^{1,2}

^{2}

^{1}

^{2}

^{1}

^{2}

This paper proposes a solid model based on four-dimensional trivariate B-spline for strain and stress analysis of ventricular myocardium. With a series of processing steps in the four-dimensional medical images, the feature points of ventricular inner and outer wall are obtained. A B-spline surface is then used to build the dynamic deformation model of the myocardial walls. With such a surface model, a hexahedron control mesh can be constructed by sweeping the cloud data, and the ventricular solid model is built by fitting the trivariate B-spline parameters. Based on these models, a method of isogeometric analysis can be applied to calculate the stress and strain continuously distributed in the ventricle. The model is represented smoothly in the cylindrical coordinate system and is easy to measure myocardium dynamics for finding abnormal motion. Experiments are carried out for comparing the stress and strain distribution. It is found that the solid model can determine ventricular dynamics which can well reflect the deformation distribution in the heart and imply early clues of cardiac diseases.

Cardiovascular diseases are currently the leading cause of death in the world, and the rate of death is increasing each year in many countries. Therefore, more and more physiologists and researchers make efforts to understand how the heart works and how to diagnosis and treat the heart diseases. Dynamics and kinetics of the left cardiac ventricle are the primary representation of the cardiac motion, as a series of systolic and diastolic motions of the left cardiac ventricle make the heart pump blood to circulate the whole body. Furthermore, the stress and strain express the characteristics of elasticity and motion of myocardial walls [

With assist of computer medical imaging technology, such as magnetic resonance imaging (MRI), computed tomography (CT), positron emission tomography, single photon emission computed tomography, ultrasound, and X-ray, these imaging techniques can give different representation of the heart. In this paper, the endocardium and epicardium shapes of a ventricle are constructed with a segmentation technique in MRI images. For constructing model, finite element (FE) models are the most commonly used in biomechanics. Especially, hexahedral and tetrahedral FE models are popular in the representation of cardiac model, for example, Figure

Finite element models of left ventricle in the literature [

The appearance of isogeometric analysis makes it possible to carry out mechanical analysis directly by a CAD model. As a new computational technique, isogeometric analysis improves on and generalizes the standard FE method and has been proven to be a powerful method exceeding the FE method [

One can get a B-spline curve of order

The definition of

A volumetric B-spline can also be defined by

(1) Coordinate transform. A

Stress and strain actually reflect elasticity of ventricular myocardial walls [

Three-dimensional stress and strain components.

The relation between strain and displacement is

We regard B-spline basis functions as the displacement function and also the basis function of the ventricular shape, to calculate the stress and strain. Solid models at adjacent time points can be reconstructed by (

By elasticity theory, the strain of one point in the model can be calculated

Then the derivatives of B-spline basis functions in

Both inside and outside data points of ventricular myocardial walls can be obtained from 3D medical images [

(a) Ventricular point distribution model (b) B-spline surface model.

A sweeping method [

(a) Left ventricular inside wall quadrilateral mesh, (b) hexahedral control mesh.

Stress and strain of a ventricle are calculated based on the steps described in the above sections. Here, for calculation of the stress results, we set myocardial elastic modulus 11 Kpa and Poisson’s ratio of 0.49. Figure

Principal strain distribution in

Principal stress distribution in

With the situation that traditional finite element methods are difficult for direct use in mechanics analysis which has continuous distribution in space, this paper proposes a solid B-spline model to construct a continuous ventricular mechanical model and applies isogeometric analysis. Stress and strain calculative formulas are derived. The proposed model features continuous, smooth, and inseparable. According to a set of ventricular hexahedral solid B-spline models sampled at different times in a cardiac cycle, strain and stress are determined for medical analysis.

Thanks are due to Z. Xu, T. T. Jiang, Y. Q. Xu, Y. H. Du, et al. for their helpful work and laboratory implementation in the project during their study in the university. This work was supported in part by the National Natural Science Foundation of China (NSFC nos. 61103140 and 61105073) and Zhejiang Provincial S&T Department (2010R10006 and 2010C33095).