# 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

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**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