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Toward an optimal distribution of b values for intravoxel incoherent motion imaging
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Magnetic Resonance Imaging 29 (2011) 766 – 776
Toward an optimal distribution of b values for intravoxel incoherent motion imaging Andreas Lemke a,⁎, Bram Stieltjes b , Lothar R. Schad a , Frederik B. Laun c b
a Department of Computer Assisted Clinical Medicine, Heidelberg University, 68167 Mannheim, Germany Department of Radiology, Section Quantitative Imaging-Based Disease Characterization, German Cancer Research Center, 69120 Heidelberg, Germany c Department of Medical Physics in Radiology, German Cancer Research Center, 69120 Heidelberg, Germany Received 13 October 2010; revised 28 February 2011; accepted 7 March 2011
Abstract The intravoxel incoherent motion (IVIM) theory provides a framework for the separation of perfusion and diffusion effects in diffusionweighted imaging (DWI). To measure the three free IVIM parameters, DWIs with several diffusion weightings b must be acquired. To date, the used b value distributions are chosen heuristically and vary greatly among researchers. In this work, optimal b value distributions for the three parameter fit are determined using Monte-Carlo simulations for the measurement of a low, medium and high IVIM perfusion regime. The first 16 b values of a b value distribution, which was optimized to be appropriate for all three regimes, are {0, 40, 1000, 240, 10, 750, 90, 390, 170, 10, 620, 210, 100, 0, 530 and 970} in units of seconds per square meter. This distribution performed well for all organs and outperformed a distribution frequently used in the literature. In case of limited acquisition time, the b values should be chosen in the given order, but at least 10 b values should be used for current clinical settings. The overall parameter estimation quality depends strongly and nonlinearly on the signal-to-noise ratio (SNR): it is essential that the SNR is considerably higher than a critical SNR. This critical SNR is about 8 for medium and high IVIM perfusion and 50 for the low IVIM perfusion regime. Initial in vivo IVIM measurements were performed in the abdomen and were in keeping with the numerically simulated results. © 2011 Elsevier Inc. All rights reserved. Keywords: Intravoxel incoherent motion imaging; Diffusion-weighted imaging; b value; Monte-Carlo simulation
1. Introduction The intravoxel incoherent motion (IVIM) theory is an advanced method to separate diffusion and perfusion effects using diffusion-weighted imaging (DWI) [1]. The IVIM theory states that the incoherent blood flow in the capillaries causes a dephasing of the blood magnetization when diffusion gradients are applied. Le Bihan [1] described the resulting signal decay by S ð bÞ = ð1 − f Þ expð−bDÞ + f expð−b ð DT + DÞÞ; S ð 0Þ
ð1Þ
⁎ Corresponding author. Tel: +49 0621 383 5120; fax: +49 0621 383 5123. E-mail address: [email protected] (A. Lemke). 0730-725X/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.mri.2011.03.004
where f is the perfusion fraction and D the diffusion coefficient. D⁎ is the pseudodiffusion coefficient and describes the additional signal void due to incoherent blood flow in the capillaries. To extract these parameters, the signal must be measured with at least four different diffusion weightings b (including b=0 s/mm²). The question of optimal data collection and robust estimation of diffusion properties in magnetic resonance DWI in the absence of perfusion effects has been discussed for almost 20 years. For isotropic diffusion, Bito et al. [2] and later Xing et al. [3] reported that only two b values with a difference of (b2 − b1)=1.1/D should be used for the optimal assessment of the apparent diffusion coefficient (ADC). They found that the distribution of averages should be about 4:1, where 4 is the number of averages using the higher value b2. Concerning anisotropic diffusion and diffusion tensor imaging (DTI), the optimal distribution of diffusion directions and the choice of b values were discussed
A. Lemke et al. / Magnetic Resonance Imaging 29 (2011) 766–776
extensively [4–8]. One difference between the reported approaches was the specified criterion the optimization of the gradient directions was based upon, e.g., Skare et al. [5] used the condition number of the transformation as a criterion, whereas Jones et al. [7] maximized the separation between points constrained to the surface of a sphere. Moreover, the optimal number of gradient directions was investigated by several authors [9,10]. Concerning the optimal distribution of b values in DTI, Correia et al. [8] reported a decreased bias of fractional anisotropy (FA) and ADC when more than two b values are applied. The biexponential signal decay due to blood flow as described in the IVIM theory was not considered in any of the above mentioned publications, mainly due to the fact that this research focused on the brain. In this organ, the reported perfusion fractions are low (fb4% [11]), and thus, the IVIM effect is minimal. Recent investigations of the IVIM effect in the abdomen revealed much higher perfusion fractions (fN20%) for several organs like the pancreas [12], the liver [13], the prostate [14] and the kidney [15]. Note that the observed f values are higher than the vascular volume fractions measured with alternative techniques, since the signal fractions of the compartments are weighted by different relaxation decay rates [16]. The observed large perfusion fractions in IVIM experiments allowed the determination of IVIM parameters in several studies, which demonstrated that the obtained IVIM parameters were superior to the commonly used ADC in differentiating abdominal disease from healthy tissue [13,17,18]. However, the calculated perfusion-related IVIM maps usually suffer from low image quality and high parameter variance. Concerning the used acquisition parameters, the number and distribution of b values varied greatly [13,18,19]. A fundamental basis for this choice is currently not available but an optimal acquisition scheme could eventually have a comparably high impact on the image quality of the IVIM maps as the optimization in DTI had on the quality of FA maps and fiber tracking. Therefore, the aim of this work was to find an optimal distribution of b values for different diffusion and perfusion properties and to find a generalized solution for heterogeneous tissue. To this end, we used Monte-Carlo simulations where the error of the fit was used as optimization criterion to find the optimal distribution. Using the found optimal b value distributions, we describe the accuracy of the estimated quantitative parameters and show initial in vivo data that are in keeping with the simulation results. 2. Methods 2.1. Simulations All simulations were implemented in C++. The signal was generated according to the IVIM theory [Eq. (1)] using three different parameter sets of the perfusion fraction f, the diffusion coefficient D and the pseudodiffusion coefficient
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D⁎. To evaluate three different ranges, namely, low, medium and high IVIM perfusion, values for these ranges were obtained from the literature: low (brain): f=5%, D=1 μm2/ms, D⁎=10 μm2/ms [20]; medium (kidney): f=30%, D=1.5 μm2/ms, D⁎=15 μm2/ms [15]; high (liver): f=30%, D=1 μm2/ms, D⁎=60 μm2/ms [13]. For each b value, Gaussian noise was added to the complex signal to simulate a Rician distribution of the magnitude signal. The normalized signal [Eq. (1)] was then fitted to the noisy signal curve using the lmfit library, which is based on the Levenberg–Marquardt algorithm. Thus, there were three parameters (f, D and D⁎); the fit of S(0) is obsolete due to the normalization. Note that this procedure requires that the b value zero is used at least once. If b=0 s/mm² is acquired several times, the signal was normalized by dividing through the mean value of S(0). Initial values were f=10%, D=0.5 μm2/ms and D⁎=50 μm2/ms. The limits of the fitted parameters were 0% and 100% for f, 0 and 1000 μm2/ms for D, 0 and 1000 μm2/ms for D⁎. The complete process was repeated N=5000 times for each set of IVIM parameters and b values. The individual relative errors ffi1 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 N N 1X 1X 2 2 ð fi − f Þ C ð Di − DÞ C B B B N i=1 C B N i=1 C B C B C; ; rD = B rf = B C C f D @ A @ A
rDT
0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 N 1X 4 4 2 Di − D C B B N i=1 C B C =B C 4 D @ A
and the relative overall error σ=σf+σD+σD⁎ were calculated. Here, fi, Di and D⁎i are the fitted results of the ith repetition and f, D and D⁎ are the values of the selected parameter set. The values f, D and D⁎ were used to calculate the individual relative errors instead of the mean values ̅ f , ̅ D and ̅ D ⁎ in order to penalize strong deviations of the fitted values from the true values. 2.2. Optimal b value distribution for low, medium and high IVIM perfusion regimes Starting from an initial b value distribution {b=0, 40 and 1000 s/mm²}, the relative overall error σ of the distributions consisting of the initial three b values plus one additional b value in the range from 0 to 1000 s/mm² were tested in steps of 10 s/mm². The distribution with the minimal σ was then chosen as optimal new b value distribution. This process was iterated consecutively to obtain distributions with up to 100 b values and was performed for the three IVIM parameter sets, yielding {blow}, {bmedium} and {bhigh}. Since the signalto-noise ratio (SNR) scales with the square root of the 1 ffiffi measurement time, the fit quality per unit time σp , was n
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calculated in order to account for the measurement time, which is proportional to the number of b values n. 2.3. Calculation of an optimal b value distribution for heterogeneous tissue The exact values of the IVIM parameters in an in vivo experiment are often unknown a priori. To find a b value distribution with reliable results for such IVIM parameter sets, the optimization process was performed as described above, but the relative overall error was summed up for the three different IVIM parameter sets of low, medium and high yielding the distribution {bsum}. To avoid stronger weighting of one parameter set due to larger fitting errors, SNRs leading to similar fitting errors were used for the different parameter sets (SNR=200/30/20 for low, medium and high, respectively). 2.4. In Vivo IVIM measurements To verify the results obtained from simulations, abdominal IVIM maps of three healthy volunteers were calculated using DWIs acquired on a 1.5-T scanner (Magnetom Avanto; Siemens Healthcare, Erlangen, Germany). An 8channel body array coil and a 24-channel spine array coil were used as receiver coils and a spin echo echo-planar imaging (SE-EPI) sequence was used for the acquisition with the following parameters: repetition time=2100 ms, echo time=56 ms, field of view=300×273 mm², matrix size=120×100, 16 slices, slice thickness/gap=5/0.25 mm, partial Fourier factor=6/8, spectral fat saturation, one average and a bandwidth of 2604 Hz/pixel. Sixteen b values were used for the calculation of the IVIM maps, and the diffusion weighting was accomplished with a single refocused spin echo diffusion preparation and the trace diffusion mode. Each b value was acquired in a single breath-hold (TA=11 s) in expiration to avoid motion artifacts [21]. The total measurement time for a complete b value distribution was about 6 min, including free breathing recovery phases and breathing commands. The IVIM-derived parameter maps were calculated according to Eq. (1) using Matlab (MathWorks, Natick, MA), with initial and limiting values equal to
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3. Results The fit quality per unit time of the optimized b value distributions as a function of the number of b values n for the three parameter sets at different SNRs are shown in Fig. 1. The characteristics of the fit quality as a function of n strongly depend on the used SNR and can be classified into three cases. In the first case, the fit quality per unit time decreases with increased number of b values at SNRs below a critical value (left). In the second case (middle), for SNRs roughly equal to the critical value, the fit quality per unit time decreases with increasing number of b values if n is small. For larger n, this trend reverses. The critical SNR is significantly higher for the low parameter set compared with the medium and high IVIM perfusion parameter sets. In the third case (right), if an SNR above the critical value is available, the fit quality per unit time increases considerably with increasing number of b values until a maximum is reached. After this point, additional b values reduce the overall measurement error with the usual pffiffiffi n dependence, which would be expected for a linear fit. This maximum is reached at 26 b values using an SNR of 200 for the low IVIM perfusion, at 36 b values using an SNR of 30 for the medium IVIM perfusion parameters and at 25 b values using an SNR of 20 for the high IVIM perfusion parameter. The corresponding optimized b value distributions {blow}, {bmedium} and {bhigh} are listed in Table 1. The order of the values indicates the order of appearance during the optimization process. The maximum for the summed fit quality per unit time using all three parameter sets is reached at 35 b values. For the determination of {bsum}, the SNR was adapted for each considered IVIM perfusion regime (200, 30 and 20 for the low, medium and high, respectively). This b value distribution is also listed in Table 1.
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Fig. 1. Fit quality per unit time of the optimized b value distributions as a function of the number of b values n for the three IVIM perfusion parameter sets (low, A; medium, B; high, C) at different SNRs. The fit quality as a function of n increases only if an SNR above the critical value is used. This critical value is significantly higher for the low perfusion regime parameters than for the medium and high IVIM perfusion regimes.
A. Lemke et al. / Magnetic Resonance Imaging 29 (2011) 766–776 Table 1 List of b value distributions Distribution of b values (s/mm2) {blow}
0, 40, 1000, 260, 560, 190, 160, 40, 170, 560, 190, 980, 40, 150, 440, 700, 180, 0, 710, 860, 40, 580, 1000, 250, 0, 150 {bmedium} 0, 40, 1000, 160, 150, 40, 680, 150, 200, 940, 170, 990, 440, 740, 40, 230, 360, 0, 270, 70, 270, 870, 0, 40, 940, 60, 320, 240, 0, 260, 60, 1000, 920, 310, 1000, 50 {bhigh} 0, 40, 1000, 10, 130, 10, 70, 980, 190, 10, 740, 170, 200, 0, 830, 240, 80, 20, 1000, 680, 10, 190, 0, 170, 880 {bsum} 0, 40, 1000, 240, 10, 750, 90, 390, 170, 10, 620, 210, 100, 0, 530, 970, 350, 40, 990, 50, 30, 100, 970, 70, 390, 290, 120, 1000, 520, 0, 60, 260, 240, 10, 0 {blit} 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 150, 200, 400, 800, 1000
The histograms of the distribution of the optimal b values {blow}, {bmedium}, {bhigh} and {bsum} are shown in Fig. 2. Here, the optimization was repeated five times for n=100, and the frequencies were summed. The higher the frequency in the histograms for a certain range, the more b values in that range should be used for an optimal measurement of the IVIM parameters. For the abdominal parameter sets, the range of 0 to 50 s/mm2 is selected most frequently by the optimization algorithm, whereas for the low IVIM perfusion, the range of 50 to 100 s/mm2 is dominant. To minimize the summed error of the parameter sets, most of the used b values should be in the range from
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0 to 50 s/mm2, whereas few b values in the range of 450 to 800 s/mm2 should be used. To compare these optimized b value distributions with the previously reported distribution of 16 b values {blit} [13], the first 16 b values of {bsum}, {blow}, {bmedium} and {bhigh} were used (Table 1), and the relative overall error σ for each distribution at different SNRs was analyzed (see Fig. 3). The respective optimized b value distribution for the specific parameter sets (low, medium and high IVIM perfusion) yield lower relative overall errors than {bsum} in the majority of cases and an almost twofold reduced error for some SNR when compared with {blit}. For all parameter sets, the error is lower for the distribution {bsum} than for {blit}. Fig. 3 also illustrates that the SNR required to obtain acceptable fitting results with an overall error below 40% differ significantly between the low IVIM perfusion parameter set (SNRN300) and the medium (SNR=52.5) or high (SNR=32.5) IVIM perfusion parameter set. In the right column of Fig. 3, the contribution of the individual relative errors σD, σD⁎ and σf to the relative overall error σ using {bsum} for the three parameter sets is shown. The diffusion coefficient D can be fitted most precisely, whereas the pseudodiffusion coefficient D⁎ has the largest error. Fig. 4 compares the abdominal IVIM maps calculated with the b value distributions {bsum} and {blit} and illustrates the fit quality in vivo. The organs can be delineated clearly
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Fig. 2. Histograms of (A) {blow}, (B) {bmedium}, (C) {bhigh} and (D) {bsum}, showing the distribution of optimal b values for n=100 at an SNR above the critical value (SNR=200/30/20 for low, medium and high IVIM perfusion regime, respectively). To minimize the summed error of the three parameter sets more b values in the range of 0 to 50 s/mm2 should be measured. Sampling of the midrange (450 to 800 s/mm2) is less relevant.
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Fig. 3. Relative overall error σ as a function of the SNR. Comparison of σ using the optimized b value distributions with a b value distribution reported in the literature for the three parameter sets (A) low, (C) medium and (E) high IVIM perfusion. Using the respective optimized b value distribution for a specific parameter set gives the lowest error, whereas for some SNRs, the error using the distribution {blit} is more than two times higher than the error of {bsum} and {blow} for the low parameter set. For all parameter sets, the error using the distribution {bsum} is lower compared with {blit}. The contribution of the individual parameter error σD, σD⁎ and σf to the relative overall error σ using {bsum} for the three parameter sets are shown in the right column (B, D and F for the low, medium and high IVIM perfusion parameter set, respectively). The diffusion coefficient D can be fitted more precisely for every parameter set, whereas the pseudodiffusion coefficient D⁎ has the largest error.
on the D and f map using the {bsum} distribution, and the image quality is slightly improved compared with the maps using {blit}. For example, the pancreas is more heterogeneous on the f map using {blit}. The most prominent difference between the two distributions can be appreciated on the D⁎ map. Using the distribution {blit}, no organ can be properly delineated, whereas the liver, at least, is well defined on the D⁎ map using {bsum}. The measured SNR in the unweighted EPI images (b=0 s/mm²) was approximately 31 in the liver, 55 in the kidney, and 60 in the spleen. Fig. 5 shows typical signal decay and the corresponding IVIM fit for the two b value distributions {bsum} and {blit}
measured with a region of interest (ROI) in the left liver of volunteer 1 (slice shown in Fig. 4). The calculated IVIM parameters for this ROI were D=0.95 μm²/ms, f=16.3%, D⁎=71.3 μm²/ms using blit, and D=0.90 μm²/ms, f=19.9%, D⁎=43.1 μm²/ms using bsum. To quantify the observed qualitative difference between the b value distributions in Fig. 4, the IVIM parameters within several ROIs were compared with regard to their S.D. and the distribution of their values. Table 2 shows the mean values and S.D. of the two b value distributions {bsum} and {blit} obtained by fitting the average signal of several ROIs placed in liver, kidney and
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Fig. 4. Comparison of the abdominal IVIM maps calculated from the b value distributions {bsum} and {blit}: (A) and (B) are the unweighted EPI (b=0 s/mm²) slices of {bsum} and {blit}, respectively. The image quality of the maps of the diffusion coefficient D, the perfusion fraction f and the pseudodiffusion coefficient D⁎ calculated from the distribution {bsum} (C), (E), and (G) show an improved image quality compared with the corresponding maps calculated from {blit} (D, F, and H). In particular, the liver in the D⁎ map is more homogenous using {bsum}.
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from the two b value distributions {bsum} and {blit}, D⁎ also decreases in the liver when using {bsum} for single voxel and average signal fitting. Fig. 6 shows the distribution of the IVIM parameter values in the liver of volunteer 1 for {blit} and {bsum}. {blit} yields more outliers than {bsum}: some extremely large D values (up to 9.7 μm²/ms), considerably more f values larger than 50% and more D⁎ values close to the upper boundary of D⁎=1000 μm²/ms are present. In addition, more values of all three IVIM parameters at the lower boundary are observed.
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4. Discussion
Fig. 5. Signal as a function of the b value and the corresponding IVIM fit (solid line) for the two b values distributions {bsum} and {blit} obtained from a ROI in the left liver of volunteer 1.
Tissue perfusion and diffusion are important biological parameters that may be used for the detection of disease or treatment response [24]. Intravoxel incoherent motion imaging is a technique that enables the measurement of perfusion additional to diffusion, and recent scanner developments have enabled its application to the abdomen [25]. Several studies have shown encouraging results using the perfusion-related IVIM parameters considering liver cirrhosis [13], noninvasive assessment of acute ureteral obstruction [17] and pancreatic lesions [18]. However, to the best of our knowledge, this article presents the first optimization of b value distributions for IVIM experiments. Our study shows that the found b value distributions can substantially minimize the overall measurement error and can be seen as a first step toward an optimal choice of b value setup for IVIM imaging. The presented optimized distributions outperformed the b value distribution frequently used in the literature, especially for the low IVIM perfusion setup. Here, the relative overall error of {blit} was in some cases more than two times higher than the error of the optimized
spleen of three volunteers: For each volunteer, 12 ROIs were placed in the left liver, excluding ducts and large vessels; 10 ROIs were placed in the cortex and medulla each; and 9 ROIs were placed in the spleen. These ROIs contained 1845, 2376 and 2014 pixels in the liver; 589, 603 and 510 in the cortex; 563, 570 and 499 in the medulla; and 1500, 1713 and 1655 in the spleen for the three volunteers, respectively. Additionally, all these pixels were also fitted individually. The results are shown in Table 3. The S.D. using {bsum} are significantly decreased for all IVIM parameters compared with using {blit} in the case of the single voxel fitting and for some IVIM parameters in the case of fitting the average signal. The mean values of D and f are almost identical comparing single voxel fitting and fitting the average signal, whereas the mean value of the pseudodiffusion coefficient D⁎ decreases considerably in all organs when the average signal is fitted. Comparing the mean values of D⁎ calculated
Table 2 Intravoxel incoherent motion imaging parameters D, f and D⁎ obtained by fitting the average signal of several ROIs placed in liver, kidney and spleen of three volunteers D (μm²/ms)
Liver 1 Liver 2 Liver 3 P Medulla 1 Medulla 2 Medulla 3 Cortex 1 Cortex 2 Cortex 3 P Spleen 1 Spleen 2 Spleen 3 P
D⁎ (μm²/ms)
f (%)
{bsum}
{blit}
{bsum}
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{bsum}
{blit}
1.00±0.05 1.06±0.09 0.99±0.06 .035 1.48±0.26 1.30±0.21 1.29±0.23 1.56±0.20 1.44±0.21 1.39±0.16 .082 0.67±0.02 0.60±0.03 0.71±0.02 .185
1.01±0.07 1.09±0.12 1.05±0.10
18.4±3.5 21.7±4.1 22.1±4.2 .17 22.9±9.8 26.0±10.2 20.7±8.9 24.2±6.5 23.5±7.2 26.6±5.8 .064 5.6±1.4 6.0±1.9 5.2±1.1 .044
17.3±3.6 20.6±5.3 21.8±4.9
78±42 65±36 108±41 .044 38±36 19±33 21±18 43±24 29±17 26±19 .049 90±53 98±70 77±48 .036
97±68 82±49 133±52
1.31±0.27 1.35±0.29 1.28±0.36 1.80±0.20 1.43±0.24 1.51±0.21 0.63±0.03 0.65±0.05 0.68±0.08
27.4±13.5 24.9±10.0 21.0±11.2 22.7±6.9 24.1±7.5 26.3±8.2 5.1±2.0 5.8±2.4 6.4±2.1
30±45 24±40 32±34 35±19 31±29 20±25 143±127 102±110 101±94
Mean values and S.D. of these ROIs are stated. The two-tailed significance level P, obtained using a paired Student's t test, testing against differences of the S.D. of the distributions {bsum} and {blit}, is also stated.
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Table 3 Mean and S.D. of the IVIM parameters D, f and D⁎ of all pixels in several ROIs placed in the liver, kidney and spleen of three volunteers D (μm²/ms)
Liver 1 Liver 2 Liver 3 P Medulla 1 Medulla 2 Medulla 3 Cortex 1 Cortex 2 Cortex 3 P Spleen 1 Spleen 2 Spleen 3 P
D⁎ (μm²/ms)
f (%)
{bsum}
{blit}
{bsum}
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{bsum}
{blit}
0.92±0.25 0.96±0.37 0.99±0.29 .014 1.30±0.52 1.36±0.44 1.48±0.51 1.51±0.34 1.48±0.48 1.71±0.29 .040 0.50±0.20 0.49±0.21 0.66±0.22 .037
0.98±0.52 1.03±0.72 0.96±0.53
23.0±14.2 26.3±16.7 24.5±14.0 .034 31.4±29.6 28.6±22.3 31.1±28.0 27.4±23.3 21.2±24.0 30.0±20.9 .049 11.4±18.8 10.9±13.6 12.5±14.2 .005
21.4±17.3 26.4±21.6 25.1±20.2
89.2±151 106±169 143±112 .028 145±189 160±178 190±201 166±165 178±190 201±188 .019 142±283 156±259 132±186 .043
120±234 190±221 168±208
1.24±0.79 1.34±0.58 1.29±0.52 1.53±0.35 1.61±0.88 1.59±0.41 0.54±0.30 0.51±0.26 0.63±0.32
29.0±31.3 27.0±27.1 33.5±27.9 31.0±25.7 22.8±25.1 25.9±30.4 18.3±23.1 15.4±19.2 16.8±19.1
159±197 178±200 189±199 164±181 203±205 215±215 143±305 168±301 131±211
The two-tailed significance level P, obtained using a paired Student's t test, testing against differences of the S.D. of the distributions {bsum} and {blit}, is also stated.
distributions. The benefit for the medium and high IVIM perfusion parameter sets was less significant but present, even when compared with a distribution that utilizes numerous b values. The contribution of the three fitted IVIM parameters to the relative overall error differs considerably, shown exemplarily for the {bsum} distribution in Fig. 3. Here, the diffusion coefficient D can be estimated with an S.D. σD below 20% at moderate SNRs (N30) for high perfusion fractions as in the abdomen. If the perfusion fraction is as low, as in the brain, an SNR of more than 80 is required to enable an S.D. σD below 20%. The relative error of the diffusion coefficient σD has the smallest contribution to the overall error for all parameter sets. The relative error of the perfusion fraction σf strongly depends on the used parameter set. For the abdominal parameter sets (medium and high IVIM perfusion), the contribution to the relative overall error is comparable to σD, whereas for the low IVIM perfusion, σf is 20-fold higher than σD. The highest contribution to the overall error for all parameter sets has σD⁎, e.g., for the high IVIM perfusion regime, σD⁎ is three times higher than σD and σf. This is in concordance with our in vivo measurements, where the high variance of the fitted D ⁎ leads to heterogeneous, nonphysiological D⁎ maps and to a high S.D. in the quantitative analysis. It is unlikely that this high variance is based on physiological heterogeneity because most of the fitted pixels reach the fitting limits, as shown in Fig. 5. Furthermore, the pseudodiffusion coefficient decreases considerably when the averaged signal is fitted (see Table 2). Consequently, the image quality of the pseudodiffusion map improves considerably and the S.D. decreases significantly using the optimized distribution {bsum} when compared with {blit}. This decrease of the S.D. is also present for the perfusion fraction f and the diffusion
coefficient D. The f and D maps enable a good delineation of the abdominal organs using {bsum}. Thus, the in vivo measurements are in keeping with the results of the simulations that indicate that {bsum} outperforms {blit} at an SNR of 30. 4.1. Proceeding for clinical applications In order to plan an experiment using the IVIM technique, the available SNR and the approximate IVIM parameters should be evaluated first to estimate the critically required SNR. If the available SNR for the experimental setup is below the critical value, e.g., as for IVIM imaging of the brain in current clinical routine, it is not feasible to calculate the three IVIM parameters in a reliable fashion. One solution could be a “segmented” analysis where the three parameters D, f and D⁎ are calculated consecutively as recently described by Patel et al. [26]. Additionally, the IVIM properties of the selected tissue may be analyzed using signal intensities averages over a ROI. Compared with pixel by pixel analysis, the uncertainty is reduced (see Table 2 vs. Table 3), and a ROI approach combined with the optimized b value distributions presented here could lead to considerable improvement in accuracy even at SNRs below the critical value. If the SNR is much higher than the critical SNR, a more reliable fitting of all IVIM parameters can be achieved using the optimized b value distributions. Here, the question arises if the distribution {bsum} or the distribution optimized for a specific organ should be used. As the error of the fit using {bsum} was comparable to the error using the distribution optimized for a specific organ, we recommend the use of {bsum}, especially if the tissue of interest has heterogeneous diffusion and perfusion properties, e.g., in tumor tissue. Regarding the question of the
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optimal number of b values in clinical routine, one has to consider the scan time available for a patient. If the measurement time is no limiting factor, the maximal number of b values given in Table 1 should be used for the optimal assessment of the IVIM parameters. However, in most cases, the acquisition of 30 or more b values is not acceptable due to scan time restrictions. If the number of b values needs to be reduced, the optimal choice of b values for any given number of b values is given in Table 1. Here, the b values are sorted in the order of appearance during the optimization process. The fit quality per unit time improves most significantly when moving from 4 to 10 b values. Thus, at least 10 of the optimal distributed b
values in Table 1 should be measured for a high-quality IVIM experiment. Data fitting and analysis of multiexponential data has already been investigated by some groups, but the reported results cannot be transferred easily to IVIM fits because of differing assumptions. For instance, Bretthorst [27] used a Bayesian view to investigate the accuracy of exponentially fitted parameters and Anastasiou and Hall [28] used the Cramer–Rao theory to optimize T2 measurements in biexponential systems. Both assumed a uniform data sampling and Gaussian noise, whereas in our study, we focused on a more general nonuniform distribution of the acquired b values, which is easily applicable here, but not necessarily to T2
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measurements with Carr–Purcell–Meiboom–Gill echo trains. Istratov and Vyvenko [29] investigated the exponential analysis of physical phenomena with an arbitrary number of exponential components. Their manuscript was focused on mathematical limitations and on the comparison of several fitting algorithms. This differs from our study, which is dedicated to biexponential decays with a priori approximately known parameters and which focuses especially on the determination of optimized sampling points (b values). The optimization approach presented here could be used for optimizing the sampling distribution for any biexponential decaying data. Our current study has some limitations. First, the algorithm presented in this study consecutively adds optimized b values. Thus, it is not necessarily certain that the obtained distribution is optimal for any—in particular small—number of b values. A better optimization would require the consideration of any possible distribution of b values but would lead to extensive computation time. Our algorithm is a tradeoff between computation time and correctness of the optimized b value distribution. In preliminary simulations, we found that for the recommended minimum of 10 b values, the consecutive algorithm outperforms, for example, a “random” algorithm, which tries out many randomly generated b value distributions. Second, this study has focused on the imprecision and inaccuracy of parameter estimation from “noise-contaminated” simulated data. Systematic errors such as organ movement or partial volume effects have not been considered in the simulations and may further influence the optimal parameter selection. Furthermore, we assumed that no third compartment such as CSF is present. Notwithstanding the above-mentioned limitations, we could show that the optimized b value distribution {bsum} decreases the S.D. and increases the image quality of the abdominal in vivo IVIM maps in comparison to {blit} and thus underscore the value of the presented optimization. In conclusion, the optimal b value distributions found using simulations can be seen as solid basis to further investigate an optimal in vivo setting and should lead to an improved quality of IVIM data in future studies.
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[5] Skare S, Hedehus M, Moseley ME, Li TQ. Condition number as a measure of noise performance of diffusion tensor data acquisition schemes with MRI. J Magn Reson 2000;147(2):340–52. [6] Hasan KM, Parker DL, Alexander AL. Comparison of gradient encoding schemes for diffusion-tensor MRI. J Magn Reson Imaging 2001;13(5):769–80. [7] Jones DK, Horsfield MA, Simmons A. Optimal strategies for measuring diffusion in anisotropic systems by magnetic resonance imaging. Magn Reson Med 1999;42(3):515–25. [8] Correia MM, Carpenter TA, Williams GB. Looking for the optimal DTI acquisition scheme given a maximum scan time: are more b-values a waste of time? Magn Reson Imaging 2009;27(2): 163–75. [9] Jones DK. The effect of gradient sampling schemes on measures derived from diffusion tensor MRI: a Monte Carlo study. Magn Reson Med 2004;51(4):807–15. [10] Papadakis NG, Murrills CD, Hall LD, Huang CL, Adrian Carpenter T. Minimal gradient encoding for robust estimation of diffusion anisotropy. Magn Reson Imaging 2000;18(6):671–9. [11] Leenders KL, Perani D, Lammertsma AA, Heather JD, Buckingham P, Healy MJ, et al. Cerebral blood flow, blood volume and oxygen utilization. Normal values and effect of age. Brain 1990;113(Pt 1): 27–47. [12] Lee SS, Byun JH, Park BJ, Park SH, Kim N, Park B, et al. Quantitative analysis of diffusion-weighted magnetic resonance imaging of the pancreas: usefulness in characterizing solid pancreatic masses. J Magn Reson Imaging 2008;28(4):928–36. [13] Luciani A, Vignaud A, Cavet M, Nhieu JT, Mallat A, Ruel L, et al. Liver cirrhosis: intravoxel incoherent motion MR imaging—Pilot study. Radiology 2008;249(3):891–9. [14] Riches SF, Hawtin K, Charles-Edwards EM, de Souza NM. Diffusionweighted imaging of the prostate and rectal wall: comparison of biexponential and monoexponential modelled diffusion and associated perfusion coefficients. NMR Biomed 2009;22(3):318–25. [15] Zhang J, Sigmund E, Rusinek H, Chandarana H, Chen Q, Storey P, et al. Quantification of renal diffusion-weighted images using a biexponential model. Proceedings of the 17th Annual Meeting of ISMRM, Honolulu (USA); 2009. p. 4111. [16] Lemke A, Laun FB, Simon D, Stieltjes B, Schad LR. An in vivo verification of the intravoxel incoherent motion effect in diffusionweighted imaging of the abdomen. Magn Reson Med 2010;64(6): 1580–5. [17] Thoeny HC, Binser T, Roth B, Kessler TM, Vermathen P. Noninvasive assessment of acute ureteral obstruction with diffusion-weighted MR imaging: a prospective study. Radiology 2009;252(3):721–8. [18] Lemke A, Laun FB, Klau M, Re TJ, Simon D, Delorme S, et al. Differentiation of pancreas carcinoma from healthy pancreatic tissue using multiple b-values: comparison of apparent diffusion coefficient and intravoxel incoherent motion derived parameters. Invest Radiol 2009;44(12):769–75. [19] Yamada I, Aung W, Himeno Y, Nakagawa T, Shibuya H. Diffusion coefficients in abdominal organs and hepatic lesions: evaluation with intravoxel incoherent motion echo-planar MR imaging. Radiology 1999;210(3):617–23. [20] Pekar J, Moonen CT, van Zijl PC. On the precision of diffusion/ perfusion imaging by gradient sensitization. Magn Reson Med 1992;23 (1):122–9. [21] Naganawa S, Kawai H, Fukatsu H, Sakurai Y, Aoki I, Miura S, et al. Diffusion-weighted imaging of the liver: technical challenges and prospects for the future. Magn Reson Med Sci 2005;4(4): 175–86. [22] Murphy BW, Carson PL, Ellis JH, Zhang YT, Hyde RJ, Chenevert TL. Signal-to-noise measures for magnetic resonance imagers. Magnetic Reson Imaging 1993;11(3):425–8. [23] Dietrich O, Raya JG, Reeder SB, Reiser MF, Schoenberg SO. Measurement of signal-to-noise ratios in MR images: influence of
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multichannel coils, parallel imaging, and reconstruction filters. J Magn Reson Imaging 2007;26(2):375–85. [24] Provenzale JM, Mukundan S, Barboriak DP. Diffusion-weighted and perfusion MR imaging for brain tumor characterization and assessment of treatment response. Radiology 2006;239(3):632–49. [25] Le Bihan D. Intravoxel incoherent motion perfusion MR imaging: a wake-up call. Radiology 2008;249(3):748–52. [26] Patel J, Sigmund EE, Rusinek H, Oei M, Babb JS, Taouli B. Diagnosis of cirrhosis with intravoxel incoherent motion diffusion MRI and
dynamic contrast-enhanced MRI alone and in combination: preliminary experience. J Magn Reson Imaging 2010;31(3):589–600. [27] Bretthorst GL. How accurately can parameters from exponential models be estimated? A Bayesian view. Concepts Magn Reson 2005;27A:73–83. [28] Anastasiou A, Hall LD. Optimisation of T2 and M0 measurements of bi-exponential systems. Magn Reson Imaging 2004;22(1):67–80. [29] Istratov AA, Vyvenko OF. Exponential analysis in physical phenomena. Rev Sci Instrum 1999;70:1233–57.
Magnetic Resonance Imaging 29 (2011) 766 – 776
Toward an optimal distribution of b values for intravoxel incoherent motion imaging Andreas Lemke a,⁎, Bram Stieltjes b , Lothar R. Schad a , Frederik B. Laun c b
a Department of Computer Assisted Clinical Medicine, Heidelberg University, 68167 Mannheim, Germany Department of Radiology, Section Quantitative Imaging-Based Disease Characterization, German Cancer Research Center, 69120 Heidelberg, Germany c Department of Medical Physics in Radiology, German Cancer Research Center, 69120 Heidelberg, Germany Received 13 October 2010; revised 28 February 2011; accepted 7 March 2011
Abstract The intravoxel incoherent motion (IVIM) theory provides a framework for the separation of perfusion and diffusion effects in diffusionweighted imaging (DWI). To measure the three free IVIM parameters, DWIs with several diffusion weightings b must be acquired. To date, the used b value distributions are chosen heuristically and vary greatly among researchers. In this work, optimal b value distributions for the three parameter fit are determined using Monte-Carlo simulations for the measurement of a low, medium and high IVIM perfusion regime. The first 16 b values of a b value distribution, which was optimized to be appropriate for all three regimes, are {0, 40, 1000, 240, 10, 750, 90, 390, 170, 10, 620, 210, 100, 0, 530 and 970} in units of seconds per square meter. This distribution performed well for all organs and outperformed a distribution frequently used in the literature. In case of limited acquisition time, the b values should be chosen in the given order, but at least 10 b values should be used for current clinical settings. The overall parameter estimation quality depends strongly and nonlinearly on the signal-to-noise ratio (SNR): it is essential that the SNR is considerably higher than a critical SNR. This critical SNR is about 8 for medium and high IVIM perfusion and 50 for the low IVIM perfusion regime. Initial in vivo IVIM measurements were performed in the abdomen and were in keeping with the numerically simulated results. © 2011 Elsevier Inc. All rights reserved. Keywords: Intravoxel incoherent motion imaging; Diffusion-weighted imaging; b value; Monte-Carlo simulation
1. Introduction The intravoxel incoherent motion (IVIM) theory is an advanced method to separate diffusion and perfusion effects using diffusion-weighted imaging (DWI) [1]. The IVIM theory states that the incoherent blood flow in the capillaries causes a dephasing of the blood magnetization when diffusion gradients are applied. Le Bihan [1] described the resulting signal decay by S ð bÞ = ð1 − f Þ expð−bDÞ + f expð−b ð DT + DÞÞ; S ð 0Þ
ð1Þ
⁎ Corresponding author. Tel: +49 0621 383 5120; fax: +49 0621 383 5123. E-mail address: [email protected] (A. Lemke). 0730-725X/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.mri.2011.03.004
where f is the perfusion fraction and D the diffusion coefficient. D⁎ is the pseudodiffusion coefficient and describes the additional signal void due to incoherent blood flow in the capillaries. To extract these parameters, the signal must be measured with at least four different diffusion weightings b (including b=0 s/mm²). The question of optimal data collection and robust estimation of diffusion properties in magnetic resonance DWI in the absence of perfusion effects has been discussed for almost 20 years. For isotropic diffusion, Bito et al. [2] and later Xing et al. [3] reported that only two b values with a difference of (b2 − b1)=1.1/D should be used for the optimal assessment of the apparent diffusion coefficient (ADC). They found that the distribution of averages should be about 4:1, where 4 is the number of averages using the higher value b2. Concerning anisotropic diffusion and diffusion tensor imaging (DTI), the optimal distribution of diffusion directions and the choice of b values were discussed
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extensively [4–8]. One difference between the reported approaches was the specified criterion the optimization of the gradient directions was based upon, e.g., Skare et al. [5] used the condition number of the transformation as a criterion, whereas Jones et al. [7] maximized the separation between points constrained to the surface of a sphere. Moreover, the optimal number of gradient directions was investigated by several authors [9,10]. Concerning the optimal distribution of b values in DTI, Correia et al. [8] reported a decreased bias of fractional anisotropy (FA) and ADC when more than two b values are applied. The biexponential signal decay due to blood flow as described in the IVIM theory was not considered in any of the above mentioned publications, mainly due to the fact that this research focused on the brain. In this organ, the reported perfusion fractions are low (fb4% [11]), and thus, the IVIM effect is minimal. Recent investigations of the IVIM effect in the abdomen revealed much higher perfusion fractions (fN20%) for several organs like the pancreas [12], the liver [13], the prostate [14] and the kidney [15]. Note that the observed f values are higher than the vascular volume fractions measured with alternative techniques, since the signal fractions of the compartments are weighted by different relaxation decay rates [16]. The observed large perfusion fractions in IVIM experiments allowed the determination of IVIM parameters in several studies, which demonstrated that the obtained IVIM parameters were superior to the commonly used ADC in differentiating abdominal disease from healthy tissue [13,17,18]. However, the calculated perfusion-related IVIM maps usually suffer from low image quality and high parameter variance. Concerning the used acquisition parameters, the number and distribution of b values varied greatly [13,18,19]. A fundamental basis for this choice is currently not available but an optimal acquisition scheme could eventually have a comparably high impact on the image quality of the IVIM maps as the optimization in DTI had on the quality of FA maps and fiber tracking. Therefore, the aim of this work was to find an optimal distribution of b values for different diffusion and perfusion properties and to find a generalized solution for heterogeneous tissue. To this end, we used Monte-Carlo simulations where the error of the fit was used as optimization criterion to find the optimal distribution. Using the found optimal b value distributions, we describe the accuracy of the estimated quantitative parameters and show initial in vivo data that are in keeping with the simulation results. 2. Methods 2.1. Simulations All simulations were implemented in C++. The signal was generated according to the IVIM theory [Eq. (1)] using three different parameter sets of the perfusion fraction f, the diffusion coefficient D and the pseudodiffusion coefficient
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D⁎. To evaluate three different ranges, namely, low, medium and high IVIM perfusion, values for these ranges were obtained from the literature: low (brain): f=5%, D=1 μm2/ms, D⁎=10 μm2/ms [20]; medium (kidney): f=30%, D=1.5 μm2/ms, D⁎=15 μm2/ms [15]; high (liver): f=30%, D=1 μm2/ms, D⁎=60 μm2/ms [13]. For each b value, Gaussian noise was added to the complex signal to simulate a Rician distribution of the magnitude signal. The normalized signal [Eq. (1)] was then fitted to the noisy signal curve using the lmfit library, which is based on the Levenberg–Marquardt algorithm. Thus, there were three parameters (f, D and D⁎); the fit of S(0) is obsolete due to the normalization. Note that this procedure requires that the b value zero is used at least once. If b=0 s/mm² is acquired several times, the signal was normalized by dividing through the mean value of S(0). Initial values were f=10%, D=0.5 μm2/ms and D⁎=50 μm2/ms. The limits of the fitted parameters were 0% and 100% for f, 0 and 1000 μm2/ms for D, 0 and 1000 μm2/ms for D⁎. The complete process was repeated N=5000 times for each set of IVIM parameters and b values. The individual relative errors ffi1 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 N N 1X 1X 2 2 ð fi − f Þ C ð Di − DÞ C B B B N i=1 C B N i=1 C B C B C; ; rD = B rf = B C C f D @ A @ A
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0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 N 1X 4 4 2 Di − D C B B N i=1 C B C =B C 4 D @ A
and the relative overall error σ=σf+σD+σD⁎ were calculated. Here, fi, Di and D⁎i are the fitted results of the ith repetition and f, D and D⁎ are the values of the selected parameter set. The values f, D and D⁎ were used to calculate the individual relative errors instead of the mean values ̅ f , ̅ D and ̅ D ⁎ in order to penalize strong deviations of the fitted values from the true values. 2.2. Optimal b value distribution for low, medium and high IVIM perfusion regimes Starting from an initial b value distribution {b=0, 40 and 1000 s/mm²}, the relative overall error σ of the distributions consisting of the initial three b values plus one additional b value in the range from 0 to 1000 s/mm² were tested in steps of 10 s/mm². The distribution with the minimal σ was then chosen as optimal new b value distribution. This process was iterated consecutively to obtain distributions with up to 100 b values and was performed for the three IVIM parameter sets, yielding {blow}, {bmedium} and {bhigh}. Since the signalto-noise ratio (SNR) scales with the square root of the 1 ffiffi measurement time, the fit quality per unit time σp , was n
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calculated in order to account for the measurement time, which is proportional to the number of b values n. 2.3. Calculation of an optimal b value distribution for heterogeneous tissue The exact values of the IVIM parameters in an in vivo experiment are often unknown a priori. To find a b value distribution with reliable results for such IVIM parameter sets, the optimization process was performed as described above, but the relative overall error was summed up for the three different IVIM parameter sets of low, medium and high yielding the distribution {bsum}. To avoid stronger weighting of one parameter set due to larger fitting errors, SNRs leading to similar fitting errors were used for the different parameter sets (SNR=200/30/20 for low, medium and high, respectively). 2.4. In Vivo IVIM measurements To verify the results obtained from simulations, abdominal IVIM maps of three healthy volunteers were calculated using DWIs acquired on a 1.5-T scanner (Magnetom Avanto; Siemens Healthcare, Erlangen, Germany). An 8channel body array coil and a 24-channel spine array coil were used as receiver coils and a spin echo echo-planar imaging (SE-EPI) sequence was used for the acquisition with the following parameters: repetition time=2100 ms, echo time=56 ms, field of view=300×273 mm², matrix size=120×100, 16 slices, slice thickness/gap=5/0.25 mm, partial Fourier factor=6/8, spectral fat saturation, one average and a bandwidth of 2604 Hz/pixel. Sixteen b values were used for the calculation of the IVIM maps, and the diffusion weighting was accomplished with a single refocused spin echo diffusion preparation and the trace diffusion mode. Each b value was acquired in a single breath-hold (TA=11 s) in expiration to avoid motion artifacts [21]. The total measurement time for a complete b value distribution was about 6 min, including free breathing recovery phases and breathing commands. The IVIM-derived parameter maps were calculated according to Eq. (1) using Matlab (MathWorks, Natick, MA), with initial and limiting values equal to
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3. Results The fit quality per unit time of the optimized b value distributions as a function of the number of b values n for the three parameter sets at different SNRs are shown in Fig. 1. The characteristics of the fit quality as a function of n strongly depend on the used SNR and can be classified into three cases. In the first case, the fit quality per unit time decreases with increased number of b values at SNRs below a critical value (left). In the second case (middle), for SNRs roughly equal to the critical value, the fit quality per unit time decreases with increasing number of b values if n is small. For larger n, this trend reverses. The critical SNR is significantly higher for the low parameter set compared with the medium and high IVIM perfusion parameter sets. In the third case (right), if an SNR above the critical value is available, the fit quality per unit time increases considerably with increasing number of b values until a maximum is reached. After this point, additional b values reduce the overall measurement error with the usual pffiffiffi n dependence, which would be expected for a linear fit. This maximum is reached at 26 b values using an SNR of 200 for the low IVIM perfusion, at 36 b values using an SNR of 30 for the medium IVIM perfusion parameters and at 25 b values using an SNR of 20 for the high IVIM perfusion parameter. The corresponding optimized b value distributions {blow}, {bmedium} and {bhigh} are listed in Table 1. The order of the values indicates the order of appearance during the optimization process. The maximum for the summed fit quality per unit time using all three parameter sets is reached at 35 b values. For the determination of {bsum}, the SNR was adapted for each considered IVIM perfusion regime (200, 30 and 20 for the low, medium and high, respectively). This b value distribution is also listed in Table 1.
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A. Lemke et al. / Magnetic Resonance Imaging 29 (2011) 766–776 Table 1 List of b value distributions Distribution of b values (s/mm2) {blow}
0, 40, 1000, 260, 560, 190, 160, 40, 170, 560, 190, 980, 40, 150, 440, 700, 180, 0, 710, 860, 40, 580, 1000, 250, 0, 150 {bmedium} 0, 40, 1000, 160, 150, 40, 680, 150, 200, 940, 170, 990, 440, 740, 40, 230, 360, 0, 270, 70, 270, 870, 0, 40, 940, 60, 320, 240, 0, 260, 60, 1000, 920, 310, 1000, 50 {bhigh} 0, 40, 1000, 10, 130, 10, 70, 980, 190, 10, 740, 170, 200, 0, 830, 240, 80, 20, 1000, 680, 10, 190, 0, 170, 880 {bsum} 0, 40, 1000, 240, 10, 750, 90, 390, 170, 10, 620, 210, 100, 0, 530, 970, 350, 40, 990, 50, 30, 100, 970, 70, 390, 290, 120, 1000, 520, 0, 60, 260, 240, 10, 0 {blit} 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 150, 200, 400, 800, 1000
The histograms of the distribution of the optimal b values {blow}, {bmedium}, {bhigh} and {bsum} are shown in Fig. 2. Here, the optimization was repeated five times for n=100, and the frequencies were summed. The higher the frequency in the histograms for a certain range, the more b values in that range should be used for an optimal measurement of the IVIM parameters. For the abdominal parameter sets, the range of 0 to 50 s/mm2 is selected most frequently by the optimization algorithm, whereas for the low IVIM perfusion, the range of 50 to 100 s/mm2 is dominant. To minimize the summed error of the parameter sets, most of the used b values should be in the range from
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0 to 50 s/mm2, whereas few b values in the range of 450 to 800 s/mm2 should be used. To compare these optimized b value distributions with the previously reported distribution of 16 b values {blit} [13], the first 16 b values of {bsum}, {blow}, {bmedium} and {bhigh} were used (Table 1), and the relative overall error σ for each distribution at different SNRs was analyzed (see Fig. 3). The respective optimized b value distribution for the specific parameter sets (low, medium and high IVIM perfusion) yield lower relative overall errors than {bsum} in the majority of cases and an almost twofold reduced error for some SNR when compared with {blit}. For all parameter sets, the error is lower for the distribution {bsum} than for {blit}. Fig. 3 also illustrates that the SNR required to obtain acceptable fitting results with an overall error below 40% differ significantly between the low IVIM perfusion parameter set (SNRN300) and the medium (SNR=52.5) or high (SNR=32.5) IVIM perfusion parameter set. In the right column of Fig. 3, the contribution of the individual relative errors σD, σD⁎ and σf to the relative overall error σ using {bsum} for the three parameter sets is shown. The diffusion coefficient D can be fitted most precisely, whereas the pseudodiffusion coefficient D⁎ has the largest error. Fig. 4 compares the abdominal IVIM maps calculated with the b value distributions {bsum} and {blit} and illustrates the fit quality in vivo. The organs can be delineated clearly
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Fig. 3. Relative overall error σ as a function of the SNR. Comparison of σ using the optimized b value distributions with a b value distribution reported in the literature for the three parameter sets (A) low, (C) medium and (E) high IVIM perfusion. Using the respective optimized b value distribution for a specific parameter set gives the lowest error, whereas for some SNRs, the error using the distribution {blit} is more than two times higher than the error of {bsum} and {blow} for the low parameter set. For all parameter sets, the error using the distribution {bsum} is lower compared with {blit}. The contribution of the individual parameter error σD, σD⁎ and σf to the relative overall error σ using {bsum} for the three parameter sets are shown in the right column (B, D and F for the low, medium and high IVIM perfusion parameter set, respectively). The diffusion coefficient D can be fitted more precisely for every parameter set, whereas the pseudodiffusion coefficient D⁎ has the largest error.
on the D and f map using the {bsum} distribution, and the image quality is slightly improved compared with the maps using {blit}. For example, the pancreas is more heterogeneous on the f map using {blit}. The most prominent difference between the two distributions can be appreciated on the D⁎ map. Using the distribution {blit}, no organ can be properly delineated, whereas the liver, at least, is well defined on the D⁎ map using {bsum}. The measured SNR in the unweighted EPI images (b=0 s/mm²) was approximately 31 in the liver, 55 in the kidney, and 60 in the spleen. Fig. 5 shows typical signal decay and the corresponding IVIM fit for the two b value distributions {bsum} and {blit}
measured with a region of interest (ROI) in the left liver of volunteer 1 (slice shown in Fig. 4). The calculated IVIM parameters for this ROI were D=0.95 μm²/ms, f=16.3%, D⁎=71.3 μm²/ms using blit, and D=0.90 μm²/ms, f=19.9%, D⁎=43.1 μm²/ms using bsum. To quantify the observed qualitative difference between the b value distributions in Fig. 4, the IVIM parameters within several ROIs were compared with regard to their S.D. and the distribution of their values. Table 2 shows the mean values and S.D. of the two b value distributions {bsum} and {blit} obtained by fitting the average signal of several ROIs placed in liver, kidney and
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Fig. 4. Comparison of the abdominal IVIM maps calculated from the b value distributions {bsum} and {blit}: (A) and (B) are the unweighted EPI (b=0 s/mm²) slices of {bsum} and {blit}, respectively. The image quality of the maps of the diffusion coefficient D, the perfusion fraction f and the pseudodiffusion coefficient D⁎ calculated from the distribution {bsum} (C), (E), and (G) show an improved image quality compared with the corresponding maps calculated from {blit} (D, F, and H). In particular, the liver in the D⁎ map is more homogenous using {bsum}.
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from the two b value distributions {bsum} and {blit}, D⁎ also decreases in the liver when using {bsum} for single voxel and average signal fitting. Fig. 6 shows the distribution of the IVIM parameter values in the liver of volunteer 1 for {blit} and {bsum}. {blit} yields more outliers than {bsum}: some extremely large D values (up to 9.7 μm²/ms), considerably more f values larger than 50% and more D⁎ values close to the upper boundary of D⁎=1000 μm²/ms are present. In addition, more values of all three IVIM parameters at the lower boundary are observed.
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Fig. 5. Signal as a function of the b value and the corresponding IVIM fit (solid line) for the two b values distributions {bsum} and {blit} obtained from a ROI in the left liver of volunteer 1.
Tissue perfusion and diffusion are important biological parameters that may be used for the detection of disease or treatment response [24]. Intravoxel incoherent motion imaging is a technique that enables the measurement of perfusion additional to diffusion, and recent scanner developments have enabled its application to the abdomen [25]. Several studies have shown encouraging results using the perfusion-related IVIM parameters considering liver cirrhosis [13], noninvasive assessment of acute ureteral obstruction [17] and pancreatic lesions [18]. However, to the best of our knowledge, this article presents the first optimization of b value distributions for IVIM experiments. Our study shows that the found b value distributions can substantially minimize the overall measurement error and can be seen as a first step toward an optimal choice of b value setup for IVIM imaging. The presented optimized distributions outperformed the b value distribution frequently used in the literature, especially for the low IVIM perfusion setup. Here, the relative overall error of {blit} was in some cases more than two times higher than the error of the optimized
spleen of three volunteers: For each volunteer, 12 ROIs were placed in the left liver, excluding ducts and large vessels; 10 ROIs were placed in the cortex and medulla each; and 9 ROIs were placed in the spleen. These ROIs contained 1845, 2376 and 2014 pixels in the liver; 589, 603 and 510 in the cortex; 563, 570 and 499 in the medulla; and 1500, 1713 and 1655 in the spleen for the three volunteers, respectively. Additionally, all these pixels were also fitted individually. The results are shown in Table 3. The S.D. using {bsum} are significantly decreased for all IVIM parameters compared with using {blit} in the case of the single voxel fitting and for some IVIM parameters in the case of fitting the average signal. The mean values of D and f are almost identical comparing single voxel fitting and fitting the average signal, whereas the mean value of the pseudodiffusion coefficient D⁎ decreases considerably in all organs when the average signal is fitted. Comparing the mean values of D⁎ calculated
Table 2 Intravoxel incoherent motion imaging parameters D, f and D⁎ obtained by fitting the average signal of several ROIs placed in liver, kidney and spleen of three volunteers D (μm²/ms)
Liver 1 Liver 2 Liver 3 P Medulla 1 Medulla 2 Medulla 3 Cortex 1 Cortex 2 Cortex 3 P Spleen 1 Spleen 2 Spleen 3 P
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1.00±0.05 1.06±0.09 0.99±0.06 .035 1.48±0.26 1.30±0.21 1.29±0.23 1.56±0.20 1.44±0.21 1.39±0.16 .082 0.67±0.02 0.60±0.03 0.71±0.02 .185
1.01±0.07 1.09±0.12 1.05±0.10
18.4±3.5 21.7±4.1 22.1±4.2 .17 22.9±9.8 26.0±10.2 20.7±8.9 24.2±6.5 23.5±7.2 26.6±5.8 .064 5.6±1.4 6.0±1.9 5.2±1.1 .044
17.3±3.6 20.6±5.3 21.8±4.9
78±42 65±36 108±41 .044 38±36 19±33 21±18 43±24 29±17 26±19 .049 90±53 98±70 77±48 .036
97±68 82±49 133±52
1.31±0.27 1.35±0.29 1.28±0.36 1.80±0.20 1.43±0.24 1.51±0.21 0.63±0.03 0.65±0.05 0.68±0.08
27.4±13.5 24.9±10.0 21.0±11.2 22.7±6.9 24.1±7.5 26.3±8.2 5.1±2.0 5.8±2.4 6.4±2.1
30±45 24±40 32±34 35±19 31±29 20±25 143±127 102±110 101±94
Mean values and S.D. of these ROIs are stated. The two-tailed significance level P, obtained using a paired Student's t test, testing against differences of the S.D. of the distributions {bsum} and {blit}, is also stated.
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Table 3 Mean and S.D. of the IVIM parameters D, f and D⁎ of all pixels in several ROIs placed in the liver, kidney and spleen of three volunteers D (μm²/ms)
Liver 1 Liver 2 Liver 3 P Medulla 1 Medulla 2 Medulla 3 Cortex 1 Cortex 2 Cortex 3 P Spleen 1 Spleen 2 Spleen 3 P
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0.92±0.25 0.96±0.37 0.99±0.29 .014 1.30±0.52 1.36±0.44 1.48±0.51 1.51±0.34 1.48±0.48 1.71±0.29 .040 0.50±0.20 0.49±0.21 0.66±0.22 .037
0.98±0.52 1.03±0.72 0.96±0.53
23.0±14.2 26.3±16.7 24.5±14.0 .034 31.4±29.6 28.6±22.3 31.1±28.0 27.4±23.3 21.2±24.0 30.0±20.9 .049 11.4±18.8 10.9±13.6 12.5±14.2 .005
21.4±17.3 26.4±21.6 25.1±20.2
89.2±151 106±169 143±112 .028 145±189 160±178 190±201 166±165 178±190 201±188 .019 142±283 156±259 132±186 .043
120±234 190±221 168±208
1.24±0.79 1.34±0.58 1.29±0.52 1.53±0.35 1.61±0.88 1.59±0.41 0.54±0.30 0.51±0.26 0.63±0.32
29.0±31.3 27.0±27.1 33.5±27.9 31.0±25.7 22.8±25.1 25.9±30.4 18.3±23.1 15.4±19.2 16.8±19.1
159±197 178±200 189±199 164±181 203±205 215±215 143±305 168±301 131±211
The two-tailed significance level P, obtained using a paired Student's t test, testing against differences of the S.D. of the distributions {bsum} and {blit}, is also stated.
distributions. The benefit for the medium and high IVIM perfusion parameter sets was less significant but present, even when compared with a distribution that utilizes numerous b values. The contribution of the three fitted IVIM parameters to the relative overall error differs considerably, shown exemplarily for the {bsum} distribution in Fig. 3. Here, the diffusion coefficient D can be estimated with an S.D. σD below 20% at moderate SNRs (N30) for high perfusion fractions as in the abdomen. If the perfusion fraction is as low, as in the brain, an SNR of more than 80 is required to enable an S.D. σD below 20%. The relative error of the diffusion coefficient σD has the smallest contribution to the overall error for all parameter sets. The relative error of the perfusion fraction σf strongly depends on the used parameter set. For the abdominal parameter sets (medium and high IVIM perfusion), the contribution to the relative overall error is comparable to σD, whereas for the low IVIM perfusion, σf is 20-fold higher than σD. The highest contribution to the overall error for all parameter sets has σD⁎, e.g., for the high IVIM perfusion regime, σD⁎ is three times higher than σD and σf. This is in concordance with our in vivo measurements, where the high variance of the fitted D ⁎ leads to heterogeneous, nonphysiological D⁎ maps and to a high S.D. in the quantitative analysis. It is unlikely that this high variance is based on physiological heterogeneity because most of the fitted pixels reach the fitting limits, as shown in Fig. 5. Furthermore, the pseudodiffusion coefficient decreases considerably when the averaged signal is fitted (see Table 2). Consequently, the image quality of the pseudodiffusion map improves considerably and the S.D. decreases significantly using the optimized distribution {bsum} when compared with {blit}. This decrease of the S.D. is also present for the perfusion fraction f and the diffusion
coefficient D. The f and D maps enable a good delineation of the abdominal organs using {bsum}. Thus, the in vivo measurements are in keeping with the results of the simulations that indicate that {bsum} outperforms {blit} at an SNR of 30. 4.1. Proceeding for clinical applications In order to plan an experiment using the IVIM technique, the available SNR and the approximate IVIM parameters should be evaluated first to estimate the critically required SNR. If the available SNR for the experimental setup is below the critical value, e.g., as for IVIM imaging of the brain in current clinical routine, it is not feasible to calculate the three IVIM parameters in a reliable fashion. One solution could be a “segmented” analysis where the three parameters D, f and D⁎ are calculated consecutively as recently described by Patel et al. [26]. Additionally, the IVIM properties of the selected tissue may be analyzed using signal intensities averages over a ROI. Compared with pixel by pixel analysis, the uncertainty is reduced (see Table 2 vs. Table 3), and a ROI approach combined with the optimized b value distributions presented here could lead to considerable improvement in accuracy even at SNRs below the critical value. If the SNR is much higher than the critical SNR, a more reliable fitting of all IVIM parameters can be achieved using the optimized b value distributions. Here, the question arises if the distribution {bsum} or the distribution optimized for a specific organ should be used. As the error of the fit using {bsum} was comparable to the error using the distribution optimized for a specific organ, we recommend the use of {bsum}, especially if the tissue of interest has heterogeneous diffusion and perfusion properties, e.g., in tumor tissue. Regarding the question of the
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optimal number of b values in clinical routine, one has to consider the scan time available for a patient. If the measurement time is no limiting factor, the maximal number of b values given in Table 1 should be used for the optimal assessment of the IVIM parameters. However, in most cases, the acquisition of 30 or more b values is not acceptable due to scan time restrictions. If the number of b values needs to be reduced, the optimal choice of b values for any given number of b values is given in Table 1. Here, the b values are sorted in the order of appearance during the optimization process. The fit quality per unit time improves most significantly when moving from 4 to 10 b values. Thus, at least 10 of the optimal distributed b
values in Table 1 should be measured for a high-quality IVIM experiment. Data fitting and analysis of multiexponential data has already been investigated by some groups, but the reported results cannot be transferred easily to IVIM fits because of differing assumptions. For instance, Bretthorst [27] used a Bayesian view to investigate the accuracy of exponentially fitted parameters and Anastasiou and Hall [28] used the Cramer–Rao theory to optimize T2 measurements in biexponential systems. Both assumed a uniform data sampling and Gaussian noise, whereas in our study, we focused on a more general nonuniform distribution of the acquired b values, which is easily applicable here, but not necessarily to T2
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measurements with Carr–Purcell–Meiboom–Gill echo trains. Istratov and Vyvenko [29] investigated the exponential analysis of physical phenomena with an arbitrary number of exponential components. Their manuscript was focused on mathematical limitations and on the comparison of several fitting algorithms. This differs from our study, which is dedicated to biexponential decays with a priori approximately known parameters and which focuses especially on the determination of optimized sampling points (b values). The optimization approach presented here could be used for optimizing the sampling distribution for any biexponential decaying data. Our current study has some limitations. First, the algorithm presented in this study consecutively adds optimized b values. Thus, it is not necessarily certain that the obtained distribution is optimal for any—in particular small—number of b values. A better optimization would require the consideration of any possible distribution of b values but would lead to extensive computation time. Our algorithm is a tradeoff between computation time and correctness of the optimized b value distribution. In preliminary simulations, we found that for the recommended minimum of 10 b values, the consecutive algorithm outperforms, for example, a “random” algorithm, which tries out many randomly generated b value distributions. Second, this study has focused on the imprecision and inaccuracy of parameter estimation from “noise-contaminated” simulated data. Systematic errors such as organ movement or partial volume effects have not been considered in the simulations and may further influence the optimal parameter selection. Furthermore, we assumed that no third compartment such as CSF is present. Notwithstanding the above-mentioned limitations, we could show that the optimized b value distribution {bsum} decreases the S.D. and increases the image quality of the abdominal in vivo IVIM maps in comparison to {blit} and thus underscore the value of the presented optimization. In conclusion, the optimal b value distributions found using simulations can be seen as solid basis to further investigate an optimal in vivo setting and should lead to an improved quality of IVIM data in future studies.
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