Maximum Likelihood Estimation of the Size Distribution of Liver Cell Nuclei from the Observed Distribution in a Plane Section

Abstract

From sections of liver biopsies, the distribution of the radii of sectioned nuclei is recorded. Under the assumption that the nuclei are spherical, the problem of inferring the real size distribution of the nuclei in the liver from the observed distribution of the profile radii in the section is studied. This is an example of the 'corpuscle problem' first studied by Wicksell. A parametric model for the distribution of the nucleus radii is proposed and ordinary maximum likelihood estimation is discussed. The small sections are clearly under-represented in the profile distribution. This problem which seems to be common in like situations, is here approached by the idea that the unrecorded nucleus sections come from nuclei whose angle of intersection with the microtome is less than some fixed angle p. It is also possible to estimate p and to test the important hypothesis that p = 27r, i.e. that the nucleus size distribution is directly observed in the section. The theory is stupplemented by detailed numerical investigations.