed by low-pass filtering procedures, aiming to remove force data that are below the load cell sensitivity. The operation fundamentally concerns data from the toe region, leading to move the zero stress condition of the specimen to an actually strained specimen length. It follows a reduction of the strain amplitude of the toe region and the subsequent shift of the high stiffness quasi-linear region to improper low strain conditions.

Aiming to actually interpret the mechanical response of the stomach, constitutive parameters have been updated. Different sets of parameters were evaluated by defining a grid around the basic set, according to a variational process by using multipliers m_C and m_α of groups of parameters, as initial stiffness C_r and non linearity α_r parameters (r = 1,4,6), respectively. Each set of parameters was evaluated assuming the same multipliers for tissues from the different stomach regions and the different layers. With regard to each set, tensile loading conditions were simulated aiming to evaluate tissues tensile stiffness values in the quasi-linear region. Among all the sets of parameters, a sub-domain of multipliers m_C and m_α was identified, which provided stiffness values in the quasi-linear regions matching the experimental quasi-linear tracts. On the other side, the sets of the sub-domain provided different strain amplitudes of the toe regions. Computational analyses of stomach inflation were performed considering sets of parameters from the sub-domain. The final set of parameters (Table 1) was identified by comparing computational results with experimental data from inflation tests (Fig. 4c).