March 10, 2013
Math is good for your health
Siv Sivaloganathan, a professor in the University of Waterloo's Department of Applied Mathematics, comments that mathematics has contributed to dramatic advances in the sciences as well as in many other fields of human endeavour; however, in a sense, he says that the biomedical sciences represent the final frontier for mathematicians, an area where the full power of mathematics has yet to be felt.
He is currently collaborating with experts across Canada to tackle problems of current interest to clinical scientists and doctors, assisting them in finding optimal ways to deal with specific problems associated with a variety of diseases.
“In this last decade, we have actually seen a dramatic increase in the numbers of mathematicians interacting with medical researchers and clinicians,’’ Dr. Sivaloganathan said.
He said mathematical analysis can assist neurosurgeons treating hydrocephalus (a buildup of cerebrospinal fluid (CSF) in the ventricles of the brain), which is the most common condition that pediatric neurosurgeons have to deal with.
The increase in fluid in the ventricles, results in an increase in pressure which leads to compression of the brain tissue against the outer skull. For infants, whose cranial bones are not as yet fused, this results in expansion of the infant’s skull and can also lead to brain damage.
“For three decades, neurosurgeons have treated this condition by inserting a ventricular shunt, which drains the CSF, when pressure builds up in the cranium. Although the volume content of the central ventricle in healthy individuals is roughly the same and does not vary from individual to individual – the actual configuration of the ventricle may vary dramatically between individuals. Thus, a shunt may become occluded, as the brain tissue bounces back, due to the decompressive action of the shunt. As a result, 50 per cent of primary shunt insertions have to be re-operated within two years due to shunt occlusion. Mathematical models can help determine what would be the optimal location for a ventricular shunt in a hydrocephalic patient. In our approach, we use the initial CT scans of a patient, as input for our mathematical/computational model, we can then run “in silico” experiments to see how the brain tissue would bounce back when fluid is drained from the central ventricle. From the predicted final configuration, it is then easy to see where the shunt should be inserted, namely in a location where the brain shows the least bounce back.”
Dr. Sivaloganathan said his group is currently involved in collaborative work with oncologists at Princess Margaret Hospital to develop more efficient and effective ways to treat cancer.
“For ovarian cancer, the gold standard has been surgery as the primary course of action, followed by adjuvant chemotherapy, followed again by surgery if necessary. However, over the last decade there has been a growing belief (partly driven by pragmatic economic necessities) that it may be more optimal to reverse the order of therapeutic modalities. Whilst long term clinical trials were running, testing this hypothesis, we used our “in silico” methods to confirm what was being borne out by the clinical data, i.e. that neo-adjuvant chemotherapy as the first line of action followed by surgery and then adjuvant chemotherapy did in fact result in better outcomes.”
Dr. Sivaloganathan said analyzing mathematical models does provide insight and guidance, assisting oncologists in treating malignant tumours in other parts of the body. He said a critical step in the development of cancerous tumours is the transition from avascular state to a fully vascularised one. This is accomplished by the excretion of chemicals (known as Vascular Endothelial Growth Factors - VEGF) which send signals that result in the sprouting of new blood vessels from existing blood vessels - these new vessels are attracted towards the avascular tumours and contribute to the vascularisation of the tumour. At this stage, the tumour undergoes a phase of dramatic growth, and metastasizes to other parts of the body. But Dr. Sivaloganathan observes that oncologists and biomedical engineers are constantly devising novel, innovative ways to combat this transition.
“Using careful sequencing of antiangiogenic and cytotoxic drugs (the former used to prune the tumour vasculature, and the latter to deliver the knockout blow), oncologists have made significant progress over the last few decades in dealing with malignant tumours.” He notes, however, that there is still room for much improvement. “Our recent work with biomedical engineers at MIT has shown that a most effective strategy is to combine both agents in a nanoparticle. Our mathematical modelling approach together with the experimental work of the MIT group confirm that administration of the drugs in this manner is far superior to the simple sequencing of antiangiogenic and cytotoxic agents.”
He also said that his Group had used mathematical and computational analysis to study the so called “cancer stem cell” hypothesis where a small subpopulation of tumour cells has been hypothesised to drive tumour growth. Focussing on therapeutic methods that specifically target this small subpopulation may lead to more effective ways to combat cancer.
“Currently we're not targeting the cancer stem cells clinically, but instead treating the after effects. More and more evidence is emerging that a small sub-population of mutated stem cells is the driving force that causes the emergence of many different types of cancers.”
Dr. Sivaloganathan mentioned epilepsy and type 2 diabetes as other diseases which are being tackled by mathematicians and physicians through a combination of mathematical analysis and medical studies/trials. And these are but a few examples of how mathematical analysis can contribute to dramatic progress in the biomedical sciences and eventually lead to the resolution of many medical problems.
“Most medical problems are amenable to mathematical and computational analysis. In addition, technological advances worldwide have meant that we can assemble appropriate expertise from anywhere in the world to tackle a particular problem. It doesn't have to be just a local endeavour.”
He added that as applied mathematicians, his group is guided by “Ockham’s principle” that a mathematical model should be as simple as possible but not too simple. The simplest models will answer the required questions if the scientist knows what aspects of the physical problem to include in the mathematical formulation (i.e. what critical features should be captured in the model in order to answer the question at hand). In many ways good mathematical modelling is as much an art as a science.
