Mathematical Modeling of Multiple Pathways in Colorectal Carcinogenesis using Dynamical Systems with Kronecker Structure

Like many other tumors, colorectal cancers develop through multiple pathways containing different driver mutations. This is also true for colorectal carcinogenesis in Lynch syndrome, the most common inherited colorectal cancer syndrome. However, a comprehensive understanding of Lynch syndrome tumor evolution which allows for tailored clinical treatment and even prevention is still lacking.We suggest a linear autonomous dynamical system modeling the evolution of the different pathways. Starting with the gene mutation graphs of the driver genes, we formulate three key assumptions about how these different mutations might be combined. This approach leads to a dynamical system that is built by the Kronecker sum of the adjacency matrices of the gene mutation graphs. This Kronecker structure makes the dynamical system amenable to a thorough mathematical analysis and medical interpretation, even if the number of incorporated genes or possible mutation states is increased.For the case that some of the mathematical key assumptions are not satisfied, we explain possible extensions to our model. Additionally, improved bio-medical measurements or novel medical insights can be integrated into the model in a straightforward manner, as all parameters in the model have a biological interpretation. Modifications of the model are able to account for other forms of colorectal carcinogenesis, such as Lynch-like and familial adenomatous polyposis cases.Graphical Abstract:From the Medical Hypothesis Over the Modeling Approach To the Mathematical Structure.The medical hypothesis of multiple pathways in carcinogenesis is widely known for various types of cancer. Left: We present a model for this phenomenon at the example of Lynch syndrome, the most common inherited colorectal cancer syndrome, with specific key driver events in the MMR genes, CTNNB1, APC, KRAS and TP53. Middle: This current medical understanding of carcinogenesis is translated into a mathematical model using a specific dynamical system, which can be represented by a graph structure, where each vertex in the graph represents a genotypic state and the edges correspond to the transition probabilities between those states. Starting with all colonic crypts in the state of all genes being wild-type and a single MMR germline mutation due to Lynch syndrome, we are interested in the distribution of the crypts among the graph at different ages of the patient in order to obtain estimates for the number of crypts in specific states, e.g. adenomatous or cancerous states. Right: The underlying matrix of the dynamical system makes use of the Kronecker sum and product. It is a sparse upper triangular matrix accounting for the assumption that mutations cannot be reverted. This allows fast numerical solving by using the matrix exponential. Each nonzero entry of the matrix represents a connection between genotypic states in the graph.

Interval-Valued Intuitionistic Fuzzy Partition Matrices

If in an interval-valued intuitionistic fuzzy matrix each element is again a smaller interval-valued intuitionistic fuzzy matrix then the interval-valued intuitionistic fuzzy matrix is called interval-valued intuitionistic fuzzy partion matrix (IVIFPMs). In this paper, the concept of interval-valued intuitionistic fuzzy partion matrices (IVIFPMs) are introduced and defined different types of interval-valued intuitionistic fuzzy partion matrices (IVIFPMs). The operations like direct sum, Kronecker sum, Kronecker product of interval-valued intuitionistic fuzzy matrices are presented and shown that their resultant matrices are also interval-valued intuitionistic fuzzy partion matrices (IVIFPMs).

The Property of Kippenhahn Curves of Kronecker Sum and Applications

In many physical applications, we must solve a system of linear equations such as . The Kronecker sum can get a convenient representation for linear equations . In this paper, let and be denoted reducible companion matrices. We study the property of the Kippenhahn curve and of the numerical rang being an elliptic disc by making use of matrices Kronecker sum and the Kippenhahn curve continually.

The Property of Kippenhahn Curves in Engineering Applications

In this paper, let and be deoted reducible companion matrices, we study the property of the Kippenhahn curve and of the numerical rang being an elliptic disc by making use of matrices Kronecker sum , Kronecker product and Hadamard product and the Kippenhahn curve continually.

On the Necessary and Sufficient Condition for a Set of Matrices to Commute and Some Further Linked Results

This paper investigates the necessary and sufficient condition for a set of (real or complex) matrices to commute. It is proved that the commutator[A,B]=0for two matricesAandBif and only if a vectorv(B)defined uniquely from the matrixBis in the null space of a well-structured matrix defined as the Kronecker sumA⊕(−A∗), which is always rank defective. This result is extendable directly to any countable set of commuting matrices. Complementary results are derived concerning the commutators of certain matrices with functions of matricesf(A)which extend the well-known sufficiency-type commuting result[A,f(A)]=0.