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To characterize the DDIF (Decay due to Diffusion in the Internal Field) method using intact animal trabecular bone specimens of varying trabecular structure and porosity, under ex vivo conditions closely resembling in vivo physiological conditions. The DDIF method provides a diffusion contrast which is related to the surface-to-volume ratio of the porous structure of bones. DDIF has previously been used successfully to study marrow-free trabecular bone, but the DDIF contrast hitherto had not been tested in intact specimens containing marrow and surrounded by soft tissue.
DDIF imaging was implemented on a 4.7 T small-bore, horizontal, animal scanner. Ex vivo results on fresh bone specimens containing marrow were obtained at body temperature. Control measurements were carried out in surrounding tissue and saline.
Significant DDIF effect was observed for trabecular bone samples, while it was considerably smaller for soft tissue outside the bone and for lipids. Additionally, significant differences were observed between specimens of different trabecular structure.
The DDIF contrast is feasible despite the reduction of the diffusion constant and of T1 in such conditions, increasing our confidence that DDIF imaging in vivo may be clinically viable for bone characterization.
Fragility fracture is a serious and costly public health issue. With the growth of the elderly segment of the United States (and global) population, there is increasing incidence of metabolic bone diseases. In the U.S. alone, approximately 1.5 million fractures occur annually while the incidence of osteoporosis (low bone mass) is estimated at 14 million patients and the incidence of osteopenia (reduced bone mass) at 30 million, with a concomitant annual projected economic impact of $20 billion (1,2). One-third of patients admitted to U.S. hospitals for hip (femoral neck) fracture die within one year, primarily due to the rapid decline in mobility and general quality of life (3). Improved methods for screening, diagnosis and treatment monitoring of metabolic bone disease are of great importance.
Bone strength is a key predictor of fracture risk (4). Direct mechanical testing of bone strength is destructive and invasive. In clinical in vivo applications, bone strength cannot be measured directly. Instead, it is estimated by the use of biomarkers. The widely accepted clinical biomarker standard for bone strength is bone mineral density (BMD), estimated by Dual-Energy X-ray Absorptiometry (DXA). In fact, the World Health Organization defines osteoporosis in terms of the DXA BMD score (5) which is used in fracture risk assessment tools such as FRAX (6). DXA-derived scores are predictive risk factors for fractures (7). DXA measurements provide two-dimensional measures of bone density (the signal is integrated along the third dimension, depth) and, for that reason, cannot distinguish trabecular architecture which is important to the overall biomechanical function of bone (8,9) and in bone disease (e.g. osteoporosis) (10). Currently, much is still not understood about bone strength and, therefore, noninvasive methods for characterizing bone mechanical strength are needed (4,11). Both trabecular and cortical bones contribute significantly to the overall ability of bones to withstand loading (12,13). Trabecular bone has a higher surface-to-volume ratio than cortical bone, which translates in higher metabolic activity and more rapid turnover (14,15). The classic technique to assess trabecular microarchitecture in patients is histomorphometry of biopsy specimens by light microscopy, most typically of the trabecular bone of the iliac crest (16), while μCT is rapidly becoming the current standard. However, biopsy is generally a method of last resort for both patients and physicians because it is a minor surgical procedure. Furthermore, biopsy may not be statistically representative of the skeleton as a whole, and cannot be performed in load-bearing bones (which are the bones of most concern in fracture risk).
The porous geometry of trabecular bone makes it amenable to study by proton magnetic resonance methods such as spectroscopy (NMR, MRS) and imaging (MRI). Diffusible water, which is present in the interstitial (pore) space of trabecular bone, can be studied using magnetic resonance methods to probe the pore space of trabecular structure. As a result, there is strong interest in applying non-invasive, clinically applicable magnetic resonance methods for the in vivo study of trabecular bone structure (17,18).
Other noninvasive imaging modalities, based on high-resolution MRI (18–25), X-ray Computed Tomography (CT) and Ultrasound (US) (26,27) seek to improve fracture risk assessment by characterizing the microarchitecture of trabecular bone and estimating bone mechanical properties by post-processing of digitized images (28–30). Considerable effort has been expended in the development of high-resolution micro-MRI to spatially resolve trabeculae (18,31–35), as well as in the development of software to derive accurate imaging-based metrics that potentially correlate with biomechanical properties of healthy, aging, or diseased bone. These methods could yield statistical parameters to describe the microarchitecture, building on classic histomorphometry results (19,21,25,28,36–45). Furthermore, solid state MRI is being developed to measure solid bone composition, another factor in bone strength and metabolic state (46). Additionally, much work has been done on the measurement of relaxation times, more specifically the reversible component of the transverse relaxation time (47–49). Relaxation times may be influenced by static microscopic magnetic susceptibility distributions (resulting from the bone structure and geometry), but also by a number of other factors such as molecular diffusion through the resulting magnetic field gradients chemical exchange, and the presence of paramagnetic ions. Therefore, relaxation times may in some cases indirectly reflect biomechanical properties such as Young’s modulus (50); however, voxel-wise values of (or ) are dependent on the voxel size and they reflect the average orientation of trabecular elements relative to B0 (51). Lastly, other MR methods of trabecular bone characterization include manipulation of multiple quantum coherences to modulate the dipolar field, but the interpretation of these results is difficult (52–54).
Here we focus on an alternative method, DDIF (Decay due to Diffusion in the Internal Field). The DDIF contrast is generated from differences in magnetic susceptibility in the tissue. Susceptibility imaging in itself is an active area of research (18,48,50,52,55), but the DDIF contrast is not the same as susceptibility imaging. DDIF was originally used in studies of porous media. It was developed as a spectroscopic tool for characterizing porous media such as the rock of a petroleum reservoir, by harnessing the spin dephasing which occurs when molecules diffuse through magnetic field gradients arising from spatial variations in magnetic susceptibility among the various structural phases of the material (56–58). In porous media the susceptibility differences at the tissue-matrix interface result in internal field gradients that influence the proton signal due to diffusion in the porous space (59). The DDIF method is based on a stimulated echo sequence (56). A particle (for example a proton) diffusing in a porous medium from location x1 to x2, in the absence of external gradients accumulates a phase factor increment Δ given from the relation
where the symbol denotes the local internal magnetic field at location x and and time t, and γ is the gyromagnetic ratio of the proton. The echo signal is proportional to the integral of that phase factor over a measure which is the diffusion propagator in pore space. One may gain a qualitative understanding of the DDIF signal in terms of diffusion propagator eigenfunctions (57,60). In order to gain an intuitive understanding, it helps to visualize the case of a simple two-component material, composed of a water-like fluid and of a solid matrix. The internal field induced within the fluid in the pore space has a complex non-uniform spatial structure. Its gradient is often largest near the tissue interface, i.e. near the solid matrix surface. As a proton diffuses, performing a random walk through the pore space, it accumulates phase in proportion to the local instantaneous internal field (Equation 1). The total phase accumulated depends on the surface-to-volume ratio of the porous material which, in turn, influences Bi. The MR signal is an integral of the phase factor over time, resulting in signal loss (DDIF weighting) as a function of diffusion time. When the diffusion or mixing time t is sufficiently long, each proton effectively samples all paths and the magnetization decays with a T1-like (slower) decay. In the context of a biomedical application (bone imaging) the DDIF contrast is the result of susceptibility-induced diffusion decay from the porous structure of the trabecular bone. The DDIF contrast does not require high-resolution MRI in order to resolve the trabeculae, and it provides a diffusion contrast that is related to the geometrical structure of the trabecular bone.
The DDIF method has been previously applied ex vivo to bovine trabecular specimens whose marrow had been removed and were immersed in saline (61,62). These ex vivo studies demonstrated that DDIF correlated with the mechanical compressive strength of ex vivo bovine bone specimens, that the measured DDIF decay constants correlated with simulated DDIF decay constants based on the trabecular bone surface geometry, and that DDIF measurements agreed well with the susceptibility differences induced by trabecular pores (approximately up to 200 μm). These ex vivo specimens approximate well the two-component idealization, but real bone is not a simple two-component material. Bone marrow is a complex material which to first approximation can be viewed as a two-component liquid itself, comprised of slowly diffusing large lipid molecules (adipocytic, yellow, marrow) and fast diffusing watery haematopoietic red marrow containing iron-carrying hemoglobin. These components differ from each other both in diffusion and in relaxation time constants (63–66). In both the watery and lipid components, water is by far the most diffusible molecule, and is the dominant source of DDIF contrast. However, the reduction of water diffusion constant and of T1 in marrow compared to bulk water may reduce the DDIF contrast and make it harder to measure experimentally. The goal of this study is to examine whether DDIF is feasible in realistic bone specimens.
This study had two major aims. First, we implemented and demonstrated a DDIF imaging sequence with reduced sensitivity to imaging gradients. Second, we examined the performance of DDIF on fresh bone specimens containing marrow and studied at physiological (body) temperature. While this is an ex vivo study, we aimed to realistically approximate the conditions of an in vivo DDIF human study in a clinical context.
Fresh animal bone specimens were acquired from a local meatpacker (Superior Farms Boston) and from local supermarkets (Shaw’s Supermarkets Inc.). The bone specimens had been refrigerated to approximately 4°C prior to purchase. Upon purchase, the bone specimens were cut into smaller pieces able to fit in a standard 50 mL Falcon tube (Becton, Dickinson, Franklin Lakes, NJ, USA), using a hand saw (hacksaw) in order to minimize local damage from the heat generated from friction. Subsequently, the specimens were stored at −15°C. Seven specimens were examined for the purposes of this paper: four BV specimens, two PV, and one BR.
Some soft tissue was intentionally left in the specimens to enable comparison of the DDIF response between soft tissue and bone. Muscle tissue was carefully excised from the specimens. Prior to scanning, each specimen was placed in a 50-mL centrifuge tube and the remaining space in the tube was filled with normal saline (0.9% NaCl). The tube was then wrapped in a continuously heated water blanket and maintained at 34°C. Temperature equilibration was achieved in typically 40 min. Following temperature equilibration the specimen was placed in the coil and the coil was inserted into the magnet with the heating blanket still wrapped on one side, in order to maintain the temperature during the experiment.
Imaging was performed in an Oxford Instruments (Oxford, UK) 4.7 T 33 cm horizontal bore magnet equipped with a Bruker BioSpin (Karlsruhe, Germany) Avance console, a Bruker gradient system capable of 40 G/cm, and a Bruker volume coil of 7 cm inner diameter and 10 cm active length. Shimming was performed using two repetitions of the automated shimming procedure, followed by manual shimming when necessary (typical water linewidth, ~15–20 Hz).
The DDIF imaging pulse sequence is schematically shown in Figure 1. It produces an image of the stimulated echo. All other possible signals are dephased. The algebraic conditions for coherence pathway selection in the stimulated echo using gradient lobes are well known (for example, chapter 10 of (67)). The gradient lobes between the second and third pulses serve to dephase unwanted coherences (gradient lobes labeled “2”). The slice-selecting gradient (gradient lobe “3”) is placed along the third pulse, as well. An additional gradient (“1”) follows after slice selection and before phase encoding, and it dephases the FID produced by the third pulse. In order to balance the coherence pathway for the stimulated echo, the same gradient is repeated (“1a”) between the first and second RF pulses. This is an external gradient that biases the DDIF by inducing a gradient-dependent decay rate, and so it must be minimized. In general, we can write the following equation for the measured DDIF decay rate:
where is the DDIF decay rate due to the internal field and is the extra decay rate induced by the sequence (i.e. by externally applied gradients), defined from the expression
where γ= 2π × 42.58MHz/T is the gyromagnetic ratio of the proton. The gradient was minimized empirically, finding the smallest possible balanced gradients (labeled “1” and “1a” in Figure 1) such that the FID from the third pulse was dephased and the stimulated echo image was formed. That optimization was carried out on a saline phantom. The gradient vector (1.8 G/cm, 1.8 G/cm, 1.3 G/cm) was played for one ms, which corresponds to a bias ( ) for water (D ≈ 2.5 × 10−5 cm2/s). A minimal phase-cycling scheme was used. Two images were averaged with relative phase π between them (ie, [0 0 0], [π π π]). The DDIF encoding time is the period TE (echo time). The DDIF mixing, or diffusion, time is the period TM. For DDIF imaging, the echo time TE is set to a fixed value and a collection of images is acquired with variable TM. Typically, in our experiment TE = 10 ms and seven to nine TM values ranging from 15 ms to 800 ms were acquired at fixed TE. The overall repetition time TR was kept constant (2000 ms). Frequency-selective imaging was achieved by making the 2nd RF pulse frequency-selective (ten-lobed sinc pulse, 600 Hz pulse bandwidth, 10.35 ms). For lipid excitation, the carrier frequency of the sinc pulse was placed at −660 Hz (−3.3 ppm) relative to the water signal. Typical image acquisition times for a 64 × 64 matrix were approximately 4 minutes for the minimum number of averages (Navg = 2 according to the phase-cycling scheme), and field of view 40 mm (0.625 mm × 0.625 mm × 1.5 mm pixel dimensions).
DDIF images were acquired for fixed encoding time TE and a set of variable mixing times TM. The set of all these images was used to produce a DDIF decay curve. Regions of interest (ROI) were drawn by hand on the images to delineate areas containing trabecular bone, saline in the 50 mL Falcon tube, and fat or other remaining tissue outside the cortical bone (MRIcroN, http://www.cabiatl.com/mricro/). Subsequent analysis was carried out using custom-designed code written in MATLAB. For each of the images in a given DDIF series, image intensities were estimated voxel-wise for each of the various ROI. The ROI-averaged intensities were used to produce the DDIF decay curve (for each specimen and for each ROI) which was used to estimate ROI-wise decay constants.
T1 was estimated using a multi-point inversion-recovery spin-echo (IRSE) pulse sequence. The repetition time (TR) was kept fixed for all images, while the inversion time, TI, was varied. T1 was estimated from parametric fits to the image-intensity curves as a function of TI. To account for the fact that TR may not be regarded as infinite relative to T1 and TI, the IRSE data (the ROI-wise averaged signal intensities S) were fitted to the following parametric equation for three unknown parameters, the background constant c, intensity a, and T1 relaxation.
Fitting was performed in MATLAB using lsqnonlin for parameter estimation. The MATLAB algorithm nlparci was used to produce confidence intervals from the parameter estimates, the residuals, and the Jacobian matrix.
Between-groups comparison was performed using repeated-measures ANOVA with Bonferroni correction for multiple comparisons. Criterion for significance was P = 0.05 corrected. Analysis was performed using SPSS (SPSS v12, SPSS Inc., Chicago, IL). Effect sizes in Table 3 were calculated as Cohen’s d.
Weighted mean and weighted variance were calculated following standard methods (69). Given a set of N measurements xi with associated uncertainties σxi, the weighted mean is given by
The weights are calculated from the uncertainties,
The pooled (weighted) variance of the mean is given by
The fitting functions were tested on synthetic a priori data that were created with the three-component formula
where η ~ N(0,σ2). The variance σ2 was calibrated so as to realize a priori (synthetic) χ2 similar to the experimental χ2. In order to mimic the experimental number of degrees of freedom, the time points t for the synthetic data were in the same range and had the same spacing as the MR time variable (mixing time TM), typically, 6–8 timepoints from TM = 10 ms to TM = 800 or 900 ms. The synthetic data were fitted with four models (Table 1): (a) a monoexponential decay (two-parameter model: one amplitude and one decay constant); (b) a monoexponential decay plus constant noise (three-parameter model); (c) a biexponential decay (four-parameter model: two amplitudes and two decay constant); and (d) a triexponential decay (six-parameter model: three amplitudes and three decay constants. A typical result of these numerical experiments (for a particular realization of the vector η) is shown in Table 1. Fitting was performed in MATLAB using lsqnonlin for parameter estimation. The MATLAB algorithm nlparci was used to produce confidence intervals from the parameter estimates, the residuals, and the Jacobian matrix). In general, it is well known that the inverse problem of how to estimate parameters by fitting sums of exponentials to a dataset (here, fitting to the DDIF decay curves) is notoriously ill-conditioned (for example see Ref (70)). The results in Table 1 quantify this for typical fits to synthetic data that approximate the experimental DDIF decay curves. However, the true, experimental, data follow a multi-exponential decay with an unknown number of terms. Therefore, experimental data were fitted with model (b).
Effect size was measured with Cohen’s d (71).
Typical DDIF images are shown in Figures 2 and and3.3. Figure 2 shows DDIF images of a phantom and of a bovine vertebral specimen. The selective-excitation experiment (Figure 2, Left) was carried out on a phantom composed of two tubes attached to each other, a 3 mL syringe tube filled with saline and a second, narrower, tube filled with olive oil. These images were acquired with three different types of RF excitation: with hard RF pulses; with water-selective RF excitation; and with lipid-selective RF excitation. The residual image, defined as the difference between the hard-pulse image and the sum of the water- and lipid-selective images, (oil/water phantom, Figure 2d, Left; bovine specimen, Figure 2d, Right) is within the noise level in each case, verifying that the two selective-excitation images sum to the hard-pulse excitation image. The trabecular area of the bovine vertebral (BV) specimens had very little lipid content. The image using hard-pulse excitation (Figure 2a, Right) is similar to that using the water-selective excitation (Figure 2b, Right); and the DDIF decay curves from the corresponding experiments are also similar (Figure 4 and Figure 6). The situation was different for the porcine vertebral (PV) specimen (Figure 3), where the trabecular ROI exhibits significant lipid content (Figure 3c). The bovine rib (BR) specimen also had a significant lipid marrow contribution (image not shown).
Typical DDIF decay curves from the trabecular ROI are shown in Figure 4. Figure 4(a) shows decays from two bovine vertebral (BV) specimens. Both BV specimens were measured with both the hard-pulse and the water-selective sequences. These two specimens exhibit low lipid content. A replicate water-selective DDIF set of data was acquired three days later on the second BV specimen. The result was consistent with the first experiment, demonstrating the repeatability of DDIF imaging. Figure 4(b) shows all the water (red marrow) trabecular decay curves from all the specimens measured for this work (BV, BR, PV). The bovine specimens fall in a range with the rib specimen being on one end of the range (the lowest curve of the group) and the vertebral specimens having slightly slower decays and being above it. The porcine vertebral specimens resulted in significantly different curves compared to the bovine specimens.
Typical experimental data from the multi-point inversion-recovery spin-echo pulse sequence used to determine T1 are shown in Figure 5. The experimental data are from specimens BV4 and PV2. They show the inversion recovery curves for the trabecular, for the saline, and for the lipid ROI, together with their parametric fits. Parametric fits were estimated as previously outlined in the Methods section.
DDIF decay curves of a typical bovine (specimen BV4) and of a typical porcine specimen (specimen PV1) are plotted in Figures 6 and and7,7, respectively. The best-fitting lines through the data are parametric fits to the monoexponential three-parameter model, as outlined in the Methods section. The hard-pulse (non-selective, NS) and water-selective (S) excitations result in identical decay curves. The saline DDIF curve is clearly very different from the trabecular DDIF curve. Similar data are shown in Figure 7 for porcine specimen PV1. Specifically, shown in Figure 7 are, the water-selective trabecular DDIF decay of PV1 (faster decaying curve) as well as the lipid-selective DDIF curves of both the fatty marrow inside the trabecular ROI, and of the lipid tissue outside the bone. The (intensity-normalized) lipid DDIF curves are identical to each other, demonstrating that there is no DDIF effect for the lipids.
The performance of the algorithms used to solve the inverse problem, and to estimate DDIF decay constants from data, is illustrated in Table 1. Table 1 lists typical numerical results on synthetic a priori data that realistically mimic the experimental DDIF data. Two standard algorithms, functions nls (R) and lsqnonlin (MATLAB) were used. Both functions implement a nonlinear least squares optimization employing the Levenberg-Marquardt algorithm (72,73). The bias in the parameter estimation is inherent to the procedure, as can be seen from the right-most columns of Table 1 (fit to data on noiseless data).
Estimated DDIF decay constants are summarized in Table 2. Data were fitted with a three-parameter monoexponential fit as described in Methods. and estimated T1 relaxation constants, arranged for tissue and ROI. The saline T1 estimate from the PV specimens is very similar to that of the BV specimens, as expected. Statistical comparisons among trabecular DDIF constants across specimens of different trabecular structure (i.e. over different specimens BV, PV, and BR) are shown in Table 3. Effect size was computed as Cohen’s d.
In summary, significant DDIF effect was observed for trabecular bone samples. The DDIF effect was considerably smaller for soft tissue outside the bone, and for lipids. Further, significant differences were observed between specimens of different trabecular structure.
Previous work on trabecular bone specimens whose marrow was removed (62) has shown a significant correlation between DDIF measurement and metrics of structural parameters such as the surface-to-volume ratio. However, such clean bone specimens are sufficiently different from in vivo bone tissue in many ways. First, the T1 values of marrow water and fat can be significantly shorter than those of pure water. Second, the water diffusion constant in marrow is lower than the self diffusion constant in pure water. Third, diffusion in tissue is further reduced due to the presence of cells and other structures. Both a shorter T1 and a reduced diffusion may reduce the DDIF contrast in vivo. Thus, it is an important step to examine the bone specimens with marrow in order to ascertain the feasibility of DDIF measurements for future in vivo applications.
Clear differences between the DDIF signal in the trabecular areas of the bone specimens and the DDIF signal in saline, were indeed observed (Figure 6). Generally, DDIF decay constants in saline are comparable to the T1 relaxation constants, whereas the water signal in the trabecular ROI decays faster, demonstrating trabecular DDIF contrast, in agreement with theoretical considerations about the DDIF (56) and with prior literature on DDIF measurements in porous media and in cleaned bone (56,58,62). In contrast, there was no difference between the DDIF decay in fatty marrow within trabecular bone and the DDIF decay in fat outside the bone (Figure 7, lipid images acquired with lipid-selective RF pulses). This is consistent with the fact that fat molecules are much larger than water and their diffusion constant is significantly smaller, and thus the DDIF effect is much smaller, too. In summary, these results demonstrate the observation of the DDIF contrast for water, and the lack thereof for lipids, in agreement with previous literature on the diffusion of water-like red marrow and of fatty marrow (lipids diffusing much more slowly than water) (63–66).
In this work, specimens of different trabecular structure (bovine vs porcine) were imaged in order to test the feasibility of DDIF on different bones. Indeed, both the DDIF decay time constant and T1 are significantly different between PV and BV and the effect size was correspondingly large (see Table 3). Typically, effect sizes of 0.2–0.3 are considered to be small; 0.5, medium; and above 0.8–1.0 large (71). Effect sizes in Table 3 are clearly large, ranging from 1.4 to 5.4.
In conclusion, an imaging DDIF sequence was applied in an ex vivo study of fresh trabecular specimens. Significantly different DDIF signals among trabecular bone marrow, soft tissue outside the bone, and water were observed. Additionally, significant differences between specimens of different trabecular structure (bovine and porcine trabecular vertebral specimens) were observed. Together, these results indicate that DDIF imaging is possible in spite of the reduction of T1 and diffusion coefficient in bone marrow, and suggest that the application of DDIF in vivo is possible for improving bone characterization.
This work was supported in part by Schlumberger-Doll Research, Cambridge, MA, National Institutes of Health grants S10-RR16811 and P41-RR14075 (National Institute of Research Resources), the Athinoula A. Martinos Center for Biomedical Imaging, and the MIND Institute.