A PDE approach of inflammatory phase dynamics in diabetic wounds

Abstract

The objective of the present paper is the modeling and analysis of the dynamics of macrophages and certain growth factors in the inflammatory phase, the first one of the wound healing process. It is the phase where there exists a major difference between diabetic and non-diabetic wound healing, an effect that we will consider in this paper.

A Turing mechanism in order to explain the patchy nature of Crohn's disease

“…As mentioned in the introduction, it is generally accepted that diffusion provides an adequate description of molecular spreading but, in the case of phagocytic cells, chemotaxis is claimed to be crucial establishing the direction of movement in the sense of the pathogen gradient. However, there are in vivo experiments that corroborate our hypothesis Lauffenburger and Kennedy (1983) and several authors have made similar assumptions Cnsul et al (2014); EL Khatib and Gnieys (2007); EL Khatib et al (2011). Nevertheless, by neglecting chemotaxis in our model we do not claim that it is an unimportant phenomenon, instead, this assumption must be seen as a simplification and an idealization of the physiological mechanism we seek to describe.…”

“…The main variables of the models introduced in the mentioned works vary according to the dynamics that the authors wish to describe, the density of phagocytic cells, pro-inflammatory cytokines, anti-inflammatory mediators and bacteria are some standard quantities that are often taken into account. As in the ordinary differential equations approach the stability of the systems is systematically studied, in Chalmers et al (2015); Cnsul et al (2014); EL Khatib and Gnieys (2007); Lauffenburger and Kennedy (1981) a vast analysis of all possible scenarios is performed depending on the values of the model parameters, the authors provide biological interpretation of such behavior as well as numerical simulations; furthermore, in EL Khatib et al (2011) the existence of travelling waves solutions is proved to be at the origin of a chronic inflammatory response.…”

“…As suggested by in vitro studies, phagocytic cells (big eaters) may move following a chemotactic impulse generated by the presence of pathogens germs, for this reason most of the authors cited above include the effect of chemotaxis by mean of the classical term first introduced by Patlak in 1953 and Keller andSegel in 1970 Keller andSegel (1970); Patlak (1953). Nevertheless, there is no consensus on this assumption, as noted in Lauffenburger and Kennedy (1983), in vivo observations more often show that the phagocytes seem to move within an infected lesion randomly, this is the case in the models introduced in Cnsul et al (2014); EL Khatib and Gnieys (2007); EL Khatib et al (2011).…”

“…The diffusion coefficient of immune cells might also vary according to the type of cell and the part of the body where they act. In the consulted literature the value of this parameter varies from 10 −12 m 2 /min to 10 −10 m 2 /min depending on the context Cnsul et al (2014);EL Khatib et al (2011); Lauffenburger and Kennedy (1983); Stickle et al (1985). In the absence of experimental data more precise information about the order of this parameter in the particular case of bacterial infection in the intestinal track, we consider this coefficient to remain within this range in damaged areas of the intestine.…”