MRI of trabecular bone using a decay due to diffusion in the internal field contrast imaging sequence


  • Dionyssios Mintzopoulos PhD,

    1. Athinoula A Martinos Center for Biomedical Imaging, Department of Radiology, Massachusetts General Hospital, Charlestown, Massachusetts, USA
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  • Jerome L. Ackerman PhD,

    1. Athinoula A Martinos Center for Biomedical Imaging, Department of Radiology, Massachusetts General Hospital, Charlestown, Massachusetts, USA
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  • Yi-Qiao Song PhD

    Corresponding author
    1. Athinoula A Martinos Center for Biomedical Imaging, Department of Radiology, Massachusetts General Hospital, Charlestown, Massachusetts, USA
    2. Schlumberger-Doll Research, Cambridge, Massachusetts, USA
    • Athinoula A Martinos Center for Biomedical Imaging, MGH Building 149, 13th Street / 5th Avenue, Charlestown, MA 02129
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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.

Materials and Methods:

DDIF imaging was implemented on a 4.7 Tesla (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. J. Magn. Reson. Imaging 2011;. © 2011 Wiley-Liss, Inc.

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 United States 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 bone 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 to 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 MR 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 noninvasive, clinically applicable MR 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 postprocessing 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 T2 of the transverse relaxation time T*2 (47–49). Relaxation times may be influenced by static microscopic magnetic susceptibility distributions (resulting from the bone structure and geometry), but also by several 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 T2 (or T*2) 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 that of 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 Δϕ given from the relation:

equation image(1)

where the symbol Bmath image denotes the local internal magnetic field at location x 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). To gain an intuitive understanding, it helps to visualize the case of a simple two-component material, composed of a water-like fluid and a solid matrix. The internal field induced within the fluid in the pore space has a complex nonuniform spatial structure. Its gradient is 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 phase accumulates in proportion to the local instantaneous internal field (Eq. [1]). The total accumulated phase 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 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 hematopoietic 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 with 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.


Animal Bone Samples

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 before 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), using a hand saw (hacksaw) 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 study: four bovine vertebra (BV) specimens, two porcine vertebra (PV), and one bovine rib (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. Before 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 to maintain the temperature during the experiment.


Imaging was performed in an Oxford Instruments (Oxford, UK) 4.7 Tesla (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 Bernstein et al) (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. 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:

equation image(2)

where Rmath image is the DDIF decay rate due to the internal field and Rmath image is the extra decay rate induced by the sequence (i.e. by externally applied gradients), defined from the expression

equation image(3)

where γ = 2π × 42.58 MHz/T is the gyromagnetic ratio of the proton. The gradient was minimized empirically, finding the smallest possible balanced gradients (labeled “1” and “1a” in Fig. 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 (1/Rmath image ≅ 6.8 s) 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 (i.e., [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 second RF pulse frequency-selective (10-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 min 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).

Figure 1.

Schematic of the frequency-selective DDIF imaging sequence. The first RF pulse is a hard pulse (flip angle 90°, duration 0.1 ms). The second RF pulse is a frequency-selective pulse, such as a sinc pulse (90° flip angle, duration 10.35 ms). For nonselective excitation, the second RF pulse is simply replaced with a hard pulse. The third pulse is slice-selective (typical values, flip angle 90°, duration 2 ms). Slice-selective gradients “(3)” are applied during the last pulse. Spoiler gradients for coherence pathway selection “(2)” are applied after the DDIF encoding period following the second pulse. A small gradient “(1)” is applied to spoil the free induction decay (FID) of the third pulse. From the coherence pathway selection rules for the Stimulated Echo, the same gradient must be applied during the encoding period (1a, following the first pulse) for balancing. Since this gradient interferes with the true DDIF signal by introducing an amount of biasing, it is empirically adjusted so as to be the smallest possible gradient spoiling the FID. The DDIF encoding time is the period TE (echo time), typically 10 ms. TM is the mixing, or diffusion period. Typically seven to nine TM values spanning 15 ms to 900 ms are acquired at fixed TE.

Image Analysis

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, 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.

Estimation of T1 and DDIF 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:

equation image(4)

Equation [4] is valid in the limit TE < TI, TR (for example see Haacke et al) (68). The DDIF decay curves were fitted to a monoexponential decay plus constant noise (three-parameter model):

equation image(5)

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.

Statistical Analysis

Between-groups comparison was performed using repeated-measures analysis of variance (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.

Computation of Weighted Averages

Weighted mean and weighted variance were calculated following standard methods (69). Given a set of N measurements xi with associated uncertainties equation image, the weighted mean is given by:

equation image(6)

The weights are calculated from the uncertainties,

equation image(7)

The pooled (weighted) variance of the mean is given by:

equation image(8)

Testing of the Fitting Algorithm

The fitting functions were tested on synthetic a priori data that were created with the three-component formula

equation image(9)

where η ∼ N(0,σ2). The variance σ2 was calibrated so as to realize a priori (synthetic) χ2 similar to the experimental χ2. 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 time points 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 constants); 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 for 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 Transtrum et al) (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).

Table 1. Fitting Algorithm: Typical Performance on Synthetic Data*
A priori parametersR(a)MATLAB(b)
  • *

    Data were synthesized using three-component formulas y = a1exp(−t/b1) + a2exp(−t/b2) + a3exp(−t/b3). Seven time points were used. Additive noise was drawn from a zero-centered Gaussian so as to mimic typical χ2 values of the actual experimental data (see the Methods section). The synthetic data were fitted to (A) a one-component (monoexponential) decay without constant term, a1exp(−t/b1); (B) a one-component (monoexponential) decay with constant term, a0 + a1exp(−t/b1); (C) a two-component (biexponential) decay without constant term, a1exp(−t/b1) + a2exp(−t/b2)+; and (D) a three-component (triexponential) decay without constant term, a1exp(−t/b1) + a2exp(−t/b2) + a3exp(−t/b3). The numerical fit was performed using two standard nonlinear-square fitting algorithms (in R and MATLAB) employing the Levenberg-Marquardt algorithm. The results were very similar, except for case (D), where the R function nls failed to converge. The same realization of additive random error was used in all fits. Given a certain realization of random error, the numerical algorithms systematically result in similar error δbi of the estimated relaxation constants, which implies that the estimated parameters bi are systematically biased as some (unknown) function of the noise.

  • (a)

    R version 2.11.1, function nls.

  • (b)

    MATLAB, function lsqnonlin.

  • (†)

    Fit algorithm preconditioned using regression on the log data to determine a starting point for the Levenberg-Marquardt algorithm.

  • (§)

    Percent difference between the estimated parameter pmath image (pmath image = amath image, bmath image) and the corresponding a priori parameter pi.

  • (¶)

    The three-component nls fit resulted in singular gradient and did not evaluate fitting parameters.

  • (☆)

    Parameter estimation for noiseless data (η = 0).

a20.3 0.60.7-- 
a30.3   --    0.7120.50.3 
b2338.3  750.8--   746.6555.564.2743.4119.7
b32728.7   --    750.272.51660.539.1

Effect Size

Effect size was measured with Cohen's d (71).


Typical DDIF images are shown in Figures 2 and 3. Figure 2 shows DDIF images of a phantom and of a bovine vertebral specimen. The selective-excitation experiment (Fig. 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, Fig. 2d, Left; bovine specimen, Fig. 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 BV specimens had very little lipid content. The image using hard-pulse excitation (Fig. 2a, Right) is similar to that using the water-selective excitation (Fig. 2b, Right); and the DDIF decay curves from the corresponding experiments are also similar (Fig. 4 and Fig. 6). The situation was different for the porcine vertebral (PV) specimen (Fig. 3), where the trabecular ROI exhibits significant lipid content (Fig. 3c). The bovine rib (BR) specimen also had a significant lipid marrow contribution (image not shown).

Figure 2.

DDIF images of a phantom (left) and of a bone specimen (right). Pulse sequences used are: (a) hard-pulse (nonselective), (b) water selective, and (c) fat selective excitation (DDIF TE = 10 ms, TM = 15 ms, TR = 2000 ms). The frequency selection is obtained by a 10-lobed sinc pulse (600 Hz BW) centered at 0 Hz (b) and −660 Hz (−3.3 ppm) (c), respectively. The sum of the selective water-only and fat-only images adds up to the total nonselective image as shown by the residual image (d) defined as the difference d = a − b − c. All images are scaled to the maximum of (a). Following image reconstruction, images were subtracted (d = a − b − c) without additional relative scaling. Left: Images from a phantom composed of one tube with saline and one with olive oil. Right: Typical images from a bovine vertebra specimen (BV4) with a piece of marrow attached, immersed in saline in a 50-mL Falcon tube. The specimen was maintained at 34°C throughout imaging. Note the small lipid content in the BV specimen in the trabecular area. The small water signals outside the Falcon tube are from the heating blanket that maintained the specimen at body temperature.

Figure 3.

Spin-echo and DDIF images of a porcine vertebra specimen (PV1). The reference image (a) is a conventional spin-echo (SE) image (TE = 10 ms, TR = 1000 ms). The DDIF water-selective image is shown in (b); the lipid-selective one is (c) (DDIF TE = 10 ms, TM = 15 ms, TR = 2000 ms). Image (a) is normalized to its maximum, same for images (b) and (c). Note that the trabecular area has much larger lipid content than the bovine specimens. The arrows point to fat tissue outside the bone. The small water signals outside the Falcon tube are from the heating blanket that maintained the specimen at body temperature. DDIF images were acquired at 64 × 64 resolution. The SE was acquired at 128 × 128 resolution. Slice orientation and FOV = 40 mm were identical for all images.

Figure 4.

DDIF data from several trabecular specimens: reproducibility of measurements and variability among specimens. a: Two trabecular decays from two different bovine vertebra specimens No. 2 (specimen BV2) and No. 4 (BV4) (experimental data without fitting the data). NS = nonselective pulse, S = water-selective pulse. The notation (S, r2) denotes a replicate measurement of specimen BV4(S) acquired 3 days later in order to demonstrate the repeatability and robustness of the DDIF decay curves. The dotted lines trace averages of each of the two data sets (no fitting). b: Experimental DDIF curves (water-selective, S) from four bovine vertebrae specimens (BV1–BV4), one bovine rib specimen (BR1) and two porcine vertebra specimens (PV1, PV2) are shown (without fitting to the data). The BV trabecular areas have low lipid content, therefore decays measured with a hard pulse (NS) are not appreciably different from those measured with a water-selective (S) excitation pulse. Notice that BR1 is close to the BV specimens, but the PV specimens are markedly different. All decay curves are normalized to their first (highest signal intensity) point.

Typical DDIF decay curves from the trabecular ROI are shown in Figure 4. Figure 4a shows decays from two 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 3 days later on the second BV specimen. The result was consistent with the first experiment, demonstrating the repeatability of DDIF imaging. Figure 4b 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 with 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.

Figure 5.

Inversion-recovery spin-echo (IRSE) data for determination of T1. The panels demonstrate typical inversion-recovery data for T1 estimation and fit to data (solid lines). Two methods of analysis are shown, averaging on the left panel, and distributional on the right panel. a: Fit to ROI-wise average IRSE intensities. The fit was done on the ROI-wise averaged intensities. These data curves are for specimen BV4 (see Fig. 2) and the data points correspond to separate ROIs drawn onto the trabecular area, the lipid tissue, and the saline in the Falcon tube (squares, trabecular ROI; circles, saline; ×, lipid). Data are plotted on a normalized intensity scale. b: Distribution histograms of T1 for specimens BV4 and PV2. Here, image intensities were fitted voxel-wise and the distribution (histogram) of the resultant T1 was plotted. The arrows indicate the T1 peaks for lipids (L) at approximately 500 ms and the T1 peaks from the trabecular ROI (T) for specimens BV4 (bovine vertebra), broadly centered around 1200 ms, and PV2 (porcine vertebra), centered at approximately 600 ms. The two methods of analysis (averaging, left panel; and distributional, right panel) agree well. Acquisition details: All data were acquired with a spin-echo pulse sequence with an extra 180° inversion pulse. Repetition time (TR) was fixed to 2000 ms for all acquisitions, and inversion time T1 was varied between 200 ms and 1600 ms. The fit to data (Eq. [4]) was designed to take into account effects of finite TR.

DDIF decay curves of a typical bovine (specimen BV4) and of a typical porcine specimen (specimen PV1) are plotted in Figures 6 and 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 (nonselective, 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.

Figure 6.

Examples of DDIF decay curves in bovine specimens. Two typical decays are shown, for trabecular ROI (squares, nonselective pulse; ×, water-selective pulse) and for an ROI drawn on the free saline in the Falcon tube, surrounding the tissue specimen (circles, nonselective pulse; stars, water-selective pulse). Data were fit to a three-parameter uniexponential decay, a0 + a1exp(−t/b1) (solid line, nonselective pulse; broken line, water-selective pulse). The DDIF decays shown here are all for bovine vertebra specimen BV4, corresponding to the images in Figure 2. Decays resulting from selective and nonselective pulses are identical due to the low-lipid content in the trabecular ROIs of those specimens. Note the difference in DDIF decay between the trabecular ROI and the free water (saline) ROI.

Figure 7.

Examples of DDIF decay curves in porcine vertebra. The trabecular DDIF decay curve for water-selective (WS) excitation (squares, solid line, number 1; DDIF decay constant 110 ms) is shown together with DDIF curves from the trabecular lipids (diamonds, solid line, number 2; DDIF decay constant 392 ms) and from the lipids outside the bone (circles, broken line, number 3; DDIF decay constant 383 ms) (normalized data). The lipid curves were acquired with fat-selective (FS) RF excitation (−660 Hz relative to the water peak). The ROIs are indicated on the WS and FS DDIF images on the right side. Note that the DDIF curve from the lipids outside of the bone specimen is identical to the lipid DDIF curve from the trabecular ROI, indicating the absence of DDIF effect for the lipids. DDIF decay constants were estimated from fitting the data to a three-parameter uniexponential decay, a0 + a1exp(−t/b1). Data are for sample PV1.

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 using 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 and estimated T1 relaxation constants, arranged for tissue and ROI, are summarized in Table 2. Data were fitted with a three-parameter monoexponential fit as described in the Methods section. 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.

Table 2. Summary Values of DDIF Decay Time Constant and of Estimated T1 Values
 Region of interest
  • All estimated parameters are measured in milliseconds. T1 estimated using a multipoint inversion recovery spin echo pulse sequence. Data fit to a0 + a1 exp(−t/b1); table entries are mean and standard deviation of decay constant b1.

  • (a)

    One specimen measured. Error bars are estimated Std. Deviations from the nonlinear fit algorithm.

  • (§)

    Nonselective pulse.

  • (†)

    Water-selective pulse (Figure 3b).

  • (¶)

    Lipid-selective pulse (centered on lipids, Figure 3c).

  • Tissue types abbreviated as follows: BR, bovine short-rib; BV, bovine vertebrae; PV, porcine vertebrae.

BV280 ± 301200 ± 701500 ± 4003100 ± 600400 ± 200(§)700 ± 200
BR(a)220 ± 60460 ± 151000 ± 10003300 ± 1700350 ± 140380 ± 14
PV110 ± 30(†)590 ± 91400 ± 1600(†)3300 ± 80500 ± 40
PV370 ± 140(¶)400 ± 200(¶)
Table 3. Statistical Comparisons Among Trabecular DDIF Time Constants
Tissue typeDDIF decay constantsComparisons
Mean ± SD (95% CI) (ms)TypeEffect sizeP-value(†)(a)P-value(†)(b)
  • (†)

    Significance is estimated at the .05 level.

  • (a)

    Adjustment for multiple comparisons: Least Significant Difference (equivalent to no adjustments).

  • (b)

    Adjustment for multiple comparisons: Bonferroni.

  • Tissue types abbreviated as follows: BR, Bovine Short-rib; BV, Bovine vertebrae; PV, Porcine vertebrae.

  • Errorbars: SD, Std. Deviation; CI, Confidence Interval.

  • Effect Size is calculated as Cohen's d.

BR220 ± 60 (60)BR - BV1.40.090.27
BV280 ± 30 (25)BR - PV2.50.02*0.05*
PV110 ± 30 (50)BV - PV5.40.05*<0.001***

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. Furthermore, 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 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 (Fig. 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 (Fig 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 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 despite 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.