Computation of a probabilistic and anisotropic failure metric on the aortic wall using a machine learning-based surrogate model
Introduction
Thoracic aortic aneurysm (TAA) is a life-threatening condition that can lead to rupture and dissection of the aorta. If left untreated, the five-year survival rate is 54% [1]. The pathology of the thoracic aorta can be eradicated by elective surgery or endovascular treatment. Aortic size-based criterion is currently used to triage patients to surgery, which is often recommended for patients with TAA diameter larger than 5–5.5 cm [2,3]. However, the aortic size may not accurately reflect the risk for all patients [4,5]: some small aneurysms (e.g., <4 cm) have been known to rupture [5]. As adverse events of aneurysms (i.e., rupture and dissection) are mechanics-driven, aortic wall stresses, computed from patient-specific finite element analysis (FEA) models, can provide more valuable insights into TAA risk [6,7]. To assess the risk of TAA rupture and dissection, the FEA-computed stress components are typically converted into a scalar-valued index (i.e., failure metric).
Isotropic failure indices are often employed in patient-specific models to assess the risk of aortic aneurysms, including the von Mises equivalent stress [8], the maximum principal stress (MPS) [9], and rupture potential index (RPI) [10]. Since failure strengths of aortic tissue are highly directional dependent (i.e., anisotropic) [[11], [12], [13], [14], [15], [16]], these isotropic metrics may not be appropriate for the aortic wall [17]. Experimental works have also revealed large variations of aortic wall strengths [11,13,14,16], which suggest that the probability distribution of wall strengths can be incorporated for a more accurate failure metric. In a recent study [18], we proposed a probabilistic and anisotropic failure metric, namely the failure probability (FP), by using tissue failure testing data of 84 ascending thoracic aortic aneurysm (ATAA) patients. Using synthetic data, the novel FP metric demonstrated a better discriminative capability than that of isotropic or deterministic failure metrics.
Since TAA dissection/rupture usually occurs under hypertensive blood pressures brought on by extreme emotional or physical stress [19,20], failure metric is typically evaluated under an elevated blood pressure in patient-specific FEA models [7,21]. In a classical patient-specific model, computation of failure metrics consists of the following steps: (1) segmentation, registration, and mesh generation of the aortic wall at multiple phases (e.g., systole and diastole) from clinical images; (2) identification of unknown patient-specific hyperelastic material parameters from multi-phase aorta geometries by using an optimization-based inverse method [[22], [23], [24], [25], [26]]; (3) computing wall stresses at an elevated blood pressure using nonlinear FEA simulation [7,21]; (4) post-processing to convert the FEA-computed stress distributions into a scalar-valued failure metric distribution; and (5) capturing the maximum failure metric on the aortic wall. However, these classical procedures may not be suitable for clinical applications that require prompt feedback because: (i) manual segmentation and mesh generation requires heavy human involvements, which are labor-intensive and time-consuming; (ii) optimization-based inverse methods are iterative in nature and thus computationally expensive, which can take hours to weeks to complete for a single patient [[22], [23], [24], [25], [26]]; (iii) FEA stress computation and post-processing can also take minutes to hours for one patient; and (iv) convergence issues can occur in nonlinear FEA simulations. Hence, the computation of failure metrics using the classical methods is impractical in time-sensitive clinical settings.
Machine learning (ML) and deep learning (DL) approaches have led to great success in many real-world applications [[27], [28], [29], [30], [31]]. Recently, DL-based algorithms have shed light on automatic image segmentation and mesh generation in real-time [29,[32], [33], [34]]. To alleviate the high computational cost of classical methods, data-driven techniques, in particular ML algorithms, have been recently employed as surrogate models to accelerate traditional computation procedures. For instance, deep learning approaches were proposed by our group and others as fast and accurate surrogates of FEA-based stress analysis [[35], [36], [37]] and computational fluid dynamics (CFD) based hemodynamic analysis [38]. ML-models were developed to predict abdominal aortic aneurysm expansion as surrogates of growth and remodeling approaches [39,40]. In a previous study [41], we developed an ML approach to rapidly (i.e., within 1 s) identify hyperelastic parameters from two-phase aorta geometries. An ML-based surrogate model can be trained to establish the nonlinear and complex relationship between inputs and outputs, which can fundamentally solve the computational cost challenge. By supervised training on a large FEA simulation dataset, an ML-surrogate model may be developed to compute failure metrics directly from multi-phase aorta geometries, bypassing the abovementioned classical steps (2)–(5). Once trained, the ML-surrogate can enable real-time prediction of the failure metric.
In this paper, we developed an ML-based surrogate model to compute the probabilistic and anisotropic FP metric. As shown in Fig. 1, the ML-model used the same inputs as in the classical patient-specific FEA model, i.e., the aorta geometries at the systolic and diastolic phases. Our previously proposed ML framework [41] was extended for direct failure metric computation, and the hyperelastic material parameter identification, as well as the nonlinear FEA simulation, can be bypassed. The systolic and diastolic geometries were encoded into shape codes using principal component analysis. The failure metrics at different quantiles were predicted from shape codes using a fully connected neural network. CT-derived geometries of 60 ATAA patients and tissue testing data of 79 patients were obtained. FEA simulations were used to generate datasets for training, validation, and testing of the ML-surrogate model. The predictive accuracy of the ML-surrogate was examined. To compare the performance of the ML-predicted FP metrics with other failure metrics, a numerical case study was performed using synthetic “baseline” data.
Section snippets
Representative ATAA geometries and material properties
To train, validate, and test the ML-surrogate model, FEA simulation data were needed. By following the same procedures established in Ref. [41], we first generated representative aorta geometries from clinical images of real patients by using a statistical shape model (SSM). We sampled representative sets of hyperelastic parameters from experimental tissue testing data by constructing a convex hull in the material parameter space. We extended the methods in Ref. [41] by incorporating CT-derived
Prediction of the FP metric
The number of units of the neural network was determined using grid search. Ten-fold cross-validations were performed to evaluate the NMAE and NSTAE with 3000 epochs, the results are shown in Table 2. The lowest NMAE was achieved by using 2 hidden layers and 256 units in each layer.
Using the 15623 training and validation cases, the neural network was trained with 10,000 epochs in about 30 min on a single GPU. Given a pair of systolic and diastolic geometries as the inputs, the trained ML-model
Discussion
In this study, we developed a novel end-to-end ML approach that enabled fast and accurate prediction of a probabilistic failure metric on the aortic wall. Comparing with classical procedures to obtain the FP metric (i.e., optimization-based material parameter identification [[22], [23], [24], [25], [26]] and nonlinear FEA [7,21]), the ML approach can fundamentally resolve the computation cost challenge. Using aorta geometries at two cardiac phases, different quantiles of the FP metric under
Conclusion
We proposed an end-to-end ML-based surrogate model to predict a probabilistic and anisotropic failure (FP) metric on the aortic wall from in vivo geometries at two cardiac phases and given blood pressures. FEA simulations were employed to generate the training, validation, and testing datasets. The proposed ML-based predictive model accelerated the computation of failure metrics, bypassing the traditional process that requires both inverse identification and forward computation. The trained
Declaration of competing interest
Dr. Wei Sun serves as the Chief Scientific Advisor of Dura Biotech. He has received compensation and owns equity in the company.
Acknowledgements
This study is supported by American Heart Association (AHA) 18TPA34230083.
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