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Uncertainty in Multiclass Classification

Dr. Rishabh Pathak
1. What is Uncertainty in Classification? Uncertainty refers to the model’s confidence or doubt in its predictions. Quantifying uncertainty is important to understand how reliable each prediction is. In multiclass classification , uncertainty estimates provide probabilities over multiple classes, reflecting how sure the model is about each possible class. 2. Methods to Estimate Uncertainty in Multiclass Classification Most multiclass classifiers provide methods such as: predict_proba: Returns a probability distribution across all classes. decision_function: Returns scores or margins for each class (sometimes called raw or uncalibrated confidence scores). The probability distribution from predict_proba captures the uncertainty by assigning a probability to each class. 3. Shape and Interpretation of predict_proba in Multiclass Output shape: (n_samples, n_classes) Each row corresponds to the probabilities of ...
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The Mathematical Models used in 3D-0D closed-loop model for the simulation of cardiac biventricular electromechanics.

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Dr. Rishabh Pathak


 

The mathematical models to the reconstruction of cardiac muscle fiber architecture in biventricular geometries and the development of a 3D cardiac electromechanical (EM) model coupled with a 0D closed-loop model for the cardiovascular system. Here is an overview of the mathematical models discussed in the document:

 1. Fiber Generation Methods: The document outlines the methods used to reconstruct the cardiac muscle fiber architecture in biventricular geometries. Specifically, Laplace-Dirichlet-Rule-Based-Methods (LDRBMs) are employed to generate realistic fiber orientations within the heart. These methods involve solving Laplace boundary-value problems to determine the orientation of myocardial fibers based on boundary conditions on the heart's surfaces.

 2. 3D Cardiac EM Model: The document presents a detailed 3D cardiac electromechanical model that captures the biophysical processes involved in heart function. This model integrates aspects of electrophysiology, active contraction of cardiomyocytes, tissue mechanics, and blood circulation within the heart chambers. By considering these components, the model can simulate the electromechanical behavior of the heart in a comprehensive manner.

 3. 0D Closed-Loop Model: In addition to the 3D cardiac EM model, the document discusses the incorporation of a 0D closed-loop model for the cardiovascular system. This model represents the hemodynamics of the entire circulatory system using lumped parameters to simulate blood flow dynamics, pressure-volume relationships, and systemic interactions. The coupling of the 3D EM model with the 0D closed-loop model enables a holistic simulation of the heart's electromechanical activity in the context of circulatory dynamics.

 4. Numerical Approximation: The document also covers the numerical discretization strategies employed to solve the coupled 3D-0D model. This includes space and time discretizations using the Finite Element Method (FEM) with different mesh resolutions to handle the varying scales of electromechanical and hemodynamic processes. The Segregated-Intergrid-Staggered (SIS) approach is utilized to sequentially solve the core models contributing to cardiac EM and blood circulation.

 Overall, the mathematical models presented in the document provide a framework for simulating biventricular electromechanics and studying the complex interactions between the heart and the circulatory system. These models enable researchers to investigate cardiac function, electromechanical behavior, and hemodynamic responses in a comprehensive and integrated manner.


Piersanti, R., Regazzoni, F., Salvador, M., Corno, A. F., Dede', L., Vergara, C., & Quarteroni, A. (2021). 3D-0D closed-loop model for the simulation of cardiac biventricular electromechanics. *arXiv preprint arXiv:2108.01907*.

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