Surface Reconstruction of in Vivo Geometry Based on Medical Images Using Multilevel Radial Basis Functions

Compactly supported radial basis functions (RBFs) were used for surface reconstruction of in vivo geometry, translated from two dimensional (2D) medical images. RBFs provide a flexible approach to interpolation and approximation for problems featuring unstructured data in three-dimensional space. Point-set data are obtained from the contour of segmented 2-D slices. Multilevel RBFs allow smoothing and fill in missing data of the original geometry while maintaining the overall structure shape. INTRODUCTION Computer simulations of blood flow in arteries and veins are widely used by biomedical and bioengineering researchers to study the importance of hemodynamics, the fluid dynamics of blood. Hemodynamics has been shown to be the key factor in the pathogenesis of atherosclerosis (Giddens et al 1993, Ross 1993). In particular, wall shear stress (WSS), has been investigated in the development of vessel lumen narrowing due to excessive tissue growth, intimal hyperplasia, and atherosclerosis. An accurate representation of in vivo geometry is required for accurate simulation. Medical images, such as computerized tomography (CT), magnetic resonance imaging (MRI), and ultrasound (US), of a patient’s blood vessels were used to obtain the data set. Regions of interest (typically the lumen) are chosen and segregated from the background. Compactly supported RBFs was chosen because of its simplicity and fast computation procedure. On the other hand, global RBFs are useful in filling in for missing data. Therefore, multilevel RBF approach is used to integrate the best aspect of 3D scattered data fitting with locally and globally supported basis functions. RADIAL BASIS FUNCTIONS The generic RBF representation of a function is of the form ! " ! " j j j f x x c # $ % ! ! (0.1) where x! is the position vector in region and the sum of extends over a specified range of basis functions, j = 0, 1, ..., . The are unknown coefficients to be determined. In principle, one may employ any set of basis functions d " n j c & ' 1 n j j # $ to represent . As the name implies, RBFs are generally of the form ( ) f x ! " , j j r x R # # ( ) $ * + , ! (0.2)