Special Ar ticles • Original Research Ashoor and Khorshidi Compartments Associated With ADC on MRI
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Special Articles Original Research
Estimation of the Number of Compartments Associated With the Apparent Diffusion Coefficient in MRI: The Theoretical and Experimental Investigations Mansour Ashoor 1 Abdollah Khorshidi2 Ashoor M, Khorshidi A
Keywords: apparent diffusion coefficient, compartment, DWI, gradient factor, Monte Carlo method DOI:10.2214/AJR.15.14497 Received January 29, 2015; accepted after revision September 13, 2015. 1 Radiation Application Research School, Nuclear Science and Technology Research Institute, Tehran, Iran. 2
Department of Medical Physics and Biomedical Engineering, Tehran University of Medical Sciences, PO Box 141556559, Tehran, Iran. Address c orrespondence to A. Khorshidi ([email protected]). Supplemental Data Available online at www.ajronline.org. AJR 2016; 206:455–462 0361–803X/16/2063–455 © American Roentgen Ray Society
OBJECTIVE. The goal of the present study was to estimate the number of compartments and the mean apparent diffusion coefficient (ADC) value with the use of the DWI signal curve. MATERIALS AND METHODS. A useful new mathematic model that includes internal correlation among subcompartments with a distinct number of compartments was proposed. The DWI signal was simulated to estimate the approximate association between the number of subcompartments and the molecular density, with density corresponding to the ratio of the ADC values of the compartments, as determined using the Monte Carlo method. RESULTS. Various factors, such as energy depletion, temperature, intracellular water accumulation, changes in the tortuosity of the extracellular diffusion paths, and changes in cell membrane permeability, have all been implicated as factors contributing to changes in the ADC of water (ADCw); therefore, one may consider them as pseudocompartments in the new model proposed in this study. The lower the coefficient is, the lower the contribution of the compartment to the net signal will be. The results of the simulation indicate that when the number of compartments increases, the signal will become significantly lower, because the gradient factor (i.e., the b value) will increase. In other words, the signal curve is approximately linear at all b values when the number of compartments in which the tissues have been severely damaged is low; however, when the number of compartments is high, the curve will become constant at high b values, and the perfusion parameters will prevail on the diffusion parameters at low b values. Therefore, normal tissues will be investigated when the number of compartments and the ADC values are high and the b values are low, whereas damaged tissues will be evaluated when the number of compartments and the ADC values are low and the b values are high. CONCLUSION. The present study investigates damaged tissues at high b values for which the effect of eddy currents will also be compensated. These b values will probably be used in functional MRI. he apparent diffusion coefficient (ADC) can identify the random molecular motion of water in the brain and reveal acute ischemic injury within minutes of the onset of stroke. The ADC, which is a measure of the mean free path of water, is consistently reduced in areas of the brain affected by ischemia in subjects affected by acute stroke; however, the acute pathophysiologic mechanisms involved in this phenomenon are not completely understood. The importance of ADC and its association with ischemic tissue have been extensively studied. Likewise, tissue microstructure, cellularity, and nerve fiber integrity may also be evaluated using the ADC [1–8]. When a restriction of the mean random molecular motion of water occurs, the ADC will change. Use of the ADC provides an opportunity to
T
obtain valuable information on the physiologic parameters associated with disease detection and tumor delineation, in addition to data that can aid in the differentiation of normal and damaged tissues. Energy depletion, temperature, intracellular water accumulation, changes in the tortuosity of the extracellular diffusion paths, and changes in cell membrane permeability have all been implicated as factors contributing to changes in the ADC of water (ADCw); therefore, they, along with the number of compartments, have been considered in the new proposed model in this study. The increase in the ADCw appears to occur concomitantly with the loss of membrane structure and the loss of barriers to diffusion. It is possible that other MRI measurements, such as cerebral blood flow, phosphorus or proton spectrosco-
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Ashoor and Khorshidi py, or a combinations of these measurements, may improve tissue characterization. Tissue metabolism results in the generation of waste products that are eliminated through the capillary network to constitute deoxygenated venous blood. Remarkably, this process depends on many parameters, such as blood pressure, blood velocity, capillary network density and geometry, capillary wall permeability, nutrient or oxygen diffusion, and extraction rates, the latter of which can be grouped into three categories: microvascular anatomic and histologic findings, blood microcirculation, and blood-tissue exchanges [9–19]. All elements in these categories may be considered as pseudocompartments. Biologic tissues are a finite and inhomogeneous medium, because they are heterogeneous and contain multiple subcompartments. Von Meerwall [19] showed that, for each measurement volume, the ADC exists as a unique parameter, because most tissues have multiple subcompartments, including, at a minimum, intracellular and extracellular compartments. Under the assumption that measurement times are short, diffusion is unrestricted in each subcompartment and there is no exchange; therefore, MRI signal attenuation is associated with both the density of the molecules diffusing in each subcompartment and the number of molecules present. Koenig et al. [20] used a method in which all subcompartments were separated by fitting the data with the use of multiexponential decay; however, they indicated that when measurement times are longer, a different situation occurs. First, restricted diffusion may appear in the smallest subcompartments. Second, molecular exchanges may be seen between communicating compartments, so the analytic solution becomes difficult. Again, this method is related to the density of the molecules diffusing in each subcompartment and the number of subcompartments. However, this approach is consistent with that used in nuclear MRI dispersion studies, which have suggested that cell membranes can be ignored on the nuclear MRI time scale. In general, the origin of the water signal and the multiexponential function in DWI are not yet fully understood [21–30]. The possible role of compartmentalization in water signal decay on DWI studies of biologic systems has been investigated using biexponential analysis, but such studies have not included internal correlation multiexponential analysis [31–35]. The goal of the present study was to estimate the number of compartments and the
456
mean ADC value with the use of the DWI signal curve. We propose a useful new mathematic model that includes internal correlation among subcompartments with a distinct number of compartments. The DWI signal is simulated to estimate the approximate association of the number of subcompartments with the molecular density, with density corresponding to the ratio of the ADC values of the compartments, as determined using the Monte Carlo method. Although this model is applied mathematically, its effects will be reflected in the ADC values and the volume fractions, in terms of how the ratio of some coefficients relates to the number of compartments. The lower the coefficient is, the lower the contribution of the compartment to the net signal will be. Materials and Methods Theory
When introducing the intravoxel incoherent motion model in 1986, Le Bihan et al. [36] concluded that diffusion measurements can also be sensitive to vascular blood motion and that a DWI sequence can be used to measure both diffusionand perfusion-related parameters. Depending on the permeability of the barriers that divide these compartments, exchange and transport between the compartments should be considered. There appears to be a correlation among these factors; therefore, the proposed model may very well be able to indicate this internal relationship on the basis of the coefficients. As the signal decays exponentially alongside the product of the diffusion coefficient and the b value, blood magnetization dominates the signal decay at low b values. Most measurements of diffusion in biologic tissues take the form of an ADC. When measurement times are short, signal attenuation is expressed as follows: N s(b) = ∑ p × e–b × D s(0) i = 1 i
i
(1),
where s is the signal strength, b is the gradient value, pi is the density of molecules diffusing in compartment i, e is the abbreviation of exponential function, Di is the associated diffusion coefficient, and N is the number of subcompartments. Measurements with low b values will be more sensitive to fast diffusion components. The ideal approach would be to separate all subcompartments by fitting the data with a multiexponential decay [35]. By applying the central limit theorem for N compartments when there are long diffusion times, one can justify the use of a single apparent diffusion constant Da: N
Da = ∑ pi × Di i=1
(2).
This approach is also consistent with the approaches used in nuclear MRI dispersion studies. When faced with intermediate diffusion times or various permeability times, one must take into account the geometric arrangement of and the diffusion coefficients for each compartment, as well as their rates of exchange [37]. As stated elsewhere, there are various parameters that play key roles in brain behaviors [38]. There are many compartments of the brain, such as neurons, astrocytes, glia, and extracellular space, that perform their duties alone but nonetheless have an effect on one another, causing the ADC values to be very complex in terms of microscopic-level measurements [39–41]. One of the reasons that we cannot expect the size of tissue compartments to remain constant is because water can shift within the compartments. When the size of a compartment changes, the ADC will also change. A nuclear MRI signal that is sensitized by the diffusion gradients and that includes many compartments is proposed in the following new purely mathematic formula: d ns d n − 1s d n − 2s + an − 1 n − 1 + a n − 2 n − 2 + dbn db db d n − 3s an − 3 n − 3 + ... + a0s = 0 db n
n
a n − 1 = − ∑ D i, an − 2 = ∑ D i × Dj,..., i=1
i = 1, j = 1 (i ≠ j)
(3),
n
a 0 = (−1) 1 s(b) = n−1 s(0)
(
n
∑e
i=1
−Di × b
n
∏D i=1
−
∑
i
n i=1
Di × e−D × b
∑
i
n i=1
Di
)
where b is the gradient factor, D1, is the ADC value in compartment i, n is the number of subcompartments, d n/db n is a symbol for the nth derivative, a is the coefficient in the series, e denotes exponential function, and j represents the change from 1 to n. Although this model is purely mathematic, some microscopic-level tissue measurements that are not yet fully understood, such as the ADC and the volume fractions, will be reflected because the ratio of some coefficients will be theoretically related to the density of the compartments. The lower the coefficient is, the lower the contribution of the compartment to the net signal will be. This model is for more than two compartments (Fig. S1, supplemental data, can be viewed in the AJR electronic supplement to this article, available at www. ajronline.org). Equation 3 indicates the internal correlation among compartments that makes known some parameters and problems related to the field of diffusion and likely also perfusion, in addition to showing the ulterior development of the density of the molecules diffusing in compartments. For instance, in a two-compartment model, one may use equation 3 to obtain the relationship that follows:
AJR:206, March 2016
Compartments Associated With ADC on MRI s(b)
=
Df Df + D s
× e−D × b + s
Ds Df + Ds
1
× e−D × b (4), f
s(b) s(0)
=
1
(
Df + Dm
2 D f + Dm + Ds
n = 2; mean ADC = 0.016 3; 0.04
where Df and Ds are the ADC values of the compartments with fast and slow diffusion, respectively; s denotes signal strength, e is the exponential function, and b represents the gradient value. Equation 4 is approximately in agreement with findings from previous studies [12, 41, 42]. Similarly, in the three-compartment model, the equation is as follows:
4; 0.0716 5; 0.1117
0.9
6; 0.1596 7; 0.2155
0.8
8; 0.2794
10; 0.4312 12; 0.6137 15; 0.9497
0.7
20; 1.6677
0.6
× e−D × b + s
(5),
30; 3.7028
0.5
40; 6.5352 50; 10.1605
where Dm defines the ADC value for the compartment with a medium rate of diffusion, s denotes signal strength, e is the exponential function, and b represents the gradient value. In general, one may estimate the number of compartments and the mean ADC value, with the use of equation 3 on the DWI signal curve.
0.4
100; 40.299 200; 160.3442
Materials
0.1
Df + Ds D f + Dm + Ds
× e−D
m
×b
+
Dm + Ds D f + Dm + Ds
)
× e−D × b f
As indicated, equation 3 has been proposed for use in estimating the number of compartments and the mean ADC, and the signal is simulated using the Monte Carlo method, with random numbers assigned to the diffusion constants. The random number distribution is gaussian, with a mean of 0.5 (range, 0–1) used for the diffusion coefficients. The simulation was performed for various values of n (2–1000), as shown in Figure 1. When n = 2, the fitted curve on the simulated data had a low curvature, indicating similar effects of diffusion and perfusion on all b values.
Animal Model
To assess the proposed model, six male Wistar rats (weight, 300–450 g) that were subjected to embolic stroke underwent imaging performed with a 7-T MRI system. DWI appears to be useful for the early detection of stroke. The method of inducing embolic stroke and administering recombinant tissue plasminogen activator as treatment has been previously described. In brief, an aged white clot (prepared 24 h before ischemia) is slowly injected into the internal carotid artery to block a major coronary artery. This model of embolic stroke provides a relatively reproducible infarct volume localized to the territory supplied by the major coronary artery [43–45].
Measurements Made Using a 7-T MRI System
MRI measurements were performed using a 7-T superconducting magnet with a 20-cm bore interfaced to a console. An actively shielded gradient
s(b)/s(0)
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s(0)
80; 25.8361
250; 250.3257 300; 360.2459
0.3
350; 490.1639 400; 640.0068 450; 809.6044
0.2
0
500; 999.3752 800; 2560
1000; 3990 31; 3.950 (Damaged) 41; 6.8629 (Normal) 0
1000 2000 3000 4000 5000 6000 7000 8000 9000 10,000 b Value (s/mm2)
Fig. 1—Simulated DWI signal curve. Various numbers of compartments (n) and various mean apparent diffusion coefficient (ADC) values (expressed as value × 10 −4) are shown. Damaged = damaged tissue; normal = normal tissue; s(b)/s(0) = signal attenuation alongside product of diffusion coefficient and b value.
coil set with a 12-cm bore that was capable of producing magnetic field gradients of up to 20 gauss/ cm was used. The radiofrequency pulses were applied using a saddle coil (diameter, 7.5 cm) that was actively decoupled by transistor-transistor logic control from the 1.4-cm surface receptor coil, which was positioned over the centerline of the animal’s skull. Stereotactic ear bars were used to minimize movement during the imaging procedure. While MRI measurements were performed, anesthesia was administered using a gas mixture of 69% nitrous oxide, 30% oxygen, and 0.75–1% halothane. The mean (± SD) rectal temperature was maintained at 37°C ± 0.5°C, with the use of a feedback-controlled water bath. A modified FLASH sequence was used to reproduce positioning of the animal in the magnet at each MRI session [46]. DWI measurements were performed 24 hours after the onset of embolization. The ADCw value was determined using the Stejskal-Tanner sequence with three b values (10, 800, and 1800 s/mm2) in each of three diffusionsensitizing directions, with use of the following imaging parameters: TR/TE, 1500/40; number of slices, 7; FOV, 32 mm2; and matrix, 128 × 64.
Each scan required an acquisition time of 3.2 minutes. The total time required for acquisition of the entire sequence was approximately 14.4 minutes.
Data Analysis
Two ROIs were selected for the measurement of MRI parameters. The mean and SD of ADCw values for the parenchymal tissue (not the CSF) in the contralateral hemisphere were measured, and the threshold values used to determine the number of pixels with abnormal values in the ipsilateral hemisphere were defined using the mean ADCw minus 1.5 SD in the contralateral hemisphere, where this value is heuristic. Homologous ROIs in the ischemic tissue were also measured in the contralateral hemisphere. A paired t test was used to compare the two ROIs, with statistical significance denoted by p = 0.05. The monoexponential intravoxel incoherent motion model was used to calculate the ADC, with the use of the following equation: s(b) = s(0) × exp(−b × ADC), where s is the signal strength and b is the b value [47]. Because the diffusion of water may be directionally dependent (anisotropic), measurement in only one di-
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Ashoor and Khorshidi 1
0.8
s(b)/s(0)
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0.9
0.7
MRM-Inglis [11], 2011 n = 7; mean ADC = 0.2152
0.6
9; 0.3509
8; 0.2797
6; 0.1595 (Optimum) 4; 0.0717
0.5
5; 0.1117 SUM_Diff (n = 7; Inglis [11])
0.4
SUM_Diff (n = 8; Inglis [11]) SUM_Diff (n = 9; Inglis [11])
0.3
SUM_Diff (n = 6; Inglis [11]) (Optimum) SUM_Diff (n = 4; Inglis [11])
0.2
SUM_Diff (n = 5; Inglis [11])
0.1 0
0
1000 2000 3000 4000 5000 6000 7000 8000 9000 10,000 b Value (s/mm2)
Fig. 2—Simulated DWI signal curve upon which data from Inglis et al. [11] are fitted to estimate number of compartments (n) and mean apparent diffusion coefficient (ADC) values (expressed as values × 10 −4 mm2 /s). Optimized values are six compartments and mean ADC value of 0.16 × 10 −4 mm2 /s. MRM = magnetic resonance microscopy; SUM_Diff = summation of differences in MRM-Inglis curve and simulated curves (optimum curve, which indicates minimum difference in comparison with MRM-Inglis curve, was acquired with use of six compartments and mean ADC value of 0.1595 mm2 /s); s(b)/s(0) = signal attenuation alongside product of diffusion coefficient and b value. Fig. 3—Apparent diffusion coefficient map created by imaging of rat brain. Normal tissues versus damaged tissues are shown, with damaged tissues depicted as bright segment on right side of map.
rection can lead to incorrect interpretation of the image. The diffusion gradient was therefore applied in each of three orthogonal directions (x, y, and z), and the mean of these measurements was calculated to determine the trace of the diffusion tensor, which is reported to minimize the effects of diffusion anisotropy [48]. Ideally, one should consider all requisites for dynamic modeling of the system—namely, fluid type and volume, thermal dynamics, and various parameters involved in capillary spaces. The ADC measurement is,
458
in fact, a weighted mean of the diffusion coefficients of the various compartments.
Results The number of compartments may also play a role similar to that of the partitioning that is produced in all phases of molecular diffusion by the gaussian functions, with various mean expectation values that are independent of each other but can act together, with each of them causing a relative decrease
in the signal fraction. These fractions are able to characterize the many various clusters in the diffusion map, resulting in differentiation of tissues with the related grades. The ratio of the coefficients of the compartments will take into account any exchanges occurring between the compartments. Figure 1 shows that, with an increased number of compartments, the mean ADC value and the signal curve will increase. In this proposed model, damaged tissues can be characterized at low b values, and the results of the simulation illustrate that when the number of compartments decreases, the signal curve will become linear at a high b value; however, when the number of compartments increases, the curve will be constant at these values. As an example, the data published by Inglis et al. [11] were mapped on simulated curves; for the optimum curve, the total number of compartments was six and the ADC value was 15.95 × 10−6 mm2/s, as shown in Figure 2. The ADC map formed using the protocols is shown in Figure 3. In the damaged tissue region on the ADC map, the values were decreased relative to the values in the normal tissue region on the ADC map, because of stroke onset. For calculation of the ADC value, suitable areas were chosen after image processing procedures were applied to reduce noise and artifacts. Therefore, the mean ADC values for the damaged and normal tissue regions were 392 ± 34.1 × 10 −6 and 659 ± 40.7 × 10 −6 mm2/s, respectively [49]. Table 1 presents the ADC values and the number of the compartments in the damaged and normal tissue regions, on the basis of the proposed model. Experimental ADC values were derived from the damaged tissue region, as shown in Figure 3, and were in good agreement with simulated data presented in Figure 1. For the damaged tissue region, 31 compartments and a mean ADC value of 3.95 × 10−4 mm2/s was found, whereas, for the normal tissue region, 41 compartments and a mean ADC value of 6.86 × 10−4 mm2/s were noted. It seems that, for the damaged tissue region, both the number of compartments and the mean ADC value were lower than the respective values noted for the normal tissue region. The lower the number of compartments is, the more tissue destruction there will be. Discussion Knowing the number of compartments allows one to possibly reduce the effects of restriction and increase the accuracy of the
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Compartments Associated With ADC on MRI
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TABLE 1: The Number of Compartments and the Mean Apparent Diffusion Coefficient (ADC) Values in the Regions of Damaged and Normal Tissue Simulated No. of Compartments
ADC Value (mm2 /s), Mean ± SD
Damaged tissue region
31
3.92 × 10 −4 ± 3.41 × 10 −5
Normal tissue region
41
6.59 ×10 −4 ± 4.07 × 10 −5
0.76
0.595 ± 0.04
Region
Ratio of damaged tissue to normal tissue
ADC value, as well as evaluate the brain at the macroscopic and microscopic levels, in which diffusion is both isotropic and anisotropic. Neeman et al. [50] and Yablonsky et al. [51] have shown synchronous changes in the ADC alongside brain activation. In addition, a comprehensive analysis of the diffusion attenuation curves obtained with different diffusion times may lead to an accurate description of the medium microstructure of capillaries. Many pathologic conditions, such as tumors, stroke, epilepsy, and diffuse axonal injury, can influence the ADCw. Again, when the number of compartments is known, one may be able to detect different degrees of tissue function. Therefore, determining the number of compartments from the DWI signal curve with the use of the proposed model can likely assist in both finding a better description and gaining an improved understanding of the problems related to this field. In the present study, we propose a new mathematic model for estimating the number of compartments that affect the DWI signal. In this model, the b value is adjusted according to the parameters of the pulse sequence. The b value is an effective and very important parameter, because when a DW gradient is used, water diffusion and the random, or Brownian, motion of water molecules can cause attenuation in the signals. Increasing the diffusion gradient amplitude (i.e., the separation of its lobes, or the pulse width of each lobe) results in a higher b value. Accurate determination of diffusion coefficients requires use of the complete attenuation factor and consideration of cross-terms between diffusion and imaging gradients. Compartmentation can be deduced using the diffusion filter effect of the imaging gradients [50, 51]. The b value used in most DWI sequences is approximately 1000 s/mm2. To achieve this b value, the DWI gradient lobe typically is several tens of milliseconds in length, which inevitably leads to a rather long TE. This, in turn, reduces the signal-to-noise ratio and introduces unwanted T2 signal con-
trast in the DW image, and it also causes the T2* signal effects to be more dependent on local microvascular architecture than on T1 signal effects [13, 35, 52–57]. There are technical limitations to producing a high b value; therefore, the proposed model may be used to evaluate damaged tissues over a number of compartments. In this situation, the signal will be significant at low b values. This model may also improve the temporal resolution in the functional maps by modeling the diffusion times in any compartment. To optimize the gradient b value, one has to consider the signal-to-noise ratio and the gradient power available, the accuracy of diffusion measurement, the type of tissues, and the desire to eliminate artifacts arising from macroscopic or microscopic nondiffusional motion. The optimization of this factor largely depends on experimental priorities. Furthermore, one may consider statistical arguments for optimizing the range of b values to be used to obtain the best accuracy in terms of diffusion. Thus, the accuracy of the maps obtained will be improved at low b values. In the present study, the ADCw was measured using three b values (10, 800, and 1800 s/mm2). In the proposed mathematic model, the results of the simulation illustrate that, by increasing the number of compartments, the signal is severely decreased at low b values and is linear with a low slope at high b values. In other words, if the number of compartments is too large, the perfusion effects dominate the diffusion effects, and the signal will be constant at a b value greater than a distinct b value. The value of Di tended to be close to the distinct b value when the number of subcompartments was increased. It seems that in the process of forming damaged tissue, the number of compartments will decrease; likewise, it seems that, in the process of forming normal tissue, the number of compartments will increase. A decrease in the number of compartments gives rise to an increase in perfusion, causing the signal to decrease at a low b value
and creating a negligible change at a high b value. This concept is well indicated in the proposed model in which the ADC values and the number of the compartments noted for the damaged tissue are lower than those noted for the normal tissue. For the normal tissues, the signal will be significant at all b values, because perfusion and diffusion will act together. The pseudodiffusion coefficient, which is an index of unused blood microcirculation in the capillary space, disappears in the model; however, its effect is nonetheless included in this model. Technical limitations notwithstanding, it can be concluded that the steep signal decay observed at low b values in DWI studies of tissue can be attributed to signals arising in the arterial vasculature. This finding supports the argument that the apparent arterial fraction must be taken into account during estimation of the diffusion coefficient and that it may be an incorrect approach to use only two b values. The present study assists in evaluating damaged tissues with low b values for which the effect of eddy currents may be compensated. Therefore, normal tissues can be evaluated at high b values and damaged tissues can be evaluated at lower values. In addition, angiogenesis can be measured indirectly by a variety of different MRI parameters, both structural and functional. These parameters include microvessel density, blood volume, blood flow, vessel size, and tissue oxygenation, among others [42, 58]. It is expected that this proposed model will be able to improve the measurement of such parameters in this particular field of research. One aspect that seems certain is that restricted diffusion must have some influence on the signal, because there is a strong correlation between diffusion-tensor imaging and the organization of white matter pathways [59]. Understanding the roles of tissue structure and the cellular-level physics of water in causing nonexponential signal attenuation is crucial to applications involving water diffusion as a tool for functional brain imaging [60–63]. It seems that when the number of compartments can be determined, temporal resolution may improve. The effect of restricted motion on the diffusion coefficients may be assessed by altering the sequence diffusion time, which is the period between the onsets of the two motion-probing gradients placed at either side of the 180° refocusing pulse in a spinecho sequence. Theoretic models of restrict-
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Ashoor and Khorshidi ed diffusion predict that the observed diffusion coefficient will decrease with diffusion time, because the motion of molecules that are close to a barrier, such as a cell membrane, will be limited. However, the existence of restriction effects in a normal living brain is still a matter of controversy. In fact, even though one recent study found evidence of restriction effects in a rat brain with the use of an effective diffusion time as short as 375 microseconds, most studies have found that restricted motion has little influence on the values of the observed diffusion coefficients for the range of diffusion times (much greater than 10–200 ms) accessible with typical DWI sequences and contemporary MRI systems [64]. Overall, it is not yet clear what determines the rates of diffusive motion in brain tissue— that is, whether the diffusion coefficients are determined by the physics of water at the cellular level, such as in the model discussed by Le Bihan et al. [1], or whether water motion constrained by subcellular microstructure is more important. These changes likely arise from small arterial networks and capillaries, and because large veins usually have much larger diameters and, thus, smaller flow changes, the changes in the ADC values for large veins would be small and insignificant. Each of the changes or parameters can be modeled as a compartment, as shown in equation 3. In conclusion, there is still considerable debate over the origin of the nonmonoexponential signal decay in DWI. Although Le Bihan et al. [1] interpreted the perfusion fraction as the partial volume of the whole capillary vascular fraction, and animal experiments conducted by Duong and Kim [65] strongly suggest that the fast-decaying component noted at low b values should be attributed solely to the arterial blood. Venous blood was shown to have an 80-fold lower pseudodiffusion coefficient, approximately equal to the diffusion coefficient for tissue. In fact, one may obtain the perfusion information from this model by using the diffusion gradients. In equation 1, the density of molecules was included without considering the internal correlation among subcompartments; however, in equation 3, this correlation was considered without directly including the density factor, which was included in the ratio of the coefficients. Conclusion We proposed a new mathematic model with which to characterize the compart-
460
ments that intervene on the DWI signal. Although the proposed model is purely mathematic, some microscopic-level tissue measurements that are not yet fully understood, such as the ADC and the volume fractions, will be reflected because the ratio of some coefficients will be theoretically related to the density of the compartments. If the coefficient is very low, the contribution of the compartment to the net signal will be very low. This signal was simulated, with the use of various numbers of compartments, by using the Monte Carlo method. The results of the simulation indicated that, although the number of compartments increases, the signal will severely become very low as the b value increases. In other words, the signal curve was approximately linear at higher b values when the number of compartments was low, but when there was a high number of compartments, the signal curve indicating that tissues were normal was constant. The lower the number of compartments is, the more tissue destruction there will be. Acknowledgments We thank Zheng Gang Zhang, Quan Jiang, and James R. Ewing, for technical assistance, and the Nuclear Magnetic Resonance Facility in the Department of Neurology at Henry Ford Health Sciences Center (Detroit, Michigan), for providing the means and support required for implementation of this study. References
1. Le Bihan D, Mangin JF, Poupon C, et al. Diffusion tensor imaging: concepts and applications. J Magn Reson Imaging 2001; 13:534–546 2. Clark CA, Le Bihan D. Water diffusion compartmentation and anisotropy at high b values in the human brain. Magn Reson Med 2000; 44:852–859 3. Kim DH, Chung S, Vigneron DB, Barkovich AJ, Glenn OA. Diffusion-weighted imaging of the fetal brain in vivo. Magn Reson Med 2008; 59:216–220 4. Benveniste H, Hedlund LW, Johnson A. Mechanism of detection of acute cerebral ischemia in rats by diffusion-weighted magnetic resonance microscopy. Stroke 1992; 23:746–754 5. Takayama H, Kobayashi M, Sugishita M, Mihara B. Diffusion-weighted imaging demonstrates transient cytotoxic edema involving the corpus callosum in a patient with diffuse brain injury. Clin Neurol Neurosurg 2000; 102:135–139 6. Duong TQ, Ackerman JJ, Ying HS, Neil JJ. Evaluation of extra- and intracellular apparent diffusion in normal and globally ischemic rat brain via
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Reson Med 2002; 48:649–657 33. Grant SC, Buckley DL, Gibbs S, Webb AG, Blackband SJ. MR microscopy of multicomponent diffusion in single neurons. Magn Reson Med 2001; 46:1107–1112 34. Goodman JA, Ackerman JJ, Neil JJ. Cs + ADC in rat brain decreases markedly at death. Magn Reson Med 2008; 59:65–72 35. Sugahara T, Korogi Y, Kochi M, et al. Usefulness of diffusion-weighted MRI with echo-planar technique in the evaluation of cellularity in gliomas. J Magn Reson Imaging 1999; 9:53–60 36. Le Bihan D, Breton E, Lallemand D, Grenier P, Cabanis E, Laval Jeantet M. MR imaging of intravoxel incoherent motions: application to diffusion and perfusion in neurologic disorders. Radiology 1986; 161:401–407 37. Zientara GP, Freed JH. Spin-echoes for diffusion in bounded, heterogeneous media: a numerical study. J Chem Phys 1980; 72:1285–1292 38. Ewing JR, Cao Y, Fenstermacher J. Single-coil arterial spin-tagging for estimating cerebral blood flow as viewed from the capillary: relative contributions of intra- and extravascular signal. Magn Reson Med 2001; 46:465–475 39. Le Bihan D, Moonen CTW, van Zijl PCM, Pekar J, DesPres D. Measuring random microscopic motion of water in tissues with MR imaging: a cat brain study. J Comput Assist Tomogr 1991; 15:19–25 40. Sehy JV, Ackerman JJ, Neil JJ. Apparent diffusion of water, ions, and small molecules in the Xenopus oocyte is consistent with Brownian displacement. Magn Reson Med 2002; 48:42–51 41. Schwarcz A, Bogner P, Meric P, et al. The existence of biexponential signal decay in magnetic resonance diffusion-weighted imaging appears to be independent of compartmentalization. Magn Reson Med 2004; 51:278–285 42. Ungersma SE, Pacheco G, Ho C, et al. Vessel imaging with viable tumor analysis for quantification of tumor angiogenesis. Magn Reson Med 2010; 63:1637–1647 43. Buckley DL, Bui JD, Phillips MI, et al. The effect of ouabain on water diffusion in the rat hippocampal slice measured by high resolution NMR imaging. Magn Reson Med 1999; 41:137–142 4 4. Jiang Q, Zhang RL, Zhang ZG, Ewing JR, Divine GW, Chopp M. Diffusion-, T2-, and perfusionweighted nuclear magnetic resonance imaging of middle cerebral artery embolic stroke and recombinant tissue plasminogen activator intervention in the rat. J Cereb Blood Flow Metab 1998; 18:758–767 45. Zhang RL, Chopp M, Zhang Z, Jiang Q, Ewing JR. A rat model of embolic focal cerebral ischemia. Brain Res 1997; 766:83–92 46. Haase A, Frahm J, Matthaei D, Hänicke W, Merboldt KD. FLASH imaging: rapid NMR im-
aging using low flip-angle pulses—1986. J Magn Reson 2011; 213:533–541 47. Jiang Q, Zhang ZG, Chopp M, et al. Temporal evolution and spatial distribution of the diffusion constant of water in rat brain after transient middle cerebral artery occlusion. J Neurol Sci 1993; 120:123–130 48. Barber PA, Darby DG, Desmond PM, et al. Prediction of stroke outcome with echoplanar perfusion- and diffusion-weighted MRI. Neurology 1998; 51:418–426 49. Ashoor M, Jiang Q, Chopp M, Jahed M. Introducing a new definition towards clinical detection of microvascular changes using diffusion and perfusion MRI. Scientia Iranica 2005; 12:109–115 50. Neeman M, Freyer JP, Sillerud LO. Effects of imaging gradients on diffusion measurements by MR imaging. In: Le Bihan D, ed. Diffusion and perfusion MRI: applications to functional MRI. New York, NY: Raven Press, 1995:73–76 51. Yablonskiy DA, Bretthorst GL, Ackerman JJ. Statistical model for diffusion attenuated MR signal. Magn Reson Med 2003; 50:664–669 52. Snedecor GW, Cochran WG. Statistical methods, 7th ed. Ames, IA: Iowa State University Press, 1980 53. McIntyre DJO, Robinson SP, Tarlo K, Griffiths JR. Comparison of T1 and T2* enhancement in MNU chemically-induced mammary carcinomas after administration of the blood pool agent NC100150 injection. In: Proceedings of the 8th annual meeting of the International Society for Magnetic Resonance in Medicine. Denver, CO: International Society for Magnetic Resonance in Medicine, 2000:1046 54. Peled S, Cory CG, Raymond SA, Kirschner DA, Jolesz FA. Water diffusion, T2, and compartmentation in frog sciatic nerve. Magn Reson Med 1999; 42:911–918 55. Bernstein MA, Zhou XJ, Polzin JA, et al. Concomitant gradient terms in phase contrast MR: analysis and correction. Magn Reson Med 1998; 39:300–308 56. Bernstein MA, King KF, Zhou XJ. Handbook of MRI pulse sequences. Burlington, MA: Elsevier Academic Press, 2004 57. Meier C, Dreher W, Leibfritz D. Diffusion in compartmental systems. Part II. Diffusion-weighted measurements of rat brain tissue in vivo and postmortem at very large b-values. Magn R eson Med 2003; 50:510–514 58. Lemke A, Laun FB, Simon D, Stieltjes B, Schad LR. An in vivo verification of the intravoxel incoherent motion effect in diffusion-weighted imaging of the abdomen. Magn Reson Med 2010; 64:1580–1585 59. Moussavi-Biugui A, Stieltjes B, Fritzsche K, Semmler W, Laun FB. Novel spherical phantoms for
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Ashoor and Khorshidi Q-ball imaging under in vivo conditions. Magn Reson Med 2011; 65:190–194 6 0. Gangstead SL, Song AW. On the timing characteristics of the apparent diffusion coefficient contrast in fMRI. Magn Reson Med 2002; 48:385–388 61. Jin T, Zhao F, Kim SG. Sources of functional apparent diffusion coefficient changes investigated by diffusion-weighted spin-echo fMRI. Magn Re-
son Med 2006; 56:1283–1292 62. Fujita N. Extravascular contribution of blood oxygenation level-dependent signal changes: a numerical analysis based on a vascular network model. Magn Reson Med 2001; 46:723–734 63. Boxerman JL, Bandettini PA, Kwong KK, et al. The intravascular contribution to fMRI signal change: Monte Carlo modeling and diffusionweighted studies in vivo. Magn Reson Med 1995;
34:4–10 64. Clark CA, Hedehus M, Moseley ME. Diffusion time dependence of the apparent diffusion tensor in healthy human brain and white matter disease. Magn Reson Med 2001; 45:1126–1129 65. Duong TQ, Kim SG. In vivo MR measurements of regional arterial and venous blood volume fractions in intact rat brain. Magn Reson Med 2000; 43:393–402
F O R YO U R I N F O R M AT I O N
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Special Articles Original Research
Estimation of the Number of Compartments Associated With the Apparent Diffusion Coefficient in MRI: The Theoretical and Experimental Investigations Mansour Ashoor 1 Abdollah Khorshidi2 Ashoor M, Khorshidi A
Keywords: apparent diffusion coefficient, compartment, DWI, gradient factor, Monte Carlo method DOI:10.2214/AJR.15.14497 Received January 29, 2015; accepted after revision September 13, 2015. 1 Radiation Application Research School, Nuclear Science and Technology Research Institute, Tehran, Iran. 2
Department of Medical Physics and Biomedical Engineering, Tehran University of Medical Sciences, PO Box 141556559, Tehran, Iran. Address c orrespondence to A. Khorshidi ([email protected]). Supplemental Data Available online at www.ajronline.org. AJR 2016; 206:455–462 0361–803X/16/2063–455 © American Roentgen Ray Society
OBJECTIVE. The goal of the present study was to estimate the number of compartments and the mean apparent diffusion coefficient (ADC) value with the use of the DWI signal curve. MATERIALS AND METHODS. A useful new mathematic model that includes internal correlation among subcompartments with a distinct number of compartments was proposed. The DWI signal was simulated to estimate the approximate association between the number of subcompartments and the molecular density, with density corresponding to the ratio of the ADC values of the compartments, as determined using the Monte Carlo method. RESULTS. Various factors, such as energy depletion, temperature, intracellular water accumulation, changes in the tortuosity of the extracellular diffusion paths, and changes in cell membrane permeability, have all been implicated as factors contributing to changes in the ADC of water (ADCw); therefore, one may consider them as pseudocompartments in the new model proposed in this study. The lower the coefficient is, the lower the contribution of the compartment to the net signal will be. The results of the simulation indicate that when the number of compartments increases, the signal will become significantly lower, because the gradient factor (i.e., the b value) will increase. In other words, the signal curve is approximately linear at all b values when the number of compartments in which the tissues have been severely damaged is low; however, when the number of compartments is high, the curve will become constant at high b values, and the perfusion parameters will prevail on the diffusion parameters at low b values. Therefore, normal tissues will be investigated when the number of compartments and the ADC values are high and the b values are low, whereas damaged tissues will be evaluated when the number of compartments and the ADC values are low and the b values are high. CONCLUSION. The present study investigates damaged tissues at high b values for which the effect of eddy currents will also be compensated. These b values will probably be used in functional MRI. he apparent diffusion coefficient (ADC) can identify the random molecular motion of water in the brain and reveal acute ischemic injury within minutes of the onset of stroke. The ADC, which is a measure of the mean free path of water, is consistently reduced in areas of the brain affected by ischemia in subjects affected by acute stroke; however, the acute pathophysiologic mechanisms involved in this phenomenon are not completely understood. The importance of ADC and its association with ischemic tissue have been extensively studied. Likewise, tissue microstructure, cellularity, and nerve fiber integrity may also be evaluated using the ADC [1–8]. When a restriction of the mean random molecular motion of water occurs, the ADC will change. Use of the ADC provides an opportunity to
T
obtain valuable information on the physiologic parameters associated with disease detection and tumor delineation, in addition to data that can aid in the differentiation of normal and damaged tissues. Energy depletion, temperature, intracellular water accumulation, changes in the tortuosity of the extracellular diffusion paths, and changes in cell membrane permeability have all been implicated as factors contributing to changes in the ADC of water (ADCw); therefore, they, along with the number of compartments, have been considered in the new proposed model in this study. The increase in the ADCw appears to occur concomitantly with the loss of membrane structure and the loss of barriers to diffusion. It is possible that other MRI measurements, such as cerebral blood flow, phosphorus or proton spectrosco-
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Ashoor and Khorshidi py, or a combinations of these measurements, may improve tissue characterization. Tissue metabolism results in the generation of waste products that are eliminated through the capillary network to constitute deoxygenated venous blood. Remarkably, this process depends on many parameters, such as blood pressure, blood velocity, capillary network density and geometry, capillary wall permeability, nutrient or oxygen diffusion, and extraction rates, the latter of which can be grouped into three categories: microvascular anatomic and histologic findings, blood microcirculation, and blood-tissue exchanges [9–19]. All elements in these categories may be considered as pseudocompartments. Biologic tissues are a finite and inhomogeneous medium, because they are heterogeneous and contain multiple subcompartments. Von Meerwall [19] showed that, for each measurement volume, the ADC exists as a unique parameter, because most tissues have multiple subcompartments, including, at a minimum, intracellular and extracellular compartments. Under the assumption that measurement times are short, diffusion is unrestricted in each subcompartment and there is no exchange; therefore, MRI signal attenuation is associated with both the density of the molecules diffusing in each subcompartment and the number of molecules present. Koenig et al. [20] used a method in which all subcompartments were separated by fitting the data with the use of multiexponential decay; however, they indicated that when measurement times are longer, a different situation occurs. First, restricted diffusion may appear in the smallest subcompartments. Second, molecular exchanges may be seen between communicating compartments, so the analytic solution becomes difficult. Again, this method is related to the density of the molecules diffusing in each subcompartment and the number of subcompartments. However, this approach is consistent with that used in nuclear MRI dispersion studies, which have suggested that cell membranes can be ignored on the nuclear MRI time scale. In general, the origin of the water signal and the multiexponential function in DWI are not yet fully understood [21–30]. The possible role of compartmentalization in water signal decay on DWI studies of biologic systems has been investigated using biexponential analysis, but such studies have not included internal correlation multiexponential analysis [31–35]. The goal of the present study was to estimate the number of compartments and the
456
mean ADC value with the use of the DWI signal curve. We propose a useful new mathematic model that includes internal correlation among subcompartments with a distinct number of compartments. The DWI signal is simulated to estimate the approximate association of the number of subcompartments with the molecular density, with density corresponding to the ratio of the ADC values of the compartments, as determined using the Monte Carlo method. Although this model is applied mathematically, its effects will be reflected in the ADC values and the volume fractions, in terms of how the ratio of some coefficients relates to the number of compartments. The lower the coefficient is, the lower the contribution of the compartment to the net signal will be. Materials and Methods Theory
When introducing the intravoxel incoherent motion model in 1986, Le Bihan et al. [36] concluded that diffusion measurements can also be sensitive to vascular blood motion and that a DWI sequence can be used to measure both diffusionand perfusion-related parameters. Depending on the permeability of the barriers that divide these compartments, exchange and transport between the compartments should be considered. There appears to be a correlation among these factors; therefore, the proposed model may very well be able to indicate this internal relationship on the basis of the coefficients. As the signal decays exponentially alongside the product of the diffusion coefficient and the b value, blood magnetization dominates the signal decay at low b values. Most measurements of diffusion in biologic tissues take the form of an ADC. When measurement times are short, signal attenuation is expressed as follows: N s(b) = ∑ p × e–b × D s(0) i = 1 i
i
(1),
where s is the signal strength, b is the gradient value, pi is the density of molecules diffusing in compartment i, e is the abbreviation of exponential function, Di is the associated diffusion coefficient, and N is the number of subcompartments. Measurements with low b values will be more sensitive to fast diffusion components. The ideal approach would be to separate all subcompartments by fitting the data with a multiexponential decay [35]. By applying the central limit theorem for N compartments when there are long diffusion times, one can justify the use of a single apparent diffusion constant Da: N
Da = ∑ pi × Di i=1
(2).
This approach is also consistent with the approaches used in nuclear MRI dispersion studies. When faced with intermediate diffusion times or various permeability times, one must take into account the geometric arrangement of and the diffusion coefficients for each compartment, as well as their rates of exchange [37]. As stated elsewhere, there are various parameters that play key roles in brain behaviors [38]. There are many compartments of the brain, such as neurons, astrocytes, glia, and extracellular space, that perform their duties alone but nonetheless have an effect on one another, causing the ADC values to be very complex in terms of microscopic-level measurements [39–41]. One of the reasons that we cannot expect the size of tissue compartments to remain constant is because water can shift within the compartments. When the size of a compartment changes, the ADC will also change. A nuclear MRI signal that is sensitized by the diffusion gradients and that includes many compartments is proposed in the following new purely mathematic formula: d ns d n − 1s d n − 2s + an − 1 n − 1 + a n − 2 n − 2 + dbn db db d n − 3s an − 3 n − 3 + ... + a0s = 0 db n
n
a n − 1 = − ∑ D i, an − 2 = ∑ D i × Dj,..., i=1
i = 1, j = 1 (i ≠ j)
(3),
n
a 0 = (−1) 1 s(b) = n−1 s(0)
(
n
∑e
i=1
−Di × b
n
∏D i=1
−
∑
i
n i=1
Di × e−D × b
∑
i
n i=1
Di
)
where b is the gradient factor, D1, is the ADC value in compartment i, n is the number of subcompartments, d n/db n is a symbol for the nth derivative, a is the coefficient in the series, e denotes exponential function, and j represents the change from 1 to n. Although this model is purely mathematic, some microscopic-level tissue measurements that are not yet fully understood, such as the ADC and the volume fractions, will be reflected because the ratio of some coefficients will be theoretically related to the density of the compartments. The lower the coefficient is, the lower the contribution of the compartment to the net signal will be. This model is for more than two compartments (Fig. S1, supplemental data, can be viewed in the AJR electronic supplement to this article, available at www. ajronline.org). Equation 3 indicates the internal correlation among compartments that makes known some parameters and problems related to the field of diffusion and likely also perfusion, in addition to showing the ulterior development of the density of the molecules diffusing in compartments. For instance, in a two-compartment model, one may use equation 3 to obtain the relationship that follows:
AJR:206, March 2016
Compartments Associated With ADC on MRI s(b)
=
Df Df + D s
× e−D × b + s
Ds Df + Ds
1
× e−D × b (4), f
s(b) s(0)
=
1
(
Df + Dm
2 D f + Dm + Ds
n = 2; mean ADC = 0.016 3; 0.04
where Df and Ds are the ADC values of the compartments with fast and slow diffusion, respectively; s denotes signal strength, e is the exponential function, and b represents the gradient value. Equation 4 is approximately in agreement with findings from previous studies [12, 41, 42]. Similarly, in the three-compartment model, the equation is as follows:
4; 0.0716 5; 0.1117
0.9
6; 0.1596 7; 0.2155
0.8
8; 0.2794
10; 0.4312 12; 0.6137 15; 0.9497
0.7
20; 1.6677
0.6
× e−D × b + s
(5),
30; 3.7028
0.5
40; 6.5352 50; 10.1605
where Dm defines the ADC value for the compartment with a medium rate of diffusion, s denotes signal strength, e is the exponential function, and b represents the gradient value. In general, one may estimate the number of compartments and the mean ADC value, with the use of equation 3 on the DWI signal curve.
0.4
100; 40.299 200; 160.3442
Materials
0.1
Df + Ds D f + Dm + Ds
× e−D
m
×b
+
Dm + Ds D f + Dm + Ds
)
× e−D × b f
As indicated, equation 3 has been proposed for use in estimating the number of compartments and the mean ADC, and the signal is simulated using the Monte Carlo method, with random numbers assigned to the diffusion constants. The random number distribution is gaussian, with a mean of 0.5 (range, 0–1) used for the diffusion coefficients. The simulation was performed for various values of n (2–1000), as shown in Figure 1. When n = 2, the fitted curve on the simulated data had a low curvature, indicating similar effects of diffusion and perfusion on all b values.
Animal Model
To assess the proposed model, six male Wistar rats (weight, 300–450 g) that were subjected to embolic stroke underwent imaging performed with a 7-T MRI system. DWI appears to be useful for the early detection of stroke. The method of inducing embolic stroke and administering recombinant tissue plasminogen activator as treatment has been previously described. In brief, an aged white clot (prepared 24 h before ischemia) is slowly injected into the internal carotid artery to block a major coronary artery. This model of embolic stroke provides a relatively reproducible infarct volume localized to the territory supplied by the major coronary artery [43–45].
Measurements Made Using a 7-T MRI System
MRI measurements were performed using a 7-T superconducting magnet with a 20-cm bore interfaced to a console. An actively shielded gradient
s(b)/s(0)
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s(0)
80; 25.8361
250; 250.3257 300; 360.2459
0.3
350; 490.1639 400; 640.0068 450; 809.6044
0.2
0
500; 999.3752 800; 2560
1000; 3990 31; 3.950 (Damaged) 41; 6.8629 (Normal) 0
1000 2000 3000 4000 5000 6000 7000 8000 9000 10,000 b Value (s/mm2)
Fig. 1—Simulated DWI signal curve. Various numbers of compartments (n) and various mean apparent diffusion coefficient (ADC) values (expressed as value × 10 −4) are shown. Damaged = damaged tissue; normal = normal tissue; s(b)/s(0) = signal attenuation alongside product of diffusion coefficient and b value.
coil set with a 12-cm bore that was capable of producing magnetic field gradients of up to 20 gauss/ cm was used. The radiofrequency pulses were applied using a saddle coil (diameter, 7.5 cm) that was actively decoupled by transistor-transistor logic control from the 1.4-cm surface receptor coil, which was positioned over the centerline of the animal’s skull. Stereotactic ear bars were used to minimize movement during the imaging procedure. While MRI measurements were performed, anesthesia was administered using a gas mixture of 69% nitrous oxide, 30% oxygen, and 0.75–1% halothane. The mean (± SD) rectal temperature was maintained at 37°C ± 0.5°C, with the use of a feedback-controlled water bath. A modified FLASH sequence was used to reproduce positioning of the animal in the magnet at each MRI session [46]. DWI measurements were performed 24 hours after the onset of embolization. The ADCw value was determined using the Stejskal-Tanner sequence with three b values (10, 800, and 1800 s/mm2) in each of three diffusionsensitizing directions, with use of the following imaging parameters: TR/TE, 1500/40; number of slices, 7; FOV, 32 mm2; and matrix, 128 × 64.
Each scan required an acquisition time of 3.2 minutes. The total time required for acquisition of the entire sequence was approximately 14.4 minutes.
Data Analysis
Two ROIs were selected for the measurement of MRI parameters. The mean and SD of ADCw values for the parenchymal tissue (not the CSF) in the contralateral hemisphere were measured, and the threshold values used to determine the number of pixels with abnormal values in the ipsilateral hemisphere were defined using the mean ADCw minus 1.5 SD in the contralateral hemisphere, where this value is heuristic. Homologous ROIs in the ischemic tissue were also measured in the contralateral hemisphere. A paired t test was used to compare the two ROIs, with statistical significance denoted by p = 0.05. The monoexponential intravoxel incoherent motion model was used to calculate the ADC, with the use of the following equation: s(b) = s(0) × exp(−b × ADC), where s is the signal strength and b is the b value [47]. Because the diffusion of water may be directionally dependent (anisotropic), measurement in only one di-
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Ashoor and Khorshidi 1
0.8
s(b)/s(0)
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0.9
0.7
MRM-Inglis [11], 2011 n = 7; mean ADC = 0.2152
0.6
9; 0.3509
8; 0.2797
6; 0.1595 (Optimum) 4; 0.0717
0.5
5; 0.1117 SUM_Diff (n = 7; Inglis [11])
0.4
SUM_Diff (n = 8; Inglis [11]) SUM_Diff (n = 9; Inglis [11])
0.3
SUM_Diff (n = 6; Inglis [11]) (Optimum) SUM_Diff (n = 4; Inglis [11])
0.2
SUM_Diff (n = 5; Inglis [11])
0.1 0
0
1000 2000 3000 4000 5000 6000 7000 8000 9000 10,000 b Value (s/mm2)
Fig. 2—Simulated DWI signal curve upon which data from Inglis et al. [11] are fitted to estimate number of compartments (n) and mean apparent diffusion coefficient (ADC) values (expressed as values × 10 −4 mm2 /s). Optimized values are six compartments and mean ADC value of 0.16 × 10 −4 mm2 /s. MRM = magnetic resonance microscopy; SUM_Diff = summation of differences in MRM-Inglis curve and simulated curves (optimum curve, which indicates minimum difference in comparison with MRM-Inglis curve, was acquired with use of six compartments and mean ADC value of 0.1595 mm2 /s); s(b)/s(0) = signal attenuation alongside product of diffusion coefficient and b value. Fig. 3—Apparent diffusion coefficient map created by imaging of rat brain. Normal tissues versus damaged tissues are shown, with damaged tissues depicted as bright segment on right side of map.
rection can lead to incorrect interpretation of the image. The diffusion gradient was therefore applied in each of three orthogonal directions (x, y, and z), and the mean of these measurements was calculated to determine the trace of the diffusion tensor, which is reported to minimize the effects of diffusion anisotropy [48]. Ideally, one should consider all requisites for dynamic modeling of the system—namely, fluid type and volume, thermal dynamics, and various parameters involved in capillary spaces. The ADC measurement is,
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in fact, a weighted mean of the diffusion coefficients of the various compartments.
Results The number of compartments may also play a role similar to that of the partitioning that is produced in all phases of molecular diffusion by the gaussian functions, with various mean expectation values that are independent of each other but can act together, with each of them causing a relative decrease
in the signal fraction. These fractions are able to characterize the many various clusters in the diffusion map, resulting in differentiation of tissues with the related grades. The ratio of the coefficients of the compartments will take into account any exchanges occurring between the compartments. Figure 1 shows that, with an increased number of compartments, the mean ADC value and the signal curve will increase. In this proposed model, damaged tissues can be characterized at low b values, and the results of the simulation illustrate that when the number of compartments decreases, the signal curve will become linear at a high b value; however, when the number of compartments increases, the curve will be constant at these values. As an example, the data published by Inglis et al. [11] were mapped on simulated curves; for the optimum curve, the total number of compartments was six and the ADC value was 15.95 × 10−6 mm2/s, as shown in Figure 2. The ADC map formed using the protocols is shown in Figure 3. In the damaged tissue region on the ADC map, the values were decreased relative to the values in the normal tissue region on the ADC map, because of stroke onset. For calculation of the ADC value, suitable areas were chosen after image processing procedures were applied to reduce noise and artifacts. Therefore, the mean ADC values for the damaged and normal tissue regions were 392 ± 34.1 × 10 −6 and 659 ± 40.7 × 10 −6 mm2/s, respectively [49]. Table 1 presents the ADC values and the number of the compartments in the damaged and normal tissue regions, on the basis of the proposed model. Experimental ADC values were derived from the damaged tissue region, as shown in Figure 3, and were in good agreement with simulated data presented in Figure 1. For the damaged tissue region, 31 compartments and a mean ADC value of 3.95 × 10−4 mm2/s was found, whereas, for the normal tissue region, 41 compartments and a mean ADC value of 6.86 × 10−4 mm2/s were noted. It seems that, for the damaged tissue region, both the number of compartments and the mean ADC value were lower than the respective values noted for the normal tissue region. The lower the number of compartments is, the more tissue destruction there will be. Discussion Knowing the number of compartments allows one to possibly reduce the effects of restriction and increase the accuracy of the
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Compartments Associated With ADC on MRI
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TABLE 1: The Number of Compartments and the Mean Apparent Diffusion Coefficient (ADC) Values in the Regions of Damaged and Normal Tissue Simulated No. of Compartments
ADC Value (mm2 /s), Mean ± SD
Damaged tissue region
31
3.92 × 10 −4 ± 3.41 × 10 −5
Normal tissue region
41
6.59 ×10 −4 ± 4.07 × 10 −5
0.76
0.595 ± 0.04
Region
Ratio of damaged tissue to normal tissue
ADC value, as well as evaluate the brain at the macroscopic and microscopic levels, in which diffusion is both isotropic and anisotropic. Neeman et al. [50] and Yablonsky et al. [51] have shown synchronous changes in the ADC alongside brain activation. In addition, a comprehensive analysis of the diffusion attenuation curves obtained with different diffusion times may lead to an accurate description of the medium microstructure of capillaries. Many pathologic conditions, such as tumors, stroke, epilepsy, and diffuse axonal injury, can influence the ADCw. Again, when the number of compartments is known, one may be able to detect different degrees of tissue function. Therefore, determining the number of compartments from the DWI signal curve with the use of the proposed model can likely assist in both finding a better description and gaining an improved understanding of the problems related to this field. In the present study, we propose a new mathematic model for estimating the number of compartments that affect the DWI signal. In this model, the b value is adjusted according to the parameters of the pulse sequence. The b value is an effective and very important parameter, because when a DW gradient is used, water diffusion and the random, or Brownian, motion of water molecules can cause attenuation in the signals. Increasing the diffusion gradient amplitude (i.e., the separation of its lobes, or the pulse width of each lobe) results in a higher b value. Accurate determination of diffusion coefficients requires use of the complete attenuation factor and consideration of cross-terms between diffusion and imaging gradients. Compartmentation can be deduced using the diffusion filter effect of the imaging gradients [50, 51]. The b value used in most DWI sequences is approximately 1000 s/mm2. To achieve this b value, the DWI gradient lobe typically is several tens of milliseconds in length, which inevitably leads to a rather long TE. This, in turn, reduces the signal-to-noise ratio and introduces unwanted T2 signal con-
trast in the DW image, and it also causes the T2* signal effects to be more dependent on local microvascular architecture than on T1 signal effects [13, 35, 52–57]. There are technical limitations to producing a high b value; therefore, the proposed model may be used to evaluate damaged tissues over a number of compartments. In this situation, the signal will be significant at low b values. This model may also improve the temporal resolution in the functional maps by modeling the diffusion times in any compartment. To optimize the gradient b value, one has to consider the signal-to-noise ratio and the gradient power available, the accuracy of diffusion measurement, the type of tissues, and the desire to eliminate artifacts arising from macroscopic or microscopic nondiffusional motion. The optimization of this factor largely depends on experimental priorities. Furthermore, one may consider statistical arguments for optimizing the range of b values to be used to obtain the best accuracy in terms of diffusion. Thus, the accuracy of the maps obtained will be improved at low b values. In the present study, the ADCw was measured using three b values (10, 800, and 1800 s/mm2). In the proposed mathematic model, the results of the simulation illustrate that, by increasing the number of compartments, the signal is severely decreased at low b values and is linear with a low slope at high b values. In other words, if the number of compartments is too large, the perfusion effects dominate the diffusion effects, and the signal will be constant at a b value greater than a distinct b value. The value of Di tended to be close to the distinct b value when the number of subcompartments was increased. It seems that in the process of forming damaged tissue, the number of compartments will decrease; likewise, it seems that, in the process of forming normal tissue, the number of compartments will increase. A decrease in the number of compartments gives rise to an increase in perfusion, causing the signal to decrease at a low b value
and creating a negligible change at a high b value. This concept is well indicated in the proposed model in which the ADC values and the number of the compartments noted for the damaged tissue are lower than those noted for the normal tissue. For the normal tissues, the signal will be significant at all b values, because perfusion and diffusion will act together. The pseudodiffusion coefficient, which is an index of unused blood microcirculation in the capillary space, disappears in the model; however, its effect is nonetheless included in this model. Technical limitations notwithstanding, it can be concluded that the steep signal decay observed at low b values in DWI studies of tissue can be attributed to signals arising in the arterial vasculature. This finding supports the argument that the apparent arterial fraction must be taken into account during estimation of the diffusion coefficient and that it may be an incorrect approach to use only two b values. The present study assists in evaluating damaged tissues with low b values for which the effect of eddy currents may be compensated. Therefore, normal tissues can be evaluated at high b values and damaged tissues can be evaluated at lower values. In addition, angiogenesis can be measured indirectly by a variety of different MRI parameters, both structural and functional. These parameters include microvessel density, blood volume, blood flow, vessel size, and tissue oxygenation, among others [42, 58]. It is expected that this proposed model will be able to improve the measurement of such parameters in this particular field of research. One aspect that seems certain is that restricted diffusion must have some influence on the signal, because there is a strong correlation between diffusion-tensor imaging and the organization of white matter pathways [59]. Understanding the roles of tissue structure and the cellular-level physics of water in causing nonexponential signal attenuation is crucial to applications involving water diffusion as a tool for functional brain imaging [60–63]. It seems that when the number of compartments can be determined, temporal resolution may improve. The effect of restricted motion on the diffusion coefficients may be assessed by altering the sequence diffusion time, which is the period between the onsets of the two motion-probing gradients placed at either side of the 180° refocusing pulse in a spinecho sequence. Theoretic models of restrict-
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Ashoor and Khorshidi ed diffusion predict that the observed diffusion coefficient will decrease with diffusion time, because the motion of molecules that are close to a barrier, such as a cell membrane, will be limited. However, the existence of restriction effects in a normal living brain is still a matter of controversy. In fact, even though one recent study found evidence of restriction effects in a rat brain with the use of an effective diffusion time as short as 375 microseconds, most studies have found that restricted motion has little influence on the values of the observed diffusion coefficients for the range of diffusion times (much greater than 10–200 ms) accessible with typical DWI sequences and contemporary MRI systems [64]. Overall, it is not yet clear what determines the rates of diffusive motion in brain tissue— that is, whether the diffusion coefficients are determined by the physics of water at the cellular level, such as in the model discussed by Le Bihan et al. [1], or whether water motion constrained by subcellular microstructure is more important. These changes likely arise from small arterial networks and capillaries, and because large veins usually have much larger diameters and, thus, smaller flow changes, the changes in the ADC values for large veins would be small and insignificant. Each of the changes or parameters can be modeled as a compartment, as shown in equation 3. In conclusion, there is still considerable debate over the origin of the nonmonoexponential signal decay in DWI. Although Le Bihan et al. [1] interpreted the perfusion fraction as the partial volume of the whole capillary vascular fraction, and animal experiments conducted by Duong and Kim [65] strongly suggest that the fast-decaying component noted at low b values should be attributed solely to the arterial blood. Venous blood was shown to have an 80-fold lower pseudodiffusion coefficient, approximately equal to the diffusion coefficient for tissue. In fact, one may obtain the perfusion information from this model by using the diffusion gradients. In equation 1, the density of molecules was included without considering the internal correlation among subcompartments; however, in equation 3, this correlation was considered without directly including the density factor, which was included in the ratio of the coefficients. Conclusion We proposed a new mathematic model with which to characterize the compart-
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ments that intervene on the DWI signal. Although the proposed model is purely mathematic, some microscopic-level tissue measurements that are not yet fully understood, such as the ADC and the volume fractions, will be reflected because the ratio of some coefficients will be theoretically related to the density of the compartments. If the coefficient is very low, the contribution of the compartment to the net signal will be very low. This signal was simulated, with the use of various numbers of compartments, by using the Monte Carlo method. The results of the simulation indicated that, although the number of compartments increases, the signal will severely become very low as the b value increases. In other words, the signal curve was approximately linear at higher b values when the number of compartments was low, but when there was a high number of compartments, the signal curve indicating that tissues were normal was constant. The lower the number of compartments is, the more tissue destruction there will be. Acknowledgments We thank Zheng Gang Zhang, Quan Jiang, and James R. Ewing, for technical assistance, and the Nuclear Magnetic Resonance Facility in the Department of Neurology at Henry Ford Health Sciences Center (Detroit, Michigan), for providing the means and support required for implementation of this study. References
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