Diffusion tensor imaging (DTI) is a widely used magnetic-resonance imaging (MRI) technique, which enables noninvasive assessment of structural integrity of fibrous tissues with a high degree of anisotropy, such as brain white matter and myocardium.1, 2 Specifically, the technique could be exploited for a dynamical follow-up of minor anisotropy alternations due to tissue structural changes arising during progressive disease development, such as schizophrenia3, multiple sclerosis4 and myocardium infarct.5 The method is gaining clinical interest also in applications to tissues with less expressive anisotropy or highly localized compartments with increased level of fiber alignment, such as articular cartilage6, 7, which is a relatively thin tissue with a thickness of up to few millimeters and has a depth-dependent collagen fiber architecture. In DTI, the basic assumption is that diffusive motion of spin bearing particles within the tissue is determined by an alignment of tissue fibers; hence their diffusional anisotropy directly corresponds to anisotropy of the restrictive fibers. The method basically consists of an imaging part, usually employing spin-echo based MRI pulse sequences8, to which a pair of diffusion sensitizing gradients (DSG) is added in order to encode magnetic resonance (MR) signal of spin bearing particles with diffusive motion, resulting into a diffusion-attenuated MR signal. In order to obtain sufficient information on anisotropy of diffusive motion, DSG must be applied in at least six non-coplanar directions to determine six independent elements of the laboratory-frame diffusion tensor.9 In the DT-MRI analysis, diffusion anisotropy is calculated by transforming the laboratory-frame diffusion tensor into the principal frame of reference using diagonalization.10
Determination of diffusion anisotropy can be biased due to instrumentation imperfections9, such as non-optimally calibrated DSG, and due to postprocessing errors.11, 12 It was shown, that a number and directionality of the applied DSG configuration play an important role in a noise propagation in DTI post-processing analysis.12 Specifically, noise propagation in DTI, resulting to noise-induced rotational variance of diffusion tensor ellipsoid, can be reduced by decreasing a condition number (CN) of the b-matrix12 as well as by increasing the signal-to-noise ratio (SNR).11 Therefore, attempts were made to find a robust measure for diffusion anisotropy, such as the lattice index.13 Among all the proposed measures fractional anisotropy (FA) became commonly accepted. Accuracy in determination of diffusion tensor is of a great importance in biomedical imaging as falsely measured tissue anisotropy could lead to clinical misinterpretations and inappropriate treatment decisions.14
Reliability of the DTI method can be efficiently tested either by using perfectly aligned fiber phantoms with an a priori known anisotropy, yielding anisotropic diffusion along the preferential fiber orientation15, 16, or by using completely isotropic materials. In both cases, overestimated apparent anisotropy could arise as an undesirable consequence of the DTI analysis. A fundamental question is, how the DTI factors, specifically SNR and a choice of a DSG configuration, influence accuracy of a diffusion tensor determination. This issue is specifically important in diffusion tensor magnetic resonance microscopy (DT-MRM), in which SNR is usually low due a high diffusion weighting and due to a small voxel size, respectively. The effect of low SNR is more pronounced in some biomaterials with anisotropic diffusion that exhibit short
The main motivation for this study is analysis of factors influencing diffusion anisotropy in DTMRM signal post-processing. The study is organized into two parts. In the first part, the effect of noise propagation from synthetic raw DTI data to the diffusion tensor eigenvalues is examined theoretically for different DSG configurations: selected commonly used, random and isotropic DSG configurations. In the second part, the theoretical results are verified experimentally. DT-MRM was performed for two different materials, tap water with isotropic diffusion and bovine articular cartilage-on-bone samples before and after compression. The study is in particular focused to unfavorable experimental conditions that often arise in DT-MRM and could result in biased diffusional anisotropy.
Single-voxel DTI data of an isotropic medium with a diffusion constant equal to
where
where
using Monte Carlo simulation approach.21 For each DSG, its transformation matrix was calculated using definition
where
and the effective gradient for a spin echo-like DTI pulse sequence including a pair of DSG gradients is equal to
The 3 x 3
The laboratory-frame diffusion tensor,
was then diagonalized to the principal-axis-frame diffusion tensor
The diffusion tensor eigenvalues
For each DSG configuration, either taken from the Table 112 or calculated as random or isotropic directions, diffusion tensor eigenvalues as well as
A list of the analyzed commonly used diffusion sensitizing gradients (DSG) configurations in DTI, adopted from12, along with the corresponding values of
CN[1] | ||||
---|---|---|---|---|
1 | Tetrahedral | 6 | 9.148 | 16.53 |
2 | Cond 6 | 6 | 5.984 | 14.19 |
3 | Decahedral | 10 | 2.748 | 10.70 |
4 | Jones noniso | 7 | 2.560 | 12.13 |
5 | Dual-gradient | 6 | 2.000 | 11.44 |
6 | Jones 10 | 10 | 1.624 | 9.67 |
7 | Jones 20 | 20 | 1.615 | 8.10 |
8 | Jones 30 | 30 | 1.594 | 7.16 |
9 | Papadakis | 12 | 1.587 | 9.29 |
10 | Jones 6 | 6 | 1.583 | 11.04 |
11 | Muthupallai | 6 | 1.581 | 11.11 |
12 | Tetraortho | 7 | 1.527 | 10.68 |
13 | DSM 6 | 6 | 1.323 | 11.41 |
14 | DSM 10 | 10 | 1.324 | 10.02 |
15 | DSM 20 | 20 | 1.668 | 8.43 |
16 | DSM 30 | 30 | 1.430 | 7.45 |
17 | DSM 40 | 40 | 1.401 | 6.87 |
the corresponding ADC and FA were calculated as a function of DA(0 ≤ DA ≤ 1) and SNR(1 ≤ SNR ≤ 100) for
The diffusion tensor quantities were then averaged over the characteristic window of the domain (0.4 ≤ DA ≤ 0.6,20 ≤ SNR ≤ 60) to obtain their representative characteristic scalar values denoted as
where
Simulations were performed using an in-house written program, developed within the Matlab programming environment (MathWorks Inc., Natick, MA, USA). Flowcharts presenting the simulation algorithm for the simulations including DSG configurations from Table 1 and random or isotropic DSG configurations are shown in Figure 1.
Additional numerical simulations were performed in order to investigate the effect of noise propagation in tissues with low diffusional anisotropy. In these simulations, diffusion tensor in the principal frame of reference was considered as a prolate spheroid, which was, for the sake of convenience, oriented with the largest dimension along the z-axis of the laboratory frame of reference (zLAB). In this direction, non-restricted diffusion was assumed,
The primary eigenvector (corresponding to the largest eigenvalue) was equal to
The transformation of the diffusion tensor into the laboratory frame of reference was calculated as
DT-MRM experiments of a tap water phantom with a cone shape were performed on a horizontal-bore 2.35-T MRI scanner (Oxford Instruments, Abingdon, United Kingdom), equipped with microimaging accessories (Bruker, Ettlingen, Germany) and controlled by a Tecmag Apollo spectrometer (Tecmag, Houston TX, USA). The gradient system had top gradients of 0.25 T/m and slew rate of 1200 mT/m/ ms. For acquisition of one-dimensional DTI profiles along cone axis, a spin-echo based 1D DT-MRM sequence was employed. For the sequence, the following imaging parameters were used: 256 acquisition points, field of view 40 mm, spatial resolution of 156 μm, dwell time 10 μs, number of averages 4 (with the half-Cyclops phase cycling scheme), echo and repetition time TE/TR = 30/1030 ms. Square-shaped DSG pulses with
Bovine cartilage-on-bone samples, containing an intact cartilage tissue and the underlying part of a subchondral bone, were carefully dissected from fresh stifle joints of bovine femur bones (provided by a local meat provider) using a commercially available bow saw and a dentist driller set (Meisinger, Neuss, Germany). Samples were cut into cylindrically shaped pieces with a diameter of 6 mm and with an average height of 8 mm, fitting to an NMR tube with an inner diameter of 7 mm. After dissection, the samples were washed in physiological phosphate buffer saline (PBS) and sealed into plastic bags for deep-freezing storage.22 Prior to DT-MRM experiments, each sample was allowed to spontaneously defreeze at temperature of 8°C. Then, the sample was inserted into an NMR tube and immersed into Fluorinert FC-70 (Sigma-Aldrich, Germany), which was used to prevent samples from desiccation. Compression of articular cartilage was obtained by loading a plastic indenter, positioned above the articular surface, with weight-induced pressure of
DT-MRM experiments on articular cartilage-on-bone samples were performed on a 7-Tesla vertical-bore superconducting magnet equipped with microimaging accessories and controlled by the Avance spectrometer (Bruker, Ettlingen, Germany). The gradient system had maximum gradients of 1 T/m and slew rate of 4000 mT/m/ms. Due to relatively fast transversal relaxation processes in cartilage tissue, DT-MRM was performed using a stimulated-echo pulse sequence. The following imaging parameters were used: imaging matrix 256 × 128 (or 128 × 64), field of view 20 × 10 mm, isotropic in-plane resolution of 78 um (or 156 um), slice thickness 2 mm, dwell time 10 μs (or 20 μs), number of averages 32, echo and repetition time. TE/TR = 40/2000 ms. A DSG configuration with
Simulated 2D maps over the DA-SNR domain of the second largest eigenvalue
Figure 3 shows DA−SNR domain-averaged values of diffusion tensor eigenvalues
Figure 4 shows maximal, average and minimal values of FA (Figure 4A) as well as the optimally fitting parameters of Eq. 12, i.e., 𝛼FA and 𝜅FA, along with the corresponding values of
Average fractional anisotropy FA (solid symbols) and the corresponding condition number CN (void symbols) as a function of
Figure 6 depicts results of numerical simulations that were performed with three different preset fractional anisotropies (FA = 0.0, 0.1, 0.3) and with two different SNRs (SNR = 5, 30). In the simulations random and isotropic DSG configurations with
Experimental results of a tap water phantom, examined by 1D DT-MRM are shown in Figure 7 with stack plots of 1D profiles of diffusion tensor quantities (ADC, FA,
Figure 8 shows the experimental DT-MRM results of two bovine cartilage-on-bone samples,
Average values of average diffusion coefficient (ADC) and fractional anisotropy (FA) in three different regions of an uncompressed and compressed cartilage sample obtained with two different spatial resolutions
ADC [10−9 m2/s] | FA[1] | ADC [10−9 m2/s] | FA[1] | |
0.99±0.13 | 0.27±0.13 | 1.12±0.14 | 0.14±0.08 | |
0.63±0.42 | 0.87±0.27 | 1.01±0.82 | 0.82±0.23 | |
1.35±0.11 | 0.24±0.06 | 1.34±0.19 | 0.11±0.04 |
ADC = average diffusion coefficient; FA = fractional anisotropy
The aim of this study is to analyze the effect of the signal-to-noise ratio and DSG configuration on noise propagation in the DT-MRM post-processing analysis for the isotropic (FA = 0) as well as for anisotropic case (FA > 0). The principal findings of the study are: i) noise propagation in the DT-MRM analysis is manifested in an increased deviation of diffusion tensor eigenvalues; ii) deviations of diffusion tensor eigenvalues result to an overestimation of fractional anisotropy, while the average diffusion coefficient remains unchanged; iii) fractional anisotropy bias could be reduced by increasing and by optimizing a DSG configuration to a small condition number at a large number of DSG directions.
The analysis of numerical simulations is based on correlating the diffusion tensor quantities of isotropic medium with the condition number of the transformation matrix and the number of DSG directions. It was evidently shown (Figures 3-5) that the extent of the fractional anisotropy overestimation is dependent of the both parameters. Interestingly, noise propagation with random DSG configurations appears in a form of a symmetric deviation of the largest and the smallest diffusion tensor eigenvalue from the expected value of (Figure 3B), while the second diffusion tensor eigenvalue remains practically unchanged over the entire range of condition numbers. The deviation contributes to an apparent fractional anisotropy considerably, while the deviation is canceled in the calculation of the average diffusion coefficient.
With random DSG configurations, an average fractional anisotropy highly correlates with a condition number, which is consistent with
Fractional anisotropy overestimation in iso-tropic water phantom was studied by DT-MRM only in 1D due to a required large set of isotropic DSG configurations with different.
In comparison to conventional MRI, in MRM, noise has additional origins. Firstly, SNR is usually low due to a much smaller voxel size. Secondly, in MRM imaging gradients are due to an increased spatial resolution large, which in turn contribute to their interaction with DSG. Interaction between DSG and possible background gradients is possible as well. If the contributions are neither compensated by flipping the signs of DSG on alternate averages9 nor properly considered in the calculation of transformation matrix elements according to Eq. 4, variations of the diffusion attenuated MR signal within an individual voxel could be misinterpreted as a noise which could lead to an overestimated fractional anisotropy. SNR can be improved by optimizing magnetization recovery during each repetition time. The echo time, however, should be set as a compromise between two competing effects, diffusion weighting that increases with the echo time and transversal relaxation that decreases with it.
A limitation of the study is that the simulations of anisotropic diffusion were performed only with two different SNR values. However, the selected SNRs were taken from DT-MRM experiment on articular cartilage presented in this study. Another limitation of the study is that the experiments on the water phantom were performed only in one dimension to save experimental time and to thus enable testing of more DSG configurations. The one dimensional DT-MRM approach would be difficult to apply with cartilage samples due to the tissue heterogeneity, which was even more pronounced after the cartilage compression.
In this study a possible overestimation of fractional anisotropy in DT-MRM was analyzed. It was shown by means of numerical simulations and DT-MRM experiments on the isotropic water phantom and low-anisotropy bovine cartilage-on-bone samples that noise propagation from raw data to diffusion tensor eigenvalues can be efficiently reduced by applying DSG configurations with small condition numbers and large numbers of DSG directions.