In this work we propose a mathematical model that describes the orientation of ventricular cardiac fibers. These fibers are commonly computed as the normalized gradient of certain harmonic potentials, so our work consisted in finding the equations that such a vector field satisfies, considering the unitary norm constraint. The resulting equations belong to the Frank-Oseen theory of nematic liquid crystals, which yield a bulk of mathematical properties to the cardiac fibers, such as the characterization of singularities. The numerical methods available in literature are computationally expensive and not sufficiently robust for the complex geometries obtained from the human heart, so we also propose a preconditioned projected gradient descent scheme that circumvents these difficulties in the tested scenarios. The resulting model further confirms recent experimental observations of liquid crystal behavior of soft tissue, and provides an accurate mathematical description of such behavior.

翻译：本文提出一种描述心室心肌纤维定向的数学模型。这些纤维通常被计算为某些调和势函数的归一化梯度，因此我们的工作在于找到满足单位范数约束条件的矢量场所满足的方程。所得方程属于向列型液晶的Frank-Oseen理论，为心肌纤维提供了丰富的数学特性，例如奇异点的表征。现有文献中的数值方法计算成本高昂，且对从人体心脏获得的复杂几何形状不够鲁棒，因此我们还提出了一种预处理投影梯度下降方案，在测试场景中克服了这些困难。所得模型进一步证实了近期关于软组织液晶行为的实验观测结果，并为该行为提供了精确的数学描述。