Mathematical Models for Therapeutic Approaches to Control Psoriasis

Psoriasis is a common chronic inflammatory human skin disorder characterized by T-Cell-mediated hyperproliferation of epidermal Keratinocytes. Cell-biological findings on the disease reveal that Helper T-Cells in the human blood as well as Dendritic Cells (DCs) play a significant role in inflicting the disease. Further, clinical research suggests that excessive generation of nitric oxide through a complex chain of biochemical events causes scaliness of Psoriatic plaques on human skin. On that basis, in this chapter, we have tried to formulate the basic mathematical model involving the densities of immune cells and Keratinocytes, where proliferation of Keratinocytes together with excessive nitric oxide production is a precursor to the Psoriatic lesions.

It is a well-known reality that during the interaction between T-Cells and Keratinocytes, cytokines are released. As a result, T-Cell population is reduced. These cytokines are transformed into Keratinocytes in the course of some cell-biological mechanisms and it facilitates the generation of the growth of Keratinocytes, which in turn originates into the disease Psoriasis. In our basic model of Psoriasis discussed earlier, there is no such conception of cytokines release during the interaction of T-Cells and Keratinocytes. Actually, the generation of the disease Psoriasis in the context of cytokines release is not yet been properly explored. Here, our aim is to analyze the Psoriatic system in the presence of cytokines release, which in turn generates Keratinocytes and also to control the over production of Keratinocytes to keep the disease under control.

An output function of any dynamical system is returned in such a way that the fluctuations in the output decrease, and a negative feedback takes place in the system. Stability of any system is usually endorsed by negative feedback approach. Negative feedback control mechanism has a significant effect on the disease dynamics like Psoriasis. With the help of a negative feedback control which is comparable to the introduction of a therapeutic drug regime, a stable control mechanism to the growth of Keratinocytes concentration can be easily furnished. This feedback strategy reduces the functioning response of cytokines and decreases activation and proliferation of Keratinocytes. As a mathematical understanding of this chapter, we have tried to expose the activation of Keratinocyte cells due to T-Cells-mediated cytokines, that is the causal effect of excessive growth of Keratinocyte cell population.

In autoimmune disorder like Psoriasis, some of the T-Cells are to be effectively enhanced. This process is identified as spontaneous proliferation. When the activation is finished, T helper cells permit to self-proliferate. This is accomplished by releasing Interleukin 2 (IL-2), which acts on itself through an autocrine manner. Thus, stimulated T-Cells must be detached by apoptosis at the end of an immune reply in order to uphold cellular homeostasis. We have considered that the growth of T-Cells has occurred in logistic fashion as T-Cells cannot proliferate unboundedly, where T-Cells are generated through the expansion of accessible T-Cells from precursors. Our aim is to observe the performance of the immune system with the maximum proliferation of T-Cells. In this chapter, we have improved our mathematical model by considering the introduction of T-Cell proliferation of existing T-Cells in the growth term of T-Cell population, which is more practical from the biological feasibility.

As Psoriasis is an autoimmune disease, mainly Keratinocytes is one of the causal effects of this disease. It is known that half-saturation constant represents the concentration at which half the maximum intake rate is reached. In context of the disease Psoriasis, the effect of half-saturation guarantees that the activation rate of Keratinocyte cells is half-maximal of T-Cell density. This rate cannot exceed the rate of T-Cells by accumulation of half-saturation. Since our aim is to reduce excess Keratinocyte growth, the half-saturation constant has a significant effect on the disease Psoriasis. Suppression taking place on Dendritic Cells along with half-saturation effect plays an effective role in Psoriatic system. Keeping this view in mind, we have developed a mathematical model introducing the half-saturation constant through T-Cells. It has been infiltrated in the activation process of Keratinocytes in course of the Psoriatic plaques. Our focus is to observe the behavioral pattern in Psoriatic cell biology for including such an effect. We have furnished our analysis in two different pathways; one is through ODE system and another is through FODE system, which has been discussed later in this book. It has been observed how T-Cells extremely stimulate a portion of Keratinocyte cells per day through their activation with half-maximal of T-Cell density. This notion of half-saturation adds superior effects for reducing the excess production of Keratinocyte cell population.

Till now, we have discussed the dynamics of Psoriasis with the help of three particular cells, viz., T-Cells (CD4\(^{+}\) T-Cells), Dendritic Cells, and Keratinocytes. But CD8\(^{+}\) T-Cells has a crucial role toward the disease Psoriasis. To measure the effect of CD8\(^{+}\) T-Cell population on Psoriatic system, we have introduced CD8\(^{+}\) T-Cell population in the mathematical model of Psoriasis that interacts with DCs in the system. This interaction leads to generate Keratinocytes, which in turn supports to expand the Keratinocytes growth. To confine this growth, we have applied drug at the interaction between CD8\(^{+}\) T-Cells and DCs. Another method to produce Keratinocytes is the interaction between T-Cells and Keratinocytes itself. We have also set the drug in that interaction to control the growth of Keratinocytes, whose surplus production generates the disease Psoriasis.

Nowadays, major attentions have been concentrated toward the models of fractional-order equations in several research fields. The nonlocal characteristics, which do not exist with the integer-order differential operators, are the significant properties of these types of models. It is understood by this feature that the subsequent phase of the model not only depends upon its present state but also upon all of its chronological situations (Elbasha et al, Nonlinear Anal: Real World Appl 12(5):2692–2705, 2011)[1].

Fractional-order models provide an interdisciplinary approach to multidimensional research domains. The nonlocal characteristic that do not take place in the integer-order differential operators, is the considerable identity of these varieties of model dynamics. The stage of the fractional-order model system not only is conditional upon its present situation but also upon all of its chronological arrangements. Based on these attributes, we have introduced the fractional-order differential equation into our proposed integer-order mathematical model, discussed in the sixth chapter (Part I).

In order to demonstrate the impact of memory on the cell-biological system, a mathematical model of Psoriasis involving CD4\(^{+}\) T-Cells, Dendritic Cells, CD8\(^{+}\) T-Cells, and Keratinocyte cell population has been developed in this chapter, using fractional-order differential equations with the effect of cytokines release, which is the extended work of Chap. 7. We have tried to explore the suppressed memory associated with the cell-biological system by incorporating fractional calculus and also to locate the position of Keratinocyte cell population considering the fact that fractional derivative possesses nonlocal property.