# Mathematics111

sclerosis
Multiple sclerosis (MS) is a serious chronic disease of the central nervous system whose cause is not understood. Some knowledge is available on mechanisms which play a role in the disease and it is believed to be autoimmune in nature. The myelin sheaths which insulate the nerve cells are damaged through being attacked by elements of the immune system. The majority view among experts seems to be that the disease is not caused by a pathogenic organism or virus. It may, however, bethat a virus (or more than one) plays an indirect role. There is, for instance, the idea of molecular mimicry. Here immune cells which recognize certain foreign proteins attack similar structures belonging to useful molecules in the body. For instance it could be that the legitimate target of the immune cells are parts of a bacterium while the analogous structures which are actually attacked belong to the myelin. Since the immune system is programmed not to attack molecules belonging to self there is a motivation for microorganisms to develop molecular mimicry – it can be more than an accident. Another indirect viral influence is the so-called innocent bystander mechanism. Here the virus causes an upregulation of the activity of the immune system. If the immune system is already defective then this may be enough to cause it to attack myelin.

What aspects of MS might usefully be modelled using mathematics? One possibility would be to model populations of immune cells, concentrations of signalling molecules or both by an ODE system. This has some similarity to what was discussed for the case of AIDS in earlier posts but might not involve any virus. It could be pure immune system dynamics. As far as I can see little has been done in this area up to now. I will discuss one example below. In fact, in looking for examples I have widened the field to include work on EAE. The background to this is that MS is confined to human beings and no other species is affected by a closely analogous naturally occurring disease. This restricts the opportunities for research. EAE is an artificial disease (typically of mice) which mimics some of the features of MS and can be used for research purposes. It played a big role in the development of one of the best presently available treatments for MS, Copaxone or Copolymer 1. This substance is also known by the name glatiramer acetate, which I find particularly ugly. It was tested on EAE with the expectation that it would lead to a worsening of the disease and actually had the opposite effect. It is now successfully used for the treatment of MS and it is clear that it is unlikely to have been developed without the help of the mice.

Returning to mathematical models, I want to discuss some work of Lev Bar-Or (Math. Biosci. 163, 35-58). To explain what it is about it is necessary to know that CD4+ T-cells or T-helper cells (the ones which were mentioned in the posts on AIDS) come in two varieties called Th1 and Th2. They can be distinguished by the different kind of cytokines (certain signalling molecules) which they produce. In fact there are two different phases of the immune system where these sets of cytokines dominate and they are also called Th1 and Th2. It seems that in general a Th2 immune response is associated with less intense disease activity in MS (or EAE) and that a Th1 response is associated with more intense activity. The immune system can shift from one phase to the other over time and it could be very valuable to understand more about the dynamics of this process. The cytokines associated with a Th1 immune response might reasonably be thought of as bad for MS patients. An example is interferon \gamma. This was once tried as a therapy for MS but the clinical trials were broken off when it was observed that it had a negative effect on the patients’ health. On the other hand similar trials with the compound interferon \beta gave positive results and led to what is probably the most effective long-term therapy for MS at present. At this point a word of caution is in order concerning trusting simplified pictures and trusting the applicability of animal models. As mentioned in the paper of Lev Bar-Or, it has been found that reducing interferon \gamma concentration can lead to a worsening of EAE in mice. A possible explanation for this is given there.

The mathematical model in the above paper is a system of four coupled ODE containing a large number of parameters. The unknowns are the cytokine production by helper T-cells and macrophages of types Th1 and Th2 respectively. In general cytokines of one of the types stimulate the cells which produce that type and inhibit the cells which produce the other type. If this were all that was contained in the model then it would probably not lead to very interesting dynamics. There is, however, an extra effect which can lead to a reversal of some of the signs in the coupling. This has to do with the antigen presentation activity of type II MHC molecules, a subject I do not want to enter into here. Depending on the choice of parameters the dynamical system has either one or two equilibria. (This conclusion is based on numerics.) Either one of the two types (Th1 or Th2) dominates for the given parameter set or there may be coexistence of Th1 and Th2 dominated equilbria. In reality there are a huge number of cytokines and different types of immune cells. A more realistic dynamical system would contain many more variables but would probably not be very useful. The system with four unknowns arises by some averaging and some process like this seems inevitable in bringing mathematics to bear on the problem of understanding the dynamics of complicated biological systems.

There are some interesting features of MS which seem to invite a dynamical systems approach. The first is that there is often a gap of many years between the first relevant symptom and the development of the full disease. This is reminiscent of the situation with HIV mentionedin a previous post but in the present case mutations of a virus cannot be the explanation. In any case it is possible to pose the question: is there a period of dormancy here or is there an active and rapid dynamics which finally leads to a qualitative change? The second feature is that there are two qualitative phases of MS, the relapsing-remitting and progressive. In the relapsing-remitting phase the symptoms repeatedly get worse and then better again while in the progressive case the symptoms steadily get worse. In many patients the disease makes the transition from relapsing-remitting to progressive at some time. This invites the comparison with a dynamical system with an attractor which changes from a limit cycle to an equilibrium point as a parameter is varied. But what should the dynamical system be?

If the case of the understanding of MS is compared with the successful example of HIV there are couple of evident differences. One is that AIDS research has attracted more money and more publicity than MS research. Here I want to concentrate on another, which is the lack of good quantitative diagnostic criteria in the case of MS. The powerful technique is the MRT scan which gives information about the number of lesions in the brain and which are active. This can at best be interpreted statistically. Moreover MRT scans are very expensive which limits the frequency with which they can be done in most cases. It seems to me that the ideal thing would be to have some chemical whose concentration in the blood can be measured relatively easily and which gives a reliable indication as to whether the state of the patient is getting better or worse. It is information of this kind which led to the breakthrough with AIDS. In principle it would be better to take samples from the central nervous system which is the scene of the action rather than the blood but doing this frequently is not practical since it requires a lumbar puncture. Perhaps it would be better to concentrate on doing a parallel analysis on some other autoimmune disease such as rheumatoid arthritis where sampling is easier so as to establish some basic theoretical understanding of the dynamical processes involved.

Balo’s concentric sclerosis, mentioned in the last post, may be a disease related to MS or may be a form of MS. For a long time the only observations of the rings of demyelination which occur there were from autopsies. Now MRT offers new possibilities and there are some indications that rings of demyelination may occur in more standard cases of MS, particularly in the very early stages. Perhaps this rare disease might offer a clue to understanding the dynamics of a much more common one – studying an extreme case may be the key to solving a scientific problem.