Maximum Likelihood Estimation for Unrooted 3-Leaf Trees: An Analytic Solution for the CFN Model.

Maximum likelihood estimation is among the most widely-used methods for inferring phylogenetic trees from sequence data. This paper solves the problem of computing solutions to the maximum likelihood problem for 3-leaf trees under the 2-state symmetric mutation model (CFN model). Our main result is a closed-form solution to the maximum likelihood problem for unrooted 3-leaf trees, given generic data; this result characterizes all of the ways that a maximum likelihood estimate can fail to exist for generic data and provides theoretical validation for predictions made in Parks and Goldman (Syst Biol 63(5):798-811, 2014). Our proof makes use of both classical tools for studying group-based phylogenetic models such as Hadamard conjugation and reparameterization in terms of Fourier coordinates, as well as more recent results concerning the semi-algebraic constraints of the CFN model. To be able to put these into practice, we also give a complete characterization to test genericity.

Lymphopenia predicts reduced survival in canine hepatocellular carcinoma.

Platelet to lymphocyte ratio (PLR) is a prognostic marker in human hepatocellular carcinoma (HCC) however, its utility in canine HCC has not been explored. The aim of the study was to determine if PLR could predict survival outcomes in 42 dogs with HCC. PLR was not a significant predictive factor (p = 0.15) but lymphopenia alone was significantly correlated with a reduced probability of survival (p = 0.024). Further studies are needed to evaluate if peripheral lymphocyte count mirrors that of the tumor microenvironment in canine HCC.