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.

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.

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.