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.
This work is supported by the Norwegian Research Council project 262203 “Flow-based interpretation of Dynamical Contrast Enhanced Imaging data”.
- 7.A. Fieselmann, M. Kowarschick, A. Ganguly, J. Horneggerand, R. Fahrig, Deconvolution-based CT and MR brain perfusion measurement: theoretical model revisited and practical implementation details. Int. J. Biomed. Imaging Article ID 467563, 20 p. (2011)Google Scholar
- 8.J.W. Forrester, Industrial dynamics: a major breakthrough for decision makers. Harv. Bus. Rev. 36(4), 37–66 (1958)Google Scholar
- 10.E. Hodneland, Å. Kjørestad, E. Andersen, J. Monssen, A. Lundervold, J. Rørvik, A. Zanna, In vivo estimation of glomerular filtration in the kidney using DCE-MRI, in Image and Signal Processing and Analysis (IEEE, Piscataway, NJ, 2011), pp. 755–761. ISSN 1845–5921Google Scholar
- 11.K. Jafari-Khouzani, K.E. Emblem, J. Kalpathy-Cramer, A. Bjørnerud, M.G. Vangel, E.R. Gerstner, K.M. Schmainda, K. Paynabar, O. Wu, P.Y. Wen, T. Batchelor, B. Rosen, S.M. Stufflebeam, Repeatability of cerebral perfusion using dynamic susceptibility contrast MRI in glioblastoma patients. Transl. Oncol. 8(3), 137–146 (2015)CrossRefGoogle Scholar
- 12.R.E. Kalman, A new approach to linear filtering and prediction problems. Trans. AMSE J. Basic Eng. (Ser. D) 82, 34–45 (1960)Google Scholar
- 16.G. Nævdal, O. Sævareid, R.J. Lorentzen, Data assimilation using MRI data, in Proceedings, VII European Congress on Computational Methods in Applied Sciences and Engineering (2016)Google Scholar