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Mathematics and Medicine: How Mathematics, Modelling and Simulations Can Lead to Better Diagnosis and Treatments

  • Erik A. Hanson
  • Erlend Hodneland
  • Rolf J. Lorentzen
  • Geir Nævdal
  • Jan M. Nordbotten
  • Ove Sævareid
  • Antonella Zanna
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)

Abstract

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.

Notes

Acknowledgements

This work is supported by the Norwegian Research Council project 262203 “Flow-based interpretation of Dynamical Contrast Enhanced Imaging data”.

References

  1. 1.
    S.I. Aanonsen, G. Nævdal, D.S. Oliver, A.C. Reynolds, B. Vallès, The ensemble Kalman filter in reservoir engineering – a review. SPE J. 14(3), 393–412 (2009)CrossRefGoogle Scholar
  2. 2.
    J.E. Aarnes, T. Gimse, K.-A. Lie, An introduction to the numerics of flow in porous media using Matlab, in Geometric Modelling, Numerical Simulation, and Optimization (Springer, Heidelberg, 2007), pp. 265–306CrossRefGoogle Scholar
  3. 3.
    G.I. Barenblatt, I.P. Zheltov, I.N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Mech. 24, 1286–1303 (1960)zbMATHGoogle Scholar
  4. 4.
    A. Bjørnerud, K.E. Emblem, A fully automated method for quantitative cerebral hemodynamic analysis using DSC-MRI. J. Cereb. Blood Flow Metab. 30(5), 1066–1078 (2010)CrossRefGoogle Scholar
  5. 5.
    G. Brix, W. Semmler, R. Port, L.R. Schad, G.L.G, W.J. Lorenz, Pharmacokinetic parameters in CNS Gd-DTPA enhanced MR imaging. J. Comput. Assist. Tomogr. 15, 621–628 (1991)CrossRefGoogle Scholar
  6. 6.
    G. Evensen, The ensemble Kalman filter: theoretical formulation and practical implementation. Ocean Dyn. 53, 343–367 (2003)CrossRefGoogle Scholar
  7. 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. 8.
    J.W. Forrester, Industrial dynamics: a major breakthrough for decision makers. Harv. Bus. Rev. 36(4), 37–66 (1958)Google Scholar
  9. 9.
    R.M. Henkelman, Does IVIM measure classical perfusion? Magn. Reson. Med. 16(3), 470–475 (1990)CrossRefGoogle Scholar
  10. 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. 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. 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
  13. 13.
    H. Larsson, M. Stubgaard, J.L. Frederiksen, M. Jensen, O. Henriksen, O.B. Paulson, Quantitation of blood-brain barrier defect by magnetic resonance imaging and gadolinium-DTPA in patients with multiple sclerosis and brain tumors. Magn. Reson. Med. 16, 117–131 (1990)CrossRefGoogle Scholar
  14. 14.
    A. Matzavinos, C.-Y. Kao, J.E.F. Green, A. Sutradhar, M. Millerand, A. Friedman, Modeling oxygen transport in surgical tissue transfer. Proc. Natl. Acad. Sci. USA 29, 12091–12096 (2009)CrossRefGoogle Scholar
  15. 15.
    P. Meier, K.L. Zierler, On the theory of the indicator-dilution method for measurement of blood flow and volume. J. Appl. Physiol. 6(12), 731–744 (1954)CrossRefGoogle Scholar
  16. 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
  17. 17.
    D.S. Oliver, Y. Chen, Recent progress on reservoir history matching: a review. Comput. Geosci. 15, 185–221 (2010)CrossRefGoogle Scholar
  18. 18.
    S. Patankar, Numerical Heat Transfer and Fluid Flow, 1st edn. (Hemisphere Publishing Corporation, Washington, 1980)zbMATHGoogle Scholar
  19. 19.
    M. Presho, S. Wo, V. Ginting, Calibrated dual porosity, dual permeability modeling of fractured reservoirs. J. Pet. Sci. Eng. 77, 326–337 (2011)CrossRefGoogle Scholar
  20. 20.
    S. Sourbron, A tracer-kinetic field theory for medical imaging, IEEE Trans. Med. Imaging 33(4), 935–946 (2014)CrossRefGoogle Scholar
  21. 21.
    S.P. Sourbron, D.L. Buckley, Trace kinetic modelling in MRI: estimating perfusion and capillary permeability. Phys. Med. Biol. 57(2), R1–R33 (2012)CrossRefGoogle Scholar
  22. 22.
    P. Tofts, A.G. Kermode, Measurement of the blood-brain barrier permeability and leakage space using dynamic MR imaging. 1. Fundamental concepts. Magn. Reson. Med. 17, 357–367 (1991)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Erik A. Hanson
    • 1
  • Erlend Hodneland
    • 2
  • Rolf J. Lorentzen
    • 3
  • Geir Nævdal
    • 3
  • Jan M. Nordbotten
    • 1
    • 4
  • Ove Sævareid
    • 3
  • Antonella Zanna
    • 1
  1. 1.Department of MathematicsUniversity of BergenBergenNorway
  2. 2.Christian Michelsen ResearchBergenNorway
  3. 3.International Research Institute of StavangerStavangerNorway
  4. 4.Department of Civil and Environmental EngineeringPrinceton UniversityPrincetonUSA

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