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# A first step towards a comprehensive model of Alzheimer's disease

Alzheimer's disease (AD) is a multifactorial disease. Therefore, an epidemiologic model of AD can hardly consider every risk factor, every lifestyle choice, and every genetic background occurring throughout life, in order to arrive at a personalized, individual prediction of future disease course. This is where a mathematical model takes all its sense. It’s a flexible tool that can take into account many factors, determines the causal relations, and hence, helps search for solutions.

Following this idea, we aimed to develop a mathematical model that puts into relation various biologic entities linked to AD and observe their evolution in a human brain during 50 (fifty) years, more precisely, between 30 (thirty) and 80 (eighty) years old.

It is this model that Éléonore Chamberland, Master's student in Applied Mathematics under the co-supervision of Nicolas Doyon and Simon Duchesne, is presenting today at the annual AD/PD conference that is being held in Gothenburg, Sweden (Éléonore will be doing a remote presentation).

She created the model with the idea that the person does not have AD. This implies that, for all our parameters, she tried to take values that would be appropriate for an average normal healthy person.

The model is composed of 19 (nineteen) differential equations, one for each component that we observe. Her model also takes account of the influence of biological sex. Indeed, it is well known that women are more affected by AD than men.

On the next image, she presents her results in a figure encompassing the concentrations of the variables as a function of age, for the four (4) combinations of sex and APOE4 status.

You will notice an important change in most of the concentrations around fifty (50) years, where we have a transition from an anti-inflammatory state to a pro-inflammatory state. Women APOE4 carriers are the first to have that switch, followed by men APOE4 positive, women APOE4 negative, and finally, men APOE4 negative.

In short, the curves that we obtain are mostly qualitatively coherent with reality. Indeed, we observe an augmentation of the concentration of the different forms of amyloid-beta, phosphorylated and hyperphosphorylated tau proteins, and intra- and extracellular NFTs. We also observe a diminution of the density of neurons, which is directly related to neuron loss since we do not consider the shrinkage of the brain. We also have an augmentation of the density of activated astrocytes.

The model has not been fitted to experimental data, which is necessary to do, and will be done. This explains why we only presented a qualitative analysis. And even with this type of analysis, some problems can be noticed, for example, the drastic change around fifty (50) years of age, which is certainly not representative of reality.

In brief, the model is on a good path but still needs validation and adjustments. It is certainly a step in the right direction, one for which Éléonore should be proud!