- Email: [email protected]
Effect of accurately calculated b matrix on the evaluation of the diffusion tensor using echoplanar diffusion tensor imaging
ITBM-RBM 2002 ; 23 : 340-344 0 2002 kditions scientifiques et m&kales
Elsevier
SAS.
Tous droits
rkrv&
Original article
Effect of accurately calculated b matrix on the evaluation of the diffusion tensor using echoplanar diffusion tensor imaging S. Boujraf‘* , R . Luypaert2,H. Eisendrath2, M. Osteaux2 ’ Institutjlir Neumradiologie, UniversitiitsSpitalZiirich, Frauenklinikstr IO, NORDI, 8091 Ziirich, Switzeland; 2Biomedical MR Unit, AZ-VUB, Laarbeeklaan 101, 1090 Brussels,Belgium Received
15 May
2000; revised
3 June 2002; accepted
7 August
2002
Abstract Diffusion-weighted magnetic resonance imaging is a recognised tool in early detection of infarction of the human brain. If diffusion weighted-images with different values of b matrix are acquired during one individual investigation, it is possible to calculate diffusion coefficient maps that are elements of the diffusion tensor: they represent the apparent diffusion coefficient of protons of water molecules in each pixel in the corresponding sample. The relationship between signal intensity in the diffusion-weighted images, diffusion tensor and b matrix is derived from the Bloch equations. Our goal is to establish the error made in the calculation of the elements of the diffusion tensor when imaging gradients are ignored in the calculation of the b matrix elements. The numerical calculation of the b matrices show that a systematic error of up to 7% is introduced in the derived diffusion tensor elements when neglecting imaging gradients. 0 2002 Editions scientifiques et mkdicales Elsevier SAS magnetic resonance
Imaging I diffusion
tensor I b matrix I echoplanar
imaging
R6sum6 - Effet de prkcision du calcul de la matrice b sur Mvaluation du tenseur de diffusion en utilisant I’imagerie kchoplanalre du tenseur de diffusion. L’imagerie par r&sonance mag&tique (IRM) participe utilement B la detection de /‘infarctus du cerveau humain. En faisant l’acquisition des images de diffusion avec des valeurs diffhrentes de la matrice 6, ii est possible de calculer une cartographic de coefficient de diffusion des protons des mol&ules d’eau dans chacun des pixels de l’image de l’&hantillon. La relation entre l’intensith du signal dans les images de diffusion, le tenseur de diffusion et la matrice b est obtenue B partir des gquations de Block. Notre but est d’estimer l’erreur faite en nbgligeant les gradients d’imageries dans le calcul des BltSments de la matrice . Le calcul numerique montre que plus de 7 % d’erreur est i!?troduite dans les 6ments du tenseur de diffusion, en &gligeant les gradients d’imagerie. 0 2002 Editions scientifiques et mt5dicales Elsevier SAS imagerie par rbsonance
* Correspondence E-mail address:
and reprints. Fax: +41 1 255 4504. [email protected] (S. Boujraf).
magnbtique
I tenseur de diffusion
I matrice bl imagerie bchopianaire
341
Evaluation of the MRI diffusion tensor using EPI
INTRODUCTION
The self-diffusion tensor [l] can be estimatedusing the relationship:
Ia!9 = exp Z(O)
(1)
b matrix Where bii containinginformation on the gradientpulses,D, are the componentsof the self-diffusion tensor [2, 31. For an accurateestimationof the diffusion tensor,we must take into account all diffusion and imaging gradientsin the sequenceas well as all their cross terms. The purpose of this paper is to present a numerical calculation of the b matrix for a pulsed gradient diffusion-weighted echo planar imaging (EPI) sequencewith sinusoidal readout gradients. The contributions of the different gradients (and their interactions)to the values of the b matrix elements arecalculatedexplicitly and compared. MATERIAL
running on a personalcomputer.The b matrix elements in equation I were calculatedusing: b
Mt)-2H(t-z)f(t)]
(2)
wheref(t) = G( t’) dt’ andH(t) is the Heaviside function [l]. 5’0 For our sequence (see figure I), trapezoidal diffusion-weighting pulseswith a variableamplitude G(t) wereappliedbeforeand after the 180” radio frequencypulse. The b, elementsof the b matrix were calculated for diffusion gradients along the phase, read and slice directions and along the bisectors phase-read,phase-sliceand read-slice.The ramp-up and ramp-down times of all the diffusion and imaging gradientswere takeninto account,including the sinusoidalreadoutgradients. The full b matrix elementswere calculatednumerically, then approximatedanalytically according to the Stejskal-Tannerformula for trapezoidalpulses, including rise and fall times Eof the diffusion gradient pulses:
AND METHOD
Our numerical calculations were performed using Mathcad Software (Math Soft, International, UK)
b = (~.G)‘[(~-3.~+~-(~.5)]
(3)
Echo 80”
Figure 1. Schematic diagram of the diffusion-weighted
single shot echoplanar imaging sequence.
342
S. Boujraf
et al.
Evaluation of the MRI diffusion tensor using EPI
where y, Sand A are respectivelythe gyromagnetic ratio for protons,the diffusion gradientpulse duration andthe delay betweenthe start times of the diffusion gradients respectively [2]. This approximation correspondsto ignoring the imaging gradients. The experimental validation of the different b matriceswas performedon a 1.5 Teslawhole body imager with a maximum gradient strength of 25 mTesla m-’ (Siemens,Erlangen,Germany),using a seriesof calibration measurementson an isotropic phantom(waterdopedwith NiSO,X6H,O at -20°C). The measurement parameters were echo time TE = 123 ms, repetition time TR = 3000 ms, slice thickness 6 mm, field of view FOV = 240 mm,
matrix size 128*.The diffusion tensorelementswere calculated using both sets of values for the b,, and compared to results found for spin echo imaging sequencefrom the literature [5]. Our MRI imager is capable of generating -+25 mTesla m-’ in 150l..ts. For this sequence,the ramps of diffusion gradients were fixed to E = 700 ps with a durationof 6= 26 ms anddiffusion time A = 59.7 ms. RESULTS Table 1 and 2 report the b matrix elementsin seconds per squaremillimetre calculated for three diffusion gradientstrengths0,ll and22 mTeslam-‘. Each line
344
S. Boujraf et al.
of table I and2 representsthe independentelements of one b matrix (this matrix being symmetrical). In table 3 we report the relative differences betweenthe full andspectroscopicb matrix elements. The five star elements correspondto a percentage that is lessthan 0.01%. For validating the b matrix resultsandfor showing their effect on the degreeof accuracyobtained for the diffusion tensorelements,thesewere calculated using quantitativediffusion analysis [3, 41. For Of, the diffusion tensorcalculatedusing b matricestaking into accountall imaging and diffusion gradients, and D, the diffusion tensor calculated using the b matricesignoring the imaging gradients,we found (in squaremillimetres per second): 2.08 - 1O-34.33 . 1O-6 1.56a1O-5
DISCUSSION AND CONCLUSIONS The numerical calculation of the b matrices show that a systematicerror of up to 7% is introducedby neglecting the imaging gradients in the b matrix elementscalculations. As expectedthe error is more important for weak diffusion gradients. The effect of this systematic error on the estimation of the diffusion tensor elementsfor anisotropic phantomwith known diffirsion coefficients can be describedas follows: the diffusion tensoris, to a good approximation,diagonal, as expectedfor an isotropic medium. In each case,we obtain diagonalelementsthat overestimate the true value of the diffusion coefficient. This overestimation is up to three times smaller when using the full b matrix. REFERENCES
Df = and
1.56.. lo-’ 4.33 1O-65.38 2.02.. 1O-6 1O-32.13 5.38 .. 1O-3 1O-6 I
2.27 - 1O-3 1.84. 1O-52.39. lo-’ D, =
1.84. lo-’ 2.14 +1O-3 1.36. lO-5 2.39 - 1O-5 1.36. 1O-52.18 . 1O-3
In order to evaluate these results, they were comparedto literature data for the diffusion coefficient of water [5], Do = (1.96 f 0.5) . 1O-3mm2 s-l (Table4). This result was obtainedusing diffusionweighted spin echotechnique.
Basser P, Matteillo J, LeBihan D. Estimation of the effective self-diffusion tensor from the NMR spin echo. J Magn Reson B 1994;103:247-54. Matteillo J, Basser P, Lebihan DJ. Analytical expression for the b matrix in NMR diffusion imaging and spectroscopy. J Magn Reson A 1994;108:131-41. Budinsky L, Jellus V, Szomolanyi P, Coremans J, Stadnik T, Boujraf S, et al. Quantitative diffusion analysis performed in clinical scanner. MAGMA 1998;6(Suppl 1): 120-l. Boujraf S, Luypaert R, Coremans J, Budinsky L, Jellus V, Eisendrath H.. The b matrix in diffusion tensor imaging using EPI. MAGMA 1998;6(Suppl 1): 117. LeBihan D, Cuenod CA, Posse S. Determination of the gradient factor b in local&d diffusion ‘H spectroscopy. Proceeding of the 11” Annual Scientific Meeting of the Society of Magnetic Resonance in Medicine, August 8-14. 1992, Berlin, Germany, Vol. 1, p. 1218.
Elsevier
SAS.
Tous droits
rkrv&
Original article
Effect of accurately calculated b matrix on the evaluation of the diffusion tensor using echoplanar diffusion tensor imaging S. Boujraf‘* , R . Luypaert2,H. Eisendrath2, M. Osteaux2 ’ Institutjlir Neumradiologie, UniversitiitsSpitalZiirich, Frauenklinikstr IO, NORDI, 8091 Ziirich, Switzeland; 2Biomedical MR Unit, AZ-VUB, Laarbeeklaan 101, 1090 Brussels,Belgium Received
15 May
2000; revised
3 June 2002; accepted
7 August
2002
Abstract Diffusion-weighted magnetic resonance imaging is a recognised tool in early detection of infarction of the human brain. If diffusion weighted-images with different values of b matrix are acquired during one individual investigation, it is possible to calculate diffusion coefficient maps that are elements of the diffusion tensor: they represent the apparent diffusion coefficient of protons of water molecules in each pixel in the corresponding sample. The relationship between signal intensity in the diffusion-weighted images, diffusion tensor and b matrix is derived from the Bloch equations. Our goal is to establish the error made in the calculation of the elements of the diffusion tensor when imaging gradients are ignored in the calculation of the b matrix elements. The numerical calculation of the b matrices show that a systematic error of up to 7% is introduced in the derived diffusion tensor elements when neglecting imaging gradients. 0 2002 Editions scientifiques et mkdicales Elsevier SAS magnetic resonance
Imaging I diffusion
tensor I b matrix I echoplanar
imaging
R6sum6 - Effet de prkcision du calcul de la matrice b sur Mvaluation du tenseur de diffusion en utilisant I’imagerie kchoplanalre du tenseur de diffusion. L’imagerie par r&sonance mag&tique (IRM) participe utilement B la detection de /‘infarctus du cerveau humain. En faisant l’acquisition des images de diffusion avec des valeurs diffhrentes de la matrice 6, ii est possible de calculer une cartographic de coefficient de diffusion des protons des mol&ules d’eau dans chacun des pixels de l’image de l’&hantillon. La relation entre l’intensith du signal dans les images de diffusion, le tenseur de diffusion et la matrice b est obtenue B partir des gquations de Block. Notre but est d’estimer l’erreur faite en nbgligeant les gradients d’imageries dans le calcul des BltSments de la matrice . Le calcul numerique montre que plus de 7 % d’erreur est i!?troduite dans les 6ments du tenseur de diffusion, en &gligeant les gradients d’imagerie. 0 2002 Editions scientifiques et mt5dicales Elsevier SAS imagerie par rbsonance
* Correspondence E-mail address:
and reprints. Fax: +41 1 255 4504. [email protected] (S. Boujraf).
magnbtique
I tenseur de diffusion
I matrice bl imagerie bchopianaire
341
Evaluation of the MRI diffusion tensor using EPI
INTRODUCTION
The self-diffusion tensor [l] can be estimatedusing the relationship:
Ia!9 = exp Z(O)
(1)
b matrix Where bii containinginformation on the gradientpulses,D, are the componentsof the self-diffusion tensor [2, 31. For an accurateestimationof the diffusion tensor,we must take into account all diffusion and imaging gradientsin the sequenceas well as all their cross terms. The purpose of this paper is to present a numerical calculation of the b matrix for a pulsed gradient diffusion-weighted echo planar imaging (EPI) sequencewith sinusoidal readout gradients. The contributions of the different gradients (and their interactions)to the values of the b matrix elements arecalculatedexplicitly and compared. MATERIAL
running on a personalcomputer.The b matrix elements in equation I were calculatedusing: b
Mt)-2H(t-z)f(t)]
(2)
wheref(t) = G( t’) dt’ andH(t) is the Heaviside function [l]. 5’0 For our sequence (see figure I), trapezoidal diffusion-weighting pulseswith a variableamplitude G(t) wereappliedbeforeand after the 180” radio frequencypulse. The b, elementsof the b matrix were calculated for diffusion gradients along the phase, read and slice directions and along the bisectors phase-read,phase-sliceand read-slice.The ramp-up and ramp-down times of all the diffusion and imaging gradientswere takeninto account,including the sinusoidalreadoutgradients. The full b matrix elementswere calculatednumerically, then approximatedanalytically according to the Stejskal-Tannerformula for trapezoidalpulses, including rise and fall times Eof the diffusion gradient pulses:
AND METHOD
Our numerical calculations were performed using Mathcad Software (Math Soft, International, UK)
b = (~.G)‘[(~-3.~+~-(~.5)]
(3)
Echo 80”
Figure 1. Schematic diagram of the diffusion-weighted
single shot echoplanar imaging sequence.
342
S. Boujraf
et al.
Evaluation of the MRI diffusion tensor using EPI
where y, Sand A are respectivelythe gyromagnetic ratio for protons,the diffusion gradientpulse duration andthe delay betweenthe start times of the diffusion gradients respectively [2]. This approximation correspondsto ignoring the imaging gradients. The experimental validation of the different b matriceswas performedon a 1.5 Teslawhole body imager with a maximum gradient strength of 25 mTesla m-’ (Siemens,Erlangen,Germany),using a seriesof calibration measurementson an isotropic phantom(waterdopedwith NiSO,X6H,O at -20°C). The measurement parameters were echo time TE = 123 ms, repetition time TR = 3000 ms, slice thickness 6 mm, field of view FOV = 240 mm,
matrix size 128*.The diffusion tensorelementswere calculated using both sets of values for the b,, and compared to results found for spin echo imaging sequencefrom the literature [5]. Our MRI imager is capable of generating -+25 mTesla m-’ in 150l..ts. For this sequence,the ramps of diffusion gradients were fixed to E = 700 ps with a durationof 6= 26 ms anddiffusion time A = 59.7 ms. RESULTS Table 1 and 2 report the b matrix elementsin seconds per squaremillimetre calculated for three diffusion gradientstrengths0,ll and22 mTeslam-‘. Each line
344
S. Boujraf et al.
of table I and2 representsthe independentelements of one b matrix (this matrix being symmetrical). In table 3 we report the relative differences betweenthe full andspectroscopicb matrix elements. The five star elements correspondto a percentage that is lessthan 0.01%. For validating the b matrix resultsandfor showing their effect on the degreeof accuracyobtained for the diffusion tensorelements,thesewere calculated using quantitativediffusion analysis [3, 41. For Of, the diffusion tensorcalculatedusing b matricestaking into accountall imaging and diffusion gradients, and D, the diffusion tensor calculated using the b matricesignoring the imaging gradients,we found (in squaremillimetres per second): 2.08 - 1O-34.33 . 1O-6 1.56a1O-5
DISCUSSION AND CONCLUSIONS The numerical calculation of the b matrices show that a systematicerror of up to 7% is introducedby neglecting the imaging gradients in the b matrix elementscalculations. As expectedthe error is more important for weak diffusion gradients. The effect of this systematic error on the estimation of the diffusion tensor elementsfor anisotropic phantomwith known diffirsion coefficients can be describedas follows: the diffusion tensoris, to a good approximation,diagonal, as expectedfor an isotropic medium. In each case,we obtain diagonalelementsthat overestimate the true value of the diffusion coefficient. This overestimation is up to three times smaller when using the full b matrix. REFERENCES
Df = and
1.56.. lo-’ 4.33 1O-65.38 2.02.. 1O-6 1O-32.13 5.38 .. 1O-3 1O-6 I
2.27 - 1O-3 1.84. 1O-52.39. lo-’ D, =
1.84. lo-’ 2.14 +1O-3 1.36. lO-5 2.39 - 1O-5 1.36. 1O-52.18 . 1O-3
In order to evaluate these results, they were comparedto literature data for the diffusion coefficient of water [5], Do = (1.96 f 0.5) . 1O-3mm2 s-l (Table4). This result was obtainedusing diffusionweighted spin echotechnique.
Basser P, Matteillo J, LeBihan D. Estimation of the effective self-diffusion tensor from the NMR spin echo. J Magn Reson B 1994;103:247-54. Matteillo J, Basser P, Lebihan DJ. Analytical expression for the b matrix in NMR diffusion imaging and spectroscopy. J Magn Reson A 1994;108:131-41. Budinsky L, Jellus V, Szomolanyi P, Coremans J, Stadnik T, Boujraf S, et al. Quantitative diffusion analysis performed in clinical scanner. MAGMA 1998;6(Suppl 1): 120-l. Boujraf S, Luypaert R, Coremans J, Budinsky L, Jellus V, Eisendrath H.. The b matrix in diffusion tensor imaging using EPI. MAGMA 1998;6(Suppl 1): 117. LeBihan D, Cuenod CA, Posse S. Determination of the gradient factor b in local&d diffusion ‘H spectroscopy. Proceeding of the 11” Annual Scientific Meeting of the Society of Magnetic Resonance in Medicine, August 8-14. 1992, Berlin, Germany, Vol. 1, p. 1218.