Analysis for quantifying microscopic diffusion anisotropy

Abstract

A method for MRI includes performing an MRI experiment on a material to acquire an echo attenuation data set using a gradient modulation scheme comprising 1D diffusion encoding and to acquire an echo attenuation data set using a gradient modulation scheme comprising 2D diffusion encoding, determining respective first-order deviations .sub.2 from mono-exponential decay in said echo attenuation data sets, and generating MRI parameter maps or MR image contrast based on said first-order deviations .sub.2.

Claims

1. A method for MRI, the method comprising: performing an MRI experiment on a material to acquire an echo attenuation data set using a gradient modulation scheme comprising 1D diffusion encoding and to acquire an echo attenuation data set using a gradient modulation scheme comprising 2D diffusion encoding; determining respective first-order deviations .sub.2 from mono-exponential decay in said echo attenuation data sets; and generating MRI parameter maps or MR image contrast based on said first-order deviations .sub.2.

2. The method according to claim 1, wherein the 1D diffusion encoding includes diffusion encoding along a single encoding gradient direction, and the 2D diffusion encoding includes diffusion encoding along two orthogonal encoding gradient directions.

3. The method according to claim 2, wherein the 1D diffusion encoding forms part of a single PGSE sequence, and the 2D diffusion encoding forms part of a double PGSE sequence.

4. The method according to claim 2, wherein each of the acquired echo attenuation data sets includes signal intensities averaged across multiple encoding directions.

5. The method according to claim 1, wherein each first order deviation .sub.2 corresponds to a second central moment of a distribution of diffusion coefficients for the respective echo attenuation data sets.

6. The method according to claim 1, wherein the parameter maps or MR image contrast is generated based on a difference between said first-order deviations .sub.2.

7. The method according to claim 6, wherein the parameter maps or MR image contrast is generated based on a mean diffusivity and a difference between said first-order deviations .sub.2.

8. The method according to claim 1, further comprising: fitting respective fitting functions to the echo attenuation data sets using the first order deviations .sub.2 as respective fit parameters.

9. A method for MRI, the method comprising: performing an MRI experiment on a material to acquire an echo attenuation data set using a gradient modulation scheme comprising anisotropic diffusion encoding and to acquire an echo attenuation data set using a gradient modulation scheme comprising isotropic diffusion encoding; determining respective first-order deviations .sub.2 and .sub.2.sup.iso from mono-exponential decay in said echo attenuation data sets; and generating MRI parameter maps or MR image contrast based on said first-order deviations .sub.2 and .sub.2.sup.iso.

10. The method according to claim 9, wherein the anisotropic diffusion encoding includes 1D diffusion encoding including diffusion encoding along a single encoding gradient direction, or 2D diffusion encoding including diffusion encoding along two orthogonal encoding gradient directions.

11. The method according to claim 10, wherein the 1D diffusion encoding forms part of a single PGSE sequence, or the 2D diffusion encoding forms part of a double PGSE sequence.

12. The method according to claim 9, wherein the acquired echo attenuation data set acquired using a gradient modulation scheme comprising anisotropic diffusion encoding includes signal intensities averaged across multiple encoding directions.

13. The method according to claim 9, wherein each first order deviation .sub.2 and .sub.2.sup.iso corresponds to a second central moment of a distribution of diffusion coefficients for the respective echo attenuation data sets.

14. The method according to claim 9, wherein the parameter maps or MR image contrast is generated based on a difference between said first-order deviations .sub.2 and .sub.2.sup.iso.

15. The method according to claim 9, wherein the parameter maps or MR image contrast is generated based on a mean diffusivity and a difference between said first-order deviations .sub.2 and .sub.2.sup.iso.

16. The method according to claim 9, further comprising: fitting respective fitting functions to the echo attenuation data sets using .sub.2 and .sub.2.sup.iso as respective fit parameters.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1A-C show schematic representations of signal decays vs. b for isotropic (dashed line) and non-isotropic (solid line) diffusion weighting for different types of materials. The inset A depicts signal attenuation curves in case of anisotropic materials with 1D or 2D curvilinear diffusion. The attenuation curves are multi-exponential for non-isotropic diffusion weighting, while they are mono-exponential for isotropic diffusion weighting. The deviation between the attenuation curves for isotropic and non-isotropic diffusion weighting provides a measure of anisotropy. The inset B depicts an example of isotropic material with several apparent diffusion contributions resulting in identical and multi-exponential signal attenuation curves for isotropic and non-isotropic diffusion weighting. The inset C depicts an example of material with a mixture of isotropic and anisotropic components resulting in multi-exponential signal decays for both isotropic and non-isotropic diffusion weighting, while the deviation between the attenuation curves for isotropic and non-isotropic diffusion weighting provides a measure of anisotropy.

(2) FIG. 2A-C show experimental results with analysis for different types of materials. Experimental results for isotropic (circles) and for non-isotropic (crosses) diffusion weighting are shown in all the insets. Experimental results and analysis are shown for a sample with free isotropic diffusion (inset A), for a sample with restricted isotropic diffusion (inset B) and for a sample with high degree of anisotropy (inset C).

(3) FIGS. 3A and 3B show a Monte-Carlo error analysis for the investigation of systematic deviations and precision as a function of the range of diffusion weighting b for estimating the degree of micro-anisotropy with the disclosed analytical method.

BACKGROUND TO THE ANALYSIS METHOD ACCORDING TO THE PRESENT INVENTION

(4) Below there will be disclosed one possible method for isotropic diffusion weighting as a background to the analysis method according to the present invention. It is important to understand that this is only given as an example and as a background for the isotropic diffusion weighting. The analysis method according to the present invention is of course not limited to this route or method. Fact is that all possible diffusion weighting methods involving one part (gradient modulation scheme) for isotropic diffusion weighting and one other part for the non-isotropic diffusion weighting are possible starting points, and thus pre-performed methods, for the analysis method according to the present invention.

(5) Assuming that spin diffusion in a microscopically anisotropic system can locally be considered a Gaussian process and therefore fully described by the diffusion tensor D(r), the evolution of the complex transverse magnetization m(r,t) during a diffusion encoding experiment is given by the Bloch-Torrey equation. Note that the Bloch-Torrey equation applies for arbitrary diffusion encoding schemes, e.g. pulse gradient spin-echo (PGSE), pulse gradient stimulated echo (PGSTE) and other modulated gradient spin-echo (MGSE) schemes. Assuming uniform spin density and neglecting relaxation, the magnetization evolution is given by

(6) $\begin{array}{cc}\frac{m\left(r,t\right)}{t}=-ig\left(t\right).Math.\mathrm{rm}\left(r,t\right)+.Math.\left[D\left(r\right).Math.m\left(r,t\right)\right],& \left(1\right)\end{array}$
where is the gyromagnetic ratio and g(t) is the time dependent effective magnetic field gradient vector. The NMR signal is proportional to the macroscopic transverse magnetization

(7) $\begin{array}{cc}M\left(t\right)=\frac{1}{V}{}_{V}m\left(r,t\right)\mathrm{dr}.& \left(2\right)\end{array}$

(8) If during the experiment each spin is confined to a domain characterized by a unique diffusion tensor D, the macroscopic magnetization is a superposition of contributions from all the domains with different D. Evolution of each macroscopic magnetization contribution can thus be obtained by solving Eqs. (1, 2) with a constant and uniform D. The signal magnitude contribution at the echo time t.sub.E is given by

(9) $\begin{array}{cc}\begin{array}{c}I\left(t_E\right)=I_0\exp \left(-\underset{0}{\overset{t_E}{}}{q}^T\left(t\right).Math.D.Math.q\left(t\right)\mathrm{dt}\right)\\ =I_0\exp \left(-q^2\underset{0}{\overset{t_E}{}}{F\left(t\right)}^2{\hat{q}}^T\left(t\right).Math.D.Math.\hat{q}\left(t\right)\mathrm{dt}\right)\end{array},& \left(3\right)\end{array}$
where I.sub.0 is the signal without diffusion encoding, g=0, and q(t) is the time-dependent dephasing vector

(10) $\begin{array}{cc}q\left(t\right)=\underset{0}{\overset{t}{}}g\left(t^{}\right){\mathrm{dt}}^{}=\mathrm{qF}\left(t\right)\hat{q}\left(t\right),& \left(4\right)\end{array}$
defined for the interval 0<t<t.sub.E. The dephasing vector in Eqs. (3) and (4) is expressed in terms of its maximum magnitude q, the time-dependent normalized magnitude |F(t)|1 and a time-dependent unit direction vector {circumflex over (q)}(t). Note that in spin-echo experiments, the effective gradient g(t) comprises the effect of gradient magnitude reversal after each odd 180 radio frequency (RF) pulse in the sequence. Eq. (3) assumes that the condition for the echo formation q(t.sub.E)=0 is fulfilled, which implies F(t.sub.E)=0. In general there might be several echoes during an NMR pulse sequence.

(11) The echo magnitude (3) can be rewritten in terms of the diffusion weighting matrix,

(12) $\begin{array}{cc}b=q^2\underset{0}{\overset{t_E}{}}{F\left(t\right)}^2\hat{q}\left(t\right).Math.{\hat{q}}^T\left(t\right)\mathrm{dt},\mathrm{as}& \left(5\right)\\ I\left(t_E\right)=I_0\exp \left(-b\text{:}D\right)=I_0\exp \left(-\underset{i}{.Math.}\underset{j}{.Math.}{b}_{\mathrm{ij}}{D}_{\mathrm{ij}}\right).& \left(6\right)\end{array}$
Integral of the time-dependent waveform F(t).sup.2 defines the effective diffusion time, t.sub.d, for an arbitrary diffusion encoding scheme in a spin-echo experiment

(13) $\begin{array}{cc}{t}_d=\underset{0}{\overset{t_E}{}}{F\left(t\right)}^2\mathrm{dt}.& \left(7\right)\end{array}$

(14) In the following we will demonstrate that even for a single echo sequence, gradient modulations g(t) can be designed to yield isotropic diffusion weighting, invariant under rotation of D, i.e. the echo attenuation is proportional to the isotropic mean diffusivity,
D=tr(D)/3=(D.sub.xx+D.sub.yy+D.sub.zz)/3.(8)

(15) In view of what is disclosed above, according to one specific embodiment of the present invention, the isotropic diffusion weighting is invariant under rotation of the diffusion tensor D.

(16) According to the present invention, one is looking for such forms of dephasing vectors F(t){circumflex over (q)}(t), for which

(17) $\begin{array}{cc}\underset{0}{\overset{t_E}{}}{F\left(t\right)}^2{\hat{q}}^T\left(t\right).Math.D.Math.\hat{q}\left(t\right)\mathrm{dt}=t_d\stackrel{\_}{D}& \left(9\right)\end{array}$
is invariant under rotation of D. If diffusion tenor D is expressed as a sum of its isotropic contribution, DI, where I is the identity matrix, and the anisotropic contribution, i.e. the deviatoric tensor D.sup.A, so that D=DI+D.sup.A, the isotropic diffusion weighing is achieved when the condition

(18) $\begin{array}{cc}\underset{0}{\overset{t_E}{}}{F\left(t\right)}^2{\hat{q}}^T\left(t\right).Math.D^A.Math.\hat{q}\left(t\right)\mathrm{dt}=0& \left(10\right)\end{array}$
is fulfilled.

(19) In spherical coordinates, the unit vector {circumflex over (q)}(t) is expressed by the inclination and azimuth angle as
{circumflex over (q)}.sup.T(t)={{circumflex over (q)}.sub.x(t): {circumflex over (q)}.sub.y(t),{circumflex over (q)}.sub.z(t)}={sin (t)cos (t), sin (t)sin (t), cos (t)}.(11)
The symmetry of the diffusion tensor, D=D.sup.T, gives
{circumflex over (q)}.sup.T.Math.D.Math.{circumflex over (q)}={circumflex over (q)}.sub.x.sup.2D.sub.xx+{circumflex over (q)}.sub.y.sup.2D.sub.yy+{circumflex over (q)}.sub.z.sup.2D.sub.zz+2{circumflex over (q)}.sub.x{circumflex over (q)}.sub.yD.sub.xy+2{circumflex over (q)}.sub.x{circumflex over (q)}.sub.zD.sub.xz+2{circumflex over (q)}.sub.y{circumflex over (q)}.sub.zD.sub.yz(12)
or expressed in spherical coordinates as
{circumflex over (q)}.sup.T.Math.D.Math.{circumflex over (q)}=sin.sup.2 cos.sup.2D.sub.xx+sin .sup.2 sin .sup.2D.sub.yy+cos.sup.2D.sub.zz+2 sin cos sin sin D.sub.xy+2 sin cos cos D.sub.xz+2 sin sin cos D.sub.yz.(13)
Equation (13) can be rearranged to

(20) $\begin{array}{cc}\hat{q}.Math.D.Math.\hat{q}=\stackrel{\_}{D}+\frac{3{\cos }^2-1}{2}\left(D_{\mathrm{zz}}-\stackrel{\_}{D}\right)+{\sin }^2\left[\frac{D_{\mathrm{xx}}-D_{\mathrm{yy}}}{2}\cos \left(2\right)+D_{\mathrm{xy}}\sin \left(2\right)\right]+\sin \left(2\right)\left(D_{\mathrm{xz}}\cos +D_{\mathrm{yz}}\sin \right).& \left(14\right)\end{array}$
The first term in Eq. (14) is the mean diffusivity, while the remaining terms are time-dependent through the angles (t) and (t) which define the direction of the dephasing vector (4). Furthermore, the second term in Eq. (14) is independent of , while the third and the forth terms are harmonic functions of and 2 , respectively (compare with Eq. (4) in [13]). To obtain isotropic diffusion weighting, expressed by Eq. (9), the corresponding integrals of the second, third and fourth terms in Eq. (14) must vanish. The condition for the second term of Eq. (14) to vanish upon integration leads to one possible solution for the angle (t), i.e. the time-independent magic angle
.sub.m=a cos(1/{square root over (3)})(15)

(21) By taking into account constant .sub.m, the condition for the third and the fourth term in Eq. (14) to vanish upon integration is given by

(22) 0$\begin{array}{cc}\begin{array}{c}\underset{0}{\overset{t_E}{}}{F\left(t\right)}^2\cos \left[2\left(t\right)\right]\mathrm{dt}=0\\ \underset{0}{\overset{t_E}{}}{F\left(t\right)}^2\sin \left[2\left(t\right)\right]\mathrm{dt}=0\\ \underset{0}{\overset{t_E}{}}F{\left(t\right)}^2\cos \left[\left(t\right)\right]\mathrm{dt}=0\\ \underset{0}{\overset{t_E}{}}{F\left(t\right)}^2\sin \left[\left(t\right)\right]\mathrm{dt}=0\end{array}.& \left(16\right)\end{array}$
Conditions (16) can be rewritten in a more compact complex form as

(23) $\begin{array}{cc}\underset{0}{\overset{t_E}{}}{F\left(t\right)}^2\exp \left[\mathrm{ik}\left(t\right)\right]\mathrm{dt}=0,& \left(17\right)\end{array}$
which must be satisfied for k=1, 2. By introducing the rate {dot over ()}(t)=F(t).sup.2, the integral (17) can be expressed with the new variable as

(24) $\begin{array}{cc}\underset{0}{\overset{t_E}{}}\exp \left[\mathrm{ik}\left(\right)\right]d=0.& \left(18\right)\end{array}$
Note that the upper integration boundary moved from t.sub.E to t.sub.d. The condition (18) is satisfied when the period of the exponential is t.sub.d, thus a solution for the azimuth angle is

(25) $\begin{array}{cc}\left(\right)=\left(0\right)+\frac{2}{t_d}n,& \left(19\right)\end{array}$
for any integer n other than 0. The time dependence of the azimuth angle is finally given by

(26) $\begin{array}{cc}\left(t\right)=\left(0\right)+\frac{2n}{t_d}\underset{0}{\overset{t}{}}{F\left(t^{}\right)}^2{\mathrm{dt}}^{}.& \left(20\right)\end{array}$
The isotropic diffusion weighting scheme is thus determined by the dephasing vector q(t) with a normalized magnitude F(t) and a continuous orientation sweep through the angles .sub.m (15) and (t) (20). Note that since the isotropic weighting is invariant upon rotation of D, orientation of the vector q(t) and thus also orientation of the effective gradient g(t) can be arbitrarily offset relative to the laboratory frame in order to best suit the particular experimental conditions.

(27) As understood from above, according to yet another specific embodiment, the isotropic diffusion weighting is achieved by a continuous sweep of the time-dependent dephasing vector q(t) where the azimuth angle (t) and the magnitude thereof is a continuous function of time so that the time-dependent dephasing vector q(t) spans an entire range of orientations parallel to a right circular conical surface and so that the orientation of the time-dependent dephasing vector q(t) at time 0 is identical to the orientation at time t.sub.E. Furthermore, according to yet another embodiment, the inclination is chosen to be a constant, time-independent value, i.e. the so called magic angle, such that =.sub.m=a cos (1/{square root over (3)}).

(28) The orientation of the dephasing vector, in the Cartesian coordinate system during the diffusion weighting sequence, spans the entire range of orientations parallel to the right circular conical surface with the aperture of the cone of 2*.sub.m, (double magic angle) and the orientation of the dephasing vector at time 0 is identical to the orientation of the dephasing vector at time t.sub.E, i.e. (t.sub.E)(0)=2**n, where n is an integer (positive or negative, however not 0) and the absolute magnitude of the dephasing vector, q*F(t), is zero at time 0 and at time t.sub.E. The isotropic weighting can also be achieved by q-modulations with discrete steps in azimuth angle , providing q(t) vector steps through at least four orientations with unique values of e.sup.i, such that modulus 2 are equally spaced values. Choice of the consecutive order and duration of the time intervals during which is constant is arbitrary, provided that the magnitude F(t) is adjusted to meet the condition for isotropic weighing (10, 16).

(29) Specific Implementations

(30) The pulsed gradient spin-echo (PGSE) sequence with short pulses offers a simplest implementation of the isotropic weighting scheme according to the present invention. In PGSE, the short gradient pulses at times approximately 0 and t.sub.E cause the magnitude of the dephasing vector to instantaneously acquire its maximum value approximately at time 0 and vanish at time t.sub.E. The normalized magnitude is in this case given simply by F(t)=1 in the interval 0<t<t.sub.E and 0 otherwise, providing t.sub.d=t.sub.E. A simplest choice for the azimuth angle (20) is the one with n=1 and (0)=0, thus

(31) $\begin{array}{cc}\left(t\right)=\frac{2t}{t_E}.& \left(21\right)\end{array}$
The dephasing vector is given by

(32) $\begin{array}{cc}{q}^T\left(t\right)=q\left\{\sqrt{\frac{2}{3}}\cos \left(\frac{2t}{t_E}\right),\sqrt{\frac{2}{3}}\sin \left(\frac{2t}{t_E}\right),\sqrt{\frac{1}{3}}\right\}.& \left(22\right)\end{array}$
The corresponding effective gradient, calculated from

(33) $\begin{array}{cc}{g}^T\left(t\right)=\frac{1}{}\frac{d}{\mathrm{dt}}{q}^T\left(t\right)\mathrm{is}& \left(23\right)\\ g^T\left(t\right)=\frac{q}{}\left[\left(0\right)-\left(t_E\right)\right]\left\{\sqrt{\frac{2}{3}},0,\sqrt{\frac{1}{3}}\right\}+\sqrt{\frac{2}{3}}\frac{2}{t_E}\frac{q}{}\left\{-\sin \left(\frac{2t}{t_E}\right),\cos \left(\frac{2t}{t_E}\right),0\right\}.& \left(24\right)\end{array}$
Here (t) is the Dirac delta function. Rotation around the y-axis by a tan(2) yields

(34) $\begin{array}{cc}{g}^T\left(t\right)=\frac{q}{}\left[\left(0\right)-\left(t_E\right)\right]\left\{0,0,1\right\}+\sqrt{\frac{2}{3}}\frac{2}{t_E}\frac{q}{}\left\{-\sqrt{\frac{1}{3}}\sin \left(\frac{2t}{t_E}\right),\cos \left(\frac{2t}{t_E}\right),-\sqrt{\frac{2}{3}}\sin \left(\frac{2t}{t_E}\right)\right\}.& \left(25\right)\end{array}$
The effective gradient in Eqs. (24, 25) can conceptually be separated as the sum of two terms,
g(t)=g.sub.PGSE(t)+g.sub.iso(t).(26)
The first term, g.sub.PGSE, represents the effective gradient in a regular PGSE two pulse sequence, while the second term, g.sub.iso, might be called the iso-pulse since it is the effective gradient modulation which can be added to achieve isotropic weighting.

(35) As may be seen from above, according to one specific embodiment of the present invention, the method is performed in a single shot, in which the latter should be understood to imply a single MR excitation.

(36) The Analysis Method According to the Present Invention

(37) Below the analysis method according to the present invention will be discussed in detail.

(38) Fractional anisotropy (FA) is a well-established measure of anisotropy in diffusion MRI. FA is expressed as an invariant of the diffusion tensor with eigenvalues .sub.1, .sub.2 and .sub.3,

(39) $\begin{array}{cc}\mathrm{FA}=\sqrt{\frac{{\left({}_1-{}_2\right)}^2+{\left({}_1-{}_3\right)}^2+{\left({}_2-{}_3\right)}^2}{2\left({}_1^2+{}_2^2+{}_3^2\right)}}.& \left(27\right)\end{array}$
In typical diffusion MRI experiments, only a voxel average anisotropy can be detected. The sub-voxel microscopic anisotropy is often averaged out by a random distribution of main diffusion axis. Here we introduce a novel parameter for quantifying microscopic anisotropy and show how it can be determined by diffusion NMR.

(40) Information about the degree of microscopic anisotropy can be obtained from comparison of the echo-attenuation curves, E(b)=/lb)/l.sub.0, with and without the isotropic weighting. Multi-exponential echo attenuation is commonly observed in diffusion experiments. The multi exponential attenuation might be due to isotropic diffusion contributions, e.g. restricted diffusion with non-Gaussian diffusion, as well as due to the presence of multiple anisotropic domains with varying orientation of main diffusion axis. The inverse Laplace transform of E(b) provides a distribution of apparent diffusion coefficients P(D), with possibly overlapping isotropic and anisotropic contributions. However, in isotropically weighed diffusion experiments, the deviation from mono-exponential attenuation is expected to originate mainly from isotropic contributions.

(41) In practice, the diffusion weighting b is often limited to a low-b regime, where only an initial deviation from mono-exponential attenuation may be observed. Such behaviour may be quantified in terms of the kurtosis coefficient K (Jensen, J. H., and Helpern, J. A. (2010). MRI quantification of non-Gaussian water diffusion by kurtosis analysis. NMR Biomed 23, 698-710.)

(42) 0$\begin{array}{cc}\ln E=-\stackrel{\_}{D}b+\frac{{\stackrel{\_}{D}}^{2}K}{6}{b}^2-\mathrm{.Math.}& \left(28\right)\end{array}$
The second term in Eq. (28) can be expressed by the second central moment of the distribution P(D).

(43) Provided that P(D) is normalized,

(44) $\begin{array}{cc}\underset{0}{\overset{}{}}P\left(d\right)\mathrm{dD}=1,& \left(29\right)\end{array}$
the normalized echo signal is given by the Laplace transform

(45) $\begin{array}{cc}E\left(b\right)=\underset{0}{\overset{}{}}P\left(D\right)e^{-\mathrm{bD}}\mathrm{dD}.& \left(30\right)\end{array}$
The distribution P(D) is completely determined by the mean value

(46) $\begin{array}{cc}\stackrel{\_}{D}=\underset{0}{\overset{}{}}\mathrm{DP}\left(D\right)\mathrm{dD}& \left(31\right)\end{array}$
and by the central moments

(47) $\begin{array}{cc}{}_m=\underset{0}{\overset{}{}}{\left(D-\stackrel{\_}{D}\right)}^{m}P\left(D\right)\mathrm{dD}.& \left(32\right)\end{array}$

(48) The second central moment gives the variance, .sub.2=.sup.2, while the third central moment, .sub.3, gives the skewness or asymmetry of the distribution P(D). On the other hand, the echo intensity can be expressed as a cumulant expansion (Frisken, B. (2001). Revisiting the method of cumulants for the analysis of dynamic light-scattering data. Appl Optics 40) by

(49) $\begin{array}{cc}\ln E=-\stackrel{\_}{D}b+\frac{{}_2}{2}{b}^2-\mathrm{.Math.}& \left(33\right)\end{array}$
The first-order deviation from the mono-exponential decay is thus given by the variance of P(D).

(50) Assuming diffusion tensors with axial symmetry, i.e. .sub.1=D.sub. and .sub.2=.sub.3=D.sub., and an isotropic distribution of orientation of the tensor's main diffusion axis, the echo-signal E(b) and the corresponding distribution P(D) can be written in a simple form. In case of the single PGSE experiment, using a single diffusion encoding direction, the distribution is given by

(51) $\begin{array}{cc}P\left(D\right)=\frac{1}{2\sqrt{\left(D-D_{}\right)\left(D_{.Math..Math.}-D_{}\right)}},& \left(34\right)\end{array}$
with the mean and variance,

(52) $\begin{array}{cc}\stackrel{\_}{D}=\frac{D_{.Math..Math.}+2D_{}}{3}\mathrm{and}{}_2=\frac{4}{45}{\left(D_{.Math..Math.}-D_{}\right)}^2.& \left(35\right)\end{array}$
The echo-attenuation for the single PGSE is given by

(53) $\begin{array}{cc}E\left(b\right)=\frac{\sqrt{}}{2}\frac{e^{-{\mathrm{bD}}_{}}}{\sqrt{b\left(D_{.Math..Math.}-D_{}\right)}}\mathrm{erf}\left(\sqrt{b\left(D_{.Math..Math.}-D_{}\right)}\right).& \left(36\right)\end{array}$

(54) For a double PGSE with orthogonal encoding gradients, the distribution P(D) is given by

(55) $\begin{array}{cc}P\left(D\right)=\frac{1}{\sqrt{\left(D_{.Math..Math.}+D_{}-2D\right)\left(D_{.Math..Math.}-D_{}\right)}},& \left(37\right)\end{array}$
with the same mean value as for the single PGSE but with a reduced variance,
.sub.2= 1/45(D.sub.D.sub.).sup.2.(38)
As in the single PGSE, also in double PGSE the echo-attenuation exhibits multi-component decay,

(56) 0$\begin{array}{cc}E\left(b\right)=\frac{\sqrt{}}{2}\frac{e^{-b\frac{D_{}+D_{.Math..Math.}}{2}}}{\sqrt{b\frac{D_{}-D_{.Math..Math.}}{2}}}\mathrm{erf}\left(\sqrt{b\frac{D_{}-D_{.Math..Math.}}{2}}\right).& \left(39\right)\end{array}$
For randomly oriented anisotropic domains, the non-isotropic diffusion weighting results in a relatively broad distribution of diffusion coefficients, although reduced four-fold when measured with a double PGSE compared to the single PGSE. On the other hand the isotropic weighting results in

(57) $\begin{array}{cc}P\left(D\right)=\left(D-\frac{D_{.Math..Math.}+D_{}}{3}\right),& \left(40\right)\\ \mathrm{with}{}_2=0& \left(41\right)\end{array}$
and a mono-exponential signal decay

(58) $\begin{array}{cc}E\left(b\right)=e^{-b\stackrel{\_}{D}}.& \left(42\right)\end{array}$

(59) The variance .sub.2 could be estimated by applying a function of the form (33) to fitting the echo attenuation data. However, in case of randomly oriented anisotropic domains, the convergence of the cumulant expansion of (36) is slow, thus several cumulants may be needed to adequately describe the echo attenuation (36). Alternatively, the distribution (34) may be approximated with the Gamma distribution

(60) $\begin{array}{cc}P\left(D\right)=D^{-1}\frac{e^{-D\text{/}}}{\left(\right){}^{}},& \left(43\right)\end{array}$
where is known as the shape parameter and is known as the scale parameter. For the Gamma distribution, the mean diffusivity is given by D=.Math., while the variance is given by .sub.2=.Math..sup.2. The Gamma distribution is an efficient fitting function. With the two parameters it can capture a wide range of diffusion distributions, with both isotropic as well as anisotropic contributions. Conveniently, the Laplace transform of the Gamma function takes a simple analytical form,

(61) $\begin{array}{cc}E\left(b\right)={\left(1+b\right)}^{-}={\left(1+b\frac{{}_2}{\stackrel{\_}{D}}\right)}^{-\frac{{\stackrel{\_}{D}}^2}{{}_2}}.& \left(44\right)\end{array}$

(62) The variance, .sub.2.sup.iso, obtained by fitting the function (44) to the isotropic diffusion weighted echo-decay is related to the isotropic diffusion contributions, since the variance is expected to vanish with isotropic weighting in a pure microscopically anisotropic system (see Eq. 41). The same fitting procedure on non-isotropically weighted data will yield the variance .sub.2 due to both isotropic and anisotropic contributions. The difference .sub.2.sub.2.sup.iso vanishes when all diffusion contributions are isotropic and therefore provides a measure of microscopic anisotropy. The mean diffusivity D, on the other hand, is expected to be identical for both isotropically and non-isotropically weighted data. The difference .sub.2.sub.2.sup.iso is thus obtained by using the .sub.2.sup.iso and .sub.2 as free fit parameters when Eq. (44) is fitted to isotropically and non-isotropically weighted data sets, respectively, while a common parameter D is used to fit both data sets.

(63) The difference .sub.2.sub.2.sup.iso along with D provide a novel measure for the microscopic fractional anisotropy (FA) as

(64) $\begin{array}{cc}\mathrm{FA}=\sqrt{\frac{3}{2}}\sqrt{\frac{{}_2-{}_2^{\mathrm{iso}}}{{}_2-{}_2^{\mathrm{iso}}+2{\stackrel{\_}{D}}^2\text{/}5}}.& \left(45\right)\end{array}$

(65) The FA is defined so that the FA values correspond to the values of the well-established FA when diffusion is locally purely anisotropic and determined by randomly oriented axially symmetric diffusion tensors with identical eigenvalues. Eq. (45) is obtained by setting FA=FA (27), assuming .sub.2.sub.2.sup.iso=.sub.2 and expressing the eigenvalues D.sub. and D.sub. in terms of D and .sub.2 (see Eq. 35). In the case of a one-dimensional curvilinear diffusion, when D.sub.>>D.sub., FA=FA=1 and in the case of two-dimensional curvilinear diffusion, when D.sub.<<D.sub., FA=FA=1/{square root over (2)}.

(66) The difference .sub.2.sub.2.sup.iso in Eq. (45) ensures that the microscopic anisotropy can be quantified even when isotropic diffusion components are present. Isotropic restrictions, e.g. spherical cells, characterised by non-Gaussian restricted diffusion, are expected to cause a relative increase of both .sub.2 and .sub.2.sup.iso by the same amount, thus providing the difference .sub.2.sub.2.sup.iso independent of the amount of isotropic contributions. The amount of non-Gaussian contributions could be quantified for example as the ratio {square root over (.sub.2.sup.iso)}/D

(67) For anisotropic diffusion with finite orientation dispersion, i.e. when local diffusion tensors are not completely randomly oriented, the D and .sub.2.sub.2.sup.iso are expected to depend on the gradient orientation in the non-isotropic diffusion weighting experiment. Furthermore, variation of the apparent diffusion coefficient (ADC), i.e. volume weighted average diffusivity, dependent on the gradient orientation and given by the initial echo decay of the non-isotropic diffusion weighting experiment, may indicate a finite orientation dispersion. Non-isotropic weighting experiment performed in several directions, similar to the diffusion tensor and diffusion kurtosis tensor measurements, performed with a range of b values to detect possibly multi-exponential decays, combined with the isotropic weighting experiment, is thus expected to yield additional information about microscopic anisotropy as well as information related to the orientation dispersion of anisotropic domains.

(68) Eq. (44) could be expanded by additional terms in cases where this is appropriate. For example, the effects of a distinct signal contribution by the cerebrospinal fluid (CSF) in brain could be described by adding a mono-exponential term with the isotropic CSF diffusivity D.sub.1 to Eq. (44),

(69) $\begin{array}{cc}E\left(b\right)={\mathrm{fe}}^{-{\mathrm{bD}}_1}+\left(1-f\right){\left(1+b\frac{{}_2}{\stackrel{\_}{D}}\right)}^{-\frac{{\stackrel{\_}{D}}^2}{{}_2}},& \left(46\right)\end{array}$
where f is the fraction of the additional signal contribution. Eq. (46) could be used instead of Eq. (44) to fit the experimental data.

(70) A further explanation directed to inter alia FA estimation and optimal range of the diffusion weighting b is given below in the section describing the figures in more detail.

(71) In relation to the description above and below it should be mentioned that also multi-echo variants of course are possible according to the present invention. Such may in some cases be beneficial for flow/motion compensation and for compensation of possible asymmetry in gradient generating equipment.

(72) Specific Embodiments of the Present Invention

(73) Below, specific embodiments of the analysis method according to the present invention will be disclosed. According to one specific embodiment, the method involves approximating the distribution of apparent diffusion coefficients by using a Gamma distribution and the signal attenuation by its inverse Laplace transform. This may increase the speed of the fitting procedure discussed below. The distribution of diffusion coefficients may contain isotropic and/or anisotropic contributions, it may arise due to a distribution of Gaussian diffusion contributions or it may be a consequence of a non-Gaussian nature of diffusion, e.g. restricted diffusion, or it may be due to orientation dispersion of anisotropic diffusion contributions (randomly oriented diffusion tensors) or it may be due to a combination of the above.

(74) One of the main advantages of the analysis method according to the present invention is that it can quantify degree of microscopic anisotropy with high precision from low b-range signal intensity data even in the presence of isotropic contributions, i.e. when they cause deviation from mono-exponential decay. Typically, isotropic contributions would bias the quantification of anisotropy from single PGSE attenuation curves, because multi-exponential signal decays due to isotropic contributions may look similar or indistinguishable to the ones caused by anisotropic contributions. The analysis according to the present invention allows to separate the influence of anisotropic diffusion contributions from the influence of isotropic diffusion contributions on the first order deviation from the mono-exponential decays, where the first order deviation may be referred to as diffusion kurtosis or the second central moment of diffusion distribution, and therefore allows for quantification of the degree of microscopic anisotropy. Therefore, according to the present invention, the method may involve using a fit function (44) comprising the parameters:

(75) initial value

(76) $\left(I_0=\underset{b.fwdarw.0}{\lim }I\left(b\right)\right),$
initial slope

(77) $(\stackrel{\_}{D}=-\underset{b.fwdarw.0}{\lim }\frac{}{b}\ln \left(\frac{I\left(b\right)}{I_0}\right),$
i.e. the volume weighted average diffusivity or the mean diffusivity of a diffusion tensor D) and curvature, i.e. the second central moment of diffusion distribution (.sub.2). See Eq. (44). Note that E(b)=I(b)/I.sub.0. Therefore, according to one specific embodiment of the present invention, the two acquired echo attenuation curves (log E vs. b, where E is echo amplitude, which may be normalized, and b is the diffusion weighting factor) are compared in terms of initial value, initial slope or curvature, and/or the ratio between the two echo attenuation curves is determined, so that the degree of microscopic anisotropy may be determined.

(78) The method may involve fitting the isotropic and non-isotropic weighted data with the fit function (44) comprising the parameters: initial values

(79) $\left(I_0=\underset{b.fwdarw.0}{\lim }I\left(b\right)\right)$
for isotropic and non-isotropic data, initial slope

(80) 0$(\stackrel{\_}{D}=-\underset{b.fwdarw.0}{\lim }\frac{}{b}\ln \left(\frac{I\left(b\right)}{I_0}\right),$
i.e. the volume weighted average diffusivity or the mean diffusivity of a diffusion tensor D) with constraint that D values are identical for both isotropic and non-isotropic diffusion weighted data and the second central moments, .sub.2.sup.iso and .sub.2 for isotropic and non-isotropic diffusion weighted data, respectively. The microscopic fractional anisotropy (FA) is then calculated from the mean diffusivity, D, and the second central moments of diffusion distribution, .sub.2.sup.iso and .sub.2 according to Eq. (45). As disclosed above, according to one embodiment a fit function comprising the parameters initial value, initial slope and curvature (zeroth, first, and second central moment of the probability distribution of diffusion coefficients), fraction of the additional diffusion contribution (f) and/or diffusivity of the additional contribution (D.sub.1) (see discussion below) are used. When the extended fitting model described in Eq. (46) is applied, then the mean diffusivity, D, the additional diffusion contribution (f) and the diffusivity of the additional contribution (D.sub.1) are constrained to be equal for the isotropic and the non-isotropic diffusion weighted data.

(81) Moreover, according to another embodiment of the present invention, the microscopic fractional anisotropy (FA) is calculated from mean diffusivity (D) and difference in second central moments of diffusion distribution (.sub.2.sup.iso and .sub.2).

(82) The method may also involve fitting the isotropic and non-isotropic weighted data, where non-isotropic weighted data is acquired separately for several directions of the magnetic field gradients with the fit function (44) comprising the parameters: initial value

(83) $\left(I_0=\underset{b.fwdarw.0}{\lim }I\left(b\right)\right),$
initial slope

(84) $({\stackrel{\_}{D}}_g=-\underset{b.fwdarw.0}{\lim }\frac{}{b}\ln \left(\frac{I\left(b\right)}{I_0}\right),$
i.e. the volume weighted average diffusivity D.sub.g generally depending on the gradient orientation) and curvature, i.e. the second central moment of diffusion distribution (.sub.2). Different fit constraints may be applied to optimize the accuracy of the estimated fit parameters. For example, the, initial slope (D.sub.g), i.e. the volume weighted average diffusivity D.sub.g estimated from the non-isotropic diffusion weighted data acquired with diffusion weighting in different directions, such to yield information required to construct the diffusion tensor, D, could be subjected to the constraint where the trace of the diffusion tensor (determined by non-isotropic diffusion weighting experiments) is identical to three times the mean diffusivity, D, obtained from the isotropic diffusion weighting data, i.e. tr(D)=3D. In such a case, the microscopic fractional anisotropy (FA) parameter could still be calculated according to Eq. (45), however, care need to be taken when interpreting results of such calculations. As an alternative to fitting the non-isotropic diffusion weighted data in different gradient directions, e.g. from diffusion tensor measurement, the signal intensities may be averaged across the gradient orientations. The resulting curve will approximate a sample having full orientation dispersion, since averaging under varying gradient orientations is identical to averaging under varying orientations of the object. Therefore, according to one specific embodiment of the present invention, the echo attenuation curves acquired with at least one of the gradient modulation scheme based on isotropic diffusion weighting and the gradient modulation scheme based on non-isotropic diffusion weighting are averaged across multiple encoding directions.

(85) The method may involve the use of additional terms in Eq. (44), such as Eq. (46), applied to the analysis described in the above paragraphs. Eq. (46) comprises two additional parameters, i.e. fraction of the additional diffusion contribution (f) and diffusivity of the additional contribution (D.sub.1). One such example may be the analysis of data from the human brain, where the additional term in Eq. (46) could be assigned to the signal from the cerebrospinal fluid (CSF). The parameter D in Eq. (46) would in this case be assigned to the mean diffusivity in tissue while the parameter D.sub.1 would be assigned to the diffusivity of the CSF. The isotropic diffusion weighting could thus be used to obtain the mean diffusivity in the brain tissue without the contribution of the CSF.

(86) In addition, valuable information about anisotropy may be obtained from the ratio of the non-isotropically and the isotropically weighted signal or their logarithms. For example, the ratio of the non-isotropically and the isotropically weighted signals at intermediate b-values, might be used to estimate the difference between the radial (D.sub.) and the axial (D.sub.) diffusivity in the human brain tissue due to the diffusion restriction effect by the axons. Extracting the information about microscopic anisotropy from the ratio of the signals might be advantageous, because the isotropic components with high diffusivity, e.g. due to the CSF, are suppressed at higher b-values. Such a signal ratio or any parameters derived from it might be used for generating parameter maps in MRI or for generating MR image contrast.

(87) It is interesting to note that the FA parameter is complementary to the FA parameter, in the sense that FA can be finite in cases when FA=0, while, on the other hand, FA tends to vanish when FA is maximized by anisotropy on the macroscopic scale. Such approach could be used to analyse microscopic anisotropy and orientation distribution in a similar way as the diffusion tensor and kurtosis tensor analysis is used. Compared to the kurtosis tensor analysis, here presented microscopic fractional anisotropy analysis is advantageous in that it can separate the isotropic diffusion components that may contribute to the values of kurtosis detected with the present methodology for kurtosis tensor measurements.

(88) The analysis method according to the present invention is applicable in many different situations. According to one embodiment, the method is performed so that mean diffusivity is constrained to be identical for both isotropic and non-isotropic diffusion weighted data. If Eq. (46) is employed, then parameters f and D.sub.1 are also constrained to be equal for isotropic and non-isotropic diffusion weighted data. Furthermore, according to another specific embodiment of the present invention, the echo attention curve acquired with the gradient modulation scheme based on isotropic diffusion is assumed to be monoexponential. This may be of interest for the sake of approximating the microscopic anisotropy.

(89) According to another embodiment, the method is performed so that a mean diffusivity for isotropic diffusion weighted data is allowed to be different from the mean diffusivity for non-isotropic diffusion weighted data. This latter case would be better in cases when the micro-domains do not have random orientations, i.e. orientation dispersion is not isotropic. In such cases the mean diffusivity depends on orientation of the non-isotropic weighting. The analysis method according to the present invention may involve different forms of experiments. In this sense it should be noted that the present analysis encompasses all diffusion weighting pulse sequences where one achieves isotropic diffusion weighting and the other with a non-isotropic diffusion weighting. According to the present invention, there are however some specific set-up alternatives that may be mentioned additionally. According to one specific embodiment, the isotropic diffusion weighting and the non-isotropic diffusion weighting is achieved by two different pulse gradient spin echos (PGSEs). According to one specific embodiment of the present invention, the gradient modulation scheme based on isotropic diffusion weighting comprises at least one harmonically modulated gradient, which removes curvature of log E vs. b originating from anisotropy. According to yet another embodiment, the method involves a single-PGSE yielding maximum curvature of log E vs. b for the non-isotropic diffusion weighting, and a single-PGSE augmented with sinusoidal isotropic gradients for the isotropic diffusion weighting.

(90) As discussed above and shown in FIGS. 1A-C and e.g. 2A-C, the analysis method according to the present invention may be performed on many different forms of materials and substances. According to one specific embodiment of the present invention, the method allows for determining the degree of anisotropy in systems with anisotropic and/or isotropic diffusion, such as in liquid crystals. This may be used to infer about the geometry of micro-domains. For example, the analysis may be used to identify cases of curvilinear diffusion in 1D with highest microscopic anisotropy (FA=1) or in 2D when diffusion is restricted to domains with planar geometry (FA=1/2), and other intermediate cases between isotropic and 1D diffusion. As may be understood, the present invention also encompasses use of an analysis method as disclosed above. According to one specific embodiment, the present invention provides use of the analysis method, for yielding an estimate of the microscopic fractional anisotropy (FA) with a value for quantifying anisotropy on the microscopic scale. Furthermore, there is also provided the use of a method according to the present invention, wherein anyone of the parameters microscopic fractional anisotropy (FA), mean diffusivity (D), {square root over (.sub.2.sup.iso)}/D, fraction of the additional diffusion contribution (f) and/or diffusivity of the additional contribution (D.sub.1) or any other parameter(s) calculated from .sub.2, .sub.2.sup.iso or mean diffusivity is used for generating parameter maps in MRI or for generating MR image contrast. Information about microscopic diffusivity may be obtained from the ratio of the non-isotropically and isotropically weighted signals or their logarithms and may be used in parameter maps in MRI or for generating MR image contrast. The difference between the radial and axial diffusivity may be extracted (see above), which is obtained from the ratio of the signals.

(91) Furthermore, intended is also the use of a method according to the present invention, wherein microscopic fractional anisotropy (FA) is used for characterizing tissue and/or diagnosing, such as for diagnosing a tumour disease or other brain or neurological disorders.

(92) Moreover, also as hinted above, the analysis method according to the present invention may also be coupled to a pre-performed method involving isotropic and non-isotropic diffusion weighting.

DETAILED DESCRIPTION OF THE DRAWINGS

(93) In FIG. 1A-C there is shown a schematic representation of signal decays vs. b for isotropic and non-isotropic diffusion weighting for different types of materials. In FIG. 1 the following is valid: A) Solid lines represent decays in a non-isotropic diffusion weighting experiment for 1D and 2D curvilinear diffusion (e.g. diffusion in reversed hexagonal phase H2 (tubes) and in lamellar phase L (planes), respectively). Dashed lines are the corresponding decays with isotropic diffusion weighting. The initial decay (D) identical for the isotropic diffusion weighting as for the non-isotropic diffusion weighting. B) The decay for a system with 70% free isotropic diffusion and 30% restricted isotropic diffusion. In this case the isotropic and non-isotropic diffusion weighting result in identical signal decays in the entire b-range. C) Decays for a system with 70% anisotropic diffusion (2D) and 30% restricted isotropic diffusion. Solid line corresponds to the non-isotropic diffusion weighting while the dashed line corresponds to the isotropic diffusion weighting. The initial decays are identical for the isotropic and for the non-isotropic diffusion weighting, while the deviation between the decays at higher b values reveals the degree of anisotropy.

(94) In relation to the analysis performed and the presented results it may also be mentioned that comparing the signal decays of the two acquired echo attenuation curves may involve analysis of the ratio and/or difference between the two acquired echo attenuation curves.

(95) In FIG. 2A-C are shown experimental results with analysis of micro-anisotropy for different types of materials. Shown are normalized signal decays vs. bD for isotropic (circles) and non-isotropic (crosses) diffusion weighting. Solid lines represent optimal fits of Eq. (44) to the experimental data, with constraint of equal initial decays, D, (shown as dashed lines) for isotropic and non-isotropic weighted data. All experiments were performed at 25 C. In all experiments, signal intensities were obtained by integration of the water peak. A) free water; data from the isotropic and non-isotropic diffusion weighting overlap and give rise to mono-exponential signal decays. The analysis gives D=2.210.sup.9 m.sup.2/s and FA=0. B) Suspension of yeast cells from baker's yeast (Jstbolaget AB, Sweden) in tap water with restricted water diffusion; data from the isotropic and non-isotropic diffusion weighting overlap and give rise to multi-exponential signal decays. The analysis gives D=1.410.sup.9 m.sup.2/s and FA=0. C) Diffusion of water in a liquid crystal material composed by the Pluronic surfactant E5P68E6 with very high microscopic anisotropy, corresponding to a reverse hexagonal phase; data from the isotropic and non-isotropic diffusion weighting diverge at higher b-values and give rise to multi-exponential signal decay in case of the non-isotropic diffusion weighting and mono-exponential signal decay in case of the isotropic diffusion weighting. The analysis gives D=4.810.sup.1 m.sup.2/s and FA=1.0.

(96) In FIGS. 3A and 3B, the results of the Monte-Carlo error analysis show systematic deviations and precision of the D(A) and FA (B) parameters estimated for the 1D (dots) and 2D (circles) curvilinear diffusion according to what has been disclosed above. The ratio of the estimated mean diffusivity to the exact values D, labelled as D/D (A) with the corresponding standard deviation values and the estimated FA values (B) with the corresponding standard deviations are shown as dots/circles and error bars, respectively, as a function of the maximum attenuation factor b.sub.maxD for signal to noise level of 30.

(97) For FA estimation, the optimal choice of the b-values is important. To investigate the optimal range of b-values, a Monte-Carlo error analysis depicted in FIGS. 3A and 3B has been performed. The echo-signal was generated as a function 16 equally spaced b-values between 0 and b.sub.max for the cases of 1D and 2D curvilinear diffusion with randomly oriented domains. The upper limit, b.sub.max, was varied and the attenuation factors bD were chosen to be identical for the 1D and 2D case. The signal was subjected to the Rician noise with a constant signal to noise, SNR=30, determined relative to the non-weighted signal. Isotropic and non-isotropic weighed attenuation data were analyzed with the protocol described herein to obtain D and FA parameters. This analysis was repeated in 1000 iterations by adding different simulated noise signals with the given SNR. The procedure yields the mean and the standard deviation of the estimated D and FA, shown as dots/circles and error bars respectively in FIG. 3B.

(98) The optimal range of the diffusion weighting b is given by a compromise between accuracy and precision of the FA analysis and it depends on the mean diffusivity. If the maximum b value used is lower than 1/D, the FA tends to be underestimated, while for maximum b values larger than 1/D the FA tends to be overestimated. On the other hand the accuracy of FA is compromised particularly at too low values of the maximum b, due to increased sensitivity to noise. See FIG. 3B.