A review of recent developments in mathematical modeling of bone remodeling

In this article, we summarize the developments in the mathematical modeling of the mechanics of bone and related biological phenomena. We will devote special attention to the results of the last 10–15 years, although we will cover some relevant classical work to better frame the more recent researches. We will propose a division of the literature based on the main aim of the model (mechanical/biomathematical) and the type of biological phenomena considered (stimulus, growth, cell population dynamics). Finally, we will suggest some possible directions for future investigations.

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Comprehensive Assessment and Mathematical Modeling of T Cell Population Dynamics and Homeostasis

The Journal of Immunology ◽

10.4049/jimmunol.180.4.2240 ◽

2008 ◽

Vol 180 (4) ◽

pp. 2240-2250 ◽

Cited By ~ 47

Author(s):

Véronique Thomas-Vaslin ◽

Hester Korthals Altes ◽

Rob J. de Boer ◽

David Klatzmann

Keyword(s):

Mathematical Modeling ◽

Population Dynamics ◽

T Cell ◽

Cell Population ◽

Comprehensive Assessment ◽

Cell Population Dynamics

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Processes on the emergent landscapes of biochemical reaction networks and heterogeneous cell population dynamics: differentiation in living matters

Journal of The Royal Society Interface ◽

10.1098/rsif.2017.0097 ◽

2017 ◽

Vol 14 (130) ◽

pp. 20170097 ◽

Cited By ~ 16

Author(s):

Sui Huang ◽

Fangting Li ◽

Joseph X. Zhou ◽

Hong Qian

Keyword(s):

Population Dynamics ◽

Cell Population ◽

Individual Cell ◽

Biochemical Reaction ◽

Reaction Networks ◽

Chemical Compositions ◽

Cell Population Dynamics ◽

Heterogeneous Cell Population ◽

Biological Phenomena ◽

A Cell

The notion of an attractor has been widely employed in thinking about the nonlinear dynamics of organisms and biological phenomena as systems and as processes. The notion of a landscape with valleys and mountains encoding multiple attractors, however, has a rigorous foundation only for closed, thermodynamically non-driven, chemical systems, such as a protein. Recent advances in the theory of nonlinear stochastic dynamical systems and its applications to mesoscopic reaction networks, one reaction at a time, have provided a new basis for a landscape of open, driven biochemical reaction systems under sustained chemostat. The theory is equally applicable not only to intracellular dynamics of biochemical regulatory networks within an individual cell but also to tissue dynamics of heterogeneous interacting cell populations. The landscape for an individual cell, applicable to a population of isogenic non-interacting cells under the same environmental conditions, is defined on the counting space of intracellular chemical compositions x = ( x 1 , x 2 , … , x N ) in a cell, where x ℓ is the concentration of the ℓth biochemical species. Equivalently, for heterogeneous cell population dynamics x ℓ is the number density of cells of the ℓth cell type. One of the insights derived from the landscape perspective is that the life history of an individual organism, which occurs on the hillsides of a landscape, is nearly deterministic and ‘programmed’, while population-wise an asynchronous non-equilibrium steady state resides mostly in the lowlands of the landscape. We argue that a dynamic ‘blue-sky’ bifurcation, as a representation of Waddington's landscape, is a more robust mechanism for a cell fate decision and subsequent differentiation than the widely pictured pitch-fork bifurcation. We revisit, in terms of the chemostatic driving forces upon active, living matter, the notions of near-equilibrium thermodynamic branches versus far-from-equilibrium states. The emergent landscape perspective permits a quantitative discussion of a wide range of biological phenomena as nonlinear, stochastic dynamics.

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