Use of Differential Equations in Medicine

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A-level Maths By: Please log in to see tutor details
Subject: Maths » A-level Maths
Last updated: 25/12/2017
Tags: a-level, biomedical science, clinical medicine, differential equations, differentiation

Positron emission tomography (PET) is an imaging technique used to image physiology of the cancer cells with the help of a radionuclide (i.e. tracer).

The tumour cells are known to consume very high levels of glucose. Therefore, a marker of glucose uptake within the cells is the most commonly used tracer in cancer, also called as 2-Deoxy-2-[18F]fluoroglucose ([18F]FDG). The fluorine atom of FDG molecule is radioactive (i.e. unstable) which decays to half its original amount every 110min by emitting gamma rays which are detected by a camera inside the PET scanner. This allows us to produce a three dimensional image of the inside of a tumour without cutting it open!

A patient is asked to lay-down on a bed inside the PET scanner and [18F]FDG injected into their veins. The tracer mixes with the blood and reaches the heart, where it is pumped out to the arteries and delivered everywhere in the body by the complex circulatory system. As the capillaries reach out to the tumour tissue, the tracer washes out from the blood to the cancer cells via extracellular space at unknown exchange rates.

In this system, blood, extracellular spaces and tumour cells are called compartments. Since the change of tracer concentration with time in one of the compartments is a linear function of the concentrations in all other compartments, the above system can be mathematically described using differential equations as:

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——— = K1*C0(t) − (k2+k3)*C1(t) + k4*C2(t)

——— = k3*C1(t) − k4*C2(t)

where, K1 and k2 are the forward and backward tracer exchange rates between blood and extracellular space; k3 and k4 are the forward and backward tracer exchange rates between extracellular space and tumour cells; C0 is the tracer concentration within the blood; C1 is the tracer concentration within extracellular compartment and C2 is the tracer concentration within tumour cell compartment.

We then solve these set of differential equations to obtain the exchange rates K1, k2, k3 and k4 to understand patient specific tumour physiological to form a meaningful clinical interpretation. For example, K1 has been shown to show tumour blood perfusion, which has been shown to be higher in aggressive tumours.

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About The Author

Dr Puri holds a PhD in mathematical modelling of medical data and a postdoc from University of Oxford. He has plenty of experience teaching Maths around Hillingdon. Dr Puri is an expert in making difficult problems look simple and understandable.

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