Susceptibility-related MR signal dephasing under nonstatic conditions: Experimental verification and consequences for qBOLD measurements

Authors


Abstract

Purpose

To experimentally verify a theoretical model describing the MR signal dephasing under nonstatic conditions in a voxel containing a vascular network, and to estimate the stability of the model for qBOLD measurements.

Materials and Methods

Measurement phantoms reflecting the properties of the theoretical model, i.e., statistically distributed and randomly oriented cylinders in a homogeneous medium were constructed by randomly coiled polyamide fibers immersed in a NiSO4 solution. The resemblance between measured and theoretical signal curves was investigated by calculation of root mean squared error maps. Simulated nonstatic dephasing data were evaluated using the static dephasing model to estimate the stability of the model and the influence of input parameters.

Results

The theoretical model describing the MR signal dephasing under nonstatic conditions was experimentally verified in phantom measurements. In simulations, it was found that, by neglecting the effect of diffusion when predicting the MR signal-time course expected in an in vivo measurement of the tissue oxygenation, errors of 10–30% would be introduced into the parameter estimation. The simulations indicate unpredictable results for simultaneous evaluation of blood oxygenation level and blood volume fraction.

Conclusion

Neglecting the effects of diffusion in quantitative BOLD measurements could give rise to substantial errors in the parameter estimation. J. Magn. Reson. Imaging 2011;33:417–425. © 2011 Wiley-Liss, Inc.

IN 1988, OGAWA and his colleagues discovered that small veins in the brain contribute to the contrast in MR images (1). The phenomenon was named blood oxygenation level-dependent (BOLD) effect, because the signal change was found to be dependent on the blood oxygenation level in the veins. The explanation for this effect is that the deoxyhemoglobin in the veins is paramagnetic, in contrast to the surrounding tissue, and therefore distorts the magnetic environment of the surrounding water molecules leading to an increased relaxation of the MR signal.

Quantitative mapping of the BOLD effect (qBOLD) in the human brain is of great interest because such maps can provide important information about tissue viability: Oxygenation is known to be a central factor for the aggressiveness and metastasis tendency of cancer tumors (2). Additionally, hypoxia is a major obstacle to tumor therapy and is associated with poor outcome for cancer patients (3). Furthermore, an increased fraction of oxygen extracted from the blood has been shown to be an indicator of succeeding stroke occurrences in patients with cerebral vascular disease (4, 5).

Although several measurement methods exist for quantification of blood oxygenation, not many are feasible for in vivo applications and hence the practical diagnostic possibilities for quantification of cerebral blood oxygenation are few. Methods that make use of oxygen electrodes are strongly invasive and are therefore almost exclusively used in animal experiments. Near infrared spectroscopy can be used for noninvasive measurements of blood oxygenation, however, only to scan cortical tissue (6). Positron emission tomography, which is the standard method today, has problems with the short half-life of the tracers used for oxygenation measurements. An MR-based method would make quantitative oxygenation measurements more accessible and is therefore of great interest.

Problems to quantify the BOLD effect arise due to lack of comprehensive tissue models with realistic vascular structures and due to difficulties in quantifying the many physiological parameters that influence the MR signal. In 1994, Yablonskiy and Haacke (7) proposed a theoretical model that predicts the MR signal dephasing in brain parenchyma in presence of deoxyhemoglobin. The signal decay is assumed to be dependent on the vascular network, which can be characterized by its geometry, its relative volume fraction, and the amount of deoxyhemoglobin in the blood. The model has been demonstrated in phantom experiments (8, 9) and has in recent years been extended to include effects arising from extracellular fluid and blood (10, 11). The model is based on the assumption that spin-motion due to self-diffusion, can be neglected during the experiment (static dephasing), which makes the MR signal relaxation independent on the capillary radius. However, phantom studies (12) have shown a significant influence of water diffusion for smaller capillaries and it has been generally known for some time that diffusion induced dynamic averaging effect does contribute significantly to the BOLD signal, especially in the close vicinity of capillary networks. These effects have been investigated and characterized by several research groups in the past studying BOLD signal changes (13–16) but have so far been neglected in qBOLD measurements. The theoretical work of including water diffusion has been previously published (17), but an independent verification of the diffusion theory has not been carried out yet. In this work, the diffusion theory was experimentally verified in phantom measurements. In addition, the stability of qBOLD measurements under nonstatic conditions is studied in simulations.

MATERIALS AND METHODS

Theory

A collection of randomly spatially distributed and oriented cylinders can be used to approximate a blood vessel network. When averaging over randomly spatially distributed and oriented cylinders, the histogram of the field distribution approaches a Lorentzian distribution and the signal decay is close to exponential. Because the extent of the field distortion is scaled down with the cylinder radius, the signal is independent on the size of the cylinders, and depends only on the susceptibility difference, Δχ, and the volume fraction, λ, occupied by the cylinders. This, however, is only valid if spin motion (diffusion) during the experiment is neglected.

Neglecting the effects of diffusion, the following expression was derived by Yablonskiy und Haacke (7), for the MR signal from a voxel containing a set of randomly spatially distributed and oriented cylinders,

equation image(1)

where S0 is a coefficient depending on numerous parameters like spin density, flip angle, and repetition time, T2 is the transverse relaxation time, and fc(δω · t) is the, so-called, characteristic function for the set of randomly distributed cylinders,

equation image(2)

The frequency shift introduced by the cylinders, δω, is directly proportional to the susceptibility difference between the cylinders and the surrounding medium,

equation image(3)

Consecutively, the susceptibility difference is related to the venous blood oxygen saturation,

equation image(4)

where Hct is the blood hematocrit level, and Δχdo is the susceptibility difference between completely deoxygenated and completely oxygenated red blood cells.

In case of diffusion, the characteristic function will depend not only on δω but also on the apparent diffusion coefficient, ADC, and the radius of the cylinders, R, (17),

equation image(5)

with

equation image(6)

Phantoms

Single straight polyamide strings were used to determine the susceptibility difference between the strings and solutions with different NiSO4 concentrations, according to the method presented by Sedlacik et al (18). Custom-built phantoms (Fig. 1) with two different string diameters (245 μm and 194 μm) were used for the single string measurements. The single string measurements were performed at 1.5 Tesla (T) using a small loop coil and with the following measurement parameters (parameters for the 194 μm string are given in parentheses): repetition time (TR) = 1000 ms, B0 to string angle = π/2, 256 × 256 matrix, slice thickness = 3 mm, field of view (FOV) = 100 (82) mm, time between two adjacent echoes = 6 (7) ms, Bw = 340 (280) Hz/px, and ζ = 30.9 (28.8)%.

Figure 1.

Single string phantom. The single string phantom was constructed of straight polyamide strings, with two different string diameters (245 μm and 194 μm), in a NiSO4 solution.

Measurement phantoms reflecting the properties of the analytical model, i.e., statistically distributed and randomly oriented cylinders in a homogeneous medium, were constructed using randomly coiled monofilamentous polyamide strings (Polyamide 6-6.6 black, Fa. Krahmer GmbH, Buchholz, Germany) immersed in a NiSO4 solution. A small string diameter (27 μm) was chosen to maximize the influence of diffusion on the signal. For comparison, measurements were carried out using phantoms with a larger string diameter (89-245 μm). Measurements were made at a volume fraction of 2–5% which is the volume fraction of capillary vessels expected in the human brain. The phantoms were constructed using a spherical geometry, which prevented distortions of the magnetic field inside the probe. Hollow plastic balls with a diameter of 150 mm were used as outer containers. Smaller plastic balls, with a diameter of 48 mm, were perforated to allow inflow of liquid and used as inner containers. The inner container was filled with the polyamide stings (Fig. 2).

Figure 2.

Measurement phantom designed to reflect the properties of the analytical model. Hollow plastic balls with a diameter of 150 mm were used as outer containers. Smaller plastic balls, with a diameter of 48 mm, were perforated to allow inflow of liquid and used as inner containers. The inner container was filled with randomly coiled polyamide stings. Phantoms with a string diameter of 27–245 μm and a relative volume fraction of strings in the inner compartment of 2–5% were constructed.

MR Experiments

All measurements were carried out on two Siemens MR Scanners, one with a magnetic field strength of 1.5T and one with a magnetic field strength of 3T (Magnetom Avanto/Trio, Siemens Medical Solutions, Erlangen, Germany) using the standard 12-channel head matrix coil. A gradient echo sampled spin echo (GESSE) sequence (19) was used to sample the MR signal intensity as a function of time in the vicinity of a spin echo. The signal was sampled only in the presence of positive readout gradients.

Data acquisition was performed using the following acquisition parameters: 32 echoes were acquired with a distance of 1.6 ms between two adjacent echoes. All phantoms were scanned using two different echo times: (i) the 7th echo was the spin echo with an echo time of 34 ms. (ii) the 16th echo was the spin echo with an echo time of 68 ms. To sample the shape of the signal curve for short echo times several echoes are required before the spin echo. With the 7th echo as the spin echo, 34 ms is the shortest possible echo time. At this echo time, a minor effect of diffusion is expected, whereas at the longer echo time of 68 ms, the diffusion is likely to have a clear influence on the MR-signal curve. An FOV of 192 mm, a 64 × 64 acquisition matrix, and a TR of 2000 ms was used for all measurements.

R2 values (=1/T2) and ADCs were determined in separate scans using a 32 echoes Carr-Purcell- Meiboom-Gill sequence and a diffusion weighted echo-planar imaging sequence, respectively. A 5-echo gradient echo (GRE) sequence with a time difference between two subsequent echoes of 9.52 ms was used to acquire high-resolved three-dimensional (3D) images used for macroscopic inhomogeneity correction as described by He and Yablonskiy (10).

Data Processing

Raw data from the Siemens scanner were imported into MATLAB for image reconstruction and data processing. The evaluation of the signal measured with the network phantoms, was performed on a pixel-by-pixel basis using MATLAB lsqcurvefit with a nonlinear Levenberg-Marquardt algorithm, and in house MATLAB code. The high-resolution 3D GRE images, also used for inhomogeneity correction, were used to define large regions of interest (ROIs) inside the string containing compartment of the phantoms. The average signal in each ROI was used for evaluation. This was performed to achieve better statistics and to compensate for inhomogeneous packing of the strings. If areas with artifacts from air bubbles were found, they were excluded from the ROI. All phantom measurements were evaluated using both the static dephasing model and the water diffusion model.

Theoretical signal curves were computed, using both the static dephasing model and the water diffusion model, by changing the relative volume fraction (ζ) from 0 to 10% in steps of 0.05% and the susceptibility difference (Δχ) from 0.5 to 2 ppm in steps of 0.0075 ppm. Those ranges cover parameter values likely to be found in the human brain. The ADC measured with the diffusion weighted echo-planar imaging sequence was used as input to the water diffusion model together with the string radius as specified from the manufacturer. Hence, both models have the same number of free parameters. The similarity between every theoretical signal curve and the measured signal curves was examined by calculation of the root mean squared error (RMSE).

Simulations

Simulations were performed to investigate the accuracy and the stability of qBOLD measurements in vivo under nonstatic conditions. The ADC was set to 0.8 · 10−9 m2/s, ζ to 3%, Δχ to 0.6 ppm and R2 to 12 s−1, which are values expected in gray matter. The vessel diameter was set to 50 μm. The simulations were carried out for a 3T system, and a signal-to-noise ratio of 200 was assumed, which is around the highest values found for in vivo measurements. Data were simulated using both the static dephasing model and the water diffusion model and were subsequently evaluated using the static dephasing model. The simulations were performed for two different echo times (TE), one shorter (TE = 34 ms) and one longer (TE = 68 ms).

In Vivo Measurements

A GESSE scan was performed at 3T using 32 echoes with an echo distance of 4 ms, the spin echo occurred at the 13th gradient echo corresponding to an echo time of 108 ms. A TR of 1500 ms and 6 averages resulted in a scan time of 19 minutes. An ROI was placed in white matter and evaluation was made pixel-by-pixel inside the ROI. R2 values, ADCs and high-resolved 3D images used for macroscopic inhomogeneity correction were obtained as described above.

RESULTS

The presence of diffusion effects is visualized in Figure 3. The shape of the signal curves measured at different echo times can be compared if the signal curves are normalized to the spin echo signal and plotted against the time from spin echo. A clear deviation in the shape of the two signal curves can be seen, which is not in agreement with the static dephasing model.

Figure 3.

Characteristic measurement using a cylinder network phantom. a,b: Magnitude and phase images of the 89 μm phantom, acquired with the high resolution 3D GRE sequence at 3T. The evaluation ROI is marked with a white circle. c: Signal as function of echo time, measured with the GESSE sequence using one shorter spin echo time (TESE) of 34 ms and one longer TESE of 68 ms. d: The measured signal (c) normalized at the spin echo signal intensity and plotted against the time from the spin echo. Absence of error bars indicates that the standard error interval was smaller than the size of the data symbol.

To characterize the network phantoms, single sting measurements were performed. Table 1 shows reference values for the volume fraction (ζ), susceptibility difference (Δχ) and relaxation rate (R2) for the measurement phantoms used. In all measurements, the ADC was very close to the self-diffusion constant of water (2 · 10−3 mm2/s).

Figure 4 shows the resemblance between measured and theoretical signal curves. A low RMSE value at a certain combination of ζ and Δχ indicates that the signal curve calculated using those parameter values is similar to the measured signal curve. A curved band of low RMSE values can be seen across the plots (Fig. 4a,b,d,e) indicating a set of signal curves very similar to the measured signal curve. Using a large string diameter and short echo time (Fig. 4a,b), the evaluation performed with the water diffusion model results in essentially the same RMSE plot as for the static dephasing model. Furthermore, the minimum RMSE at the expected relative volume fraction of 3% (Fig. 4c) is located at a Δχ of 0.99 ppm and 1.00 ppm for the static dephasing method and the water diffusion method, in that order. Those values are close to the measured reference value of 1.02 ppm, indicating that both models manage to describe this system well. In contrast, for small string diameters and longer spin echo times (Fig. 4d,e) a large discrepancy is seen between the RMSE plots calculated using the static dephasing model and the water diffusion model. At the expected volume fraction of 2%, the static dephasing model has a RMSE minimum at a Δχ of 0.53 ppm, while the water diffusion model has its minimum at a Δχ of 1.22 ppm (Fig. 4f). The latter is in good agreement with the result from the single string measurement, 1.25 ppm, which can be seen as the reference value.

Table 1. Reference Values of ζ, Δχ, and R2 for the Measurement Phantoms
String diameterζΔχR2
  1. ζ = volume fraction (%), Δχ = susceptibility difference (ppm), R2 = transverse relaxation rate (s−1).

27 μm21.2513.56
245 μm31.0210.28
Figure 4.

RMSE plots illustrating the features of the tissue models. (a,b,d,e) Natural logarithm of the root mean square error (logRMSE) between the measured signal curve and theoretical signal curves. Calculated using the static dephasing model (a:) and using the water diffusion model (b) for the 245 μm phantom at 3 T using an echo time of 34 ms. Calculated using the static dephasing model (d:) and using the water diffusion model (e) for the 27 μm phantom at 1.5T using an echo time of 68 ms. The expected set of Δχ and ζ is marked with a white cross. c,f: Profiles of logRMSE (a,b,d,e) at the expected volume fraction, for the static dephasing model (solid line) and the water diffusion model (dotted line).

The results from the evaluation of the simulated data is shown in Figure 5 and summarized in Table 2. Relative parameter errors of approximately 20–30% are obtained when both Δχ and ζ are simultaneously fitted, also for the static dephasing data. Using a fixed ζ the relative parameter error in Δχ is approximately 3–6%. The parameter error in Δχ when nonstatic data are evaluated and both ζ and R2 have fixed values is 1.7% for the shorter spin echo time compared with a parameter error of 3.7% for the longer spin echo time.

Figure 5.

Evaluation of data simulated with and without diffusion using parameters expected in vivo. A signal-to-noise ratio of 200, a Δχ of 0.6 ppm, a ζ of 3%, and an R2 of 12 s−1 was used for the simulations. All evaluations were performed without taking the effect of diffusion into account to estimate the error introduced when signal curves that have a diffusion effect are evaluated using the static dephasing model. a: Distribution of Δχ and ζ when three parameters were used as fit parameters and a shorter spin echo time was used. b: Distribution of Δχ and ζ when three parameters were used as fit parameters and a longer spin echo time was used. c: Distribution of Δχ and R2 when ζ was fixed at a value of 3% and a shorter spin echo time was used. d: Distribution of Δχ and R2 when ζ was fixed at a value of 3% and a longer spin echo time was used.

Table 2. Δχ and ζ Values Obtained From Evaluation of the Simulated In Vivo Data
 ModelTEΔχζ
  1. Data were simulated using the WD model and the SD model and was subsequently evaluated using the SD model.

  2. WD = water diffusion, SD = static dephasing, TE = Spin Echo Time (ms), Δχ = susceptibility difference (ppm), ζ = volume fraction (%).

Free fitWD340.67 ± 0.172.04 ± 0.69
 SD340.59 ± 0.083.03 ± 0.61
 WD680.39 ± 0.104.11 ± 1.58
 SD680.61 ± 0.122.85 ± 0.77
Fix ζWD340.52 ± 0.03 
 SD340.60 ± 0.03 
 WD680.51 ± 0.02 
 SD680.60 ± 0.02 
Fix ζ and R2WD340.60 ± 0.01 
 WD680.54 ± 0.02 

Figure 6a shows the spin echo magnitude image from an in vivo GESSE scan on a healthy subject. The ROI was positioned in the white matter of the brain where homogeneous ζ and Δχ are expected. Figure 6b shows the sets of ζ and Δχ obtained at evaluation of each pixel inside the ROI in Figure 6a without taking the effect of diffusion into account. The evaluation results in a distribution of parameter sets, similar to the distributions observed during the simulations (Figs. 4, 5), corresponding to the valley of low RMSE values. In the magnified view, it can be seen that the ζ values in the range of the expected 1–2 %, corresponds to Δχ values in the expected range of 0.3–0.5 ppm. When ζ is fixed to 1.5% the obtained Δχ values are almost exclusively found in a range of probable in vivo values (Fig. 6c). Such small Δχ value as the 0.17 ppm obtained when ζ and Δχ are fitted simultaneously to the mean signal from the ROI is not realistic in vivo. The value of 0.37 ppm obtained when ζ is fixed at 1.5% is in more agreement with the expected susceptibility difference. The plotted curves in Figure 6d illustrate the disadvantageous feature that several sets of ζ and Δχ results in more or less identical signal curves. Even though Δχ differs with 0.2 ppm, it is not possible to distinguish between the two curves.

Figure 6.

Parameter estimation in vivo. a: Magnitude image from the GESSE sequence at the spin echo, with the evaluation ROI marked in white. b: Sets of Δχ and ζ values obtained by individual evaluation of every pixel in the ROI marked in (a). The smaller plot offers a magnified view of ζ values from 0 to 5%. c: Histogram of the obtained Δχ values when using both Δχ and ζ as fit parameters (filled diamonds) and when ζ is fixed to 1.5% (squares). d: Mean signal from the ROI marked in (a) and the fitted signal curves when both Δχ and ζ are used as fit parameters (black solid line) and when a fixed ζ of 1.5% is used (red dashed line).

DISCUSSION

According to the static dephasing theory, the signal decay around a spin echo should be independent of the echo time. As seen in Figure 3d, there is a clear difference in signal decay between signal curves measured with different echo times. The difference is more pronounced before the spin echo. This effect was seen for all network phantoms and was more distinct for smaller string diameters. The diffusion model predicts very similar unsymmetrical signal decay dependent on the echo time. In the motional narrowing regimen, the characteristic diffusion time is on the order of the time that it takes for a water molecule to diffuse a distance comparable with the field-creating particle's radius. At longer echo times, the diffusion distance is increased and the dynamic averaging is extended to larger vessels. The signal deviates more from the static dephasing regimen behavior with increasing SE time and decreasing vessel size.

The static dephasing model and the water diffusion model, both assume statistically distributed and randomly oriented cylinders in a homogeneous medium. Monofilamentous polyamide strings are suitable for phantom construction, because they are homogeneous and available in diameters that cover a large range of the capillary diameters. However, even though a relatively small string filled compartment (48 mm diameter) was used in the phantoms, the length of the string required for a reasonable volume fraction of 2–5% is several kilometers for a string with a diameter of 27 μm. To coil the strings, obeying the properties of the model, i.e., statistically distributed and randomly orientated, is not straightforward. Hence, errors are likely to be present due to dissimilarities between the measurement and the theoretical geometry.

Because the evaluation of the phantom measurements is performed in ROIs covering a large part of the string-filled compartment, the effect of inhomogeneously coiled strings is expected to be balanced out. However, this may be true for the static dephasing situation but in case of water diffusion, the result is dependent on the exact vessel radius distribution, and the effect of inhomogeneously coiled strings will not average out by using the mean signal in a ROI. Hence, the errors introduced in the measurements due to inhomogeneous coiling of the strings are likely to be more prominent for the measurements where a strong water diffusion effect (longer echo time, smaller string diameter) is present.

The diffusion model used in this work only considers the diffusion effect near the static dephasing regimen, that is, when the diffusion length is much shorter than the radius of the vessel that causes the field distortion (17). Kiselev and Posse (17) concluded that the model can be used with reasonable accuracy for cylinder diameters down to 25 μm with a maximum echo time of 100 ms. This gives a limitation for the string diameters and the echo times that can be used in the experiment. The smallest string diameter, used to point out the diffusion effect, was chosen just above this boundary. The shorter echo time was selected as short as possible while still allowing enough echoes before the spin echo to reveal the signal evolution around the spin echo. The longer echo time was chosen so that the last gradient echo was acquired within 100 ms. The two spin echo times, 34 and 68 ms, correspond to a diffusion length of 5.8 and 8.2 μm, respectively, which should be small enough to assume static dephasing in case of the 245 μm string phantom.

The effects seen in the phantom measurements are similar to the effects predicted by the water diffusion model. Using a large string diameter and short echo time (Fig. 4a,b,c), no large diffusion effect is expected. Not surprisingly, the RMSE plots for the static dephasing model and the water diffusion model are almost identical. For the phantom measurements performed with a small string diameter and a long echo time (Fig. 4d,e,f), a large discrepancy is present between the RMSE plots calculated with the static dephasing model and the water diffusion model. For the static dephasing model, none of the low RMSE values correspond to a signal curve calculated using the reference values of λ and Δχ. This shows that the static dephasing model does not manage to describe the measured curve.

The water diffusion model is able to predict the measured signal curve. However, studying the RMSE plots, a dark band, representing low RMSE values appears across the plots. All signal curves located in this valley are very similar to the measured signal curve. Furthermore, the RMSE intensity along this track is not continuous but darker spots can be found along the line, indicating local RMSE minimum. The position of the overall minimum will be dependent on the image noise and, hence, the value obtained at evaluation can turn up at any value located in or in the vicinity of the valley. This band of low RMSE values seen across the plot has been previously reported (21) and demonstrates the difficulties to achieve simultaneous estimation of ζ and Δχ. However, the band of low RMSE values indicating similar signal curves is relatively narrow. Hence, if ζ or Δχ is previously known, it should be possible to accurately estimate the other of the two parameters. The same conclusions were recently drawn by Sedlacik and Reichenbach (9) who observed that the global minimum not always corresponded to the correct parameter values and concluded that simultaneous estimation of the two parameters by the cylinder network model is problematic. Furthermore, they found that, if one of the two parameters was fixed to it independently predetermined value, the other was estimated correctly by the local RMSE minimum. This could also be observed during in vivo measurements where simultaneous estimation of ζ or Δχ led to large variations of both parameters.

The experiments show that diffusion has a clear effect on the MR signal in string network phantom experiments. At a relevant echo time of 68 ms, a diffusion effect could be found also for a large string diameter of 245 μm. The self-diffusion of water is smaller in vivo than in the phantoms used and, hence, the effect is expected to be somewhat smaller. However, seeing an effect for the 245 μm string phantom one would expect an evident effect in vivo where the radius distribution of the capillary network ranges from 20 μm down to 4 μm.

Modeling the human brain is not easy due to its great complexity. The two-compartment model, describing venous blood vessels as infinitely long paramagnetic cylinders immersed in a homogeneous tissue matrix, is undoubtedly an oversimplification. As previously mentioned, several attempts have been made to find models that are more realistic. Intravascular signal as well as a long T2 component, presumably originating from interstitial fluid (ISF) and cerebrospinal fluid (CSF), has been implemented with promising results (10, 11). However, in addition to signal from ISF and CSF and the intravascular space, diffusion is present and it will most likely change the outcome of the experiment if it is not accounted for.

The major difficulty in modeling diffusion effects is that it depends on the size and distribution of the objects that are causing the field distortions. The true vessel size distribution is, in general, not known during an in vivo measurement, which complicates the attempt to correct for diffusion effects. The diffusion rate depends on system geometry rather than on the external magnetic field strength. For this reason, high external magnetic field, large susceptibility differences and large size susceptibility inclusions should favor the static dephasing regimen. At higher field strengths SNR is increased and the susceptibility effect is enlarged because the frequency shift is direct proportional to the magnetic field strength. However, both R2* and unwanted susceptibility effects also increase with the field strength, which may cause problems with the correction for macroscopic field inhomogeneities and difficulties due to rapid signal decay.

During the simulations using parameters expected in vivo it could be found that the relation between ζ and Δχ is more well determined when a longer spin echo time is used (cf. Fig. 5a,b). On the contrary, when evaluating the signal curves with shorter echo time a smaller parameter error (i.e., a narrower distribution) is generally obtained than for the signal curves with the longer echo time. This is due to the more prominent deviation from the static dephasing model before the spin echo when water diffusion is present. For the shorter echo time, fewer echoes are sampled before the spin echo, and hence, a smaller deviation from the modeled curve is expected when comparing this curve to the static dephasing model than for the longer echo time. This, however, does not mean that the obtained results agree with the true values. For a spin echo experiment, the diffusion gives the only contribution of susceptibility effects to the relaxation rate. At longer echo times, the diffusion distance is increased and in vivo this would result in that the dynamic averaging is extended to larger vessels.

The practical influence of diffusion effects was investigated through simulations using parameters expected in vivo. It was found that by neglecting the effect of diffusion when predicting the MR signal-time course as expected in an in vivo measurement, the GESSE measurement with the shorter spin echo time overestimated Δχ with 12% while the GESSE measurement with the longer spin echo time underestimated Δχ with 35%. Hence, there could exist an optimal spin echo time where the obtained mean value agrees with the true mean value. This spin echo time, however, would depend on the radius and the apparent diffusion coefficient, and still the relative parameter error would be large when Δχ and ζ is fitted simultaneously. When ζ is fixed but R2 is used as fit parameter, the obtained Δχ is approximately 15% smaller than the true value for both echo times; R2 is correspondingly increased to fit the static dephasing model to the nonstatic dephasing data.

Kiselev et al (20) estimated the vessel caliber in healthy human brain tissue to be in the range of 21–51 μm. This is slightly higher than expected from the known microvasculature anatomy. During the simulations, a vessel diameter of 50 μm was used. Hence, the effect of diffusion seen in the simulations is presumably even stronger in vivo. Consequently, modeling and consideration of diffusion effects are essential for an accurate estimation of the tissue oxygenation using the model presented in this work. However, considering the effect of diffusion requires prior knowledge of the vessel radius distribution within the voxel. The simulations performed here considered a uniform vessel size. Further studies are required where the influence of different radius distributions is examined.

Use of a more realistic tissue model naturally results in a larger amount of unknown parameters. Inclusion of intravascular signal, signal from ISF/CSF and diffusion gives a signal expression composed of three exponential functions with 11 free parameters. It is straightforward to conclude that it will be difficult to obtain stable results when fitting such an expression to a signal-decay as the one shown in Figure 3. Thus, for the method presented in this work to be a reliable oxygenation measurement method, previous knowledge of the exact value of ζ and presumable several other parameters is required. Methods available to measure ζ include PET and MRI using contrast agent. Studies combining blood volume measurements using contrast agent-based MRI with qBOLD measurements would be of great interest.

In conclusion, in this work the prediction of the MR signal as a function of cylinder radius and diffusion coefficient, as proposed by Kiselev and Posse (17), was verified in network phantom measurements. The practical influence of diffusion effects was investigated through simulations using parameters expected in vivo. It was found that by neglecting the effect of diffusion when predicting the MR signal-time course expected in an in vivo measurement, errors of approximately 10–30% would be introduced into the parameter estimation. The exact error, however, could potentially be much larger and is dependent on many factors including the sequence parameters used as well as the radius distribution. Further studies are required where the influence of different radius distributions and sequence parameters is examined. However, the error caused by not taking the effect of diffusion into account is too large to disregard, regardless of the other parameters. The results also revealed that a simultaneous evaluation of Δχ and ζ only can be achieved under very restricted conditions. The possibility to separate the two parameters is presumably exclusively determined by the image noise. However, if separated, the qBOLD technique offers a unique possibility for none invasive estimation of the tissue oxygenation level for therapy planning and monitoring.

Acknowledgements

We thank Jan Sedlacik for practical help with single string measurements and Valerij Kiselev for helpful discussions on statistical modeling and diffusion.

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