“…Figure partially adapted from Vander Linden and colleagues. 19 Figure E2 Histogram showing the PWS ii values resulting from the probabilistic approach ( blue ) with the fitted probability density function ( solid line ) and the deterministic wall strength Y ii ( dotted line ). The corresponding probabilistic failure risk is −56.0 and −13.8 for i=θ ( left ) and i=z ( right ), respectively.…”
Section: Appendix E4 Results Of the Probabilistic Failure Risk Assess...mentioning
confidence: 99%
“…Calculation of the intraluminal maximal wall stress requires information on the material behavior and in vivo wall thickness of the tissue. The material parameters s and wall thickness at diastole H dias were estimated using a material fitting approach explained in ( 18 , 19 ), by minimizing the difference between (1) the experimental ( f ii ) and model ( ) reaction forces in the direction i=θ,z of the planar biaxial test, (2) the measured ( H 0 ) and model ( ) ex vivo thickness and (3) the measured ( p sys ) and model ( ) systolic blood pressure ( Figure 3 ). Knowing the material parameters and in vivo thickness, the PWS ii , first Piola–Kirchhoff peak wall stress in the direction i at supra-systolic blood pressure, was calculated while approximating the aneurysm as a thin-walled cylinder.…”
Section: Methodsmentioning
confidence: 99%
“…F biax = F LC F p , where F p is the deformation caused by the preloading of the sample before the loading cycle, see Vander Linden and colleagues. 19 No shear deformation during biaxial testing is assumed: …”
Section: Mechanical Analysismentioning
confidence: 99%
“…λ p,θθ and λ p,zz are derived from the condition that at each ratio r , using the nonlinear equation solver ‘fsolve’ in Matlab R2023a. 19 …”
“…Figure partially adapted from Vander Linden and colleagues. 19 Figure E2 Histogram showing the PWS ii values resulting from the probabilistic approach ( blue ) with the fitted probability density function ( solid line ) and the deterministic wall strength Y ii ( dotted line ). The corresponding probabilistic failure risk is −56.0 and −13.8 for i=θ ( left ) and i=z ( right ), respectively.…”
Section: Appendix E4 Results Of the Probabilistic Failure Risk Assess...mentioning
confidence: 99%
“…Calculation of the intraluminal maximal wall stress requires information on the material behavior and in vivo wall thickness of the tissue. The material parameters s and wall thickness at diastole H dias were estimated using a material fitting approach explained in ( 18 , 19 ), by minimizing the difference between (1) the experimental ( f ii ) and model ( ) reaction forces in the direction i=θ,z of the planar biaxial test, (2) the measured ( H 0 ) and model ( ) ex vivo thickness and (3) the measured ( p sys ) and model ( ) systolic blood pressure ( Figure 3 ). Knowing the material parameters and in vivo thickness, the PWS ii , first Piola–Kirchhoff peak wall stress in the direction i at supra-systolic blood pressure, was calculated while approximating the aneurysm as a thin-walled cylinder.…”
Section: Methodsmentioning
confidence: 99%
“…F biax = F LC F p , where F p is the deformation caused by the preloading of the sample before the loading cycle, see Vander Linden and colleagues. 19 No shear deformation during biaxial testing is assumed: …”
Section: Mechanical Analysismentioning
confidence: 99%
“…λ p,θθ and λ p,zz are derived from the condition that at each ratio r , using the nonlinear equation solver ‘fsolve’ in Matlab R2023a. 19 …”
“…Our constitutive neural network consistently discovers second-invariant models. For decades, the gold standard in constitutive modeling has been to first select a constitutive model and then fit the model to data [5,6,17,32,60,69]. Attempts to improve the goodness of fit have slightly adjusted the terms of the model, and gradually modified or replaced individual terms [11].…”
For more than half a century, scientists have developed mathematical models to understand the behavior of the human heart. Today, we have dozens of heart tissue models to choose from, but selecting the best model is limited to expert professionals, prone to user bias, and vulnerable to human error. Here we take the human out of the loop and automate the process of model discovery. Towards this goal, we establish a novel incompressible orthotropic constitutive neural network to simultaneously discover both, model and parameters, that best explain human cardiac tissue. Notably, our network features 32 individual terms, 8 isotropic and 24 anisotropic, and fully autonomously selects the best model, out of more than 4 billion possible combinations of terms. We demonstrate that we can successfully train the network with triaxial shear and biaxial extension tests and systematically sparsify the parameter vector withL1-regularization. Strikingly, we robustly discover a four-term model that features a quadratic term in the second invariantI2, and exponential quadratic terms in the fourth and eighth invariantsI4f,I4n, andI8fs. Importantly, our discovered model is interpretable by design and has parameters with well-defined physical units. We show that it outperforms popular existing myocardium models and generalizes well, from homogeneous laboratory tests to heterogeneous whole heart simulations. This is made possible by a new universal material subroutine that directly takes the discovered network weights as input. Automating the process of model discovery has the potential to democratize cardiac modeling, broaden participation in scientific discovery, and accelerate the development of innovative treatments for cardiovascular disease.Our source code, data, and examples are available athttps://github.com/LivingMatterLab/CANN.
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