Non-Hermitian Operator Modelling of Basic Cancer Cell Dynamics

Abstract

Abstract:We propose a dynamical system of tumor cells proliferation based on operatorial methods. The approach we propose is quantum-like: we use ladder and number operators to describe healthy and tumor cells birth and death, and the evolution is ruled by a non-hermitian Hamiltonian which includes, in a non reversible way, the basic biological mechanisms we consider for the system. We show that this approach is rather efficient in describing some processes of the cells. We further add some medical treatment, described by adding a suitable term in the Hamiltonian, which controls and limits the growth of tumor cells, and we propose an optimal approach to stop, and reverse, this growth.

One-directional quantum mechanical dynamics and an application to decision making

“…In particular, if we have no medical treatment acting on the cells, it is natural to expect that the healthy cells become sick, but not viceversa. This effect was well described with our choice of H in [26], which is the first time, in our knowledge, in which the time evolution of a biological system is given in terms of such an Hamiltonian operator. Something along the same lines, but in an economical context, can be found in [6], where, however, ladder operators play no role.…”

“…Creation and annihilation operators appear again, in constructing non self-adjoint Hamiltonians used in Finance, in [27] and in [28]. Going back to biology, our conclusions in [26] suggest that non self-adjoint Hamiltonians of the kind considered there works well, and this would open many possible lines of research in the future. However, before considering more complicated (and useful) applications, we prefer to study in details what happens when our system S is attached to some special non self-adjoint Hamiltonian, to understand the basic mechanisms which then we can try to adapt in the analysis of more complicated and more realistic systems.…”

“…More recently, [26], we have considered a different choice of non self-adjointness for H, in the biological context of cells proliferation. The Hamiltonian ceases to be self-adjoint not because of the presence of some complex parameters, but because some operator in H is not paired to its adjoint counterpart: if A and B are (ladder) operators used in the analysis of the cells, H contains terms like A † B, but not its adjoint, B † A.…”

Modeling epidemics through ladder operators

“…This is actually the first attempt to use operatorial methods in application to infection processes: they have been successfully used in various biological contexts, to fit real data by modeling population-resources dynamics [14] , to model basic cancer cell dynamics [15] , [16] , to reproduce biological aspects of the bacterial dynamics [17] , and to model a basic epigenetic evolution [18] (see [19] , [20] for other fields of application). Our main goal is to provide a reliable long time dynamics capable to capture the final stage of the infection when the various densities of the classes have reached some sort of equilibrium values.…”

Spreading of Competing Information in a Network

“…The Hamiltonian H contains all the operators describing the interactions occurring between the different agents of the system, in particular the diffusion of fake and good news, and the ability to modify the nature of news from good to fake and vice versa. We stress that the use of operational methods has proven successful in describing the dynamics of several macroscopic systems arising in decision making, [ 5 ], population dynamics, [ 6 ], basic cancer cell dynamics, [ 7 , 8 ], biological aspects of the bacterial dynamics, [ 9 ], and epigenetic evolution, [ 10 ] (see [ 11 , 12 ] for other fields of application).…”