# A Data-Driven Learning Method for Constitutive Modeling: Application to Vascular Hyperelastic Soft Tissues

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## Abstract

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## 1. Introduction

## 2. Material and Methods

#### 2.1. A GENERIC Approach to the Learning Procedure

#### 2.2. Treatment of Dispersion and Noise in Data

#### 2.3. Pseudo-Experimental Data—Learning a Visco-Hyperelastic Response

**A mean GENERIC model**. A regression procedure is then accomplished for each one of the 50 different experiments, so as to determine their precise GENERIC expression. With the obtained values, we first compute the mean GENERIC model by simply taking mean values for each one of the GENERIC model components. This “mean” GENERIC model is compared to the noise-free numerical experiment, taken as ground truth.

**Extracting the topology of data: GENERIC-TDA model**. Instead of just computing the mean values of each term of the GENERIC model, it seems judicious to employ TDA to unveil the topology of data and to determine the final GENERIC model by interpolating from the right neighboring experimental results. To this end, we considered a data set composed by 20 different loading states (all consisting of a load-relaxation-load-relaxation sequence) and the addition of noise (10% sdv) to each one of these processes, so as to obtain 50 different tests for each one of the 20 loading processes. This makes a total of one thousand different tests.

#### 2.4. Learning the Constitutive Model of Porcine Carotid Tissue

**Experimental tests**. To introduce to the reader the most significant details in the experimental models used for our numerical analysis, here we show a brief description of the sample’s harvesting and tensile test protocols performed in [14]. The interested reader is referred to this article for a precise description of the experimental campaign.

## 3. Results

#### 3.1. Numerical Fitting of the Pseudo-Experimental Data Set

#### 3.2. Numerical Fitting of Porcine Carotid Tissue

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Kirchdoerfer, T.; Ortiz, M. Data-driven computational mechanics. Comput. Methods Appl. Mech. Eng.
**2016**, 304, 81–101. [Google Scholar] [CrossRef] [Green Version] - Brunton, S.L.; Proctor, J.L.; Kutz, J.N. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl. Acad. Sci. USA
**2016**. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Liu, Z.; Bessa, M.; Liu, W.K. Self-consistent clustering analysis: An efficient multi-scale scheme for inelastic heterogeneous materials. Comput. Methods Appl. Mech. Eng.
**2016**, 306, 319–341. [Google Scholar] [CrossRef] - Ibañez, R.; Borzacchiello, D.; Aguado, J.V.; Abisset-Chavanne, E.; Cueto, E.; Ladeveze, P.; Chinesta, F. Data-driven non-linear elasticity: Constitutive manifold construction and problem discretization. Comput. Mech.
**2017**, 60, 813–826. [Google Scholar] [CrossRef] - Latorre, M.; Montáns, F.J. What-You-Prescribe-Is-What-You-Get orthotropic hyperelasticity. Comput. Mech.
**2014**, 53, 1279–1298. [Google Scholar] [CrossRef] - Bessa, M.; Bostanabad, R.; Liu, Z.; Hu, A.; Apley, D.W.; Brinson, C.; Chen, W.; Liu, W.K. A framework for data-driven analysis of materials under uncertainty: Countering the curse of dimensionality. Comput. Methods Appl. Mech. Eng.
**2017**, 320, 633–667. [Google Scholar] [CrossRef] - Ibanez, R.; Abisset-Chavanne, E.; Aguado, J.V.; Gonzalez, D.; Cueto, E.; Chinesta, F. A manifold learning approach to data-driven computational elasticity and inelasticity. Arch. Comput. Methods Eng.
**2018**, 25, 47–57. [Google Scholar] [CrossRef] [Green Version] - Latorre, M.; Peña, E.; Montáns, F.J. Determination and Finite Element Validation of the WYPIWYG Strain Energy of Superficial Fascia from Experimental Data. Ann. Biomed. Eng.
**2017**, 45, 799–810. [Google Scholar] [CrossRef] [Green Version] - Cilla, M.; Martinez, J.; Pena, E.; Martínez, M.Á. Machine Learning Techniques as a Helpful Tool Toward Determination of Plaque Vulnerability. IEEE Trans. Biomed. Eng.
**2012**, 59, 1155–1161. [Google Scholar] [CrossRef] - Kevrekidis, I.G.; Gear, C.W.; Hummer, G. Equation-free: The computer-aided analysis of complex multiscale systems. AIChE J.
**2004**, 50, 1346–1355. [Google Scholar] [CrossRef] - Kirchdoerfer, T.; Ortiz, M. Data Driven Computing with noisy material data sets. Comput. Methods Appl. Mech. Eng.
**2017**, 326, 622–641. [Google Scholar] [CrossRef] [Green Version] - Ayensa-Jiménez, J.; Doweidar, M.H.; Sanz-Herrera, J.A.; Doblaré, M. A new reliability-based data-driven approach for noisy experimental data with physical constraints. Comput. Methods Appl. Mech. Eng.
**2018**, 328, 752–774. [Google Scholar] [CrossRef] [Green Version] - García, A.; Martínez, M.A.; Peña, E. Viscoelastic properties of the passive mechanical behavior of the porcine carotid artery: Influence of proximal and distal positions. Biorheology
**2012**, 49, 271–288. [Google Scholar] [CrossRef] [PubMed] - García, A.; Peña, E.; Laborda, A.; Lostalé, F.; De Gregorio, M.; Doblaré, M.; Martínez, M. Experimental study and constitutive modelling of the passive mechanical properties of the porcine carotid artery and its relation to histological analysis: Implications in animal cardiovascular device trials. Med. Eng. Phys.
**2011**, 33, 665–676. [Google Scholar] [CrossRef] - Holzapfel, G.A.; Gasser, T.C. A new constitutive framework for arterial wall mechanics and a comparative study of material models. J. Elast.
**2000**, 61, 1–48. [Google Scholar] [CrossRef] - Pandolfi, A.; Holzapfel, G.A. Three-Dimensional Modeling and Computational Analysis of the Human Cornea Considering Distributed Collagen Fibril Orientations. J. Biomech. Eng.
**2008**, 130, 061006. [Google Scholar] [CrossRef] [Green Version] - Peña, J.; Martínez, M.; Peña, E. A formulation to model the nonlinear viscoelastic properties of the vascular tissue. Acta Mech.
**2011**, 217, 63–74. [Google Scholar] [CrossRef] - Holzapfel, G.A.; Sommer, G.; Gasser, C.T.; Regitnig, P. Determination of layer-specific mechanical properties of human coronary arteries with nonatherosclerotic intimal thickening and related constitutive modeling. Am. J. Physiol.-Heart Circ. Physiol.
**2005**, 289, H2048–H2058. [Google Scholar] [CrossRef] [Green Version] - Gasser, T.C.; Ogden, R.W.; Holzapfel, G.A. Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. J. R. Soc. Interface
**2005**, 3, 15–35. [Google Scholar] [CrossRef] - Holzapfel, G.A.; Ogden, R.W. Constitutive modelling of arteries. Proc. R. Soc. A Math. Phys. Eng. Sci.
**2010**, 466, 1551–1597. [Google Scholar] [CrossRef] - Ibañez, R.; Abisset-Chavanne, E.; Gonzalez, D.; Duval, J.; Cueto, E.; Chinesta, F. Hybrid constitutive modeling: Data-driven learning of corrections to plasticity models. Int. J. Mater. Form.
**2018**, in press. [Google Scholar] - Eggersmann, R.; Kirchdoerfer, T.; Reese, S.; Stainier, L.; Ortiz, M. Model-free data-driven inelasticity. Comput. Methods Appl. Mech. Eng.
**2019**, 350, 81–99. [Google Scholar] [CrossRef] [Green Version] - Zhang, M.; Benítez, J.M.; Montáns, F.J. Capturing yield surface evolution with a multilinear anisotropic kinematic hardening model. Int. J. Solids Struct.
**2016**, 81, 329–336. [Google Scholar] [CrossRef] - Liang, L.; Liu, M.; Sun, W. A deep learning approach to estimate chemically-treated collagenous tissue nonlinear anisotropic stress-strain responses from microscopy images. Acta Biomater.
**2017**, 63, 227–235. [Google Scholar] [CrossRef] - Raissi, M.; Perdikaris, P.; Karniadakis, G.E. Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations. arXiv
**2017**, arXiv:1711.10561. [Google Scholar] - Wang, J.X.; Wu, J.L.; Xiao, H. Physics-informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data. Phys. Rev. Fluids
**2017**, 2, 034603. [Google Scholar] [CrossRef] [Green Version] - Russo, A.; Durán-Olivencia, M.A.; Kevrekidis, I.G.; Kalliadasis, S. Deep learning as closure for irreversible processes: A data-driven generalized Langevin equation. arXiv
**2019**, arXiv:1903.09562. [Google Scholar] - Grmela, M.; Öttinger, H.C. Dynamics and thermodynamics of complex fluids. I. Development of a general formalism. Phys. Rev. E
**1997**, 56, 6620–6632. [Google Scholar] [CrossRef] - Öttinger, H.C. Beyond Equilibrium Thermodynamics; John Wiley Sons, Inc.: Hoboken, NJ, USA, 2005. [Google Scholar]
- Pavelka, M.; Klika, V.; Grmela, M. Multiscale thermodynamics; De Gruyter: Berlin, Germany, 2018. [Google Scholar]
- González, D.; Chinesta, F.; Cueto, E. Thermodynamically consistent data-driven computational mechanics. Contin. Mech. Thermodyn.
**2018**. submitted. [Google Scholar] - Gonzalez, D.; Chinesta, F.; Cueto, E. Learning corrections for hyperelastic models from data. Front. Mater.
**2019**, 6, 14. [Google Scholar] [CrossRef] [Green Version] - Weinan, E. A Proposal on Machine Learning via Dynamical Systems. Commun. Math. Stat.
**2017**, 5, 1–11. [Google Scholar] [CrossRef] - Li, Q.; Chen, L.; Tai, C.; Weinan, E. Maximum Principle Based Algorithms for Deep Learning. J. Mach. Learn. Res.
**2018**, 18, 1–29. [Google Scholar] - Español, P. Statistical Mechanics of Coarse-Graining. In Novel Methods in Soft Matter Simulations; Karttunen, M., Lukkarinen, A., Vattulainen, I., Eds.; Springer: Berlin/Heidelberg, Germany, 2004; pp. 69–115. [Google Scholar] [CrossRef]
- Lopez, E.; Gonzalez, D.; Aguado, J.V.; Abisset-Chavanne, E.; Cueto, E.; Binetruy, C.; Chinesta, F. A Manifold Learning Approach for Integrated Computational Materials Engineering. Arch. Comput. Methods Eng.
**2016**, 1–10. [Google Scholar] [CrossRef] [Green Version] - Romero, I. A characterization of conserved quantities in non-equilibrium thermodynamics. Entropy
**2013**, 15, 5580–5596. [Google Scholar] [CrossRef] [Green Version] - Wasserman, L. Topological Data Analysis. Annu. Rev. Stat. Its Appl.
**2018**, 5, 501–532. [Google Scholar] [CrossRef] [Green Version] - Munch, E. A User’s Guide to Topological Data Analysis. J. Learn. Anal.
**2017**, 4, 47–61. [Google Scholar] [CrossRef]

**Figure 1.**Hypothesis about the existence of a constitutive manifold in which the experimental results live. Despite the noise in the data, Topological Data Analysis (TDA) techniques will help us in unveiling the true geometry of the manifold in the high-dimensional setting, which will later be embedded onto a low dimensional space for the ease of computations.

**Figure 2.**Interpretation of TDA. Orange circles have diameter R, the topology parameter. Below each simplicial complex, the bar code corresponding to dimension 1 holes) is represented. (

**a**) For a sufficiently small R parameter, say, 0.1, only a collection of data points is visible, with no topology at dimension 1. (

**b**) Increasing R to 1.0 makes the first circular topology appear. This hole is visible for R > 0.8, hence the bar in the diagram from R = 0.8 to R = 1.0. (

**c**) If we increase R, no perceptible changes are observed. Only one hole persists and it is reflected in the bar code below. (

**d**) The hole disappears by formation of big triangles at about R = 2.8. From the observation of the bar code, we notice that the data set has the topology of a circle, with one single interior hole. In general, those holes or voids that persist the most reflect the persistent topology of the data.

**Figure 3.**Experimental setup. Instron Microtester 5548 System with two clamps holding the sample. The samples are subjected to the tensile test under a humidity-controlled environment to prevent sample drying.

**Figure 4.**Experimental (

**a**) distal-circumferential, (

**b**) distal-longitudinal, (

**c**) proximal-circumferential, and (

**d**) proximal-longitudinal stress–stretch curves. The continuous blue line represents the mean values of the 14 experimental results, while the dashed lines represent the neighboring experiments, as found by TDA techniques.

**Figure 5.**Comparison of (

**a**) horizontal and (

**b**) vertical displacement predicted by a General Equation for the Nonequilibrium Reversible-Irreversible Coupling (GENERIC) model obtained as the mean of 50 different noisy GENERIC models. Comparison with the noise-free reference solution in continuous blue line.

**Figure 6.**Comparison of (

**a**) horizontal and (

**b**) vertical displacement predicted by a GENERIC model obtained as the mean of 50 different noisy GENERIC models. Comparison with the noise-free reference solution in continuous blue line and the solutions obtained by Kriging interpolation between neighbors predicted by Topological Data Analysis.

**Figure 7.**Comparison of (

**a**) distal-circumferential and (

**b**) distal-longitudinal models predicted by mean GENERIC values, or by Kriging interpolation of those samples neighboring the reference solution.

**Figure 8.**Comparison of (

**a**) proximal-circumferential and (

**b**) proximal-longitudinal models predicted by mean GENERIC values, or by Kriging interpolation of those samples neighboring the reference solution.

Mean GENERIC | 2.24% |

Simple Kriging | 16.46% |

Ordinary Kriging | 1.82% |

Local Kriging | 0.38% |

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**MDPI and ACS Style**

González, D.; García-González, A.; Chinesta, F.; Cueto, E.
A Data-Driven Learning Method for Constitutive Modeling: Application to Vascular Hyperelastic Soft Tissues. *Materials* **2020**, *13*, 2319.
https://doi.org/10.3390/ma13102319

**AMA Style**

González D, García-González A, Chinesta F, Cueto E.
A Data-Driven Learning Method for Constitutive Modeling: Application to Vascular Hyperelastic Soft Tissues. *Materials*. 2020; 13(10):2319.
https://doi.org/10.3390/ma13102319

**Chicago/Turabian Style**

González, David, Alberto García-González, Francisco Chinesta, and Elías Cueto.
2020. "A Data-Driven Learning Method for Constitutive Modeling: Application to Vascular Hyperelastic Soft Tissues" *Materials* 13, no. 10: 2319.
https://doi.org/10.3390/ma13102319