To monitor the development of cancer, the volume of a human organ often needs to be estimated. However, given current technology, data can only be collected on cross sections of an organ. This naturally leads to this question: how can an organ volume be accurately estimated based on its cross section information? This paper describes such an algorithm, based on cubic spline interpolation applied to a known formula developed in the 1980s.
In a bit more detail, a three-dimensional (3D) volume can be computed using the line integral V = |∫C s “ dc| = |∫[u(t)x’(t) + v(t)y’(t) + w(t)z’(t)]dt|, where C is a curve consisting of centroids of an object’s cross section represented by vector c = (x(t), y(t), z(t)), a radius vector from the origin to a point on C, and s = (u(t), v(t), w(t)) is a normal vector to the plane defined by the cross section, with length being the area of the cross section.
The algorithm works as follows: for each cross section indexed i (i = 1, &, N), first construct cubically a spline curve to approximate the boundary of each sampled cross section, based on a given set of discrete points on the boundary. Second, compute (using spline boundary representation) the area Si and the centroid of the cross section ci. Third, to approximate curves s and c, use cubic spline curves again, to interpolate sets {Si} and {ci}, respectively. Finally, use the spline approximations to s and c in the above volume calculation. Numerical examples are given in the paper.
I think the content of the paper provides an excellent real-world example for the applications of line integrals and curve approximation by splines (perfectly appropriate at the undergraduate level). Those who are seeking projects for their calculus or numerical analysis courses should read this paper.
