Quantitation of the susceptibility difference between trabecular bone and bone marrow: experimental studies

Imaging modalities in the assessment of osteoporosis

Imaging can be helpful in the diagnosis and treatment of osteoporosis. Several imaging modalities have become available to assess bone mass in the peripheral, axial, or entire skeleton. The basic principles, indications, and limitations of each imaging method are presented.

Subchondral bone and cartilage thickness from MRI: effects of chemical-shift artifact

Magnetic resonance imaging (MRI) is the modality of choice for visualizing and quantifying articular cartilage thickness. However, difficulties persist in MRI of subchondral bone using spoiled gradient-echo (SPGR) and other gradient-echo sequences, primarily due to the effects of chemical-shift artifact. Fat suppression techniques are often used to reduce these artifacts, but they prevent measurement of bone thickness. In this report, we assess the magnitude of chemical-shift effects (phase-cancellation and misregistration artifacts) on subchondral bone and cartilage thickness measurements in human femoral heads using a variety of pulse sequence parameters. Phase-cancellation effects were quantified by comparing measurements from in-phase images (TE=13.5 ms) to out-of-phase images (TE=15.8 ms). We also tested the assumption of the optimal in-phase TE by comparing thickness measures at small variations on TE (13.0, 13.5 and 14.0 ms). Misregistration effects were quantified by comparing measurements from water+fat images (water-only+fat-only images) to the measurements from in-phase (TE=13.5) images. A correction algorithm was developed and applied to the in-phase measurements and then compared to measurements from water+fat images. We also compared thickness measurements at different image resolutions. Results showed that both phase-cancellation artifact and misregistration artifact were significant for bone thickness measurement, but not for cartilage thickness measurement. Using an in-phase TE and correction algorithm for misregistration artifact, the errors in bone thickness relative to water+fat images were non-significant. This information may be useful for developing pulse sequences for optimal imaging of both cartilage and subchondral bone.

Quantitative magnetic resonance imaging in the calcaneus and femur of women with varying degrees of osteopenia and vertebral deformity status

Quantitative magnetic resonance imaging (QMRI) allows measurement of two parameters that are related to the integrity of the trabecular bone: R2*, the rate constant of the free induction signal, and trabecular bone volume fraction (BVF), the counterpart of apparent density. In this work, R2* and BVF were measured in 68 women (mean age, 58.2 +/- 9.5 years) of varying spinal bone mineral density (BMD) T scores (mean, -1.37 +/- 1.54) and vertebral fracture status on a commercial 1.5 T whole-body imager using customized image acquisition and processing techniques. Twenty-five of the patients had vertebral fractures, characterized by the total cumulative deformity burden exceeding 200%. R2* was measured in the calcaneus and proximal femur and BVF could be measured in the calcaneus only. On a pixel-by-pixel basis, calcaneal R2* and BVF within each subject were highly positively correlated (r2 = 0.61 +/- 0.11) but the correlation of region-of-interest (ROI) means for different calcaneal sites among patients was weaker (r2 = 0.34; p < 0.0001). The strongest discriminator of vertebral deformity was R2* of the calcaneus, which was lower in the fracture group, consistent with lower trabecular density. Among the calcaneal sites examined, the subtalar region, a location characterized by dense nearly horizontal trabeculae that transmit the stresses imparted by body weight from the tibia to the heel, best discriminated the two groups (p = 0.0001), with 77% diagnostic accuracy as determined from the area under the receiver operating characteristic (ROC) curve (compared with 66% for vertebral BMD). The cavum calcanei, an anterior site of low trabecular density, and the tuber calcanei (the location ordinarily used for ultrasound measurements) also had significantly reduced R2* in the fracture group (p < 0.005 and p = 0.01, respectively). The R2av*, computed as the average of all pixels in the calcaneus, was a strong discriminator as well (p < 0.005). On the other hand, calcaneal BVF was only marginally discriminating (p = 0.05). Among the BMD sites examined, the lumbar spine (average L1-L4) was significant (p = 0.005, 66% diagnostic accuracy), as was the femoral neck (p = 0.01). The data suggest the calcaneus to be suited as a surrogate site to assess vertebral osteoporosis and that R2* is sensitive to alterations in bone quality not captured by density.

Multiple spin echoes for the evaluation of trabecular bone quality

We report a simple and efficient MR method for the evaluation of trabecular bone quality. This technique is based on detection and imaging of Multiple Spin-Echoes (MSE), a manifestation of the dipolar field generated by residual intermolecular dipolar couplings in liquids. In the particular implementation we have used, originally proposed by Bowtell [J. Magn. Reson. 100 (1992) 1; J. Magn. Reson. 88 (1990) 643; Phys. Rev. Lett. 76 (1996) 4971], multiple spin echoes (MSE) are refocused in a two-pulse experiment in the presence of a correlation linear magnetic field gradient G(c). This gradient generates a magnetisation helix and results in the spatial modulation of the sample magnetisation. In heterogeneous systems, the amplitude of the MSE signal depends on sample heterogeneity over a distance d=pi/(gammaG(c)tau) which is half a cycle of the magnetisation helix, thus providing a novel contrast mechanism that can be tuned to a specific length scale. We have exploited this mechanism to study young bovine trabecular bone samples ex-vivo. We show that MSE images present a different contrast from conventional MR images, and that, by varying the experimental parameters, the image contrast can be related to specific trabecular pore sizes. The potential of this technique for the early diagnosis of osteoporotic diseases is discussed.

Characterization of trabecular bone by dipolar demagnetizing field MRI

A multiple spin-echo (MSE) sequence has been applied for the first time to study trabecular bone ex vivo. The second echo generated by the demagnetizing field presents discrete drops in signal intensity for certain values of the pitch of the magnetization helix created by the correlation gradient. These dips may reflect characteristic pore sizes in the trabecular bone specimens. This hypothesis is supported by similar experiments performed on a phantom with uniform pore size distribution. Trabecular bone images weighted in the MSE contrast mechanism are reported.

Trabecular bone volume fraction mapping by low-resolution MRI

Trabecular bone volume fraction (TBVF) is highly associated with the mechanical competence of trabecular bone. TBVF is ordinarily measured by histomorphometry from bone biopsies or, noninvasively, by means of high-resolution microcomputed tomography and, more recently, by micro-MRI. The latter methods require spatial resolution sufficient to resolve trabeculae, along with segmentation techniques that allow unambiguous assignment of the signal to bone or bone marrow. In this article it is shown that TBVF can be measured under low-resolution conditions by exploiting the attenuation of the MR signal resulting from fractional occupancy of the imaging voxel by bone and bone marrow, provided that a reference signal is available from a marrow volume devoid of trabeculation. The method requires accurate measurement of apparent proton density, which entails correction for various sources of error. Key among these are the spatial nonuniformity in the RF field amplitude and effects of the slice profile, which are determined by B(1) field mapping and numerical integration of the Bloch equations, respectively. By contrast, errors from variations in bone marrow composition (hematopoietic vs. fatty) between trabecular and reference site are predicted to be small and usually negligible. The method was evaluated in phantoms and in vivo in the distal radius and found to be accurate to 1% in marrow volume fraction. Finally, in a group of 12 patients of varying skeletal status, TBVF in the calcaneus was found to strongly correlate with integral bone mineral density of the lumbar vertebrae (r(2) = 0.83, p < 0.0001). The method may fail in large imaging objects such as the human trunk at high magnetic field where standing wave and RF penetration effects cause intensity variations that cannot be corrected. Magn Reson Med 46:103-113, 2001.

Theory of FID NMR signal dephasing induced by mesoscopic magnetic field inhomogeneities in biological systems

A theory of the NMR signal dephasing due to the presence of tissue-specific magnetic field inhomogeneities is developed for a two-compartment model. Randomly distributed magnetized objects of finite size embedded in a given media are modeled by ellipsoids of revolution (prolate and oblate spheroids). The model can be applied for describing blood vessels in a tissue, red blood cells in the blood, marrow within trabecular bones, etc. The time dependence of the dephasing function connected with the spins inside of the objects, s(i), is shown to be expressed by Fresnel functions and creates a powder-type signal in the frequency domain. The short-time regime of the dephasing function for spins outside the objects, s(e), is always characterized by Gaussian time dependence, s(e) approximately exp[-zeta(k)(t/tc)2], with zeta being a volume fraction occupied by the objects, t(c) being a characteristic dephasing time, and the coefficient k depending on the ellipsoid's shape through the aspect ratio of its axes (a/c). The long-time asymptotic behavior of s(e) is always "quasispherical"-linear exponential in time, s(e) approximately exp(-zetaCt/tc), with the same "spherical" decay rate for any ellipsoidal shape. For long prolate spheroids (a/c)<<1, there exists an intermediate characteristic regime with a linear exponential time behavior and an aspect-ratio-dependent decay rate smaller than (zetaC/tc).

Cross-sectional study of osteopenia with quantitative MR imaging and bone densitometry

PURPOSE

To evaluation the cancellous bone-induced intravoxel spin dephasing rate (R2') and its relationship to bone mineral density and marrow fat and to examine these parameters as predictors of vertebral fracture status.

MATERIALS AND METHODS

R2' and R2, the rate constants for reversible and irreversible spin dephasing, and marrow fat fraction were measured in the lumbar vertebrae and proximal femur. One hundred thirty-nine subjects (mean age, 62.4 years +/- 11.4 [SD]; 33 men, 106 women) had spinal dual-energy x-ray absorptiometric bone mineral density (BMD) T scores ranging from +3 to -5. R2', BMD, and bone marrow composition as determinants of vertebral fracture status were examined.

RESULTS

Strongest single predictors of fracture status for BMD and R2' were the Ward triangle (r(2) = 0.48) and trochanter (r(2) = 0.37), respectively. Combined, the two parameters and sites increased fracture prediction (r(2) = 0. 62), whereas the combination of multiple BMD sites did not. Multivariate regression involving marrow fat fraction further improved fracture status prediction. R2' was correlated with BMD at all sites, although slopes differed by a factor of up to 2.5, which reflected differences in trabecular orientation relative to the static field. R2, the true transverse relaxation rate, was negatively correlated with marrow fat fraction. A non-age-related increase in marrow fat fraction in osteoporosis parallels earlier findings in animal models.

CONCLUSION

Cancellous bone marrow R2' measured in the proximal femur provides information, which, with BMD, improves prediction of vertebral fracture status.

Quantitative measurements of cerebral blood oxygen saturation using magnetic resonance imaging

A quantitative estimate of cerebral blood oxygen saturation is of critical importance in the investigation of cerebrovascular disease because of the fact that it could potentially provide information on tissue viability in vivo. In the current study, a multi-echo gradient and spin echo magnetic resonance imaging sequence was used to acquire images from eight normal volunteer subjects. All images were acquired on a Siemens 1.5T Symphony whole-body scanner (Siemens, Erlangen, Germany). A theoretical signal model, which describes the signal dephasing phenomena in the presence of deoxyhemoglobin, was used for postprocessing of the acquired images and obtaining a quantitative measurement of cerebral blood oxygen saturation in vivo. With a region-of-interest analysis, a mean cerebral blood oxygen saturation of 58.4%+/-1.8% was obtained in the brain parenchyma from all volunteers. It is in excellent agreement with the known cerebral blood oxygen saturation under normal physiologic conditions in humans. Although further studies are needed to overcome some of the confounding factors affecting the estimates of cerebral blood oxygen saturation, these preliminary results are encouraging and should open a new avenue for the noninvasive investigation of cerebral oxygen metabolism under different pathophysiologic conditions using a magnetic resonance imaging approach.

Normal lumbar vertebrae: anatomic, age, and sex variance in subjects at proton MR spectroscopy--initial experience

Fifty-seven subjects underwent proton magnetic resonance (MR) spectroscopy of the second lumbar vertebra to evaluate single-voxel and multivoxel techniques. Measurements included lipid-to-water ratios, lipid fractions, and line width. These data provide information about vertebral fat content. There was an age-dependent linear increase in fat content and sex dependence. A higher fat concentration was found in men. The observed spectra provide a basis for future study to determine clinical utility of vertebral proton MR spectroscopy.

Strong field behavior of the NMR signal from magnetically heterogeneous tissues

A theory for the behavior of the nuclear magnetic resonance (NMR) signal obtained from magnetically heterogeneous tissues is developed for the limit of a strong external magnetic field. If BO is the magnitude of the external magnetic field, it is found that a free-induction signal decays in a time scaling as 1/Bo, a single-spin echo signal decays in a time scaling as 1/Bo(2/3), and a multiple-spin echo signal decays in a time scaling as 1/Bo(2). Moreover, it is shown that the form of the signal decay for a multiple-spin echo sequence may deviate significantly from an exponential. Numerical results for a model consisting of randomly distributed magnetic spheres are used to confirm the theory. In addition, good agreement is demonstrated between the theory and experimental measurements obtained with particle suspensions. The validity and application of the theory to biological tissues are discussed.

Analysis of ovariectomy and estrogen effects on body composition in rats by X-ray and magnetic resonance imaging techniques

Resistance of bone to fracture--bone strength--has been shown to depend on both the amount of bone and its architectural spatial organization. In vivo magnetic resonance (MR) techniques have the capability of imaging bone tissue, including the trabecular microarchitecture and the marrow composition. We have applied in vivo and ex vivo MR methods to the tibia in an ovariectomized rat model of osteoporosis. Specifically, in vivo high-resolution three-dimensional MR imaging and localized MRS were facilitated by specialized coils and high field magnets, resulting in enhanced sensitivity of detection. As a result, in vivo and ex vivo differences in marrow composition were found between sham-ovariectomized, ovariectomized, and ovariectomized animals treated with 17-beta-estradiol. Estrogen effects were detected in vivo 7 days after surgery (3 days into treatment) as a decrease in the tibial fat signal level. The in vivo effects of ovariectomy were observed 56 days after surgery as an increase in MR image fat signal level and spectral fat/water ratio in the proximal tibia. Ex vivo measurements of tibial marrow water signal discriminated clearly between the sham and ovariectomized groups and showed increased individual variations in the treatment group. Imaging further showed that the highest fat content is observed in the epiphysis. Computed tomography confirmed ovariectomy-induced loss of bone in the proximal tibial metaphysis compared with the sham group. This loss of cancellous bone with ovariectomy is consistent with the MR observations of increases in both fat and water in the metaphysis. These data showed that MR techniques complement X-ray techniques in the bone, water, and fat compositional analysis of the appendicular skeleton in response to ovariectomy and pharmacological treatment.

Relationship between cancellous bone induced magnetic field and ultrastructure in a rat ovariectomy model

The site-dependent variations in trabecular bone morphology were studied in the rat tibia by magnitude and phase difference three-dimensional nuclear magnetic resonance microscopy and image processing, and the implications of ovariectomy were evaluated. Specimens excised from the proximal tibial metaphysis in ovariectomized (n = 7) and intact control (n = 4) rats were imaged at 9.4T with their anatomic axes parallel to the direction of the magnetic field. An echo-offset 3D rapid spin-echo excitation pulse sequence was used to generate phase difference maps, from which the standard deviation of the phase difference, sigma(delta psi), was calculated. In addition, a fictitious rate constant, R2', was calculated from the slope of the exponential portion of the Fourier transform of the phase difference histogram. Trabecular bone volume fraction was also determined in the same volume of interest. The results show strong correlations between bone volume fraction and both sigma(delta psi) and R2', suggesting that these parameters could be useful for nondestructive assessment of trabecular bone volume.

Macroscopic susceptibility in ultra high field MRI

PURPOSE

Magnetic susceptibility provides the basis for functional studies and image artifacts in MRI. In this work, magnetic susceptibility and the associated artifacts were analyzed at 8 T in phantoms and in the human head.

METHOD

A mineral oil phantom was constructed in which three cylindrical air-filled tubes were inserted. This phantom was analyzed with gradient-recalled echo and SE imaging techniques acquired using varying TEs and receiver bandwidths. To visualize the presence of magnetic susceptibility artifacts in the head at 8 T, near axial, coronal, and sagittal GE images were also acquired from human volunteers.

RESULTS

The use of gradient-recalled echo imaging resulted in the production of significant magnetic susceptibility artifacts. These artifacts could be readily visualized in phantom samples containing air-filled cylindrical tubes. In the human head, susceptibility artifacts produced significant image distortion in the skull base region. In this area, susceptibility artifacts often resulted in the complete loss of MR signal. Magnetic susceptibility artifacts were manifested as bands of varying signal intensity in the frontal lobe and temporal bone region. In addition, they produced clear distortions in the appearance of brain vasculature and seemed to accentuate the relative size of venous structures within the brain.

CONCLUSION

When using gradient-recalled echo imaging in combination with relatively long TE values, magnetic susceptibility artifacts can be severe at 8 T. These artifacts could be reduced by increasing receiver bandwidths and by lowering effective TEs. As ultra high field MRI provides a fertile ground for the study of susceptibility artifacts in MRI, improvements obtained at this field strength will have a direct impact on studies performed at lower field strengths.

Adaptation of the human calcaneus to variations of impact forces during running

OBJECTIVE

The influence of varied forces under the heel induced by changes in midsole hardness on adaptations of the human calcaneus during running training was investigated.

DESIGN

A longitudinal study was conducted over a period of 20 weeks with subjects training for 50 km per week on average.

BACKGROUND

The skeletal systems' metabolism acts highly dynamic, governed by mechanical factors. The amount of running training has been shown to increase the bone mineral density in the calcaneus. Mechanical factors have not been controlled in former investigations.

METHODS

Bone quality parameters were determined before and after the training by use of an ultrasound system and quantitative MRI while the mechanics of foot-ground contact were controlled. The total group of 26 subjects was divided into three subgroups based upon different magnitude of forces under the heel inside the shoe.

RESULTS

The biomechanical testing demonstrate no relationship between midsole hardness and external or in-shoe impacts. Bone parameters showed specific differences for all groups which are pronounced in runners with intermediate impacts.

CONCLUSIONS

The observed variations reflect metabolic changes in bone marrow which appear to be effected by the impact magnitude and cannot be characterised as negative.

RELEVANCE

The current data imply that no negative changes of impacts on calcaneal bone were produced by high amounts of training in distance running. The mechanical testing indicates that the potential of modifying calcaneal adaptation directly by varying midsole hardness is limited.

Computer simulations for the optimization of magnetic resonance phase imaging applied in the study of trabecular bone

A new technique for the evaluation of bone trabeculation using magnetic resonance (MR) phase images has been recently presented. This technique calculates the phase variance in a region of interest (ROI) on the phase images of a gradient echo sequence. In this study, a computer program was developed which simulates the phase distribution in gradient echo acquired phase images of a structure that mimics trabecular bone, consisting of a three-dimensional connected network of orthogonal bone struts. Several tests were performed in order to assess the influence of imaging parameters such as the echo time, the pixel size and the slice width on phase variance. The results from this work show that with selection of appropriate imaging parameters, phase variance strongly reflects variations in trabecular bone density. Representative MR experiments were performed in the distal radius to verify the simulation results.

Applications of high-resolution echoplanar spectroscopic imaging for structural imaging

Echoplanar spectroscopic imaging (EPSI) was introduced as a fast alternative for spectroscopic imaging and has been recently implemented on clinical scanners. With further advances in gradient hardware and processing strategies, EPSI can be used to obtain spectroscopic images whose spatial resolution parallels that of conventional anatomic images within clinically acceptable acquisition time. The present work demonstrates that high-resolution EPSI can be used to derive structural images for applications in which spectroscopic information is beneficial. These applications are chemical shift (fat-water) imaging, narrow bandwidth imaging, and T2* mapping. In this paper, the EPSI sequence design and processing strategies are detailed and experimental results in normal volunteers are presented to illustrate the potential of using EPSI in imaging anatomic structures.

Multinuclear solid-state three-dimensional MRI of bone and synthetic calcium phosphates

Multinuclear three-dimensional solid-state MRI of bone, tooth, and synthetic calcium phosphates is demonstrated in vitro and in vivo with a projection reconstruction technique based on acquisition of free induction decays in the presence of fixed amplitude magnetic field gradients. Phosphorus-31 solid-state MRI provides direct images of the calcium phosphate constituents of bone substance and is a quantitative measurement of the true volumetric bone mineral density of the bone. Proton solid-state MRI shows the density of bone matrix including its organic constituents, which consist principally of collagen. These solid-state MRI methods promise to yield a biological picture of bone richer in information concerning the bone composition and short range-crystalline order than the fluid-state images provided by conventional proton MRI or the density images produced by radiologic imaging techniques. Three-dimensional solid-state projection reconstruction MRI should be readily adaptable to both human clinical use and nonmedical applications for a variety of solids in materials science.

Effects of trabecular bone on marrow relaxation in the tibia

The effects of bone on marrow relaxation in the trabecular volume of the most proximal 3 cm in the left tibia were studied with a RF-spoiled gradient echo MRI protocol on a 1.0 T MR unit. The MR measurements were performed on six healthy volunteers, and repeated within one month in order to assess the precision of the method. In the same subjects, the area bone mineral density (BMD, g/cm2) was measured at the left proximal femur using dual-energy X-ray absorptiometry. The calcaneus of the same side was examined with quantitative ultrasound. The marrow T2* relaxation deviated from a mono-exponential decay, and resembled the decay of subcutaneous fat. The shape of the relaxation curve reflected the presence of several spectral components in bone marrow, and was further influenced by the amount and structure of the surrounding trabecular bone. The bone marrow decays showed substantially reduced inter-subject variability after normalisation of the marrow data fit parameters to corresponding values for s.c. fat. This suggests the use of an internal adipose tissue reference in order to correct for diet-related variations of marrow T2* estimates. The mean relative precision of the MR measurements was between 5% and 10% depending on the data fit model. Moderate-to-strong correlations between DXA BMD indices in the proximal femur and MR parameters were found (r(max)=-0.96; p < 0.01), while ultrasound-derived measures of bone strength measured on the calcaneus demonstrated significantly weaker correlations to the MR parameters (r(max)=-0.78; p > 0.05). The method employed in this study showed reasonable precision and a moderate to good correlation compared to other bone parameters derived at the same extremity, and is a promising tool for the use on patients.

Quantitation of intrinsic magnetic susceptibility-related effects in a tissue matrix. Phantom study

A theoretical background and experimental method that allows a separation of intrinsic, tissue-matrix-specific magnetic-field inhomogeneity effects from both macroscopic (large compared with voxel dimensions) and microscopic (on the order of molecular dimensions) inhomogeneities is proposed. Such separation allows one to take full advantage of these tissue-matrix-specific magnetic field inhomogeneity effects to extract information about tissue structure. A method to measure the volume fraction occupied by the susceptibility-perturbing component in a tissue matrix, the R2' relaxation rate constant, and the susceptibility difference between the bulk component and the susceptibility-perturbing component in a tissue matrix has been developed and tested on phantoms. This method offers the potential to assess a variety of tissue parameters, including cerebral blood volume, blood volume and blood oxygenation-level changes in functional MRI, the structure of trabecular bone, and other physiologically important issues.

Comparison of computer simulated and phantom measured phase variance in the study of trabecular bone

A new method using magnetic resonance phase images for the assessment of trabecular bone structure has recently been proposed. To test this method, a mathematical model is developed which calculates the phase distribution in gradient echo acquired phase images of a structure of Pyrex glass rods immersed in a copper sulfate solution. Several experiments were performed using a phantom built in the same way as the structure used in the mathematical model. The results from the model are compared with those from the phantom tests, and the influence of resolution and bone area fraction on the phase dispersion is studied. The good correlation between theoretical and experimental results shows that phase variance increases with increasing resolution and bone density. However, the dependence of variance on bone density is less prominent for large pixel sizes.

A comparative study of trabecular bone properties in the spine and femur using high resolution MRI and CT

The purpose of this study was to use high resolution (HR) magnetic resonance (MR) and computed tomography (CT) images combined with texture analysis to investigate the trabecular structure of human vertebral and femoral specimens and to compare these techniques with bone mineral density (BMD) in the prediction of bone strength. Twenty-nine bone cubes were harvested from 12 proximal femur cadaver specimens and 29 from 8 spines. HR MR and CT images were obtained, and texture analysis techniques were used to assess trabecular structure. Additionally, BMD, elastic modulus (EM), and maximum compressive strength were determined. R2 for EM versus texture measures computed in the MR images was higher (R2 = 0.27-0.64, p < 0.01) in the spine than in the femur specimens (R2 = 0.12-0.22, p < 0.05). R2 values were similar in the CT images. R2 for EM versus BMD was 0.66 (p < 0.01) in the spine and 0.61 (p < 0.01) in the femur specimens. In the MR images, texture measures combined with BMD in a multivariate-regression model significantly increased R2, while improvement was less significant in the CT images. Thus, texture analysis may provide additional information needed to analyze bone strength and quality.

MR protocols for imaging the guinea pig knee

Magnetic resonance images of the femorotibial joints of male Dunkin-Hartley guinea pigs were obtained in two and three dimensions at 2.35 T using a wide range of T1- and T2-weighted imaging sequences. The effect of slice position on visualisation of articular cartilage, bone and periarticular tissues in sagittal and coronal sections was investigated along with the resolution and signal/noise ratio achievable. Based on that survey, a two-dimensional spin echo sequence (repetition time = 1500 ms, echo time = 40 ms) was found to give optimum visualisation of the normal joint anatomy with in-plane resolution of 75 x 150 microns and a 1 mm slice thickness in an imaging time of 25 min. This protocol was also found to be highly effective in distinguishing many features of the spontaneous, osteoarthritic-like pathology found in the joints of older animals compared to juveniles and therefore provides a means of monitoring disease progression longitudinally. Three-dimensional spin echo imaging methods demonstrated focal changes in signal intensity in the articular cartilage of the medial tibial plateau in older animals. The resulting imaging times of several hours, however, precludes their routine use in vivo.

Peripheral measurement techniques for the assessment of osteoporosis

Peripheral measurement techniques have been the first to be developed for the assessment of osteoporosis, and they remain useful. Besides traditional approaches such as radiographic absorptiometry (RA), radiogrammetry, and single-photon absorptiometry (SPA), new peripheral approaches have been developed that offer powerful ways to assess skeletal status in osteoporosis. These include single x-ray absorptiometry (SXA), peripheral dual x-ray absorptiometry (pDXA), peripheral quantitative computed tomography (pQCT), quantitative ultrasound (QUS) techniques, and magnetic resonance imaging (MRI) approaches. This review describes the current role of peripheral imaging techniques vis-à-vis their central imaging counterparts. Peripheral measurement techniques are attractive because equipment cost is substantially lower, radiation exposure is small, and the devices require less space and sometimes are even portable. Additionally, QUS and MRI offer the potential to measure aspects of bone status beyond the limits of bone densitometry. Peripheral techniques represent important diagnostic methods for the assessment of osteoporosis.

Experimental correlation between T2* and ultimate compressive strength in lumbar porcine vertebrae

RATIONALE AND OBJECTIVES

The authors used magnetic resonance (MR) imaging to investigate the correlation between T2* measurements of trabecular bone and the ultimate compressive strength of lumbar porcine vertebrae.

METHODS

Five pigs that weighted 25-32 kg were sacrificed and imaged with a 1.5-T MR system. T2* of the lumbar vertebrae was measured from gradient-echo images. The vertebrae were individually compressed at a fixed speed in the direction of the spine until crushed. The maximum load a vertebra could resist was recorded.

RESULTS

T2* ranged from 7.1 to 14.5 msec. T2* determined from 5-mm coronal sections differed from that determined from axial and sagittal sections (P < .05). Between 2.9 and 5.4 kN of force (296-550 kg) was needed to crush a vertebra. A linear correlation between the ultimate compressive strength and T2* of all vertebrae was observed for all imaging planes and section thicknesses (P < .001, except for 10-mm sagittal images, for which P < .002). The T2* determined for the axial plane showed the best correlation with the ultimate compressive strength (r = -0.83).

CONCLUSION

The correlation between T2* values and vertebral strength indicates that MR imaging may potentially be used to predict fracture risks in patients.

T2 relaxation time of bone marrow water and lipid: correlation with serum ferritin in normal individuals

PURPOSE

This study investigated the correlation between the iron content of bone marrow and the transverse relaxation times (T2 values) of bone marrow water and lipid on MRI.

METHOD

The T2 of the water and lipid fractions of bone marrow was selectively measured in the L1-3 vertebral bodies using the chemical shift misregistration effect on MRI. Results were compared to the serum ferritin levels in 27 healthy subjects.

RESULTS

The 1/T2 of the water fraction ws strongly correlated with the serum ferritin, but there was no correlation between the 1/T2 of the lipid fraction and serum ferritin.

CONCLUSION

Selective T2 measurement of the water fraction in bone marrow allows precise estimation of the marrow iron content.

Magnetic susceptibility measurement of insoluble solids by NMR: magnetic susceptibility of bone

Measurements of the volume magnetic susceptibility of solids using the classical Gouy balance approach are hampered by variations in apparent density of the packed powder. In this paper a quantitative NMR measurement of the volume magnetic susceptibility of powdered solids is described and the volume susceptibility of bone is reported. The technique is based on the measurement of changes in incremental linewidth (1/pi T2') induced in a marker fluid whose susceptibility can be predictably modified by changing the composition, such as by addition of a soluble diamagnetic compound. The spectroscopic linewidth of the marker fluid is determined by the susceptibility difference between the fluid and the suspended solid. Changes in the linewidth are accompanied by bulk magnetic susceptibility induced frequency shifts in the fluid resonance. Correlating the two dependencies allows measurement of the absolute volume susceptibility of the solid. The susceptibility of bovine rib bone was found to be -0.9 +/- 0.02 x -10(-6) (CGS) confirming previous estimates which suggested bone to be more diamagnetic than the marrow constituents. Knowledge of the susceptibility of bone is relevant in view of the growing interest in MRI osteodensitometric techniques.

Field strength and angle dependence of trabecular bone marrow transverse relaxation in the calcaneus

The contribution from reversible dephasing, R2', to the effective transverse relaxation rate, R2*, in trabecular bone marrow, is governed by the magnetic field inhomogeneity induced by the difference in diamagnetic susceptibility between bone and bone marrow and should, therefore, scale linearly with field strength. Measurement of R2' in the calcaneus at 1.5 and 4 T by means of the GESFIDE (gradient-echo sampling of free induction decay and echo) pulse sequence showed R2' to increase by a factor of 2.73 at the higher field, which is close to the ratio of the field strengths (2.67). At both field strengths, R2' dominates R2*, contributing more than 90% to the total relaxation rate at 4 T. The data further indicate large regional variations within the calcaneus and a notable dependence of R2' on the angle of the foot relative to the direction of the static field. The findings have implications on the choice of the calcaneal site for assessment of trabecular density.

A single-scan imaging technique for measurement of the relative concentrations of fat and water protons and their transverse relaxation times

A two-component chemical-shift-imaging technique is described from which fat and water images can be obtained in a single scan and in the presence of an inhomogeneous field. In addition, the method provides transverse relaxation rates R2 and R2' separately for each of the spectral components. The method is a combination and extension of the GESFIDE [gradient echo sampling of FID and echo, J. Ma and F. W. Wehrli, J. Magn. Reson. B 111, 61 (1996)] and the multipoint Dixon techniques. It is based on sampling the descending and ascending portions of a Hahn spin echo with a train of gradient echoes which are spaced at one-half of the chemical-shift modulation period. Processing of the complex echo data, involving an automated phase unwrapping algorithm, affords relative amplitudes and transverse relaxation rates of the two spectral components. An additional benefit of the method is its superior signal-to-noise ratio resulting from echo summation. Applications targeted and illustrated involve MRI osteodensitometry of trabecular bone in the presence of varying fractions of hematopoietic and fatty bone marrow.

Quantitation of T2' anisotropic effects on magnetic resonance bone mineral density measurement

In this paper, the authors quantitate the anisotropy of susceptibility effects in an uniaxial trabecular bone model and show its relevance to clinical MR bone mineral density measurements. A physical model is described that quantitates the anisotropic MR behavior of uniaxial trabecular bone. To test the model; a phantom of parallel polyethylene filaments was scanned every 15 degrees between 0 degrees and 90 degrees with respect to the system's main magnetic field (B0). The distal radial metaphysis of a healthy female volunteer was scanned in orthogonal projections. The signal from each phantom image and each radial image was separated in a pixel-wise fashion into R2 and R2' maps. As predicted, R2' relaxation showed anisotropic behavior and changed according to sin2 (theta), confirming that columnar structures parallel with B0 will cause no MR susceptibility effects. Scans of the distal radius showed that R2' relaxation was twice as great with the forearm perpendicular to B0 as when it was parallel to it, demonstrating different contributions from struts and columns. For both phantom and radial bone scans, R2 relaxation was isotropic and did not change with object orientation.

Gradient-echo magnetic resonance signal decay in a porcine vertebral body model: influence of chemical shift

RATIONALE AND OBJECTIVES

This study investigates how magnetic resonance (MR) signal and T2* of trabecular bone are affected by chemical shift.

METHODS

Five pigs were sacrificed, and 150 gradient-echo MR images with increasing echo times (TEs) were obtained of the lumbar spine. Two vertebrae were excised, defatted, and imaged. Commercial fat-protein emulsions with 40%, 27%, and 15% concentrations of fat were studied. Regions of interest in subcutaneous fat (n = 3), bladder (n = 4), vertebral body (n = 10), and defatted vertebral body (n = 10) were used to study decay of signal intensity.

RESULTS

MR signal intensity of the vertebrae decreased with a superimposed modulation. The periodicity was 4.65 msec (range, 4.60-4.68 msec). At a TE of 0 msec, a phase shift of 24 degrees (range, 14 degrees-37 degrees), which corresponds to a shift in TE of 0.31 msec at 1.5 T, was present. In the fat-protein emulsions, the amplitude of the modulation increased with the amount of fat.

CONCLUSION

Chemical shift and the amount of fat affects T2* measurements.

Effect of ovariectomy on magnetic resonance T2* in rat femur

Measurements have been made in the rat femur in vivo and ex vivo by using an asymmetric spin echo technique of T2(1), the susceptibility contribution to T2*. The trabecular spacing in this study in rat bone is considerably less than in previous studies in the human. A significant increase in T2(1) was seen in vivo 3 mm proximal to the growth plate with ovariectomy (a model of osteopenia), from 8.1 +/- 0.7 to 10.0 +/- 0.6 ms. Parallel changes in trabecular bone mineral density measured by quantitative computed tomography were found. T2(1) was higher in living bones than in the same bones measured post mortem.

Investigation of MR decay rates in microphantom models of trabecular bone

MR measurements of transverse relaxation time, T2*, in trabecular bone may provide both structural and density-related information for assessment of bone mineral status in osteoporosis. Using submillimeter scale glass phantoms as simplified models of trabecular bone, we have made a quantitative investigation of the dependence of T2* decay on modeled trabecular microstructure and MR image resolution. The experimental MR data are in excellent agreement with predictions from a computer simulation. Decreasing the modeled trabecular bone volume fraction, sigma, decreases the decay rate, as expected. However, if trabecular width and spacing are both increased without changing sigma, the decay rate is unchanged. The measured decay curves closely follow the predicted dependence on trabecular orientation. The decay rates are independent of image resolution, provided that the pixel dimensions are larger than the intertrabecular spacing. For smaller pixel sizes, the decay rate decreases with decreasing pixel size.

Determination of 1H relaxation times of water in human bone marrow by fat-suppressed turbo spin echo in comparison to MR spectroscopic methods

Relaxation times of water were measured in human vertebral bodies by a fat-suppressed dual-echo turbo spin echo/turbo inversion recovery MRI sequence. Comparison was made with T1 and T2 values obtained by localized 1H-MR spectroscopy. The accuracy of the results and the diagnostic potential of the fast quantitative MRI technique were evaluated in 20 volunteers, 11 patients with osteoporosis, 6 patients with lymphoma, and 6 patients with bone marrow metastasis. No significant alterations of T1 and T2 relaxation times of water and fat were found in osteoporosis. With both methods, an increase in the T1 values of the water resonance by 16% was observed in lymphomas, which was highly significant (P < .001) in the MRS measurements, and an elevation by the same amount was obtained by the MRI sequence for the metastases (P = .040). A strong reduction of fat fraction was quantified by MRS in the tumorous cases. T2 of the water resonance increased by more than 30% (P < .003) in metastases. Water T2 values obtained by the MRI sequence showed systematic deviations from the MRS results, especially at short echo spacings.

Magnetic resonance imaging of the calcaneus: preliminary assessment of trabecular bone-dependent regional variations in marrow relaxation time compared with dual X-ray absorptiometry

RATIONALE AND OBJECTIVES

Marrow transverse relaxation time (T2*) in magnetic resonance (MR) imaging may be related to the density and structure of the surrounding trabecular network. We investigated regional variations of T2* in the human calcaneus and compared the findings with bone mineral density (BMD), as measured by dual X-ray absorpiometry (DXA). Short- and long-term precisions were evaluated first to determine whether MR imaging would be useful for the clinical assessment of disease status and progression in osteoporosis.

METHODS

Gradient-recalled echo MR images of the calcaneus were acquired at 1.5 T from six volunteers. Measurements of T2* were compared with BMD and (for one volunteer) conventional radiography.

RESULTS

T2* values showed significant regional variation; they typically were shortest in the superior region of the calcaneus. There was a linear correlation between MR and DXA measurements (r = .66 for 1/T2* versus BMD). Differences in T2* attributable to variations in analysis region-of-interest placement were not significant for five of the six volunteers. Sagittal MR images had short- and long-term precision errors of 4.2% and 3.3%, respectively. For DXA, the precision was 1.3% (coefficient of variation).

CONCLUSION

MR imaging may be useful for trabecular bone assessment in the calcaneus. However, given the large regional variations in bone density and structure, the choice of an ROI is likely to play a major role in the accuracy, precision, and overall clinical efficacy of T2* measurements.

Quantification of trabecular structure in the distal femur using magnetic resonance phase imaging

A new approach for quantifying trabecular bone tissue using the phase images of a simple gradient-echo sequence is presented. The proposed method is based on the hypothesis that the differences in susceptibility between bone and bone marrow cause magnetic field (i.e., precession phase) variations between the image voxels. Phase images of the distal femur were obtained in vivo and characterised with the use of the phase variance. Computer simulations and experimental results indicate that the distribution of the phases varies with echo time and image resolution, as expected. Keeping these fixed, however, the phase variance is found to strongly reflect variations in trabecular structure.

Evaluation of technical factors affecting the quantification of trabecular bone structure using magnetic resonance imaging

High resolution magnetic resonance (MR) techniques combined with standard techniques of stereology and texture analysis have been used to quantify trabecular structure. Using dried excised specimens from the tibia (n = 10) and radius (n = 2) we evaluate the impact of using volumetric gradient-echo (GE) and spin-echo (SE) MR imaging sequences, the relative importance of echo time in gradient-echo MR imaging, and the impact of different threshold values to segment the bone and bone marrow on the estimation of trabecular bone structure. We also investigate the inter-relationships between the different structural parameters derived from MR images. Images were obtained using fast gradient-echo and spin-echo imaging sequences, with TE values ranging from 7 to 17 ms using 4.7 and 1.5 Tesla imaging systems. In-plane image resolution ranged from 128 to 156 microns, and slice thickness ranged from 128 to 1000 microns. We derived stereological measures such as the mean intercept length, trabecular width, fractional area of trabecular bone, trabecular number, and trabecular spacing, the fractal dimension as a texture-related parameter and the Euler number as a measure of connectivity from these images. We found that the mean intercept length as a function of angle traced an ellipse with the orientation of the principal axis of the ellipse, a measure of trabecular orientation, identical when measured from the spin-echo or gradient-echo MR images. Absolute measures such as the fractional area, trabecular width, trabecular number, and fractal dimension as measured from gradient echo images were 28%, 30%, 1.3%, and 0.6% greater, respectively, than those calculated from spin-echo images, while the trabecular spacing was 14% less when calculated from gradient-echo images compared to spin-echo images. The structural parameters also depended on the echo time used to obtain the MR image. The choice of the threshold used to segment the high resolution images also affected the estimated structural parameters significantly. Our results indicate that MR may be used to visualize and quantify trabecular bone architecture; however, the different technical factors that could affect the appearance of MR images must be understood and considered in the data analysis and interpretation.

A review of the recent advances in magnetic resonance imaging in the assessment of osteoporosis

Osteoporosis is a common metabolic disorder with considerable associated morbidity and mortality. The loss of bone mineral integrity and the resultant occurrence of atraumatic fractures are typically symptomatic of the disease. Currently skeletal status is commonly assessed using non-invasive conventional radiography and scintigraphy as well as densitometric techniques such as quantitative computed tomography and dual-energy X-ray absorptiometry. But, apart from gross bone mineral density, the fine structure of trabecular bone also plays an important role in defining the biomechanical competence of the skeleton. Recently attention has been focused on deriving measures that provide information about not only trabecular bone density but also microstructure. Magnetic resonance imaging (MRI) is one such new technique which potentially may provide information pertaining to bone density and structure as well as to occult fracture detection. Cortical bone produces a signal void in MR images, due to the fact that it contains very few mobile protons that give rise to a signal in MRI; also the MR relaxation time T2 of these protons is very short which produces a very fast decay of the MR signal during image acquisition. However, the trabecular bone network affects the MR properties of bone marrow. The difference in the magnetic properties of trabecular bone and bone marrow generates local imperfections in the magnetic field. The MR signal from bone marrow is modified due to these imperfections and the MR relaxation time T2 of marrow is shortened. The extent of relaxation time shortening and hence loss of signal intensity is proportional to the density of trabecular bone and marrow interfaces and their spatial architecture. Recent investigation in this area include studies aimed at quantifying marrow relaxation times and establishing their relationship to trabecular bone density and structure. In addition, with advances in imaging software and hardware, MR images at in-plane resolutions of 78-200 microns may be obtained. The trabecular bone structure is clearly revealed in such images and studies aimed at the development of high-resolution MRI techniques combined with quantitative image analysis techniques are currently under way. These potentially useful techniques for assessing osteoporosis and predicting fracture risk are reviewed in this paper.

Theory of NMR signal behavior in magnetically inhomogeneous tissues: the static dephasing regime

This paper is devoted to a theory of the NMR signal behavior in biological tissues in the presence of static magnetic field inhomogeneities. We have developed an approach that analytically describes the NMR signal in the static dephasing regime where diffusion phenomena may be ignored. This approach has been applied to evaluate the NMR signal in the presence of a blood vessel network (with an application to functional imaging), bone marrow (for two specific trabecular structures, asymmetrical and columnar) and a ferrite contrast agent. All investigated systems have some common behavior. If the echo time TE is less than a known characteristic time tc for a given system, then the signal decays exponentially with an argument which depends quadratically on TE. This is equivalent to an R2* relaxation rate which is a linear function of TE. In the opposite case, when TE is greater than tc, the NMR signal follows a simple exponential decay and the relaxation rate does not depend on the echo time. For this time interval, R2* is a linear function of a) volume fraction sigma occupied by the field-creating objects, b) magnetic field Bo or just the objects' magnetic moment for ferrite particles, and c) susceptibility difference delta chi between the objects and the medium.

Localized MR 1H spectroscopy reveals alterations of susceptibility in bone marrow with hemosiderosis

A noninvasive investigation of the structure of hemopoietic bone marrow is based on the determination of the magnetic field distribution within small volume elements in vertebral bodies by localized 1H MR spectroscopy. In patients with hematological diseases the status of the bone marrow was found to considerably influence the homogeneity of the magnetic field in trabecular bone in vivo. The line widths of the 1H signals were evaluated in follow-up studies during initial chemotherapy of eight patients with leukemia. Intraindividual comparison revealed significant broadening of the field distribution after a few weeks of cytotoxic treatment in five of the patients. Additionally, 19 patients after bone marrow transplantation showed significantly broader field distributions in the lipid signals than 13 matched healthy volunteers. These alterations of the microscopic field homogeneity were not caused by trabecular density effects. Iliac crest biopsies revealed high amounts of hemosiderin in the cases with broadened line widths. Ten of the 19 patients after bone marrow transplantation showed high amounts of hemosiderin and broad lines in the spectra. The content of hemosiderin of the other patients was not significantly increased.

Bone scintigraphy and multimodality imaging in bone neoplasia: strategies for imaging in the new health care climate

The integration of multiple imaging modalities in the assessment of musculoskeletal neoplasia is complex. Although no two instances are identical, certain guidelines can be gleaned from our experience as well as that reported in the literature. Assessment of most soft tissue masses is best carried forth with a combination of conventional radiography and magnetic resonance imaging (MRI). Screening skeletal scintigraphy without localizing symptomatology that includes axial and appendicular skeleton is best carried out initially with bone scintigraphy. Screening the axial skeleton in the presence of clinical symptomatology or a strong suspicion of axial skeletal metastases or pathology is best implemented as a total spine screening examination with MRI and specialized pulsing sequences. Computed tomography is reserved primarily for assessment of cortical and juxtacortical lesions, fracture fragment positioning and/or configuration, and characterization of lesion matrix calcification or ossification when conventional radiographs are indeterminate. Although physical examination and conventional radiography still remain the initial medical algorithms used to evaluate possible musculoskeletal neoplasia, primary skeletal tumors may require multimodality imaging to segregate aggressive and nonaggressive processes. In this multimodality scenario, bone scintigraphy has a critical role in assisting with differentiation between malignant and benign neoplasms.

Phantom studies simulating the impact of trabecular structure on marrow relaxation time, T2'

Phantom studies were conducted to investigate the impact of trabecular structure on the T2' signal measured by MRI. For a separation of density from structural effects, several phantoms were built. They consisted of parallel polyethylene strings arranged in a variety of different patterns to simulate a) a constant uniform trabecular distribution with increasing trabecular thickness and b) different structures with identical overall trabecular density. An asymmetric spin echo sequence was used to determine the apparent relaxation time T2'. Changes in T2' are induced by susceptibility differences between the polyethylene strings simulating trabeculae and Gd-DTPA doped saline simulating bone marrow. The results showed an increasing T2' decay rate with a) decreasing spacing while the string density was constant and b) with increasing string density while the string arrangement was constant. The results demonstrate that the T2' signal is affected not only by density but also by spatial distribution. However, the results also indicate that a separation of the two effects is not possible from a T2' measurement alone, but that e.g., a matching CT slice that would provide purely density information is additionally needed. Theoretical simulations confirmed these results.

Relationship between NMR transverse relaxation, trabecular bone architecture, and strength

Structure, biomechanical competence, and incremental NMR line broadening (R'2) of water in the intertrabecular spaces of cancellous bone were examined on 22 cylindrical specimens from the lumbar vertebral bodies of 16 human subjects 24-86 years old (mean, 60 years old). A strong association (r = 0.91; P < 0.0001) was found between Young's modulus of elasticity and R'2 for a wide range of values corresponding to cancellous bone of very different morphologic composition. NMR line broadening is caused by the inhomogeneity of the magnetic field induced as a consequence of the coexistence of two adjacent phases of different diamagnetic susceptibility--i.e., mineralized bone and water in the marrow spaces. Structural analyses performed by means of NMR microscopy and digital image processing indicated that the variation in R'2 is closely related to the trabecular microstructure. Mean trabecular plate density measured along the direction of the magnetic field was found to play a major role in predicting R'2 (r = 0.74; P < 0.0001). This behavior was confirmed when the plate density was varied in individual specimens, which was achieved by rotating the specimen, making use of the bone's structural anisotropy. It is concluded that the NMR transverse relaxation rate in human cancellous bone of the spine is significantly determined by trabecular structural parameters relevant to biomechanical strength. The results further underscore the important role played by the transverse trabeculae in contributing to cancellous bone strength. The work has implications on possible in vivo use of quantitative magnetic resonance for the assessment of fracture risk in osteoporotic patients.

Magnetic field distribution in models of trabecular bone

A magnetostatic model consisting of a tetragonal lattice of struts of diamagnetic material, mimicking vertebral trabecular bone, was developed. The model allows estimation of the magnetic field histogram within the lattice's unit cell as a function of geometric parameters. The field was computed analytically from the induced magnetic surface charge density on the faces of the struts. The contribution from the induced magnetic field to the effective transverse relaxation rate, R2', was obtained as the mean decay rate of the Fourier transformed histograms, for both fixed and randomly oriented lattices. The model predicts the field distribution to increase with both strut thickness and density, paralleling material density. Finally, significant changes in R2' are predicted at constant material density, in that the field distribution widens with simultaneously increasing strut number density and decreasing strut thickness.

High-resolution variable flip angle 3D MR imaging of trabecular microstructure in vivo

Two conceptually related variable-flip-angle 3D spin-echo pulse sequences were designed for imaging at voxel sizes of 2-5 x 10(-3) mm3 corresponding to pixel areas of less than 100 x 100 microns2 and section thicknesses on the order of 300-400 microns on a conventional 1.5 T MR imaging system equipped with 1 G/cm imaging field gradients, providing 12 sections in 10 min imaging time. The pulse sequences make use of the concept of restoring longitudinal magnetization inverted by the 180 degrees phase reversal pulse and are derivatives of pulse sequences previously dubbed "FATE" and "RASEE." It is shown that even in the small-voxel regime (< 10(-2) mm3 voxel size) and at echo times on the order of 10 ms, gradient echo images are sensitive to intrinsic fields causing artifactual boundary effects, including signal loss from intravoxel phase scrambling and spatial mismapping. At this resolution the variable flip-angle spin-echo pulse sequences are demonstrated to be better suited for imaging magnetically heterogeneous systems such as trabecular bone microstructure in vivo. These pulse sequences are found to be substantially less sensitive to distortions from magnetic dipole fields occurring at the boundaries of two phases of different magnetic permeability.