Letter to the editor Received: 23 March 2014,
Revised: 02 May 2014,
Accepted: 07 May 2014,
Published online in Wiley Online Library: 5 June 2014
(wileyonlinelibrary.com) DOI: 10.1002/nbm.3145
Sufficiency of diffusion tensor in characterizing the diffusion MRI signal to leading order in diffusion weighting Jens H. Jensena,b* Sir,
NMR Biomed. 2014; 27: 1005–1007
t0
where g(t) is the applied diffusion gradient as a function of time t, r(t) is the diffusion path, σ(t) is the spin flip function, t0 is the initial time, T is the final time and γ is the gyromagnetic ratio for the nucleus of interest. The signal is normalized to unity for g(t) = 0. The angled brackets indicate an averaging over the ensemble of diffusing molecules (water in most applications). Any 180° refocusing pulses are idealized as being instantaneous and incorporated into Equation [1] through σ(t), which has a magnitude of unity and changes sign at the time of each refocusing pulse (9). The diffusion gradient is assumed to satisfy the condition: T
0 ¼ ∫ dt σðtÞ gðtÞ
[2]
t0
so that there is no dephasing for stationary spins. Different dMRI methods correspond to different choices for g(t) and σ(t). The product g(t)σ(t) may be regarded as an ‘effective gradient’ for the pulse sequence. Integration by parts in Equation [1] leads to: S¼
T exp iγ ∫ dt f ðtÞvðt Þ
[3]
t0
where v(t) ≡ dr(t)/dt and:
* Correspondence to: J. H. Jensen, Center for Biomedical Imaging, Department of Radiology and Radiological Science, Medical University of South Carolina, 96 Jonathan Lucas Street, MSC 323, Charleston, SC 29425-0323, USA. E-mail: [email protected] a J. H. Jensen Department of Radiology and Radiological Science, Medical University of South Carolina, Charleston, SC, USA b J. H. Jensen Center for Biomedical Imaging, Medical University of South Carolina, Charleston, SC, USA Abbreviations used: dMRI, diffusion MRI; DTI, diffusion tensor imaging; MPdMRI, multiple-pulsed diffusion MRI; OG-dMRI, oscillating gradient diffusion MRI.
Copyright © 2014 John Wiley & Sons, Ltd.
1005
Diffusion MRI (dMRI) is a rich and varied field, with a multitude of distinct approaches, including diffusion tensor imaging (DTI) (1), multiple pulsed dMRI (MP-dMRI) (2,3) and oscillating gradient dMRI (OG-dMRI) (4). These methods differ in terms of the pulse sequences employed for data acquisition and the post-processing algorithms utilized for data analysis. However, they all share a common basis in diffusion and NMR physics. This commonality leads to important connections that link measurements obtained with seemingly disparate techniques. Two examples have been discussed in recent publications. First, Jespersen (5) demonstrated that single- and double-pulsed dMRI are equivalent for low diffusion weightings. Second, Novikov and Kiselev (6) derived a rigorous quantitative relationship between the short-time behavior of the diffusion coefficient and the high-frequency behavior of the OG-dMRI signal in the Gaussian approximation. Moreover, Novikov and Kiselev (6,7) emphasized the equivalence of the information provided by the diffusion coefficient, as can be measured with a classical Stejskal–Tanner sequence (8), and the frequency-dependent diffusivity determined by OG-dMRI. Here, we briefly note that these two observations are closely related and are aspects of a more general property of dMRI. In particular, we show that a knowledge of the diffusion tensor as a function of time is, in principle, sufficient to predict the dMRI signal intensity to leading order in the diffusion weighting (e.g. the b value) for any dMRI method. As the diffusion tensor is measurable with DTI, one may regard DTI as being comprehensive to this leading order. Thus, alternative methods, such as MP-dMRI and OG-dMRI, only provide additional information, beyond that which is available from DTI, in their higher order contributions to the signal. This is not to say that alternative methods are not useful, even to leading order, but that the interpretation of their results should be informed by the interconnections with other dMRI techniques. It is from this conceptual perspective that the diffusion tensor is sufficient for low diffusion weightings, with dMRI approaches other than DTI certainly having important practical advantages for specific applications. The theoretical connection between the diffusion tensor and the dMRI signal to leading order has already been presented by Jespersen (5), but was discussed primarily in the context of double-pulsed dMRI and a detailed derivation was not provided. Below, we give a self-contained demonstration of this fundamental fact, in order to clarify and highlight its physical basis and generality.
To understand how the diffusion tensor predicts the dMRI signal to leading order in the diffusion weighting, consider the phase integral expression for the dMRI signal: T S ¼ exp iγ ∫ dt σ ðtÞgðtÞrðt Þ [1]
J. H. JENSEN
t f ðt Þ≡ ∫ dt ′ σ t′ g t′
[4]
t0
For a purely diffusive system, one has the no net flow condition of 〈v(t)〉 = 0. A formal series in powers of γ for ln(S) may then be used to organize the contributions to the signal, which is equivalent to a cumulant expansion for S and yields (10,11): lnðSÞ ¼
3 T T γ2 X ∫ dt 1 f i ðt 1 Þ ∫ dt 2 f j ðt 2 Þ vi ðt 1 Þ vj ðt 2 Þ þ O γ3 [5] 2 i;j¼1 t0 t0
where fi indicates a component of f and vi indicates a component of v. Hence, the leading order term is proportional to γ2 and is determined by the velocity autocorrelation function. This expression for the dMRI signal is equivalent to the Gaussian approximation (10). For systems in statistical equilibrium, the velocity autocorrelation function is time translation invariant. This property can be expressed as:
vi ðt 1 Þ vj ðt 2 Þ ¼ vi ðt1 þ τ Þ vj ðt 2 þ τ Þ
[6]
where τ is an arbitrary time shift. In addition, for most systems of physical interest, the diffusion dynamics would obey the principle of microscopic reversibility, which implies: vi ðt1 Þ vj ðt 2 Þ ¼ vi ðt 1 Þ vj ðt 2 Þ
[7]
lnðSÞ ¼
3 T T γ2 X ∂2
∫ dt 1 f i ðt 1 Þ ∫ dt 2 f j ðt 2 Þ jt2 t 1 jDij ðjt2 t 1 jÞ 2 i;j¼1 t0 ∂t 1 ∂t2 t0 3 [12] þO γ
Therefore, a full knowledge of the diffusion tensor as a function of time is sufficient to determine the dMRI signal to the leading order of γ2 in the diffusion weighting. Equation (12) is equivalent to equation [2] of ref. (5). For the special case of a time-independent diffusion tensor, Equation [12] reduces to the familiar expression for the DTI signal in terms of the b matrix (16), whereas, for unidirectional gradients, it is equivalent to the results discussed by Novikov and Kiselev (6). More generally, Equation [12] holds for OG-dMRI with arbitrary gradient wave forms and for MP-dMRI with arbitrary numbers and orientations of wave vectors (5). It should be emphasized, however, that the measurement of the full time dependence of the diffusion tensor is often difficult in practice, which is a primary reason why a variety of dMRI methods have been developed. We also note that the order γ3 term in Equation [12] is purely imaginary (10) and therefore contributes a phase to the dMRI signal. As most dMRI experiments only utilize the signal magnitude, the next contributions of practical significance are of order γ4. In many cases, these order γ4 terms can be related to the kurtosis of a diffusion displacement probability density function (9,17,18).
From these two properties, it follows that:
vi ðt 1 Þ vj ðt2 Þ ¼ vi ðt1 þ t1 þ t 2Þ vj ðt 2 þ t 1 þ t2 Þ ¼ vi ðt2 Þ vj ðt 1 Þ ¼ vj ðt 1 Þ vi ðt2 Þ
REFERENCES [8]
This identity 〈vi(t1) vj(t2)〉 = 〈vj(t1) vi(t2)〉 is closely related to the Onsager reciprocal relationships, which play a fundamental role in statistical physics (12–14). The velocity autocorrelation function can be calculated from the displacement autocorrelation function using:
∂2 ½r i ðt 2 Þ r i ðt1 Þ r j ðt2 Þ r j ðt1 Þ ∂t 1∂t2 ¼ vi ðt1 Þ vj ðt2 Þ þ vj ðt1 Þ vi ðt2 Þ ¼ 2 vi ðt 1 Þ vj ðt 2 Þ
[9]
where Equation [8] has been invoked and ri indicates a component of r. This may be recast in terms of the diffusion tensor as: ∂2
vi ðt1 Þ vj ðt 2 Þ ¼ jt2 t1 jDij ðjt2 t 1 jÞ ∂t 1 ∂t2
[10]
with the diffusion tensor being defined by (15): Dij ðtÞ≡
1 ½r i ðt þ t 0 Þ r i ðt 0 Þ r j ðt þ t 0 Þ r j ðt0 Þ 2t
[11]
1006
Because of time translation invariance, Dij does not depend on the initial time t0. By combining Equations [5] and [10], we finally arrive at:
wileyonlinelibrary.com/journal/nbm
1. Basser PJ, Mattiello J, LeBihan D. MR diffusion tensor spectroscopy and imaging. Biophys. J. 1994; 66: 259–267. 2. Shemesh N, Özarslan E, Komlosh ME, Basser PJ, Cohen Y. From single-pulsed field gradient to double-pulsed field gradient MR: gleaning new microstructural information and developing new forms of contrast in MRI. NMR Biomed. 2010; 23: 757–780. 3. Avram AV, Özarslan E, Sarlls JE, Basser PJ. In vivo detection of microscopic anisotropy using quadruple pulsed-field gradient (qPFG) diffusion MRI on a clinical scanner. Neuroimage 2013; 64: 229–239. 4. Gore JC, Xu J, Colvin DC, Yankeelov TE, Parsons EC, Does MD. Characterization of tissue structure at varying length scales using temporal diffusion spectroscopy. NMR Biomed. 2010; 23: 745–756. 5. Jespersen SN. Equivalence of double and single wave vector diffusion contrast at low diffusion weighting. NMR Biomed. 2012; 25: 813–818. 6. Novikov DS, Kiselev VG. Surface-to-volume ratio with oscillating gradients. J. Magn. Reson. 2011; 210: 141–145. 7. Novikov DS, Kiselev VG. Effective medium theory of a diffusionweighted signal. NMR Biomed. 2010; 23: 682–697. 8. Stejskal EO, Tanner JE. Spin diffusion measurements: spin echoes in the presence of a time-dependent field gradient. J. Chem. Phys. 1965; 42: 288–292. 9. Jensen JH, Helpern JA, Ramani A, Lu H, Kaczynski K. Diffusional kurtosis imaging: the quantification of non-Gaussian water diffusion by means of magnetic resonance imaging. Magn. Reson. Med. 2005; 53: 1432–1440. 10. Stepišnik J. Validity limits of Gaussian approximation in cumulant expansion for diffusion attenuation of spin echo. Physica B, 1999; 270: 110–117. 11. Kiselev VG. The cumulant expansion: an overarching mathematical framework for understanding diffusion NMR. In: Jones DK (ed.). Diffusion MRI: Theory, Methods and Applications. Oxford University Press: Oxford; 2010, Chapter 10.
Copyright © 2014 John Wiley & Sons, Ltd.
NMR Biomed. 2014; 27: 1005–1007
SUFFICIENCY OF DIFFUSION TENSOR 12. Reif F. Fundamentals of Statistical and Thermal Physics. McGraw-Hill: New York; 1965, pp. 594–600. 13. Onsager L. Reciprocal relations in irreversible processes. I. Phys. Rev. 1931; 37: 405–426. 14. Onsager L. Reciprocal relations in irreversible processes. II. Phys. Rev. 1931; 38: 2265–2279. 15. Basser PJ. Relationships between diffusion tensor and q-space MRI. Magn. Reson. Med. 2002; 47: 392–397.
16. Mattiello J, Basser PJ, LeBihan D. Analytical expressions for the b matrix in NMR diffusion imaging and spectroscopy. J. Magn. Reson. A, 1994; 108: 131–141. 17. Jensen JH, Helpern JA. MRI quantification of non-Gaussian water diffusion by kurtosis analysis. NMR Biomed. 2010; 23: 698–710. 18. Jensen JH, Hui ES, Helpern JA. Double-pulsed diffusional kurtosis imaging. NMR Biomed. 2014; 27: 363–370.
1007
NMR Biomed. 2014; 27: 1005–1007
Copyright © 2014 John Wiley & Sons, Ltd.
wileyonlinelibrary.com/journal/nbm
Revised: 02 May 2014,
Accepted: 07 May 2014,
Published online in Wiley Online Library: 5 June 2014
(wileyonlinelibrary.com) DOI: 10.1002/nbm.3145
Sufficiency of diffusion tensor in characterizing the diffusion MRI signal to leading order in diffusion weighting Jens H. Jensena,b* Sir,
NMR Biomed. 2014; 27: 1005–1007
t0
where g(t) is the applied diffusion gradient as a function of time t, r(t) is the diffusion path, σ(t) is the spin flip function, t0 is the initial time, T is the final time and γ is the gyromagnetic ratio for the nucleus of interest. The signal is normalized to unity for g(t) = 0. The angled brackets indicate an averaging over the ensemble of diffusing molecules (water in most applications). Any 180° refocusing pulses are idealized as being instantaneous and incorporated into Equation [1] through σ(t), which has a magnitude of unity and changes sign at the time of each refocusing pulse (9). The diffusion gradient is assumed to satisfy the condition: T
0 ¼ ∫ dt σðtÞ gðtÞ
[2]
t0
so that there is no dephasing for stationary spins. Different dMRI methods correspond to different choices for g(t) and σ(t). The product g(t)σ(t) may be regarded as an ‘effective gradient’ for the pulse sequence. Integration by parts in Equation [1] leads to: S¼
T exp iγ ∫ dt f ðtÞvðt Þ
[3]
t0
where v(t) ≡ dr(t)/dt and:
* Correspondence to: J. H. Jensen, Center for Biomedical Imaging, Department of Radiology and Radiological Science, Medical University of South Carolina, 96 Jonathan Lucas Street, MSC 323, Charleston, SC 29425-0323, USA. E-mail: [email protected] a J. H. Jensen Department of Radiology and Radiological Science, Medical University of South Carolina, Charleston, SC, USA b J. H. Jensen Center for Biomedical Imaging, Medical University of South Carolina, Charleston, SC, USA Abbreviations used: dMRI, diffusion MRI; DTI, diffusion tensor imaging; MPdMRI, multiple-pulsed diffusion MRI; OG-dMRI, oscillating gradient diffusion MRI.
Copyright © 2014 John Wiley & Sons, Ltd.
1005
Diffusion MRI (dMRI) is a rich and varied field, with a multitude of distinct approaches, including diffusion tensor imaging (DTI) (1), multiple pulsed dMRI (MP-dMRI) (2,3) and oscillating gradient dMRI (OG-dMRI) (4). These methods differ in terms of the pulse sequences employed for data acquisition and the post-processing algorithms utilized for data analysis. However, they all share a common basis in diffusion and NMR physics. This commonality leads to important connections that link measurements obtained with seemingly disparate techniques. Two examples have been discussed in recent publications. First, Jespersen (5) demonstrated that single- and double-pulsed dMRI are equivalent for low diffusion weightings. Second, Novikov and Kiselev (6) derived a rigorous quantitative relationship between the short-time behavior of the diffusion coefficient and the high-frequency behavior of the OG-dMRI signal in the Gaussian approximation. Moreover, Novikov and Kiselev (6,7) emphasized the equivalence of the information provided by the diffusion coefficient, as can be measured with a classical Stejskal–Tanner sequence (8), and the frequency-dependent diffusivity determined by OG-dMRI. Here, we briefly note that these two observations are closely related and are aspects of a more general property of dMRI. In particular, we show that a knowledge of the diffusion tensor as a function of time is, in principle, sufficient to predict the dMRI signal intensity to leading order in the diffusion weighting (e.g. the b value) for any dMRI method. As the diffusion tensor is measurable with DTI, one may regard DTI as being comprehensive to this leading order. Thus, alternative methods, such as MP-dMRI and OG-dMRI, only provide additional information, beyond that which is available from DTI, in their higher order contributions to the signal. This is not to say that alternative methods are not useful, even to leading order, but that the interpretation of their results should be informed by the interconnections with other dMRI techniques. It is from this conceptual perspective that the diffusion tensor is sufficient for low diffusion weightings, with dMRI approaches other than DTI certainly having important practical advantages for specific applications. The theoretical connection between the diffusion tensor and the dMRI signal to leading order has already been presented by Jespersen (5), but was discussed primarily in the context of double-pulsed dMRI and a detailed derivation was not provided. Below, we give a self-contained demonstration of this fundamental fact, in order to clarify and highlight its physical basis and generality.
To understand how the diffusion tensor predicts the dMRI signal to leading order in the diffusion weighting, consider the phase integral expression for the dMRI signal: T S ¼ exp iγ ∫ dt σ ðtÞgðtÞrðt Þ [1]
J. H. JENSEN
t f ðt Þ≡ ∫ dt ′ σ t′ g t′
[4]
t0
For a purely diffusive system, one has the no net flow condition of 〈v(t)〉 = 0. A formal series in powers of γ for ln(S) may then be used to organize the contributions to the signal, which is equivalent to a cumulant expansion for S and yields (10,11): lnðSÞ ¼
3 T T γ2 X ∫ dt 1 f i ðt 1 Þ ∫ dt 2 f j ðt 2 Þ vi ðt 1 Þ vj ðt 2 Þ þ O γ3 [5] 2 i;j¼1 t0 t0
where fi indicates a component of f and vi indicates a component of v. Hence, the leading order term is proportional to γ2 and is determined by the velocity autocorrelation function. This expression for the dMRI signal is equivalent to the Gaussian approximation (10). For systems in statistical equilibrium, the velocity autocorrelation function is time translation invariant. This property can be expressed as:
vi ðt 1 Þ vj ðt 2 Þ ¼ vi ðt1 þ τ Þ vj ðt 2 þ τ Þ
[6]
where τ is an arbitrary time shift. In addition, for most systems of physical interest, the diffusion dynamics would obey the principle of microscopic reversibility, which implies: vi ðt1 Þ vj ðt 2 Þ ¼ vi ðt 1 Þ vj ðt 2 Þ
[7]
lnðSÞ ¼
3 T T γ2 X ∂2
∫ dt 1 f i ðt 1 Þ ∫ dt 2 f j ðt 2 Þ jt2 t 1 jDij ðjt2 t 1 jÞ 2 i;j¼1 t0 ∂t 1 ∂t2 t0 3 [12] þO γ
Therefore, a full knowledge of the diffusion tensor as a function of time is sufficient to determine the dMRI signal to the leading order of γ2 in the diffusion weighting. Equation (12) is equivalent to equation [2] of ref. (5). For the special case of a time-independent diffusion tensor, Equation [12] reduces to the familiar expression for the DTI signal in terms of the b matrix (16), whereas, for unidirectional gradients, it is equivalent to the results discussed by Novikov and Kiselev (6). More generally, Equation [12] holds for OG-dMRI with arbitrary gradient wave forms and for MP-dMRI with arbitrary numbers and orientations of wave vectors (5). It should be emphasized, however, that the measurement of the full time dependence of the diffusion tensor is often difficult in practice, which is a primary reason why a variety of dMRI methods have been developed. We also note that the order γ3 term in Equation [12] is purely imaginary (10) and therefore contributes a phase to the dMRI signal. As most dMRI experiments only utilize the signal magnitude, the next contributions of practical significance are of order γ4. In many cases, these order γ4 terms can be related to the kurtosis of a diffusion displacement probability density function (9,17,18).
From these two properties, it follows that:
vi ðt 1 Þ vj ðt2 Þ ¼ vi ðt1 þ t1 þ t 2Þ vj ðt 2 þ t 1 þ t2 Þ ¼ vi ðt2 Þ vj ðt 1 Þ ¼ vj ðt 1 Þ vi ðt2 Þ
REFERENCES [8]
This identity 〈vi(t1) vj(t2)〉 = 〈vj(t1) vi(t2)〉 is closely related to the Onsager reciprocal relationships, which play a fundamental role in statistical physics (12–14). The velocity autocorrelation function can be calculated from the displacement autocorrelation function using:
∂2 ½r i ðt 2 Þ r i ðt1 Þ r j ðt2 Þ r j ðt1 Þ ∂t 1∂t2 ¼ vi ðt1 Þ vj ðt2 Þ þ vj ðt1 Þ vi ðt2 Þ ¼ 2 vi ðt 1 Þ vj ðt 2 Þ
[9]
where Equation [8] has been invoked and ri indicates a component of r. This may be recast in terms of the diffusion tensor as: ∂2
vi ðt1 Þ vj ðt 2 Þ ¼ jt2 t1 jDij ðjt2 t 1 jÞ ∂t 1 ∂t2
[10]
with the diffusion tensor being defined by (15): Dij ðtÞ≡
1 ½r i ðt þ t 0 Þ r i ðt 0 Þ r j ðt þ t 0 Þ r j ðt0 Þ 2t
[11]
1006
Because of time translation invariance, Dij does not depend on the initial time t0. By combining Equations [5] and [10], we finally arrive at:
wileyonlinelibrary.com/journal/nbm
1. Basser PJ, Mattiello J, LeBihan D. MR diffusion tensor spectroscopy and imaging. Biophys. J. 1994; 66: 259–267. 2. Shemesh N, Özarslan E, Komlosh ME, Basser PJ, Cohen Y. From single-pulsed field gradient to double-pulsed field gradient MR: gleaning new microstructural information and developing new forms of contrast in MRI. NMR Biomed. 2010; 23: 757–780. 3. Avram AV, Özarslan E, Sarlls JE, Basser PJ. In vivo detection of microscopic anisotropy using quadruple pulsed-field gradient (qPFG) diffusion MRI on a clinical scanner. Neuroimage 2013; 64: 229–239. 4. Gore JC, Xu J, Colvin DC, Yankeelov TE, Parsons EC, Does MD. Characterization of tissue structure at varying length scales using temporal diffusion spectroscopy. NMR Biomed. 2010; 23: 745–756. 5. Jespersen SN. Equivalence of double and single wave vector diffusion contrast at low diffusion weighting. NMR Biomed. 2012; 25: 813–818. 6. Novikov DS, Kiselev VG. Surface-to-volume ratio with oscillating gradients. J. Magn. Reson. 2011; 210: 141–145. 7. Novikov DS, Kiselev VG. Effective medium theory of a diffusionweighted signal. NMR Biomed. 2010; 23: 682–697. 8. Stejskal EO, Tanner JE. Spin diffusion measurements: spin echoes in the presence of a time-dependent field gradient. J. Chem. Phys. 1965; 42: 288–292. 9. Jensen JH, Helpern JA, Ramani A, Lu H, Kaczynski K. Diffusional kurtosis imaging: the quantification of non-Gaussian water diffusion by means of magnetic resonance imaging. Magn. Reson. Med. 2005; 53: 1432–1440. 10. Stepišnik J. Validity limits of Gaussian approximation in cumulant expansion for diffusion attenuation of spin echo. Physica B, 1999; 270: 110–117. 11. Kiselev VG. The cumulant expansion: an overarching mathematical framework for understanding diffusion NMR. In: Jones DK (ed.). Diffusion MRI: Theory, Methods and Applications. Oxford University Press: Oxford; 2010, Chapter 10.
Copyright © 2014 John Wiley & Sons, Ltd.
NMR Biomed. 2014; 27: 1005–1007
SUFFICIENCY OF DIFFUSION TENSOR 12. Reif F. Fundamentals of Statistical and Thermal Physics. McGraw-Hill: New York; 1965, pp. 594–600. 13. Onsager L. Reciprocal relations in irreversible processes. I. Phys. Rev. 1931; 37: 405–426. 14. Onsager L. Reciprocal relations in irreversible processes. II. Phys. Rev. 1931; 38: 2265–2279. 15. Basser PJ. Relationships between diffusion tensor and q-space MRI. Magn. Reson. Med. 2002; 47: 392–397.
16. Mattiello J, Basser PJ, LeBihan D. Analytical expressions for the b matrix in NMR diffusion imaging and spectroscopy. J. Magn. Reson. A, 1994; 108: 131–141. 17. Jensen JH, Helpern JA. MRI quantification of non-Gaussian water diffusion by kurtosis analysis. NMR Biomed. 2010; 23: 698–710. 18. Jensen JH, Hui ES, Helpern JA. Double-pulsed diffusional kurtosis imaging. NMR Biomed. 2014; 27: 363–370.
1007
NMR Biomed. 2014; 27: 1005–1007
Copyright © 2014 John Wiley & Sons, Ltd.
wileyonlinelibrary.com/journal/nbm
