Introduction

Material and methods

Results

Discussion

Conclusions

References

Abstract

Multiple sclerosis (MS) is a demyelinating disease which characteristically shows repeated relapses and remissions irregularly in the central nervous system. At present, the pathological mechanism of MS is unknown and we do not have any theories or mathematical models to explain its disseminated patterns in time and space. In this paper, we present a new theoretical model from a viewpoint of complex system with chaos model to reproduce and explain the non-linear clinical and pathological manifestations in MS. First, we adopted a discrete logistic equation with non-linear dynamics to prepare a scalar quantity for the strength of pathogenic factor at a specific location of the central nervous system at a specific time to reflect the negative feedback in immunity. Then, we set distinct minimum thresholds in the above-mentioned scalar quantity for demyelination possibly causing clinical relapses and for cerebral atrophy. With this simple model, we could theoretically reproduce all the subtypes of relapsing-remitting MS, primary progressive MS, and secondary progressive MS. With the sensitivity to initial conditions and sensitivity to minute change in parameters of the chaos theory, we could also reproduce the spatial dissemination. Such chaotic behavior could be reproduced with other similar upward-convex functions with appropriate set of initial conditions and parameters. In conclusion, by applying chaos theory to the three-dimensional scalar field of the central nervous system, we can reproduce the non-linear outcome of the clinical course and explain the unsolved disseminations in time and space of the MS patients.

Introduction

Autoimmune-related diseases are likely to have elevated immune activity and abnormal immune response, though whether they are primary or secondary are not necessarily clear [1]. Such abnormal immune strength is sometimes difficult to be measured with a single laboratory biomarker when the pathological mechanism is uncertain. In the complex system in immunity, many types of blood cells (e.g. lymphocytes) and tissue cells (e.g. microglia) play complex roles with mutual interactions. Large numbers of many other factors like cytokines, chemokines, and permeability of blood-brain barrier make the complex interactions even more complicated [2–۶]. In addition to these numerous players of immune system, countless numbers of endogenous and exogenous factors (e.g. sex, age, race, food, stress, infection, vaccination, tobacco, medications, pregnancy, etc.) also affect the system [7–۱۲]. At present, in the field of clinical neurology, one of the most mysterious autoimmune-related diseases with unknown causes is multiple sclerosis (MS). MS is a famous demyelinating disease in the central nervous system (CNS) with irregular clinical relapses and disseminated CNS lesions. The pathogenesis of MS is not fully known, but it has been suggested to be multifactorial (e.g. auto-immunity, diet, vitamin D, higher latitude, Epstein-Barr virus infection, and smoking) with possible causal cascades [13–۱۶]. There are at least three subtypes as its clinical courses: primary-progressive MS (PPMS), relapsing-remitting MS (RRMS), and secondary-progressive MS (SPMS) transitioned from RRMS [17]. Characteristic conventional subtype of MS is RRMS, in which both of subclinical cerebral atrophy and clinical relapses take place with dissemination in time and space [18,19]. PPMS is also a worrisome phenotype; it shows almost no clinical relapses but shows faster cerebral atrophy than RRMS from an early stage [20–۲۲].