The numerous applications of surface interpolation include the modeling and visualization of physical phenomena. A tetrahedrization is the one of pre-processing steps for 4D surface interpolation. The quality of a piecewise linear interpolation in 4D space depends not only on the distribution of the data points in R3, but also on the data values. One can improve the quality of an approximation by using data dependent criteria. This paper discusses Delaunay tetrahedrization method (sphere criterion) and one of the data dependent tetrahedrization methods (least squares fitting criterion). This paper also discusses new data dependent criteria: (1) gradient difference, and (2) jump in normal direction derivatives.