Research Topic : Mathematical
Modelling and Numerical Simulation in Magnetic Resonance Imaging
The general framework of this research topic is the mathematical modelling
and the numerical simulation of various phenomena occurring in Magnetic
Resonance Imaging (MRI). This work was conducted in collaboration on the
one hand with G. Caloz (IRMAR, UMR 6625, Université de Rennes I), J. de
Certaines and G. Cathelineau (Laboratory of Magnetic Resonance in Biology
and Medicine, Université de Rennes I), one the other hand with H.
Benoit-Cattin and C. Odet (CREATIS, UMR 5515, INSA Lyon) and finally with
L. Chupin (Institut Camille Jordan, UMR 5208, INSA Lyon).
This research topic is mainly concerned with the study of perturbations in
the image acquisition process in MRI and their various consequences. A MRI
image is obtained by the use of various magnetic fields (static and
radio-frequency) and any disturbance of this magnetic field (due to
apparatus itself or to metallic implanted devices in the patient body)
results in a distortion of the image (called an artefact). The
mathematical study of magnetic susceptibility artefacts related to the
presence of a static magnetic field disturbance induced by an implanted
medical device worn by the patient (a major problem in MRI) that started
during my PhD thesis is now complete. Two complementary aspects related to
this problem have been addressed:
- to propose a method for calculating the disturbance of the static
magnetic field created by the presence of a metallic medical implant
in an MRI experiment;
- to modelise how the inhomogeneities of the static magnetic field
generate distortions of the MRI image.
The above mentioned aspects of the work have been the subject of the
publications [6], [7], [4], [3], [2] and [1]. We point out that some
methods for the correction of the magnetic susceptibility artefacts have
been investigated by B. Belaroussi in his PhD thesis
[a] defended
in October 2005 at INSA de Lyon (CREATIS laboratory).
A software (SIMRI) has been developed in the CREATIS laboratory during the
period 2002-2005 to simulate the whole MR imaging process, including the
effects of magnetic field inhomogeneities. I have been working on this
software project for some mathematical aspects connected with the
redevelopment of proper numerical methods to take into account magnetic
field inhomogeneities in the MR imaging process. This collaboration has
resulted in the publication [7].
We have also addressed the computation of the magnetic field induced by a
cluster of small metallic particles in MRI. Because of the difficulty of
the problem from a numerical point of view, the simplifying assumption
that the field due to each particle interacts only with the main magnetic
field but does not interact with the fields due to the other particles is
usually made. We have investigated from a mathematical point of view the
relevancy of this assumption and provided an error estimate for the scalar
magnetic potential in terms of the key parameter that is the minimal
distance between the particles. When the "non-interacting assumption" is
deficient, we have proposed a numerical method to compute a better
approximation of the magnetic potential by taking into account
pairwise magnetic field interactions between particles that enters in a
general framework for computing the scalar magnetic potential as a series
expansion.This work has been conducted with
Laurent
Chupin (Laboratoire de Mathématiques Blaise Pascal, Université
Clermont Auvergne) and
Sébasitien
Martin (MAP5, Université Paris Descartes). This work is presented in
publication [9].
Another part of the work concerns the effects of radio-frequency (RF)
disturbances in the MR imaging process.
On the one hand, RF field perturbations are liable for a particular type
of image distortion known as RF artefacts. The mathematical modelling and
the numerical simulation of this type of artefact is more delicate than
magnetic susceptibility artefacts because it requires to solve the Bloch
equation (that describes the evolution of the macroscopic magnetic moment
in time) with time varying magnetic fields. An explicit solution to the
Bloch equation does not exist anymore for time varying magnetic fields and
numerical methods are required. Standard numerical methods for ordinary
differential equations such as Runge-Kutta methods are expensive here
since the Bloch equation has to be solved for each voxel (up to 256^3
voxels may be considered). We have developed an original numerical method
to solve the Bloch equation in this context. From the initial
spatial distribution of magnetisation our method computes directly the
spatial distribution of magnetisation at the end of the RF pulses sequence
with an arbitrary given accuracy without any need for a discretisation of
the time interval such as in the Runge-Kutta methods. The method is
presented in publication [8].
On the other hand, RF fields in MRI can be responsible for a temperature
increase of the biological tissues in the vicinity of elongated metallic
objects such as catheters or cable used in interventional MRI
(micro-surgery under MRI). The mathematical modelling and the numerical
simulation of this phenomenon is relatively complicated. It requires to
take into account the coupling between some electromagnetic phenomenon
(such as eddy currents) with heat diffusion. A first study of this problem
has been conducted by H. Bouk'hil, with whom we have collaborated, in his
PhD thesis
[b] defended in 2003 at the Laboratory of
Magnetic Resonance in Biology and Medicine at the University of Rennes 1.
This study has been prolonged in the PhD of P. Boissoles
[c]
defended in 2005 at IRMAR, University of Rennes 1 where a numerical method
for the computation of the RF magnetic field generated in a birdcage coil
is presented and a detailed mathematical analysis of the problem of the
computation of RF fields induced by an elongated metallic object is
achieved.
[a] Boubakeur Belaroussi, Correction par traitement
d?images de l?artefact de susceptibilité magnétique dans les images IRM,
thèse de l?INSA de Lyon, 2005.
[b] Hind Bouk?hil, Contribution à la caractérisation des
effets thermiques liés aux biomatériaux métalliques en imagerie par
résonance magnétique, Thèse de l?Université de Rennes 1, Faculté de
Médecine, 2003.
[c] Patrice Boissoles, Problèmes mathématiques et
numériques issus de l'imagerie par résonance magnétique, Thèse de
l?Université de Rennes 1, IRMAR, 2005.
Publications related to this research topic
9- S. Balac, L. Chupin and S. Martin. Computation of the magnetic
potential induced by a collection of spherical particles using series
expansions. Technical report, 2019.
hal-02072281.
8- S. Balac and L. Chupin. Fast approximate solution of Bloch equation for
simulation of RF artifacts in Magnetic Resonance Imaging.
Mathematical
and
Computer
Modelling (48 : 1901-1913 (2008))
7- S. Balac, H. Benoit-Cattin, T. Lamotte and C. Odet. Analytic solution
to boundary integral computation of susceptibility induced magnetic field
inhomogeneities. Mathematical and Computer Modelling, 39(4-5):437?455,
2004.
6- S. Balac and G. Caloz. Induced magnetic field computations using a
boundary integral formulation. Applied Numerical Mathematics,
41(3):345?367, 2002.
5- S. Balac, G. Caloz, G. Cathelineau, B. Chauvel and J.D. De Certaines.
An integral representation method for numerical simulation of MRI
artifacts induced by metallic implants. Journal of Magnetic Resonance in
Medicine, 45(4):724?727, 2001.
4- S. Balac and G. Caloz. Mathematical modeling and numerical simulation
of magnetic susceptibility artifacts in Magnetic Resonance Imaging.
Computer Methods in Biomechanics and Biomedical Engineering, 3:335?349,
2000.
3- S. Balac. Simulation numérique des artefacts de susceptibilité
magnétique en IRM. Innovation et Technologie en Biologie et Médecine
(ITBM), 19(5):369?379, 1998.
2- S. Balac and G. Caloz. Magnetic susceptibility artifacts in Magnetic
Resonance Imaging : calculation of the magnetic field disturbances. IEEE
Trans. on Magnetics, 32(3):1645?1648, 1996.
1- B. Chauvel, G. Cathelineau, S. Balac, J. Lecerf and J.D. De Certaines.
Cancellation of metalinduced MRI artifacts with dual-component and
diamagnetic material : mathematical modelization and experimental
verification. Journal of Magnetic Resonance Imaging, 6(6)::936-938 (1996)
