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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
 
Magn Reson Med. Author manuscript; available in PMC 2010 August 1.
Published in final edited form as:
PMCID: PMC2753239
NIHMSID: NIHMS119697

Accuracy and Precision of MR Blood Oximetry Based On the Long Paramagnetic Cylinder Approximation of Large Vessels

Abstract

An accurate non-invasive method to measure hemoglobin oxygen saturation (%HbO2) of deep-lying vessels without catheterization would have many clinical applications. Quantitative MRI may be the only imaging modality that can address this difficult and important problem. MR susceptometry-based oximetry for measuring blood oxygen saturation in large vessels models the vessel as a long paramagnetic cylinder immersed in an external field. The intravascular magnetic susceptibility relative to surrounding muscle tissue is a function of HbO2 and can be quantified with a field mapping pulse sequence. In this work, the method’s accuracy and precision was investigated theoretically on the basis of an analytical expression for the arbitrarily oriented cylinder, as well as experimentally in phantoms and in vivo in the femoral artery and vein at 3T field strength. Errors resulting from vessel tilt, non-circularity of vessel cross-section, and induced magnetic field gradients were evaluated and methods for correction designed and implemented. Hemoglobin saturation was measured at successive vessel segments, differing in geometry such as eccentricity and vessel tilt but constant blood oxygen saturation levels, as a means to evaluate measurement consistency. The average standard error and coefficient of variation of measurements in phantoms were less than 2% with tilt correction alone, in agreement with theory, suggesting that high accuracy and reproducibility can be achieved while ignoring non-circularity for tilt angles up to about 30°. In vivo, repeated measurements of %HbO2 in the femoral vessels yielded a coefficient of variation of less than 5%. In conclusion, the data suggest that %HbO2 can be measured reproducibly in vivo in large vessels of the peripheral circulation on the basis of the paramagnetic cylinder approximation of the incremental field.

INTRODUCTION

Accurate non-invasive measurement of %HbO2 of deep-lying vessels inaccessible to pulse oximetry would have many clinical applications. Among these are quantification of global cerebral metabolic rate of oxygen (CMRO2) and assessment of congenital heart defects in pediatric patients. Generally, both currently require catheterization procedures, which are invasive and carry risks. In the case of jugular bulb oximetry, the risks (1) include arterial puncture (between 1 – 4.5% has been reported in the literature), jugular vein occlusion from misplacement of catheter and subclinical thrombosis, whose incidence rate is as high as 40%. Furthermore, accuracy is highly dependent on the position of the catheter against the vessel wall, and simultaneous measurement of both the left and right sides is not possible but important due to asymmetric venous drainage (2).

Non-invasive quantification of blood oxygenation can be achieved with MRI by measuring T2 (3) or T2* (4), or from a measurement of the relative susceptibility (57) of intravascular blood since all three quantities depend on HbO2, which determines the blood’s paramagnetism. T2 and T2* approaches require the intravascular signal intensity to be measured as a function of echo time, which in turn, is used to calculate HbO2. In T2-approach the relaxation rate is quantified with a CPMG-based pulse sequence and it is assumed that 1/T2 scales as (1-HbO2)2. T2*, the time constant for signal decay from static dephasing due to local spatial magnetic field variations within and in the vicinity of erythrocytes, is typically obtained with a multi-echo gradient-echo sequence. In order to measure arterial saturation, relaxation-rate approaches require individual calibration consisting of measuring relaxation rates of fully oxygenated blood, T2o or T2o *, which are hematocrit (Hct) and magnetic field dependent. Calibration of T2o can be obviated by using population-based values (8). Finally, the accuracy of T2 or T2* measurement must be within 5% (9) in order to achieve a precision of 3%, which is difficult to achieve. Accurate measurement of T2 is further complicated by patient motion, blood flow and partial volume effects (10).

Magnetic susceptibility-based %HbO2 quantification, on the other hand, exploits the relative susceptibility of intravascular blood with respect to surrounding reference tissue, measured by means of field mapping. The magnetic susceptibility of blood is governed by the paramagnetic deoxyhemoglobin in red blood cells and depends linearly on the volume fraction of deoxyhemoglobin. The magnetic susceptibility of most tissues is very close to that of water (11), the predominant component of tissue. Therefore, muscle tissue serves as a calibration-free phase reference relative to which the phase of either the arterial or venous blood proton signal can be measured. Quantification is further facilitated by modeling the blood vessel as a long paramagnetic cylinder. Gating is unnecessary because the phase of the signal is not affected by inflow effects so long as flow is compensated in the readout direction to avoid phase contribution from in-plane motion caused by vessel tilt. Further, it can safely be assumed that the flow velocity does not change significantly ( < 5% during systole (12)) while the slice-selection and readout gradients are operating, which is on the order of 4 ms.

In this work, we investigate the effects of geometry on accuracy and precision of the “infinitely long cylinder” model of MR susceptometry theoretically and with controlled phantom experiments. Reproducibility is investigated through in vivo quantification of blood oxygenation at different locations of the same vessel. It is a simple consistency check for reproducibility because different segments of a vessel offer varying tilt and/or non-circularity at a given %HbO2 since oxygen extraction takes place in the capillary bed.

MATERIALS AND METHODS

Principles of MR Susceptometry-based Oximetry

In MR susceptometry-based oximetry, a vessel is approximated as a long paramagnetic circular cylinder (length [dbl greater-than sign] diameter) surrounded by a uniform medium. The incremental field in the vessel relative to its surrounding (muscle tissue) is proportional to the corresponding difference in the magnetic susceptibility (SI units) Δχ (13):

equation M1
(1)

where θ is the vessel tilt angle with respect to the main magnetic field, [B with right harpoon above]o. The susceptibility of blood relative to tissue can be expressed as Δχ = Δχ do Hct·(1−HbO2), where Δχdo = 4π (0.27 ± 0.02) ppm (14) is the susceptibility difference between fully deoxygenated and fully oxygenated erythrocytes, hematocrit (Hct) is the volume fraction of the packed erythrocytes in the whole blood, and HbO2 represents the volume fraction of oxyhemoglobin (Hb). The expression for Δχ is an approximation based on equating the magnetic susceptibility of plasma with that of fully oxygenated erythrocytes. According to (14) the average difference gives rise to a 5% error for Δχdo, which translates to σ%HbO2 ~ 5Δχ /(Hct· Δχdo), i.e. less than ±2 %HbO2 for venous saturation of 65 %HbO2 (the corresponding susceptibility is Δχ ~ 0.49 ppm) and Hct = 0.41).

The field shift ΔB can be obtained for each pixel of the image from the phase difference, Δ[var phi] = γΔBΔTE, between two successive echoes separated by ΔTE. Therefore, %HbO2 can be quantified once Hct is known and if we further assume that the susceptibility of the reference muscle tissue is equal to that of plasma. The susceptibility difference between muscle tissue and plasma is approximately 1% considering that muscle is composed of 80% water and 20% protein (15) and that all diamagnetic proteins have approximately the same susceptibility (16).

The susceptibility difference of 0.27 ppm (cgs) reported by Spees et al (14) between fully deoxygenated and fully oxygenated erythrocytes used in the present work differs from previously reported values 0.20 ppm (17,18) and 0.18 ppm (5). We have particular confidence in the 0.27 ppm value since it is based on very thorough work. Further, the authors’ NMR-derived susceptibility agreed with the result obtained from measurements with a superconducting quantum interference device as well as a semi-theoretical analysis. Lastly, the mean erythrocyte magnetophoretic mobility measured by (19) agreed with values derived based on the magnetic susceptibility data and the model of (14).

Potential Sources of Error

A. Vessel Eccentricity

The higher distending pressure in arteries compared to that in veins leads to negligible eccentricity, which qualitatively agrees well with observation. Thus arteries will be taken to be circular for the rest of the article. Veins, on the other hand, collapse easily leading to non-circular cross-section due to lower distending pressure (~ 10% of systolic pressure) and elasticity. Approximating the vein as an elliptic cylinder, Eq. 1 is replaced with (for derivation see Appendix):

equation M2
(2)

To simplify the final expression, we have defined the eccentricity as equation M3, where a and b are the major and minor axes of an ellipse. Whereas the present definition of eccentricity is not the standard one it unambiguously defines an ellipse’s major and minor axes. ψ is the angle between major axis a and the component of [B with right harpoon above]o that lies in the cross-sectional plane of the elliptic cylinder. The direction of the “axial” component of [B with right harpoon above]o can be estimated from a series of axial scout images since it is perpendicular to the axis of the elliptic cylinder. We note from Eq. 2 that if the vein is parallel to the main field (θ = 0) Eq. 2 reverts to Eq. 1, i.e. the incremental field does no longer depend on eccentricity.

B. Incremental Magnetic Field Outside Vessel Boundaries

In general the vessel tilt angle is non-zero and the dipolar field outside the vein therefore affects the effective field outside its boundaries. Since in the extremities artery and vein are in close proximity, errors may result from phase induced by intravenous paramagnetism in the adjacent artery. The incremental field at a point in the artery can be expressed as (11)

equation M4
(3)

The first term in Eq. 3 can be non-zero since arterial blood is not necessarily fully oxygenated (typically, arterial saturation is 95–98% (20)), thus Δχa <0, typically holds for the susceptibility difference between arterial blood and reference tissue. However, in some pathologic conditions, such as in hypoxemia (insufficient oxygenation of arterial blood), for instance due to pulmonary insufficiency, arterial oxygen saturation is less than 90% (21). The angular and position dependence of the incremental field outside the tilted vein is described by the second term of Eq. 3. Δχa and Δχv are the relative susceptibilities of the arterial and venous blood, ρ is the distance measured from the center of the vein to a point in the artery, and rv is the “radius” of the vein. Tubes had inner radius of 0.75 cm and were placed adjacent to each other. Typical in vivo values of rv were 0.3 – 0.4 cm with ρ varying approximately from rv to rv + Da, where Da is the diameter of the artery, i.e. each pixel in the artery was summed and averaged. Note that we have set e = 0 for the artery. Eq. 3 may overestimate the error because the field in the reference tissue is also affected by the vein’s proximity and no longer uniform even under ideal circumstances, i.e. in the absence of large-scale field gradients.

C. Large-Scale Magnetic Field Perturbations from the Air-Tissue Interface

The most significant source of error results from low-frequency modulations of static magnetic field produced by the interface between air and tissue or between adjacent tissue types (for example bone and soft tissue (22)). Details of an effective method to remove the slowly-varying part of the field inhomogeneity have been discussed previously (23). In short, data is acquired without scanner-implemented default shimming, and fitting, after appropriate weighting and masking, the static field inhomogeneity to a second-order polynomial.

Simulations

In order to evaluate the size of the error in measuring blood oxygenation level resulting from vessel eccentricity e, ψ ≠ 0 and vessel tilt θ, %HbO2 was plotted as a function of (θ,ψ)e or (e,ψ)θ while the parameter in the subscript was held constant; in both plots ψ was varied from 0° to 180°. Equation (3) was plotted with the same range of θ and ψ to gauge the influence on arterial saturation by adjacent tilted vein whose incremental field outside the boundary is non-zero. For the simulations, a venous saturation of 76% was assumed, corresponding to Δχ = 0.34 ppm (SI) relative to distilled water, equivalent to 1 mM Gd-DTPA used throughout the phantom experiments described below.

Experimental Evaluation of Errors

Pulse Sequence

All images were acquired with a spoiled multi-echo gradient-echo (GRE) sequence written in Sequence Tree (Version 3.1) (24), a custom-designed pulse-sequence design and editing tool. The pulse sequence, shown in Figure 1, includes fat suppression and first-moment nulling of the slice selection gradient (i.e. direction of blood flow) direction. The phase difference is taken between echoes of equal polarity to minimize off-resonance errors. We note that the first moment of the readout gradient at the center of the equal-polarity echoes is the same for all echoes. Hence, even with vessel obliquity the phase associated with flow imparted by the readout gradient will not cause any error in the phase difference attributed to %HbO2.

Figure 1
RF- and gradient spoiled multi-echo GRE with fat suppression and flow compensation in the slice direction.

All phantom and in vivo experiments were performed on a 3T Siemens Trio scanner using an eight-channel knee array coil (Invivo Inc., Pewaukee, WI) with the following parameters to: voxel size = 1 × 1 × 5 mm3, FOV = 128×128 mm2, BW = 488 Hz/pixel, TE1 = 3.75 ms, inter-echo spacing = 2.32 ms, TR = 39.1 ms and flip angle = 13°. These parameters achieve sufficient SNR in the reference tissue and a high temporal resolution of 5 s, which can be used to obtain a time-course of %HbO2 in response to a physiologic challenge. The raw data was saved and processed offline as described in (23).

Phantom Experiments

In order to investigate errors associated with vessel tilt and non-circularity, plastic tubes (BD Biosciences Falcon tubes, Franklin Lakes, NJ), of circular and elliptic cross-section containing 1 mM Gd-doped water (corresponding to blood oxygenation of about 76 % (20) for Hct = 0.42) were scanned individually. Oval tubes (to approximate elliptic cylinders) were made by applying a clamp to the whole length of the tube overnight after heating them in boiling water. Each tube was immersed inside a container of distilled water, which served as a reference medium. Tilt angles θ from 0° to about 30° and eccentricities e ranging from 0.0 to 0.11 were evaluated (values found to be typical of those of the femoral and popliteal veins). The tilt angle was determined from the centroid’s coordinates of the tube from axial slices separated by 30 mm. The following configurations of tubes were examined: (1) parallel and circular, (2) tilted and circular, (3) parallel and non-circular, and (4) tilted and non-circular. For non-circular tubes, the cylindrical container was also rotated to vary ψ.

To investigate the phase contamination in the arterial blood from the proximity of a vein, a circular tube containing distilled water was positioned next to a circular tube containing 1 mM Gd-doped water and placed inside the cylindrical container to mimic the artery/vein vessel pair having 100 and 76 %HbO2. Subsequently, the susceptibility measurement was repeated with the arterial tube only in place. The tilt angle was varied from 0° to 30°.

In Vivo Quantification of Oxygenation in Femoral Vessels

Oxygen extraction takes place at the capillary bed; hence both arterial and venous saturation level must be independent of the location along the vessel. On the other hand, vessel tilt as well as eccentricity can be a function of the slice position. By quantifying blood oxygenation at different locations along the femoral vessels, the magnitude of the error associated with non-circularity and vessel tilt can be studied. For each of two healthy subjects (33 yrs-old male and 23 yrs-old female) the experiment was repeated on three different days and at three locations along the femoral vessels to evaluate reproducibility. The data acquisition during a session takes less than two minutes once the desired axial slices are identified; hence, possible temporal variation of %HbO2 can be assumed to be negligible. On the other hand, intra-individual variations between different scan sessions are expected (25). Therefore, as a measure of reproducibility, we derive average and standard deviation of blood oxygenation values from the same scan session from multiple locations. Written informed consent was obtained prior to the human study following an institutional review board-approved protocol. The volunteers were scanned in supine position, feet first.

RESULTS

Simulations

The dependence of venous saturation with respect to e and ψ at a vessel tilt of 20° (values typical for the femoral or popliteal vein) predicted from Eq. 2 is shown in Figure 2a. At θ = 20° oxygen saturation deviates less than 2 %HbO2 from the expected value if non-circularity is ignored. Figure 2b predicts that error associated with non-circularity becomes appreciable (~5%) only for θ ~ 30° and ψ = 0 but the resulting error is corrigible using Eq. (2). We note that we have not found θ ≥ 30° in approximately 50 legs that we have scanned.

Figure 2
a) Surface plot of %HbO2 as a function of vessel eccentricity and ψ for a vessel tilt of 20°. b) Same but as a function of vessel tilt and ψ with eccentricity e = 0.20, corresponding to a = 2.3b. c) Effect on arterial saturation ...

According to Eq. (3), the field produced by a tilted vein will perturb the field in the adjacent artery; however, the effect is negligible as shown in Figure 2c. For instance, at θ ~ 35° and ψ = 0°, the derived arterial saturation changes to approximately 97 from 98 %.

Phantom Study

Table 1 summarizes the results of the phantom experiments designed to experimentally assess the effect of vessel geometry and its corrections. Derived relative susceptibility values are listed for each tube configuration labeled by tilt angle θ, eccentricity e and projection angle ψ. The first row gives the relative susceptibility Δχ of a circular tube positioned parallel to the main field. The experimentally measured value is equal to the calculated value (0.34 ppm). The last two columns show that derived Δχ has negligible dependence on non-circularity for projection angle up to about 30° tilt angle, consistent with the simulations of Figure 2a and b. Comparison between the last two columns also shows that Δχ values are accurately (average discrepancy of 4% from the actual value of 0.34 ppm) derived with tilt correction alone. Sample magnitude and phase difference images are shown in Figure 3.

Figure 3
Magnitude (a) and phase difference (b) image of Gd-doped water immersed in distilled water; e = 0.078.
Table 1
Summary of the phantom data. e, ψ and θ were varied as indicated. Relative susceptibility (in SI units) was computed with and without tilt angle correction (w/Tilt and No Tilt, respectively). The column labeled “w/Tilt, e, ψ” ...

No significant trend was observed for the error incurred at the site of the “artery” by an adjacent “vein” as predicted by the simulation in Figure 2c. The susceptibility error at the arterial site when converted to %HbO2 was less than 1 %HbO2 .

In Vivo Study

Magnitude and phase difference images acquired at three axial locations during the same scan session are shown in Figure 4 (a – f). SNR in the reference muscle tissue was about 20 giving standard error of about ± 3 %HbO2 for in vivo blood oxygenation quantification. This error was estimated by substituting the uncertainty of the phase measurement (σ[var phi] = 1/SNRmagn) in the error propagation of Δ[var phi] = γΔBΔTE (23). SNR of the reference tissue is the dominant contributor to the random error in Δχ, since the intravascular SNR is considerably greater due to the inflow effect (> 40). We also note that the pulsatile flow artifacts are minimal and do not interfere with the phase measurements. Greater phase accumulation of the femoral vein is apparent on the phase difference images. Further, no phase contrast is apparent between the femoral artery and nearby muscle, consistent with negligible difference in susceptibility between reference muscle tissue and fully oxygenated blood. The dashed circle in Figure 4d represent the area in which the retrospective static field inhomogeneity correction was applied; noticeably larger field inhomogeneity is seen outside of this region. The data are summarized in Table 2. For each scan session %HbO2 values from the three distinct slices separated by about 20 mm are listed along with eccentricity and local vessel tilt. The coefficient of variation was less than 5% between intra-session values for each subject.

Figure 4
Corresponding magnitude (a – c) and phase difference (d – f) images of three axial slices. The locations of the corresponding axial locations are shown in the oblique sagittal image (g). The window level of the phase difference images ...
Table 2
Quantified arterial and venous saturation (%SaO2 and %SvO2, respectively) in three different femoral vessel segments and imaging sessions for two volunteers. Derived oxygen saturation levels were corrected for tilt angle.

DISCUSSION AND CONCLUSIONS

Our theoretical and experimental data suggest the long cylinder model of MR susceptometry to yield accurate results. Reproducibility of in vivo %HbO2 quantification was on the order of 5% in the femoral vessels. Simulation and experiments suggest the effect of vessel non-circularity to be below the standard error even at 30° with tilt correction alone. Based on observations in the present and previous (23) work the femoral vessel tilt is below this value for a properly positioned subject in the scanner. The phantom data summarized in Table 1 demonstrate the validity of the long-cylinder approximation. If we translate the susceptibility differences to %HbO2 the observed discrepancy was within ±2 %HbO2. Lastly, the data indicate that non-circularity, although corrigible, can be ignored.

Even though the length of a typical “vessel segment” exceeds its diameter by a factor of 6 only, the long-cylinder approximation reproduces the susceptibility predicted on the basis of the gadolinium concentration with excellent accuracy and precision. These requirements are met by the peripheral vessels studied and should be equally applicable to other large vessels such as the internal jugular vein or carotid artery. Further, the angular dependence of the incremental field is accurately described by Eq. (1) suggesting that a high spatial-resolution multi-slice protocol should ensure accurate local tilt measurement. The uncertainty in the tilt angle is a function of spatial resolution. If Δx and Δy are the in-plane displacements of the vessel’s centroid in slices separated by Δz ~ 3 cm, the tilt angle is given as equation M5. Based on error propagation one predicts an uncertainty in the angle of σθ = 1° at a spatial resolution of 0.5 × 0.5 mm2 and θ = 20°. A 1° error at vessel tilt of 20°, according to Eq. (1) then results in uncertainty of less than ±2 %HbO2 for typical range of venous saturation levels (65 – 75 %) and Hct.

Simulation and results from the phantom and in vivo studies suggest that phase contamination at the site of the artery by the adjacent vein to be well below the precision of the method. The phase images also reflect negligible effect of phase contamination at the site of artery and proximal tissue by an adjacent vein. We also note that Eq. (3) does not include the field perturbation in proximal reference tissue whose phase is subtracted from the arterial phase in deriving the %HbO2. It follows that the effect described by Eq. (3), already small, is further reduced.

The validity of our approach is also supported by the in vivo results. Vessel eccentricity causes an error of less than 1 %HbO2, so no corrections for this effect were made in the data of Table 2. Even though the vessel tilt ranged from 13° to 26° and the eccentricity was as high as 0.21 in the subjects studied the average coefficient of variation of venous and arterial %HbO2 values was less than 5%. Finally, the MR oximetry-derived %HbO2 values are in good agreement with those determined from blood sampled directly from the femoral vein (25). Our results provide a significant level of confidence for the validity of the simple cylinder model underlying MR susceptometry. Lastly, the vessels of peripheral circulation are ideal for in vivo quantification of %HbO2 due to their adjacency of muscle tissue serving as phase reference.

There are limitations of the presented embodiment of susceptibility-based oximetry. The requirement of a straight vessel segment with nearby muscle tissue would exclude %HbO2 quantification at the level of the ascending aorta due to vessel curvature and lack of adjacent reference tissue. Additional anatomically challenging regions of interest include the internal jugular vein and carotid artery due to their proximity to the trachea and skin. The effects of field globally induced magnetic inhomogeneity can be reduced by acquiring a greater number of interleaved echoes with shorter inter-echo spacing. In this case the dynamic range for the phase measurement would decrease but its adverse effect on precision would be compensated by the increased number of echoes from which the phase is computed. Additional errors may arise from heterogeneous distribution of reference tissue that does not yield the averaging effect of a uniform reference as in the lower extremities, for example. Also, the use of fat suppression leads to intermuscular “voids” where the weighted least-squares fit must be interpolated (23), which may lead to less effective correction of the static field inhomogeneity. In the absence of fat suppression the chemical shift can cause a phase discontinuity at the at muscle-lipid interface unless an appropriate echo-spacing is used, e.g. multiples of the period for the fat-water frequency difference (~2.4 ms at 3T). Finally, we have not provided independent verification of the MR-derived oxygen saturation values. Such a direct validation would require blood sampling, which is not possible in femoral vessels but would be feasible at the antecubital fossa. It is apparent that the error caused by field inhomogeneity, which is inherent to field mapping, is least predictable and most significant in anatomical challenging regions. One approach to evaluate field inhomogeneity is to numerically calculate the field resulting from spatially varying magnetic susceptibility of different tissue types (26) classified on the basis of magnitude images. The computed field distribution then can be subtracted from the actual field map to isolate the field inhomogeneity. An alternative but more direct method consists of inverting the dipole field to quantify the relative susceptibilities of field sources (27).

Acknowledgment

The work was supported by the grants NIH T32 EB000814 and NIH R01 HL 75649.

Appendix

Derivation of Incremental Field Inside an Elliptic Cylinder with Isotropic Susceptibility

For an ellipsoid with isotropic susceptibility χ = χe + Δχ, the internal field [B with right harpoon above]in is uniform and proportional to the sum of magnetization [M with right harpoon above], applied [H with right harpoon above]o and demagnetizing [H with right harpoon above]dm field,

equation M6
(A1)

where µo is the magnetic permeability of free space, H→dm = −αM→ and α is the demagnetizing factor that is a geometry-dependent number between zero and one. Ellipsoids have three independent principal axes and three demagnetizing factors that always add up to one. The magnetization field vector is defined as [M with right harpoon above] = χ[H with right harpoon above] = (χe + Δχ)[H with right harpoon above], where Δχ is the magnetic susceptibility of the ellipsoid relative to the external medium. Here, we ignore the first term (χe) since we are only concerned with the relative internal field [B with right harpoon above]in with respect to the external medium. If the applied field [B with right harpoon above]o is parallel to a principal axis of an ellipsoid, the internal field reduces to

equation M7
(A2)

In theory, the demagnetizing factor can be obtained by solving the Laplace equation for the magnetic potential Φm, [nabla]2Φm = 0, whose unique solution Φm can be derived analytically once the boundary conditions are defined, i.e. the behavior of Φm at infinity and at the interface between the body and its surrounding medium. For example, the solution for a long circular cylinder oriented parallel to the applied field ([B with right harpoon above]o = Boz) yields αxy = ½ and αz = 0 (28). In this case A2 reduces to

equation M8
(A3)

In the general case when none of the principle axes of the cylinder is aligned with the applied field the internal field can be computed by taking the vector superposition of the applied field resolved along the principal axes(11). For a long circular cylinder it is given by

equation M9
(A4)

The appropriate demagnetizing factors along each principle axis are equation M10 and αz = 0. We note that from A4 that Bin is not parallel to the applied field even though the susceptibility is isotropic. In terms of principal axes, the components of the applied field are Bx,o = Bo sin θ cos ψ, By,o = Bo sin θ sin ψ and Bz,o = Bo cos θ; the azimuthal angle ψ is defined with respect to the x-axis.

For a long elliptic cylinder the expression for the internal field is as given by A4 except that αx ≠ αy. If we set the x-axis to coincide with the major axis, then αx =b/(a+b) and αy = a/(a+b) (28), where a and b are the major and minor axes of the ellipse, respectively, and since |Δχ|[double less-than sign]1 holds, the incremental field ΔB = [B with circumflex]o·(Bin[B with right harpoon above]o) ([B with circumflex]o is a unit vector that points along the applied field [H with right harpoon above]o = [B with right harpoon above]oo) reduces to

equation M11

where equation M12 . In short, incremental field ΔB inside a material is the change in the field along the direction of the applied field.

Above result includes the Lorentz correction (−2ΔχBo / 3) that takes into account the discreteness of sources near the point of interest where the field contribution from each magnetic moment within a spherical region is summed. The spherical region is referred to as a Lorentz sphere; its radius is macroscopic compared to atomic scales but microscopic compared to the size of the object immersed in the external field.

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