## Abstract

Aortic Dissection (AD) is a disease that affects the aorta. It develops with the formation of a secondary volume in which blood is collected. This condition is often undiagnosed because of its rapid development and degeneration. Patients with AD often die between one day and one week. In mathematical and computational models of this disease, the level of uncertainty is extremely high. One of the leading causes of this uncertainty is the lack of useful datasets and experiments that allow a better understanding of the genetic and development of AD.

Uncertainties reside mainly in the model's input parameters. They result from errors in the measurement, or precisely, from a lack of information necessary for the development of the model. Therefore, assumptions and hypotheses come to our aid, but they also bring with them a price to pay: uncertainty.

The primary task of Uncertainty Quantification (UQ) is allocating uncertainty, and understanding how and which one is meant to be reduced. This tool is also able to modify models toward a patient-specific approach, that is, the ability to extract the right information from a few selected inputs that will be provided to the model.

This project aims to use the structural reliability analysis technique for the reconstruction of a dissected aorta cross-section. The objective of structural reliability analysis is to determine the probability of failure of a system concerning a performance criterion. This criterion is influenced by the uncertainty in the input parameters to the model. Given a function g(x) that represents the model, the limit state surface can be determined as g(x) = 0, which divides the output domain between success and failure response with respect to a limit-state parameter. The probability of failure is therefore defined as the integral of the joint input PDF with respect to the input parameters, on the input domain that produces a failure status.

Some of the techniques most used to determine this integral are Monte Carlo Simulation, which, however, require a large number of simulations to obtain adequate accuracy; FORM and SORM methods based on the approximation of the model near the limit-state function; or the use of surrogate models such as Polynomial-Chaos Kriging (PCK).

The goal is to use reliability analysis to identify the points to be added to the best-fit ellipse of the dissected cross-section, which will recreate a reasonable healthy aorta cross-section. In this project, the existing algorithm developed by Echard et al. in 2011 is revisited. It can actively enrich a reduced set of model evaluations in the vicinity of the limit-state function. Furthermore, the use of Kriging methods will provide information about the variance of approximation of the model, which could be used as a form of local error. The purpose of the enrichment algorithm of the initial experimental design (ED) is to provide, together with the PCK metamodel, information on the reliability of this reconstruction by using the statistic of significant interest.

The adaptive algorithm is structured as follows. Initially, a random sampling technique generates a small ED of the input parameter, and the corresponding response is computed. A metamodel is then calibrated from this dataset. A large sampling is created, and the responses are calculated from the metamodel. The limit state surface is estimated based on the metamodel. An enrichment criterion of the initial ED based on a misclassification function will define the best candidate in the input space that has the lowest value of misclassification function; its response is also added to the initial ED. The algorithm starts again but with a new initial dataset from which the PCK is rebuilt. The stopping criterion terminates the algorithm based on the convergence of the statistic of interest of the metamodel.

The reference cross-section of the aorta is collected from a patient who was diagnosed with AD. The assumption is that the aorta best-fit geometry is an ellipse. The area and the tilt of the reference ellipse are then used as geometric criterions to set the reliability analysis, so to locate the safe and failure reconstruction responses. Physiological properties of the aorta in its longitudinal direction fix both geometric constraints. A set of points initially define the dissected aorta cross-section, which represents its segmentation. By adding one random point in space to its best-fit ellipse, its shape and orientation will change. This change will produce a new ellipse with a different area and tilt. Then, the limit-state parameter on the two geometric properties will restrict the selection of the new random point to a safe response status, avoiding the failure domain. The quantile estimation is computed after the division of the domain in safe and failure conditions and by setting the limit-state parameter equal to the quantile of interest.

In conclusion, the usage of the adaptive algorithm together with the metamodel technique of Polynomial-Chaos Kriging will define the areas in the cross-section plane in which a point can be used to enrich the dissected segmentation for its reconstruction.

Uncertainties reside mainly in the model's input parameters. They result from errors in the measurement, or precisely, from a lack of information necessary for the development of the model. Therefore, assumptions and hypotheses come to our aid, but they also bring with them a price to pay: uncertainty.

The primary task of Uncertainty Quantification (UQ) is allocating uncertainty, and understanding how and which one is meant to be reduced. This tool is also able to modify models toward a patient-specific approach, that is, the ability to extract the right information from a few selected inputs that will be provided to the model.

This project aims to use the structural reliability analysis technique for the reconstruction of a dissected aorta cross-section. The objective of structural reliability analysis is to determine the probability of failure of a system concerning a performance criterion. This criterion is influenced by the uncertainty in the input parameters to the model. Given a function g(x) that represents the model, the limit state surface can be determined as g(x) = 0, which divides the output domain between success and failure response with respect to a limit-state parameter. The probability of failure is therefore defined as the integral of the joint input PDF with respect to the input parameters, on the input domain that produces a failure status.

Some of the techniques most used to determine this integral are Monte Carlo Simulation, which, however, require a large number of simulations to obtain adequate accuracy; FORM and SORM methods based on the approximation of the model near the limit-state function; or the use of surrogate models such as Polynomial-Chaos Kriging (PCK).

The goal is to use reliability analysis to identify the points to be added to the best-fit ellipse of the dissected cross-section, which will recreate a reasonable healthy aorta cross-section. In this project, the existing algorithm developed by Echard et al. in 2011 is revisited. It can actively enrich a reduced set of model evaluations in the vicinity of the limit-state function. Furthermore, the use of Kriging methods will provide information about the variance of approximation of the model, which could be used as a form of local error. The purpose of the enrichment algorithm of the initial experimental design (ED) is to provide, together with the PCK metamodel, information on the reliability of this reconstruction by using the statistic of significant interest.

The adaptive algorithm is structured as follows. Initially, a random sampling technique generates a small ED of the input parameter, and the corresponding response is computed. A metamodel is then calibrated from this dataset. A large sampling is created, and the responses are calculated from the metamodel. The limit state surface is estimated based on the metamodel. An enrichment criterion of the initial ED based on a misclassification function will define the best candidate in the input space that has the lowest value of misclassification function; its response is also added to the initial ED. The algorithm starts again but with a new initial dataset from which the PCK is rebuilt. The stopping criterion terminates the algorithm based on the convergence of the statistic of interest of the metamodel.

The reference cross-section of the aorta is collected from a patient who was diagnosed with AD. The assumption is that the aorta best-fit geometry is an ellipse. The area and the tilt of the reference ellipse are then used as geometric criterions to set the reliability analysis, so to locate the safe and failure reconstruction responses. Physiological properties of the aorta in its longitudinal direction fix both geometric constraints. A set of points initially define the dissected aorta cross-section, which represents its segmentation. By adding one random point in space to its best-fit ellipse, its shape and orientation will change. This change will produce a new ellipse with a different area and tilt. Then, the limit-state parameter on the two geometric properties will restrict the selection of the new random point to a safe response status, avoiding the failure domain. The quantile estimation is computed after the division of the domain in safe and failure conditions and by setting the limit-state parameter equal to the quantile of interest.

In conclusion, the usage of the adaptive algorithm together with the metamodel technique of Polynomial-Chaos Kriging will define the areas in the cross-section plane in which a point can be used to enrich the dissected segmentation for its reconstruction.

Original language | English |
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Publication status | Published - 28 Oct 2019 |

Event | Ninth International Conference on Sensitivity Analysis of Model Output - Universitat Oberta de Catalunya, Edifici B3, Parc Mediterrani de la Tecnologia, Avinguda Carl Friedrich Gauss, 5, 08860 Casteldefels, Barcelona, Barcelon, Spain Duration: 28 Oct 2019 → 31 Oct 2019 http://symposium.uoc.edu/23220/detail/ninth-international-conference-on-sensitivity-analysis-of-model-output.html |

### Conference

Conference | Ninth International Conference on Sensitivity Analysis of Model Output |
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Abbreviated title | SAMO |

Country/Territory | Spain |

City | Barcelon |

Period | 28/10/19 → 31/10/19 |

Internet address |

## Keywords

- sensitivity analysis
- Aortic dissection (AD)
- pce
- reliability analysis