Addressing the multiplicity of optimal solutions to the Clonal Deconvolution and Evolution Problem

The Clonal Deconvolution and Evolution Problem consists on unraveling the clonal structure and phylogeny of a tumor using estimated mutation frequency values obtained from multiple biopsies containing mixtures of tumor clones. In this article, we tackle the problem from an optimization perspective and we explore the number of optimal solutions for a given instance. Even in ideal scenarios without noise, we demonstrate that the Clonal Deconvolution and Evolution Problem is highly under-determined, leading to multiple solutions. Through a comprehensive analysis, we examine the factors contributing to the multiplicity of solutions. We find that as the number of samples increases, the number of optimal solutions decreases. Additionally, we explore how this phenomenon operates across various tumor topology scenarios. To address the issue of the existence of multiple solutions, we present sufficient conditions under which the problem can have a unique solution, and we propose a linear programming-based algorithm that leverages mutation orderings to generate instances with a single solution for a given topology. This algorithm encounters numerical challenges when applied to large instance sizes so, to overcome this, we propose a heuristic adaptation that enables the algorithm’s use for instances of any size.