Mathematics and Medicine: How mathematics, modelling and simulations can lead to better diagnosis and treatments

Starting with the discovery of X-rays by Röntgen in 1895, the progress in medical imaging has been extraordinary and immensely beneficial to diagnosis and therapy. Parallel to the increase of imaging accuracy, there is the quest of moving from qualitative to quantitative analysis and patient-tailored therapy. Mathematics, modelling and simulations are increasing their importance as tools in this quest. In this paper we give an overview of relations between mathematical modelling and imaging and focus particularly on the estimation of perfusion in the brain. In the forward model, the brain is treated as a porous medium and a two compartment model (arterial/venous) is used. Motivated by the similarity with techniques in reservoir modelling, we propose an ensemble Kalman filter to perform the parameter estimation and apply the method to a simple example as an illustrative example.acceptedVersio

Alterations in anatomic and functional imaging parameters with repeated FDG PET-CT and MRI during radiotherapy for head and neck cancer: a pilot study

Background: The use of imaging to implement on-treatment adaptation of radiotherapy is a promising paradigm but current data on imaging changes during radiotherapy is limited. This is a hypothesis-generating pilot study to examine the changes on multi-modality anatomic and functional imaging during (chemo)radiotherapy treatment for head and neck squamous cell carcinoma (HNSCC). Methods: Eight patients with locally advanced HNSCC underwent imaging including computed tomography (CT), Fluorine-18 fluorodeoxyglucose (FDG) positron emission tomography (PET)-CT and magnetic resonance imaging (MRI) (including diffusion weighted (DW) and dynamic contrast enhanced (DCE)) at baseline and during (chemo)radiotherapy treatment (after fractions 11 and 21). Regions of interest (ROI) were drawn around the primary tumour at baseline and during treatment. Imaging parameters included gross tumour volume (GTV) assessment, SUVmax, mean ADC value and DCE-MRI parameters including Plasma Flow (PF). On treatment changes and correlations between these parameters were analysed using a Wilcoxon rank sum test and Pearson’s linear correlation coefficient respectively. A p-value <0.05 was considered statistically significant. Results: Statistically significant reductions in GTV-CT, GTV-MRI and GTV-DW were observed between all imaging timepoints during radiotherapy. Changes in GTV-PET during radiotherapy were heterogeneous and non-significant. Significant changes in SUVmax, mean ADC value, Plasma Flow and Plasma Volume were observed between the baseline and the fraction 11 timepoint, whilst only changes in SUVmax between baseline and the fraction 21 timepoint were statistically significant. Significant correlations were observed between multiple imaging parameters, both anatomical and functional; 20 correlations between baseline to the fraction 11 timepoint; 12 correlations between baseline and the fraction 21 timepoints; and 4 correlations between the fraction 11 and fraction 21 timepoints. Conclusions: Multi-modality imaging during radiotherapy treatment demonstrates early changes (by fraction 11) in both anatomic and functional imaging parameters. All functional imaging modalities are potentially complementary and should be considered in combination to provide multi-parametric tumour assessment, to guide potential treatment adaptation strategies. Trial Registration: ISRCTN Registry: ISRCTN34165059. Registered 2nd February 2015