J. Biomedical Science and Engineering, 2009, 2, 96-101
Published Online April 2009 in SciRes. http://www.scirp.org/journal/jbise JBiSE
The effect of different number of diffusion
gradients on SNR of diffusion tensor-derived
measurement maps
Na Zhang1, Zhen-Sheng Deng1*, Fang Wang1, Xiao-Yi Wang2
1Institute of Biomedical Engineering, School of Info-physics and Geomatics Engineering, Central South University, Changsha, Hunan, P.R. China
(410083) ; 2Department of Radiology, XiangYa Hospital of School of Medicine, Central South University, Changsha, Hunan, P.R. China
(410008);*Corresponding author: Zhensheng Deng ([e-mail] or [e-mail])
Received Dec. 12th, 2008; revised Feb. 12th, 2009; accepted Feb. 16th, 2009
ABSTRACT
Diffusion tensor imaging (DTI) is mainly applied
to white matter fiber tracking in human brain,
but there is still a debate on how many diffusion
gradient directions should be used to get the
best results. In this paper, the performance of 7
protocols corresponding to 6, 9, 12, 15, 20, 25,
and 30 noncollinear number of diffusion gradi-
ent directions (NDGD) were discussed by com-
paring signal-noise ratio (SNR) of tensor- de-
rived measurement maps and fractional ani-
sotropy (FA) values.
All DTI data (eight healthy volunteers) were
downloaded from the website of Johns Hopkins
Medical Institute Laboratory of Brain Anatomi-
cal MRI with permission. FA, apparent diffusion
constant mean (ADC-mean), the largest eigen-
value (LEV), and eigenvector orientation (EVO)
maps associated with LEV of all subjects were
calculated derived from tensor in the 7 proto-
cols via DTI Studio. A method to estimate the
variance was presented to calculate SNR of
these tensor-derived maps. Mean ± standard
deviation of the SNR and FA values within re-
gion of interest (ROI) selected in the white mat-
ter were compared among the 7 protocols.
The SNR were improved significantly with
NDGD increasing from 6 to 20 (P<0.05). From 20
to 30, SNR were improved significantly for LEV
and EVO maps (P<0.05), but no significant dif-
ferences for FA and ADC-mean maps (P>0.05).
There were no significant variances in FA val-
ues within ROI between any two protocols (P>
0.05).
The SNR could be improved with NDGD in-
creasing, but an optimum protocol is needed
because of clinical limitations.
Keywords: Diffusion Tensor Imaging, Diffusion
Gradient, Signal Noise Ratio, Estimating Vari-
ance
1. INTRODUCTION
Diffusion tensor imaging (DTI) has emerged as a nonin-
vasive magnetic resonance imaging (MRI) modality ca-
pable of providing in vivo fundamental information of
the white matter structure, which is required for viewing
structural connectivity in the human brain [1,2]. It is
commonly used to demonstrate subtle abnormalities in a
variety of diseases (including stroke, multiple sclerosis,
dyslexia, and schizophrenia) and is currently becoming
part of many routine clinical protocols [3]. The principle
of DTI is based on diffusion anisotropy of water mo-
lecular. By acquiring diffusion weighted (DW) images
with diffusion gradients oriented in at least six noncol-
linear directions (The tensor has 6 independent parame-
ters, that is why the minimal number of diffusion gradi-
ent directions (NDGD) for DTI measurement is 6), it is
possible to measure the diffusion tensor modeled by a 3
dimension (3D) ellipsoid in each voxel [4,5]. The ten-
sor-derived matrices, like diffusion anisotropy maps and
color-coded orientation maps, which could characterize
specific features of the diffusion process, can be calcu-
lated from tensor via DTI Studio [6]. DTI always oper-
ates under the assumption of a single ellipsoid. For esti-
mation of more complex geometries, high angular reso-
lution diffusion imaging (HARDI) or Q-Ball Imaging
needs to be used [7].
NDGD is one of the most important factors for DW
images acquisition. As NDGD increasing, more DW
images are used to calculate the diffusion tensor, result-
ing in more accurate tensor estimation but much longer
imaging time. Considering signal-noise ratio (SNR), if
the same amount of time is used to acquire DW images,
one 6-diffusion gradients is used, and the other
12-diffusion gradients, in the former, more images are
acquired in each direction, which results in more aver-
ages, but in the latter, less averages. So which is better
for single 3D ellipsoid estimation is still open to debate.
Some researchers [8,9] claimed that more than 6 dif-
fusion gradients can provide better measurements of the
tensor than the conventional 6-diffusion gradients. A
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N. Zhang et al. / J. Biomedical Science and Engineering 2 (2009) 96-101 97
SciRes Copyright © 2009 JBiSE
recent study with Monte Carlo simulations[10] con-
cluded that at least 20 NDGD were necessary for a ro-
bust estimation of diffusion anisotropy, whereas at least
30 NDGD were required for a robust estimation of tensor
orientation and mean diffusivity. Ni et al [11] found that
NDGD = 6 and number of excitations (NEX) = 10 were
sufficient for estimation of FA values from region of
interest (ROI) calculations. All these researchers men-
tioned above did their studies with the constraint of con-
stant imaging time. Previous work by Poonawalla [12]
concluded that when the acquisition time was held con-
stant, the sum of the diffusion tensor variances decreased
as NDGD increased, and signal averaging may not be as
effective as increasing NDGD, especially when NDGD
is small (e.g., NDGD < 13).
In this paper, the SNR of fractional anisotropy (FA)
maps, apparent diffusion constant mean (ADC-mean)
maps, the largest eigenvalue λ1 (LEV) maps, and eigen-
vector orientation (EVO) maps associated with λ1 de-
rived from diffusion tensor and the FA values calculated
from ROI were compared in 7 protocols corresponding
to different NDGD ( 6, 9, 12, 15, 20, 25, and 30 noncol-
linear). Unlike in previous work where the NEX varied
to keep imaging time constant, the number of images
averaged (NEX = 3) is fixed in this work so as to all the
7 protocols have the same original SNR. So the imaging
time for the 7 protocols was not held constant and the
higher NDGD protocols would be expected to perform
better given that the imaging time was greater.
The purpose of this work is to independently deter-
mine the effect of NDGD on SNR of these ten-
sor-derived measurement maps mentioned above with
the fixed NEX.
2. MATERIALS AND METHODS
All DTI data used in this paper were downloaded from
the website of Johns Hopkins Medical Institute Labora-
tory of Brain Anatomical MRI with permission.
2.1. Subjects
All images were acquired in eight healthy volunteers
(three females, five males; range, 21–29 years). The sub-
jects did not have any history of neurological diseases.
Institutional review board approval was obtained for the
study, and informed consent was obtained from all sub-
jects.
2.2. Data Acquisition
A 1.5T MR scanner (Gyroscan NT; Philips Medical
Systems, Best, the Netherlands) was used. DTI data
were acquired by using a single-shot echo-planar imag-
ing sequence with 7 protocols corresponding to different
NDGD (6, 9, 12, 15, 20, 25, and 30 noncollinear), and
the b value was 700 smm-2. The image matrix was 256 ×
256 pixels, with a field of view of 246 × 220 mm
(nominal resolution, 2.2 mm). Transverse sections of 2.2
mm thickness were acquired parallel to the anterior
commissure–posterior commissure line. A total of 55
sections covered the entire hemisphere and brainstem
without gaps. The acquisition time per dataset was ap-
proximately 6 minutes. All DW imaging were repeated 3
times, so they have the same original SNR. Five addi-
tional images for each slice with minimal DW (b_0=33
smm-2) were also acquired, and all 7 DTI acquisitions
have the same b_0 images.
2.3. Definitions of DTI Measurements
The ADC, which is used to characterize the water diffusion,
can be calculated from the following Equation (1) [6].
b
SS
ADC k
k
)/ln( 0
=, (k = 1, 2… K; K ≥ 6) (1)
where, the constant b is the diffusion-weighting factor,
S0 is the signal obtained without diffusion gradient, and
Sk is the signals corresponding to the different gradient
directions (k=1, 2…K; K ≥ 6). ADC-mean can be calcu-
lated by averaging the set of ADCk.
From the diffusion tensor, three eigenvalues, λ1>λ2>λ3,
which define the diffusion magnitude, can be determined
by diagonalizing the tensor for each voxel. Three eigen-
vectors (associated with three eigenvalues), which de-
scribe the diffusivity in the three directions, can be calcu-
lated. Based on these three diffusivities, the FA commonly
used for anisotropy definitions is calculated to yield val-
ues between 0 and 1 by the following Equation (2) [6].
(
)()()
2
3
2
2
2
1
2
3
2
2
2
1
2
3
λλλ
λλλ
++
−+−+−
=
DDD
FA
(2)
where，
3
321
λλλ ++
=D
2.4. SNR Calculation
SNR measures the roughness or granularity of diffusion
tensor-derived measurement maps, it should be equal to
the ratio of power spectrum of signal to that of noise.
But in general, spectrum analyze is not recommended to
estimate the SNR of magnetic resonance (MR) images
because it is actually re-created from frequency-signal or
k-space. In addition, spectrum analyze is good for ran-
dom signal (include random noise) analyze, this is not
the case for the measurements derived from tensor cal-
culation because the only resource of the random error
during this tensor calculation comes from the finite bit
length of the computers.
The method presented by Mouyan Zou [13] can be
used to estimate approximately SNR of an image, which
is the variance of signal divided by that of noise. Ac-
cording to the theory, local variance of all pixels of an
image should be calculated, the maximum of the local
variance which stands for the signal variance is divided
by the minimum which stands for the noise variance, and
the result (see Equation (3)) as the approximate SNR
should be amended by empirical formula.

98 N. Zhang et al. / J. Biomedical Science and Engineering 2 (2009) 96-101
SciRes Copyright © 2009 JBiSE
SNR= 2
min
2
max
σ
σ
(3)
where, σ2 is an estimated value of the local variance.
Since “local variance” affects the SNR measurement,
the local neighborhood included 10×10 pixels as an
kernel was used to calculate the local variance in this
study. The kernel of larger or smaller than 10×10 pixels
was not suggested because the former resulted in lower
SNR and the latter resulted in higher SNR.
The SNR calculated for the base images, which were
averaged thrice, is the same for all 7 protocols (SNR=
67.85) by using this method.
3. DATA PROCESSING
The DW images were transferred to a workstation and
processed by employing DTI Studio [6] developed for
diffusion tensor images calculating and fiber tracking.
For each DTI dataset, the six independent elements of
the 3×3 diffusion tensor were calculated for each voxel.
After diagonalization, three eigenvalues, λ1>λ2>λ3, and
three eigenvectors were calculated for each voxel, and in
turn, LEV maps and EVO maps associated with λ1 were
obtained. Then FA maps and ADC-mean maps were also
obtained by using Equations (2) and (1). EVO maps as-
sociated with λ1 were used as an indicator of fiber orien-
tation. On the EVO maps, red, green, and blue colors
were assigned to right-left, anterior-posterior, and supe-
rior- inferior orientations, respectively [14].
Based on MatLab platform, estimating variance as an
approach was used to provide a global estimate of SNR
for these tensor-derived measurement maps that charac-
terizes the uncertainty of the DTI measurements men-
tioned above in the 7 protocols corresponding to 6, 9, 12,
15, 20, 25 and 30 noncollinear NDGD with a 700
mm2/sec b value. To illustrate the effect of NDGD on FA
values, we also calculated FA values within ROI (about
30 pixels) selected in the white matter from FA maps in
Figure 1. An FA map corresponding to 30 noncollinear diffusion
gradient directions, with an ROI (about 30 pixels) in the white
matter marked with a circle.
Figure 2. Tensor-derived measurement maps. The first, second, third, and forth rows are FA maps, ADC-mean maps, LEV maps,
and color-coded maps for the eigenvectors associated with λ1, respectively, which corresponding to 6, 9, 12, 15, 20, 25, and 30
noncollinear NDGD arranged from left to right.

N. Zhang et al. / J. Biomedical Science and Engineering 2 (2009) 96-101 99
SciRes Copyright © 2009 JBiSE
all 7 protocols. An FA map corresponding to 30 noncol-
linear NDGD with an ROI was shown in Figure 1.
An unpaired T-test for the SNR of these ten-
sor-derived measurement maps and the FA values within
ROI in the 7 protocols was performed by using
SPSS11.5 software, a P value less than 0.05 for a meas-
urement was considered as statistically significant.
4. RESULTS
The tensor-derived measurement maps of one subject
acquired from tensor calculation in the 7 protocols were
shown in Figure 2. The improvement of the SNR with
the NDGD increasing could be observed visually. The
mean ± standard deviation for the SNR of tensor-derived
measurement maps and the FA values within ROI in the
7 protocols were listed in Table 1. The SNR of these
tensor-derived measurement maps varied with the
NDGD increasing were shown in Figure 3. The correla-
tions between FA values within ROI and NDGD were
fitted by linear lines and shown in Figure 4.
In order to predigest the results, the P values for the
SNR of tensor-derived measurement maps and FA values
within ROI in the 4 protocols corresponding to different
NDGD (6, 12, 20, and 30 noncollinear) instead of all the
7 protocols were listed in Table 2.
From the curves in Figure 3, which demonstrate the
SNR as a function of the NDGD for tensor-derived
measurement maps, we note that the SNR of the ten-
sor-derived measurement maps could be improved with
more NDGD. Also it is obvious that the more NDGD
were used, the higher SNR could be obtained from Fig-
ure 2. This is perfectly accordant with the findings of
D.K. Jones et al (i.e., when the NDGD is more than 6,
which happens frequently in practice in order to improve
the SNR and reduce the bias of tensor estimation) [15].
Both Figure 4 and all P values (>0.05) of FA values
within ROI listed in Table 2 demonstrate that there are
no significant variances in the FA values within ROI
between any 2 protocols with the NDGD increasing,
which are accordant with the conclusion of the previous
original research by Ni et al [11].
5. DISCUSSION
Ni et al [11] found that NDGD = 6 and NEX = 10 were
sufficient for estimation of FA from ROI calculations.
But in this paper, the FA maps shown in the first column
in Fig 2 look bad, it is considered that the NDGD is too
low (NDGD=6)，which results in low SNR for these
tensor-derived measurement maps. Because our results
were based on the same original SNR, i.e. all the DW
images used in this study were averaged thrice, so the
SNR of these tensor-derived measurement maps varied
in the 7 protocols only is dependent on the NDGD.
The functional dependence of the SNR of each ten-
sor-derived measurement on the NDGD is the most in-
teresting result for this paper. The curves in Figure 3 and
Figure 4 indicated that the SNR of FA maps, ADC-mean
maps, LEV maps, and EVO maps associated with λ1 were
improved with the NDGD increasing, but there were no
significant variances in the FA values within ROI among
these 7 protocols (refer to the last column in Table 1).
Table 1. Mean ± standard deviation for the SNR of tensor-derived measurement maps and the FA values within ROI in the 7 protocols.
*NDGD represents number of diffusion gradient directions.
NDGD* SNR of Tensor-derived Measurement FA values
FA Maps ADC-mean LEV Maps EVO Maps
6 35.78 ± 1.39 49.08 ± 3.39 34.39 ± 1.20 40.30 ± 0.53 0.79 ± 0.06
9 51.65 ± 1.43 60.96 ± 1.87 39.96 ± 1.54 51.08 ± 0.54 0.73 ± 0.08
12 61.62 ± 2.24 66.55 ± 1.64 44.09 ± 1.73 58.22 ± 0.23 0.75 ± 0.08
15 63.98 ± 1.67 68.67 ± 1.44 45.95 ± 1.38 59.65 ± 0.46 0.76 ± 0.08
20 69.39 ± 1.66 72.63 ± 1.29 48.22 ± 1.57 61.27 ± 0.43 0.74 ± 0.09
25 71.15 ± 2.46 73.28 ± 1.47 53.16 ± 1.26 64.28 ± 0.41 0.74 ± 0.08
30 72.45 ± 7.52 73.68 ± 1.77 57.44 ± 1.65 69.40 ± 0.49 0.74 ± 0.08
Table 2. P values for the SNR of tensor-derived measurement maps and the FA values within ROI in the 4 protocols corresponding to
different NDGD (6, 12, 20, and 30 noncollinear). *NDGD represents number of diffusion gradient directions.
NDGD* P values for the SNR of tensor-derived measurement maps and the FA values within ROI
FA values ADC-mean Maps LEV Maps EVO Maps FA values
6 vs 12 <<0.005 <<0.005 0.001 <<0.005 0.287
6 vs 20 <<0.005 <<0.005 <<0.005 <<0.005 0.177
6 vs 30 <<0.005 <<0.005 <<0.005 <<0.005 0.150
12 vs 20 0.010 0.011 0.046 0.050 0.725
12 vs 30 0.011 0.009 0.001 <<0.005 0.693
20 vs 30 0.089 0.376 0.006 0.011 0.977

100 N. Zhang et al. / J. Biomedical Science and Engineering 2 (2009) 96-101
SciRes Copyright © 2009 JBiSE
(a) (b)
(c) (d)
Figure 3. SNR of tensor-derived measurement maps vs NDGD.
Figure 4. FA values within ROI vs NDGD.
In the previous studies, with the constant imaging
time, some researchers investigated various protocols
(different NDGD) in terms of the variance of FA meas-
urements and demonstrated that a protocol employing 24
or 30 NDGD outperformed a protocol with only 6
NDGD [8, 9]. Papadakis NG et al considering three dif-
fusion anisotropy maps concluded that the minimum
NDGD required for robust anisotropy estimation was
between 18 and 21[16].
In our study, we mainly focused on the effect of dif-
ferent NDGD on the SNR of tensor-derived measure-
ment maps (FA maps, ADC-mean maps, LEV maps, and
EVO maps associated with λ1) in the 7 protocols with
fixed NEX=3. The curves in Figure 3 (a) and (b) show
that, for FA maps and ADC maps, there is a remarkable
and linear improvement in the SNR when the NDGD
increases from 6 to 12. Further improvement in the SNR
(albeit less remarkable) is observed when the NDGD is
further increased. Especially, there is no considerable
difference occurred in the SNR when the NDGD in-
creases from 20 to 30, which means that it has little con-
tribution to SNR of FA and ADC-mean maps when
NDGD is more than 20. P values (20 vs 30) = 0.089 for
FA maps and 0.376 for ADC maps in the Table 2 are
much more than 0.05, which also demonstrates that there
is no significant improvement in SNR by increasing the
NDGD from 20 to 30. This is consistent with the con-
clusion suggesting that 20-diffusion gradients be proba-
bly sufficient for in vivo human study of diffusion ani-
sotropy [10].
For LEV maps and EVO maps associated with λ1,
seen from the curve in Figure 3 (c) and (d), there is a
significant improvement in SNR by increasing the
NDGD from 6 to 12 and from 20 to 30. The curve in-
creases almost linearly when NDGD increases from 6 to
30 for SNR of LEV maps. P values (20 vs 30) = 0.006
for LEV maps and 0.011 for EVO maps associated with
λ1 in the Table 2, which are less than 0.05, also demon-
strate that the SNR of LEV maps and EVO maps associ-
ated with λ1 are significant different between 20 and 30
diffusion gradients. This suggests that the NDGD less
than 30 be not enough for tensor-orientation estimation,
which also is consistent with the results from Monte
Carlo study [10].
Therefore, the more NDGD are used, the higher SNR
would be obtained. But in clinical applications of DTI,
the total scanning time could not be too long because of
the artifacts caused by patients’ motion. So an optimum
NDGD for DTI data acquisition is needed due to both
the requirements and limitations mentioned above. There
would be a trade-off between the NDGD and clinical
limitations.

N. Zhang et al. / J. Biomedical Science and Engineering 2 (2009) 96-101 101
SciRes Copyright © 2009 JBiSE
6. CONCLUSION
7 different types of results derived from 6, 9, 12, 15, 20,
25, and 30 noncollinear NDGD, respectively, have been
compared in terms of SNR of tensor-derived measure-
ment maps and FA values within ROI based on Matlab
platform. The SNR of FA maps and ADC-mean maps
increased linearly as the NDGD increased from 6 to 20.
And the curves were almost level as the NDGD increas-
ing from 20 to 30. For SNR of LEV and EVO maps, the
curves were linearly direct ratio to the NDGD. FA values
within ROI were independent of NDGD. This study pro-
vides insight into the effect of NDGD on SNR and may
be useful in understanding the tradeoffs involved in DTI
acquisition design.
ACKNOWLEDGMENTS
We are grateful to Dr. Hangyi Jiang who works in Johns Hopkins Uni-
versity School of Medicine, Dr. Maolin Qiu who works in Yale Uni-
versity, and Dr. Bob L. Hou from Memorial Sloan Kettering Cancer
Center for their helpful discussions and encouraging comments during the
course of this study. In addition, we thank Dr. Hangyi Jiang again for
supplying the data and the software (DTI Studio) for our study. Finally, we
would like to thank the reviewers for their valuable remarks.
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