Magnetic Resonance in Medicine 54:419 – 428 (2005)
Limitations of Apparent Diffusion Coefficient-Based
Models in Characterizing Non-Gaussian Diffusion
Chunlei Liu,1,2 Roland Bammer,1 and Michael E. Moseley1*
Diffusion in complex heterogeneous structures, for example,
the neural fiber system, is non-gaussian. Recently, several
methods have been introduced to address the issue of nongaussian diffusion in multifiber systems. Some are based on
apparent diffusion coefficient (ADC) analysis; and some are
based on q-space analysis. Here, using a simple mathematic
derivation, ADC-based models are shown to be mathematically
self-inconsistent in the presence of non-gaussian diffusion.
Monte Carlo simulation on restricted diffusion is applied to
demonstrate the poor data fitting that can result from ADCbased models. Specific comparisons are performed between
two generalized diffusion tensor imaging methods: one of them
is based on ADC analysis, and the other is shown to be consistent with q-space formalism. The issue of imaging asymmetric
microstructures is also investigated. Signal phase and spin
exchange are necessary to resolve multiple orientations of an
asymmetric structure. Magn Reson Med 54:419 – 428, 2005.
© 2005 Wiley-Liss, Inc.
Key words: magnetic resonance imaging; diffusion; diffusion
tensor imaging; high angular resolution; probability density
function; GDTI; HOT; ADC
Diffusion tensor imaging (DTI) is a powerful technique for
studying molecular diffusion processes in tissues (1).
When the diffusion process is gaussian, the MR signal
attenuates exponentially as a function of b-value. This
signal behavior can be characterized by a second order
diffusion tensor that is proportional to the variance of the
molecular displacement during the MR experimental time.
By measuring this diffusion tensor, DTI offers a useful tool
for detecting tissue disease states, for example, the early
stage of acute cerebral ischemia (2). However, the development of DTI-based tractography in the context of complex biologic structures has in turn revealed the limitations of DTI. Specifically, DTI can only resolve a single
fiber direction within each imaging voxel (3– 6). However,
at current MR resolution capabilities, it is not likely that a
voxel contains a single fiber. Rather, it is more likely that
a voxel contains a distribution of fiber orientations in one
voxel. In this situation, the gaussian diffusion assumption
is no longer valid.
1
Lucas MRS/I Center, Department of Radiology, Stanford University, Stanford, California, U.S.A.
2
Department of Electrical Engineering, Stanford University, Stanford, California, U.S.A.
Grant sponsor: National Institute of Health; Grant numbers: NIH1R01NS35959, NIH-1R01EB2711; Grant sponsor: Oak Foundation; Grant
sponsor: Center of Advanced MR Technology of Stanford; Grant number:
NCRR P41 RR 09784; Grant sponsor: Lucas Foundation.
*Correspondence to: Michael E. Moseley, Radiological Science Laboratory at
the, Richard Lucas MRS/I Center, Department of Radiology, Stanford University, 1201 Welch Road, Stanford, CA 94305-5488, U.S.A. E-mail:
[email protected]
Received 25 October 2004; revised 4 February 2005; accepted 8 March 2005.
DOI 10.1002/mrm.20579
Published online in Wiley InterScience (www.interscience.wiley.com).
© 2005 Wiley-Liss, Inc.
To resolve the issue of multiple fiber orientations, several methods have been proposed: (i) high angular resolution diffusion weighted imaging (HARD) methods based
on ADC analysis (7–11); (ii) diffusion spectrum imaging
(DSI) (12,13); (iii) a generalized diffusion tensor imaging
(GDTI) using higher order tensor (HOT) statistics introduced by Liu et al. (which will be referred to here as
GDTI-1) (14 –16); (iv) a GDTI method for mapping diffusivity profiles introduced by Özarslan et al. (which will be
referred to here as GDTI-2) (17), and (v) q-ball imaging
(QBI) (18,19). These methods can be tentatively divided
into two basic categories according to their data analysis
techniques. One relies on the analysis of apparent diffusion coefficient (ADC) and the other is based on q-space
analysis. Note that q-space is the reciprocal spatial space
defined through the Fourier transform of a probability
density function (PDF) (20). Among the various aforementioned techniques, DSI and QBI are examples of the qspace approach; spherical harmonic decomposition of
HARD data (9) and GDTI-2 are based on ADC analysis.
Although the two GDTI methods are called the same, the
underlying analytic concept is fundamentally different.
GDTI-1 has been shown to be consistent with q-space
imaging (14,15), whereas GDTI-2 relies on the analysis of
ADC. One intrinsic difference between these two categories of techniques is that ADC-based methods rely on the
assumption of Gaussian diffusion; whereas q-space analysis does not.
All these methods require data acquisitions with high
b-values (usually larger than 1000 s/mm2) to sensitize possible boundary effects on diffusion and, hence, differentiate between the fast and slow components in a fiber system. Moreover, data are acquired for many diffusion directions, typically on the order of 100 (7–11,15). The
diffusion directions are usually uniformly distributed. By
exploring the orientation-dependent diffusion information, these methods attempt to detect and model the underlying non-gaussian diffusion process. For example,
using diffusion weighted images acquired at b ⫽
1077 s/mm2, Tuch et al. showed that ADC profiles in
regions of multiple fibers exhibited multiple local maxima
or minima as a function of diffusion gradient orientation
(7,10). Frank proposed an approach to characterize diffusion by decomposing the ADC profiles into spherical harmonics of various orders (9). He suggested that isotropic
diffusion could be described by zeroth- and second-order
harmonics and that multiple-fiber diffusion could be characterized by including fourth- and higher-order harmonics
(9). In GDTI-2, on the other hand, the angular distribution
of the ADC profile was fitted by elements of one higherorder tensor. Özarslan et al. also established a relationship
between the higher-order tensor elements and the coefficients of spherical harmonic decomposition (17).
419
420
Liu et al.
constant G during the diffusion encoding periods, Eq. [1]
can be simplified to
S ⫽ S 0具exp共⫺j␥␦G 䡠 共x៮ 2 ⫺ x៮ 1 兲兲典.
FIG. 1. A schematic diagram of the diffusion-weighted spin-echo
sequence. The two gradient pulses have a magnitude of G and a
duration of ␦. The diffusion time is ⌬.
x៮ 1 and x៮ 2 are the time-averaged initial and final positions
of a given spin during the first and second diffusionencoding periods, respectively. Let q ⫽ ␥␦G, r ⫽ x៮ 2 ⫺ x៮ 1
and denote function p(r) the PDF of random variable r,
then
S ⫽ S0
Although tremendous progress has been made toward
achieving better angular resolution of fiber structures,
there has been little physical justification for those ADCbased techniques, such as spherical harmonic decomposition and GDTI-2, in the situation of non-gaussian diffusion. Here we evaluate the possible physical and mathematical foundations of the diffusion models that rely on
ADC measurements. We demonstrate that a fundamental
mathematical problem exists in the ADC-based approaches. For instance, self-contradiction can arise in the
modeling of non-gaussian diffusion if the analysis is based
on ADC measurement. We also show that in the presence
of non-gaussian diffusion ADC measurements alone are
insufficient to characterize the complex diffusion property. In addition to the mathematical proof, Monte Carlo
simulation and numerical computation for restricted diffusion are also applied to illustrate the poor data fit of the
ADC models. Furthermore, we investigate the possibility
of imaging asymmetric fiber structures with GDTI-1.
Asymmetric fibers can give rise to a unique type of nongaussian diffusion. As a result, this type of diffusion, in
particular, cannot be characterized by ADC-based methods.
THEORY
Background
The diffuse motion of water molecules in an inhomogeneous magnetic field causes signal dephasing, and averaging over the displacement distribution of all the spins
results in signal attenuation. More specifically, if this diffusion is treated as a stochastic process, it can be described
by the PDF of the spin displacement. For the most commonly used spin echo sequence as depicted in Fig. 1, the
signal S can be described as an ensemble average of all
spins (21):
冓 冉冕
TE/2
S ⫽ S 0 exp j␥
0
G共t兲 䡠 x共t兲 dt ⫺ j␥
冕
TE
TE/2
冊冔
G共t兲 䡠 x共t兲 dt
.
[1]
In this equation, S0 is the echo amplitude in the absence of
the diffusion gradient; G(t) is the applied diffusion gradient vector; x(t) is a given spin’s position at time t; ␥ is the
gyromagnetic ratio. Given that the amplitude of G(t) is a
[2]
冕
p共r兲exp共⫺jq 䡠 r兲 dr.
[3]
Note that in this interpretation of q-space formalism, it is
not required that the duration ␦ of diffusion-encoding gradient pulses be much smaller than the separation time ⌬
between two diffusion gradients (Fig. 1). Nevertheless, the
finite duration of diffusion gradients causes p(r) to be the
distribution function of a time-averaged displacement
rather than the intrinsic spin displacement. When the gradient pulses approach delta functions, p(r) becomes the
autocorrelation function of the spin density as described
by Callaghan (20).
Equation [3] shows that S/S0 is the characteristic function of p(r). It can be expanded using the cumulants of the
random variable r (15,16,22):
冉
共⫺j兲Qi共1兲
qi 1 共⫺j兲2 Qi共2兲
q q
S
1
1i 2 i 1 i 2
⫽ exp
⫹
⫹···
S0
1!
2!
⫹
冊
q q . . . qi n
共⫺j兲n Qi共n兲
1i 2 . . . i n i 1 i 2
⫹··· .
n!
[4]
This is also a generalization of the Kramers–Moyal expansion in a multivariate case (23,24). The expansion coeffi. . . i are the nth order cumulants of the random
cients Q i(n)
1i 2
n
variable r. The Einstein’s summation rule is utilized to
simplify the formula. Equation [4] can also be written as
(14 –16)
冉冘
⬁
m共b兲 ⫽ m共0兲exp
n⫽2
冊
jn Di共n兲
b共n兲
,
1i 2 . . . i n i 1i 2 . . . i n
[5]
. . . i and b (n) . . . i are both tensors of order n. If
where D i(n)
i 1i 2
1i 2
n
n
r is gaussian distributed, only the second-order cumulant
(2)
Q i 1i 2 or the second-order diffusion tensor D i(2)
is nonzero.
1i 2
This results in an exponential decay of signal intensity.
However, if r is non-gaussian distributed, in general, no
such exponential behavior exists. Furthermore, if r is distributed asymmetrically, the acquired MR signal is complex-valued according to Eq. [3]. The asymmetry information is contained in the odd-order cumulants. Although an
asymmetrical PDF appears to be a clear violation of the
principle of microscopic detailed balance, a possible
breakdown of this principle in voxel imaging is studied in
the following sections.
ADC and Non-Gaussian Diffusion
421
be reduced by characterizing the surrogate ADC profile,
using a finite number of tensor elements as given by (17)
ADC-Based Diffusion Models
ADC was originally defined for each voxel as (25)
ADC ⫽ ⫺
1
ln共S/S0 兲.
b
[7]
where ␪ and ␸ are the azimuthal and polar angles of the
diffusion-encoding direction in a spherical coordinate, respectively. The measured ADC profile can be further analyzed through spherical harmonic decomposition (9),
冘冘
⬁
ADC共␪, ␸兲 ⫽
l
a lmY lm共␪, ␸兲,
[8]
l⫽0 m⫽⫺l
where Ylm represents the spherical harmonics, and alm
represents the decomposition coefficient.
The GDTI-2 recently introduced by Özarslan et al. utilizes a higher-order tensor to describe the signal equation
for the diffusion experiment as
冘冘 冘
3
ln S ⫽ ln S0 ⫺ b ⫻
3
3
3
...
D i1i2 . . . ilg i1g i2 . . . g il.
[10]
il⫽1
Özarslan et al. have shown that there is a direct relationship between the components of D i 1i 2 . . . i land the coefficients of spherical harmonic decomposition (17). By including only up to the second-order terms, both the spherical harmonic decomposition method and the GDTI-2
converge to the DTI model (9,17). Furthermore, it is assumed that when the diffusion is non-gaussian, higherorder terms (alm for spherical harmonic decomposition
and D i 1i 2 . . . i l for GDTI-2) are required to describe the nonellipsoid-shape ADC profile, which can be extracted by
solving Eq. [8] or [9].
Spherical harmonic decomposition and GDTI-2 are considered ADC-based diffusion models because they both
require the existence of ADC. In the subsequent sections,
we will show, however, that ADC-based models cannot
characterize non-gaussian diffusion.
ADC and Non-Gaussian Diffusion
ADC-based models are based on the assumption that MR
signal decays monoexponentially as a function of the bvalue. However, monoexponential behavior does not exist
for every diffusion-encoding direction when the underlying diffusion process is non-gaussian. Consequently, ADC,
generally, is not well defined for non-gaussian diffusion.
Specifically, we will show that Eq. [7] implies that the
projection of displacement r in any direction defined by (␪,
␸) is gaussian. Furthermore, we will show that if its projection in any direction is gaussian, then r itself must be a
gaussian random vector.
To proceed, write G ⫽ (G, ␪, ␸) in the spherical coordinate, and r ⫽ (x, y, z) in the Cartesian coordinate; then Eq.
[2] becomes
S ⫽ S 0具exp共⫺j␥␦G共x cos ␸ sin ␪ ⫹ y sin ␸ sin ␪
⫹ z cos ␪兲兲典.
[11]
r g ⫽ G 䡠 r/G ⫽ x cos ␸ sin ␪ ⫹ y sin ␸ sin ␪ ⫹ z cos ␪,
[12]
Let
3
...
i 1 ⫽1 i 2 ⫽1
ADC共␪, ␸兲 ⫽
i1⫽1 i2⫽1
The word “apparent” in ADC was used in recognition of
the fact that the measurement of diffusion may be distorted
due to restricted molecular motion caused by cell membranes and other tissue compartments. If diffusion is the
only motion present, ADC is equal to the self-diffusion
coefficient D.
When anisotropic diffusion is present, a scalar ADC is
not sufficient to quantify the diffusion process. However, if
the underlying diffusion process is gaussian, the signal
behavior can be described by a diffusion tensor (a secondorder tensor). To determine the diffusion tensor, a minimum of six measurements is required. Note that if the
diffusion process is non-gaussian, a second-order diffusion tensor is not sufficient because higher-order statistics,
namely higher-order moments and cumulants, are required to characterize a non-gaussian random variable
(26). With a second-order diffusion tensor, DTI cannot
resolve multiple fiber orientations. In order to improve the
angular resolution of fiber structure, Tuch et al. propose to
measure ADC as a function of orientation (7,8),
1
ADC共␪, ␸兲 ⫽ ⫺ ln共S/S0 兲,
b
冘冘 冘
3
[6]
Di 1i 2 . . . i lgi 1gi 2 . . . gi l,
[9]
i l ⫽1
where D i 1i 2 . . . i l is an lth order tensor, and gi is the ith
component of the unit gradient vector. This signal behavior is fundamentally different from that of GDTI-1 as given
in Eq. [5]. In GDTI-1, the signal is a function of a series of
. . . i n , whereas GDTI-2 ashigher-order b-tensors b i(n)
1i 2
sumes that there is a monoexponential relationship between signal amplitude and the b-value. This assumption
implies that the last term on the right-hand side of the Eq.
[9] represents a measurement of ADC and that the redundancy generated by measuring ADC in each direction can
which is the projection of r in the direction of diffusion
gradient G. Suppose Eq. [7] is the correct formula (i.e.,
suppose signal decays exponentially as a function of bvalue.), then, according to Eq. [4], all cumulants of rg of
order three or more are zero. In other words, rg is a 1D
gaussian random variable (26). Its variance (the same as
the second-order cumulant) is ␴ r2g ⫽ 2 䡠 ADC(␪, ␸) 䡠 (⌬ ⫺
␦/3).
To show that the random displacement r is a 3D gaussian random vector, we observe that
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Liu et al.
␴ x2cos 2 ␸ sin2 ␪ ⫹ ␴2y sin2 ␸ sin2 ␪ ⫹ ␴z2 cos2 ␪
⫹ 2␴xy cos ␸ sin ␸ sin2 ␪ ⫹ 2␴xz cos ␸ sin ␪ cos ␪
⫹ 2␴yz sin ␸ sin ␪ cos ␪ ⫽ ␴r2g.
[13]
2
Here, ␴ x2 is the variance of x; ␴ xy
is the covariance of x and
y; and so on. Since ␴ r2g ⬎ 0, and Eq. [11] is true for any
direction defined by (␪, ␸), the matrix
⌺⫽
冋
2
␴ x2 ␴ xy
2
␴ yx ␴ 2y
2
2
␴ zx
␴ zy
2
␴ xz
2
␴ yz
␴ z2
册
Smaller Voxel Size
[14]
is positive definite, which forms the covariance matrix of
the random vector r.
Furthermore, since all higher-order (greater than 2) cumulants of rg are zero, higher-order cumulants of r are also
zero due to the linear relationship between rg and the
components of r (27). To simplify the mathematical notations, Eq. [12] can be rewritten as
r g ⫽ a ir i.
[15]
Here, ai is a trigonometric function of ␪ and ␸, as given in
Eq. [12], for example, a 1 ⫽ cos ␸ sin ␪, and ri is the ith
component of vector r. It has previously been shown that,
under linear transformation, cumulants transform in the
same way as contravariant tensors (27). Applying this
transformation law for cumulants to Eq. [15], one obtains
Q r共n兲
⫽ a i1a i2 . . . a inQ i共n兲
,
g
1i2 . . . in
angle between each two adjacent branches is 120o. The
cross sections of the tubes in both phantoms were square.
The ends of the tubes were closed. The boundaries of the
tubes were assumed to be impermeable. Within the tube,
spins could diffuse freely. At the boundary, spins were
elastically reflected. To evaluate the effect of voxel size in
reconstructing the phantom structures— especially the Yshaped tube—two types of voxel size were simulated: one
of 80 ⫻ 80 ⫻ 80 ␮m3 and one of 500 ⫻ 500 ⫻ 500 ␮m3.
[16]
where Q r(n)
is the nth order cumulant of random variable
g
rg. Because Q r(n)
⫽ 0 for all n ⬎ 2, and Eq. [16] holds for
g
. . . i must be zero for
every direction defined by (␪, ␸), Q i(n)
1i 2
n
all n ⬎ 2.
Hence, we have shown that r is a gaussian random
vector with covariance matrix ⌺.
In other words, for non-gaussian diffusion, generally,
the signal does not decay exponentially as a function of the
b-value. Monoexponential signal behavior is only the result of gaussian distributed random displacement (28).
Because ADC-based models are developed based on the
existence of an ADC along the diffusion direction, they are
only suitable for analyzing gaussian diffusion. Moreover,
for gaussian diffusion, the angular distribution of ADC has
only six degrees of freedom, which is fully determined by
a second-order symmetric tensor. Therefore, it is redundant to include higher-order terms in spherical harmonic
decomposition or in GDTI-2. If experimental data are
forced into this model, poor data fit would only result in
the case of non-gaussian diffusion.
Computer Simulation
In order to illustrate the poor data fitting that could result
from diffusion models based on ADC analysis, Monte
Carlo simulation was performed on two phantoms: (1) a
perpendicularly crossing tube (Phantom 1); and (2) a Yshaped tube (Phantom 2). In Phantom 1, the horizontal
tube was along the x axis. In Phantom 2, the separation
The width of the cross sections of both phantoms was
10 ␮m. In both phantoms, each branch of the tube had a
length of 80 ␮m (Figs. 5a and 6a). A total of 6.5 ⫻ 106 spin
trajectories with uniformly distributed starting positions
were simulated by using a random walk technique (15).
The time increment of the random walk was 0.01 ms. For
an unrestricted diffusion, the diffusion coefficient D was
set to be 2 ⫻ 10⫺3 mm2/s. The following MR parameters
were used: ␦ ⫽ 30 ms, ⌬ ⫽ 40 ms, and TE ⫽ 80 ms. The
maximum diffusion gradient strength was 40 mT/m. MR
diffusion experiments were simulated on 400 uniformly
distributed diffusion-encoding directions. The directions
were generated by maximizing the least separation of 400
randomly selected points on the surface of a unit sphere
and provided an optimal tessellation of a sphere. For each
direction, 15 different b-values were used with a maximum of 2344 s/mm2.
During the diffusion encoding period, which began at
the first diffusion gradient and ended at the second gradient, the spins were allowed to diffuse anywhere within the
tubes. At the echo time, the signal from the central voxel
(80 ⫻ 80 ⫻ 80 ␮m3, excluding the ends of tubes) was
calculated, by summing only the signal contribution from
spins located within the voxel at that moment. The voxel
location was illustrated in the boxes of Figs. 5a and. 6a
(15). Analyzing one voxel instead of the whole phantom
simulates the effect of intervoxel diffusion.
The simulated data were first analyzed using GDTI-2 as
proposed by Özarslan et al. Using the standard least mean
square algorithm a tensor of order 8 was estimated (17).
The root mean square error (RMS) between the fitted curve
and simulated data was also computed for each diffusion
direction. The diffusivity profile was visualized with a
parameterized 2-surface (17). A comparative evaluation
was also performed on the same data set using GDTI-1 as
proposed by Liu et al. Generalized diffusion tensors up to
order 5 were estimated from which the PDF skewness map
(29) was computed. The skewness map was computed as
the difference between the PDF constructed by GDTI-1 and
the corresponding gaussian PDF (14,15). Isosurface plots
of the skewness map were generated using a value that
equaled 5% of the maximum of the map.
The true distribution of the spin’s random displacement
was approximated by computing a 3D histogram of the
random displacement. 1D histograms of the displacement
projected along two selected directions were also computed. To assess whether the underlying diffusion is gaussian along these directions, these 1D histograms were fitted with a gaussian distribution function. The gaussian
distribution function had the same variance as the spin’s
displacement along these directions.
ADC and Non-Gaussian Diffusion
423
FIG. 2. Phantom 1: two representative sets of fitted curves by GDTI-1 (as proposed by Liu et al.) and GDTI-2 (as proposed by Özarslan
et al.) and the corresponding displacement distribution. In (a and b) the diffusion direction is (0.88, 0.078, 0.46). In (c and d) the direction
is (⫺0.63, 0.53, ⫺0.57). (a) The simulated data does not follow exponential behavior. (b) The distribution of the spin displacement is
non-gaussian. (c) The simulated data has an exponential behavior in this direction, and (d) the distribution of the spin displacement is
gaussian.
Larger Voxel Size
The asymmetry information of a fiber structure is contained in the signal phase. Although voxel size does not
affect the phase of a symmetric structure, it does affect that
of an asymmetric structure. To evaluate the effect of voxel
size on signal phase, a voxel size that was comparable to
what is currently achievable (30) was also simulated for
Phantom 2. The cross section of the tube had a width of
20 ␮m; each branch had a length of 500 ␮m (Fig. 7a). A
total of 9.5 ⫻ 106 spin trajectories with uniformly distributed starting positions were simulated. The rest of the
parameters were kept the same.
RESULTS
Smaller Voxel Size
Figure 2 shows the fitted signal decay curves using both
GDTI methods and the corresponding spin displacement
distribution for Phantom 1 in two representative diffusion directions. In Fig. 2a and b, the data are acquired in
the direction of (0.88, 0.078, 0.46); in Fig. 2c and d, the
direction is (⫺0.63, 0.53, ⫺0.57). Figure 2a and c plots
the signal behavior. The simulated data are marked with
a cross. The curve fitted by GDTI-1 is indicated by a
solid line; and the curve fitted by GDTI-2 is represented
by a dotted line. In Fig. 2a, the simulated data are not an
exponential function of the b-value. The curve fitted by
GDTI-2 clearly does not agree with the data. The corresponding RMS is 0.23. GDTI-1 produces an overall good
fit except for some b-values greater than 1500 s/mm2 and
the RMS is 0.030. In Fig. 2c, both GDTI methods fit the
simulated data well since the displacement distribution
along this direction is nearly gaussian. GDTI-1 has an
RMS of 0.021, while GDTI-2 has an RMS of 0.026, respectively.
Figure 2b and d plots the distributions of the displacement projected along the two diffusion directions, respectively. The computed histogram is marked with a cross,
and the fitted gaussian distribution is drawn using a solid
line. Figure 2b shows a non-gaussian distributed spin displacement, which is consistent with the nonmonoexponential signal behavior plotted in Fig. 2a. This signal behavior also explains the poor fitting produced by GDTI-2.
Figure 2d reveals a gaussian distributed random displacement, which is consistent with the exponential signal behavior plotted in Fig. 2c.
Figure 3 shows both the magnitude and the signal
phase simulated for Phantom 2. The diffusion direction
is (⫺0.25, 0.83, ⫺0.49). The curve fitted by GDTI-1 is
drawn in a solid line. Similar to Phantom 1, the magnitude data can be fit well by GDTI-1 (Fig. 3a). On the
contrary, the magnitude data cannot be fit well by
GDTI-2 as shown by the dotted line (Fig. 3a). Figure 3b
plots the phase data, which are also fit by GDTI-1. The
imperfection of the fit at larger b-values is caused by the
limited number of higher-order tensors used (only tensors up to the fifth order were used.). Figure 3c illustrates that the distribution of the random displacement
is non-gaussian and asymmetric.
Figure 4 compares the RMS of both GDTI fits over all 400
diffusion directions for Phantom 1. The mean RMS over all
the directions is 0.029 and 0.41 for GDTI-1 and GDTI-2,
respectively. For Phantom 2, the mean RMS is 0.031 and
0.39 for GDTI-1 and GDTI-2, respectively.
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Liu et al.
using GDTI-1 with and without considering the signal
phase. The histogram reveals the structure of the tube (Fig.
5b), whereas the shape of the diffusivity profile is inconsistent with the underlying phantom structure. The orientation of the maximum diffusivity does not agree with the
directions of the tube (Fig. 5c and d). Both skewness maps
reveal the same structure and are consistent with the phantom shape (Fig. 5e and f).
Figure 6 shows the same set of plots for Phantom 2 with
a voxel size of 80 ⫻ 80 ⫻ 80 ␮m3. In this case, the isosurface plot of the histogram does not completely resemble
the Y-shape phantom; small “knobs ” appear in the opposite direction of the extending tubes (Fig. 6b). ADC profiles
have symmetric shapes and differ from the geometry of the
phantom (Fig. 6c and d). The skewness map computed
considering signal phase resembles the structure of the
phantom, although some subtlety is lost (Fig. 6e). However, the skewness map reconstructed by using only the
magnitude of the signal completely fails to show any orientation information (Fig. 6f).
Larger Voxel Size
Figure 7 shows simulation results obtained for Phantom 2
with a voxel size of 500 ⫻ 500 ⫻ 500 ␮m3. Figure 7a plots
a cross section of the phantom in the x–y plane, and the
corresponding location of the voxel of interest. Fig. 7b
shows an isosurface of the reconstructed skewness map.
Note that Fig. 7b plots only the positive PDF difference.
Positive PDF difference illustrates the location where
probability increases.
DISCUSSION
We have shown that ADC-based models are mathematically self-inconsistent in the case of non-gaussian diffusion. The nonexponential signal behavior resulting from
non-gaussian diffusion cannot be sufficiently described by
ADC. Consequently, the non-gaussian diffusion process
existing in complex biologic tissues cannot be fully characterized by ADC-based models. In this case, higher-order
tensor statistics, namely higher-order cumulants of molecular random displacement, are required. Those higher order cumulants can be obtained by means of either GDTI-1
or q-space techniques.
FIG. 3. Phantom 2: one representative set of fitted curves by
GDTI-1 (as proposed by Liu et al.) and GDTI-2 (as proposed by
Özarslan et al.) and the corresponding displacement distribution.
The diffusion direction is (⫺0.25, 0.83, ⫺0.49). (a) The magnitude of
the signal does not follow exponential behavior. (b) The phase of the
signal is nonzero and is a smooth function of b-value. The imperfect
fit in GDTI-1 is caused by the limited number of higher order tensors
used. GDTI-2 only uses the magnitude of the signal; therefore, there
is no phase available from GDTI-2 fit. (c) The distribution of the spin
displacement is non-gaussian and asymmetric.
Figure 5 shows the 3D histogram of spin displacement
simulated for Phantom 1 with a voxel size of 80 ⫻ 80 ⫻
80 ␮m3, the ADC angular distribution, the diffusivity profile generated using GDTI-2, and skewness maps generated
Limitations of ADC-Based Models
The assumption of the existence of an ADC implies that
the underlying diffusion process has to be at least approximately gaussian. Since a gaussian distribution has only
six degrees of freedom, it is not necessary to include higher-order (greater than 2) terms in spherical harmonic decomposition, which requires more than six free variables,
or in the signal equation of GDTI-2. Those higher-order
terms, obtained from measured ADCs, are only the results
of poor data fitting and do not bear much physical meaning. The poor quality of the data fit has been demonstrated
here by Monte Carlo simulation. One may argue that with
smaller b-value, the fitting can become acceptable. However, calculations based on data with b ⬍1000 s/mm2 also
reveal poor fitting.
ADC and Non-Gaussian Diffusion
FIG. 4. The root mean square errors (RMS) between the fitted curves and simulated data over
400 diffusion directions for Phantom 1. The signal
intensity is normalized by S0. The mean RMS in all
the directions is 0.029 for GDTI-1. The mean RMS
over all the directions is 0.41 for GDTI-2, which is
about 14 times higher than that of GDTI-1.
FIG. 5. Comparison of results for Phantom 1. (a)
A cross section of the phantom in the x–y plane.
Dimensions are in micrometers. (b) Isosurface of
the 3D histogram of spin displacement. (c) ADC
angular distribution. (d) The diffusivity profile
computed using GDTI-2. The maximum diffusivity
does not correspond to the directions of the
tubes. (e) Skewness map computed with the
complex signal using GDTI-1. Only positive skewness is plotted. (f) Skewness map computed with
the magnitude of the signal using GDTI-1. The
two skewness maps are similar because the
phase of the signal is negligible. As expected, the
maximum skewness (computed by cumulants up
to fifth order) appears along the direction of the
tubes.
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Liu et al.
FIG. 6. Comparison of results for Phantom 2. (a)
A cross section of the phantom. Dimensions are
in micrometers. (b) Isosurface of the 3D histogram
of spin displacement. (c) ADC angular distribution. (d) The diffusivity profile computed using
GDTI-2. The maximum diffusivity does not correspond to the directions of the tubes and suggests
a completely wrong shape of the phantom. (e)
Skewness map computed with the complex signal using GDTI-1 reflects the underlying shape of
the phantom. Only positive skewness is plotted.
(f) Skewness map computed with the magnitude
of the signal using GDTI-1. With only the magnitude signal, the orientation information is completely lost. The phase is significant because of
the asymmetry of the phantom.
The main value of ADC-based diffusion models, such as
the spherical harmonic decomposition method and
GDTI-2, is providing a qualitative indication of fiber heterogeneity in white matter (18,31). Although ADC has no
physical significance in the case of non-gaussian diffusion,
it remains to be determined whether the geometric shape
FIG. 7. Results from Phantom 2 with voxel size
500 ⫻ 500 ⫻ 500 ␮m3. (a) A cross section of the
phantom. Dimensions are in micrometers. (b)
Skewness map computed with the complex signal
using GDTI-1 reflects the underlying shape of the
phantom. However, the result is less accurate because of the larger voxel size.
of an ADC profile corresponds exactly to the shape of the
underlying fiber structure or to the PDF of the displacement. So far, no systematic research has been conducted to
establish the existence or nonexistence of such a relationship. One difficulty in establishing such a theoretic framework is the fact that the measured ADC is not unique for
ADC and Non-Gaussian Diffusion
non-gaussian diffusion. The geometry of the ADC profile is
affected by the amount of diffusion-weighting applied (31)
and the physical dimension of the fiber. However, if such
a one-to-one correspondence indeed exists, then transformation rules can potentially be constructed to recover the
fiber structure.
Signal Equation for Non-Gaussian Diffusion
The limitation of an ADC-based model stems from the
assumption of an exponentially decaying signal. On the
other hand, q-space imaging and GDTI-1 do not assume
any specific signal behavior. In GDTI-1 specifically, the
Bloch–Torrey signal equation is expanded by a series of
higher order b-tensors (15,16), which is completely different from the GDTI-2 approach, in which the Bloch–Torrey
signal is expanded as a simple exponential function of the
. . . i in GDTI-1
b-value. The expansion coefficients D i(n)
1i 2
n
are tensors of increasing orders, which have well-defined
physical and statistical meanings. More specifically, the
coefficients are proportional to the higher-order cumulants
of the random displacement (15,16). In comparison, the
expansion coefficients ADC(␪, ␸) in GDTI-2 are, in fact,
ADCs measured in different diffusion directions.
Our simulations demonstrated that higher-order b-tensors and higher-order diffusion tensors are required to
characterize non-gaussian signal behavior (Fig. 2a and b).
For a given non-gaussian diffusion and in some specific
diffusion directions, the signal may decay exponentially as
a function of the b-value. This decay happens only when
the PDF projection of the random displacement is a gaussian function in those specific diffusion directions (Fig. 2c
and d). In this simulation, only tensors up to an order of 5
are estimated using the GDTI-1 method. The quality of
curve-fitting is good and can be improved by including
higher-order terms.
Imaging Asymmetric Structures
The issue of resolving multiple fiber orientations in regions where fibers cross or merge has become increasingly
important in fiber tractography (3). Currently, most diffusion imaging techniques, including techniques based on
ADC models and the current implementation of q-space
imaging, cannot resolve the orientations of asymmetric
microstructures. The GDTI-1 formalism as given in Eq [5]
reveals the fact that the asymmetric property of a microstructure is contained in the signal phase (16). By operating on the magnitude data, this information is lost and
cannot be recovered in ADC-based models. The q-space
principle can be extended to complex data and can potentially be used to image asymmetric microstructures. However, the current implementation of q-space imaging only
utilizes the magnitude data (12,32), which automatically
implies symmetric displacement profiles.
In principle, GDTI-1 is a physically correct formalism
with the capability of resolving the orientations of an
asymmetric structure. In actual experiments, however, the
accuracy of GDTI-1 is limited by signal-to-noise ratio and
the ability of MRI systems to accurately measure the signal
phase (15). As demonstrated in Figs. 5 and 6, the phase of
the signal is crucial in the reconstruction of an asymmetric
427
PDF. When the phantom is symmetric, the signal phase is
negligible; therefore, ignoring the phase does not affect the
reconstructed phantom structure (Fig. 5e and f). On the
other hand, when the phantom is asymmetric, ignoring the
signal phase results in a complete lost of the asymmetry
information (Fig. 6e and f). However, it is known that the
signal phase is difficult to measure accurately in an in vivo
diffusion experiment, since the diffusion-encoding gradient not only sensitizes the diffusion process but also any
bulk physiologic motion incurred during the encoding
period. The inaccurate signal phase problem can be alleviated by correcting for the phase error with a navigator
echo (33,34) and averaging the signal over multiple acquisitions. Without the correct phase information, GDTI-1
cannot fully identify the asymmetric structure existing in
the phantom. It should also be noted that in vivo q-space
imaging suffers from the same problem and, therefore,
operates on magnitude data as well. Overall, the preliminary simulation results presented in this research contribute to conveying an important insight and might be particularly important to better understand potential problems and limitations of diffusion MRI in the context of
non-gaussian diffusion.
Furthermore, to resolve an asymmetric microstructure,
intervoxel diffusion must occur between imaging voxels
during the diffusion encoding periods. This requirement
usually is satisfied in brain tissues since neural fibers are
interconnected. If the imaging voxel contains a structure
with closed boundaries, then the PDF of the spin displacement will always be an even function because of the conservation of mass; and in the long time limit the PDF is the
autocorrelation function of the underlying structure (20).
In this situation, the asymmetry property is completely
lost in the PDF. This lost of information is an intrinsic
disadvantage of MR diffusion measurements as well as a
common limitation of all the aforementioned methods.
The amount of phase accumulated through the intervoxel diffusion is affected by the number of spins that
experience this process. The more spins that remain
within the same voxel during the encoding periods; the
less phase can be accrued. Consequently, the signal phase
is less significant for larger voxels, which can potentially
compromise the ability of GDTI-1 to resolve asymmetric
fiber structures. The exact condition under which the effect of intervoxel diffusion is nonnegligible is beyond the
scope of this study. Nevertheless, based on limited simulation studies, GDTI-1 seems to be sensitive enough to
resolve asymmetric structures in an experimentally realizable voxel size as well (Fig. 7b).
CONCLUSION
In this study, we have shown that exponential signal behavior in a diffusion-weighted MR experiment is the result
of gaussian diffusion. ADC is only well defined for gaussian diffusion. However, ADC has no physical significance
when the underlying diffusion is non-gaussian. To be
mathematically consistent, only six independent variables
are required in an ADC-based model. Because of the complex signal behavior, models relying on the ADC computation cannot correctly describe non-gaussian diffusion.
Therefore, the application of ADC-based models should be
428
limited to gaussian diffusion. We have also seen that
GDTI-1 is capable of resolving the orientations of asymmetric microstructures by using both the magnitude and
the phase information of the signal.
ACKNOWLEDGMENTS
The authors appreciate the editorial assistance of David
Clayton, Ph.D.
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