×

zbMATH — the first resource for mathematics

Mathematical methods and tools of kinetic theory towards modelling complex biological systems. (English) Zbl 1093.82016
Based on methods of kinetic theory of interacting particles and nonequilibrium statistical mechanics, the paper designs a mathematical framework to model the dynamics of large systems of living cells. State variables employed to describe individual cells comprise both mechanical and biological parameters; the latter effectively describe various aspects of cell function - a distinctive feature of living systems.
The formalism deals with short- and long-range conservative interactions, which do not modify the numbers of cells of different types, as well as proliferative and destructive interactions. The evolution of the system is described in terms of a distribution function that satisfies a generalized version of the Boltzmann equation. Specific models are designed, which are consistent with the above general framework and concentrate on particular cases of short-range, mean field, dominant mechanical and prevailing biological interactions.
The model is applied to study tumour dynamics and the existence theorems are given for the solutions in three particular cases: (i) spatially homogeneous mass conservative cellular dynamics, (ii) spatially homogeneous destructive and proliferative cellular dynamics and (iii) cellular dynamics with predominant mechanical interactions.
The last section outlines directions of further developments of mathematical models in which the distribution function over the microscopic states is continuous over continuous variables.

MSC:
82C40 Kinetic theory of gases in time-dependent statistical mechanics
92D25 Population dynamics (general)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1007/978-0-8176-8119-7 · doi:10.1007/978-0-8176-8119-7
[2] Arlotti L., Math. Mod. Meth. Appl. Sci. 12 pp 579–
[3] DOI: 10.1142/5359 · doi:10.1142/5359
[4] DOI: 10.1080/10273660290015170 · Zbl 1050.92027 · doi:10.1080/10273660290015170
[5] DOI: 10.1142/S0218202504003799 · Zbl 1060.92029 · doi:10.1142/S0218202504003799
[6] DOI: 10.1007/s00033-003-3057-9 · doi:10.1007/s00033-003-3057-9
[7] DOI: 10.1016/0895-7177(94)90223-2 · Zbl 0811.92014 · doi:10.1016/0895-7177(94)90223-2
[8] DOI: 10.1007/978-1-4612-0513-5 · doi:10.1007/978-1-4612-0513-5
[9] DOI: 10.1142/S0218202504003544 · Zbl 1083.92032 · doi:10.1142/S0218202504003544
[10] Cercignani C., Theory and Application of the Boltzmann Equation (1993) · Zbl 0803.35151
[11] DOI: 10.1142/S0218202504003738 · Zbl 1057.92036 · doi:10.1142/S0218202504003738
[12] DOI: 10.1142/S0218202503002453 · Zbl 1043.92012 · doi:10.1142/S0218202503002453
[13] Greller L., Invasion Metastasis 16 pp 177–
[14] DOI: 10.1038/35011540 · doi:10.1038/35011540
[15] DOI: 10.1016/S0895-7177(03)80018-3 · Zbl 1043.92013 · doi:10.1016/S0895-7177(03)80018-3
[16] DOI: 10.1201/9780203494899 · Zbl 1039.92022 · doi:10.1201/9780203494899
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.