MATHEMATICS IN MODERN SCIENCE AND TECHNOLOGY

MATHEMATICS IN MODERN SCIENCE AND TECHNOLOGY

Nowadays, we have got great potential in studying brain function in vivo, without putting electrodes inside the brain, i.e. non-invasively. Some of the existing imaging methods have advantages, while some others have disadvantages, but in most of the cases, they complete each other. 

The brain is the most complicated material in the known universe. It contains 10 to the power of 11, i.e. 100 billion neurons. If we assume that each neuron is one millimeter and put these 100 billion neurons the one next to the other, the overall length is two and a half times the length of the equator, so we have got such a length of neurons within our head. Each one of these neurons communicates on average with 10.000 neighbors, and people try to understand how this machine works and how to detect the damage when it is damaged, so in case of surgery to intervene in the smallest area possible.

Brain research has received a great boost in the last 30-40 years. The 80’s was, indeed, the decade of the brain, and such decades have been repeated. In general, it seems that the knowledge we acquire about the human brain is doubled every 10-20 years. Nowadays there are many ways to obtain images of the brain. Two of them, which are those we are working on, are electroencephalography (EEG) and magnetoencephalography (MEG). The effort focuses on the possibility to combine these techniques to exploit their complementary advantages and disadvantages. We have concluded, and this is a very important result, that if we receive information, complete information, only from electroencephalography we can, in the best case, identify the 33% of the electrical activity of the brain. If we have both electroencephalography and magnetoencephalography we can identify 33% from the electroencephalography and another 33%, i.e. another 1/3, from magnetoencephalography. In total, we can calculate 2/3.

So, with any modern and complete information of these two systems, which is very difficult to have at the first place, 1/3 of the brain activity cannot be measured. If the brain is, for example, modeled as a viscous liquid, this liquid is conductive. That means that if a small current runs into a neuron, a field will be created which will generate, as a result, a current throughout the brain which is the well-known ‘induction current’. So, what we record out of the brain with electroencephalography and magnetoencephalography, is not only the activation of the specific neuron, which has taken excited by neurophysiological mechanisms but all the rest as well, i.e. the induction current. And all the difficulty lies on how to extract the induction current, which was the result of activation, to identify the stimulation itself. Prof. Dassios develops models for the calculation of current sources that generate EEG, based on the assumption not of the spherical shape of the brain, but of the elliptical model which is closer to reality. This offers a greater degree of freedom in comparison to the spherical model. Its value is that it helps us, with a simple recording, to figure out what is going on deep in the brain. So this is a sphere, but no brain is a sphere. The realistic shape is an ellipsoid which is like this, i.e. it is not the same not even by rotation.

It is flattened here, elongated here and if the eyes are here and the brain in the back, it is oriented obliquely. When we write a computer program and we use mathematical algorithms that have been developed to interpret how we can find the source from the information, using geometry, we make a mistake. If we assume that the brain is a sphere, then it may guide us at a particular point. But when we interpret the ellipsoid as a sphere, the source may be three centimeters further. You could think that this is not a big deal. But when we intervene surgically, in this case we would deduct 3 additional centimeters of gray matter. Deterioration of brain tissue again alters the measurements. Here, for instance, we know from computational experiments that the existence of such lesions, which may have come up from any disease, even cancer, have a significant effect on identifying the source. It may be up to 3 centimeters, which is too big. Thinking in terms of surgery even a millimeter is important. For example, the epileptic zone of an epileptic patient is in a particular area deep in the brain. We can’t estimate roughly that it is located in the electrodes at the side of the head or from the frontal part. Where is this useful? It is useful in diagnosis, to be able to see the type of epilepsy, in the pharmacological treatment, and even more useful in surgical treatment.

There are epilepsies that cannot be improved by any medication and we are considering the case of surgically removing those parts of the brain which are responsible for epilepsy. So, in those cases, it is very important for the surgeon to know where the problem is, in order to perform the surgery in the optimum way. Regarding the mathematics, our work is as follows. We want to track down a stream that is created within the brain, the area in which it is created. We know that the stream has got three components, the three dimensions of space. Therefore we are looking for 3 functions that describe this stream. The involvement of mathematics is intended to help identify the basic mechanisms which are involved in the biological phenomenon, to enhance some parameters and through more light to the whole study of the biological phenomenon. We are trying to understand how a machine works and all we have at our disposal is the machine itself. Hence, the obvious question is: how far can we go? and if this is true, then why we are doing all this? The answer is simple since we are still in infancy on what we know about the brain there is a lot to gain until we arrive at this philosophical barrier.

The “Marie Curie” European program includes 12 actions in total. Starting from the most simple which are the student scholarships, it goes further to conferences, research projects and ends up at the last action, which includes 15 honorary Chairs of Excellence funded by the EU, for any scientific area, from anywhere in the world who would be invited to visit another university or research center within the European Union. Along this line, Cambridge University submitted a proposal for me to spend three years with a Marie Curie Chair in Cambridge working with Prof. Fokas, and personally, I considered this invitation very honorable. It felt even better when I was finally selected. So, I went over there for three years. It was a paradise from the scientific point of view but also for concentration and thinking. At present we are involved in a project from a Program of Excellence funded by the General Secretariat for Research and Technology where we work on the functional brain. We continue this research where we want to identify the differences when we move from the simplistic model of the sphere to the real geometry of the human brain. The aim of the research group is to develop fast, reliable algorithms, which will assist in the accurate identification of the sources in the brain, in the presence of traumatic brain injuries and diseases of the brain tissue. These may be Alzheimer's, Parkinson's, cancer diseases and so on.

Another research activity which is funded by the “Heraclitus” program of the Ministry of Education, involves the study of the evolution of the boundary of a tumor, considering various shapes as the geometry of the tumor. An additional issue that we are studying using mathematical methods, analytical or numerical, involves the tumor growth, i.e. the growth of a tumor colony into normal tissue, a living tissue. Another research activity involves the study of the blood plasma flows around blood cells. We study this considering the flow as a Stoke flow. The innovation in this research is that while the cells, the blood cells are usually considered spherical or spheroidal, we now consider them like inverted spheroids, which is very close to the reality because the blood cells are biconcave discs. Anyone who tries to solve a problem creates another hundred problems. It doesn’t matter if this person will solve the problem or not. It is much easier to ‘create problems’ than ‘solve problems. Consequently, it does not mean that if there are many people who are involved in a scientific area, the problems will end and that there is no further scientific future for young people. On the contrary, the future is quiet open.