Significant Precision Improvement forTemperature Mapping

Lin Li,1* Erik M. Shapiro,2 and John S. Leigh1

MR temperature measurements are important for applicationssuch as the evaluation of thermal therapies and radiofrequency(RF) coil heating effects. In this work the spherical mean value(SMV) method has been applied to significantly improve theprecision of MR temperature mapping in a homogeneous gelphantom. Temperature-increase maps of the phantom wereobtained with three-dimensional (3D) MR phase differencemapping after heating with the RF coil. The temperature-in-crease distribution in most regions in the phantom is a har-monic function with the mean value property. Based on this prop-erty, the precision of temperature-increase maps was improvedup to sixfold with the SMV method. Comparison of this methodwith conventional smoothing, further precision improvement, andthe in vivo application of the SMV method are discussed. MagnReson Med 46:678–682, 2001. © 2001 Wiley-Liss, Inc.

Key words: temperature mapping; precision improvement;Laplace; spherical mean value; RF heating

Temperature measurements are important in many areasof physical and medical sciences. In particular, with med-ical interventions such as laser surgery and thermal ther-apy, the direct monitoring of temperature in tissues isnecessary for accurate evaluation of the treatment load andeffectiveness, and understanding the tissue’s or organ’sphysiological responses (1,2). The water proton resonancefrequency has a temperature-dependent chemical shift co-efficient (TDCSC) of –0.01 ppm/K (parts per million perdegree Kelvin). Based on this phenomenon, MR phase andspectroscopic imaging techniques have been developed tomap the temperature distribution with a typical precisionlimit of 0.5 K at 1.5 T (3,4). New methods to map temper-ature have been designed which use paramagnetic dopedsubstances, with TDCSCs of –0.04 to –0.09 ppm/K. Thesemethods have been used to investigate the effects of RFheating from MRI coils in homogeneous phantoms. Usingvery high power, temperature increases of nearly 8 K (5)have been observed with surface coils, while large bird-cage coils have caused a 1.5-K temperature increase (6).

Li and Leigh (7,8) have developed a spherical meanvalue (SMV) method that improves the precision of MRfield mapping by several orders of magnitude for magneticfields in homogeneous phantoms and in vivo. In this work

we describe the application of this method to MRI-basedtemperature-increase maps to obtain higher precision. TheSMV method is based on the principle that the magneticfield distributions are harmonic functions (9) defined assatisfying the Laplace equation. Harmonic functions havea mean value property that the SMV—the mean functionvalue on a spherical region—is equal to the center value.The SMV can be used to replace the center value andeffectively reduce the noise level, because a greater num-ber of points are averaged to yield the SMV. This averagingprocess is different from conventional smoothing because theprocess maintains the true field value for the center voxel.

There is one scenario in which this requirement of har-monic function distribution might not be satisfied every-where in space. This would be for experiments involvingstrongly focused energy deposition in tissues, such as inthermal therapy with laser and focused ultrasound. In thissituation, temperature distributions may deviate from theLaplace equation near the focal point of heat energy dep-osition.

MATERIAL AND METHODS

Theory

For a region of homogeneous thermal diffusivity (k), withno heat sources and sinks, the temperature distributionT(rY, t) (rY and t are spatial and temporal variables, respec-tively) satisfies (10):

]T]t

2 k¹2T 5 0. [1]

In the situation of a steady state (]T/]t 5 0), we have

¹2T 5 0. [2]

Therefore, T is a harmonic function, and the SMV methodcan be directly applied to effectively reduce the noise levelof the steady-state temperature measurement.

Experiment

A transmit/receive surface coil underneath a sample wasused to induce heating in the sample as well as to performthe MRI on a 4-T General Electric scanner. RF heating wasperformed as previously described (6), with the exceptionthat the current method was expanded to a 3D acquisition.The sample used was a 2-L, 160-mM CoCl2/4% agarosecylindrical block (diameter 5 17.5 cm; height 5 7 cm),which had a TDCSC of –0.041 ppm/K (6). A 3D fast-

1Department of Radiology, Metabolic Magnetic Resonance Research andComputing Center, University of Pennsylvania, Philadelphia, Pennsylvania.2Department of Chemistry, Metabolic Magnetic Resonance Research andComputing Center, University of Pennsylvania, Philadelphia, Pennsylvania.Grant sponsor: NIH; Grant number: RR02305.*Correspondence to: Lin Li, B1, Stellar-Chance Labs/6100, Dept. of Radiol-ogy, MMRRCC, University of Pennsylvania, Philadelphia, PA. E-mail:lin@mail.mmrrcc.upenn.eduReceived 6 April 2001; revised 21 June 2001; accepted 11 July 2001.

678© 2001 Wiley-Liss, Inc.

Magnetic Resonance in Medicine 46:678–682 (2001)COMMUNICATIONS

gradient-recalled-echo (FGRE) sequence was used to ac-quire the raw data, both before (baseline) and after (probe)a very high power scan. The parameters were: pulse rep-etition time (TR) 5 6 ms, echo time (TE) 5 4 ms, flipangle 5 10°, 256 3 128 matrix in transverse plane, band-width 5 62.5 kHz, field of view (FOV) 5 24 3 24 cm,40 axial slices with a 5-mm slice thickness, and imagingtime 5 31 s. The high-power scan was identical except thatit lasted 10 min and used 10 times the power (30W).Because the high-power scan time was long and the acqui-sition time of the baseline and probe images was relativelyshort, we assumed that a quasi-steady-state temperaturedistribution had been established over a substantial por-tion of the FOV by the high-power scan, and this conditiondid not change much during the acquisition sequence.Thus, Eq. [2] was valid.

Analysis

Temperature-increase maps were calculated from thephase difference between the baseline and probe MR sig-nals, as described by Shapiro et al. (6). Considering thedecreased sensitivity in the far field of the surface coil, wemasked out the image voxels with a raw MR signal ampli-tude below 5% of the maximum signal amplitude of thewhole 3D data set. We then proceeded to identify regionsin which the temperature distribution satisfied the Laplaceequation so that SMV processing could be applied in theseregions. Using the previously developed methodology (8),a 3D Laplacian operator L was constructed. The tempera-ture-increase maps were then convoluted with L to obtainthe Laplacian of the temperature-increase maps (L-maps).In local regions in which the mean of the L-map was very

close to zero (less than 2 3 10–3 K in this work) and witha standard deviation (SD) comparable to or smaller thanthe raw map noise, it was assumed that the temperaturedistributions satisfied the Laplace equation within thenoise level. Those regions in which the temperature dis-tribution was a harmonic function are hereafter called“harmonic regions.”

Next, the SMV method was applied to the harmonicregions as previously described (8). To prove the meanvalue property for the temperature distribution in the har-monic regions, SMV temperature values over sphericalshells centered at a point of interest were obtained. Be-cause the images had a nonisotropic voxel size, the spatiallength was scaled by the image unit (i.u.)—the width of thevoxel in the x dimension (0.94 mm in this study). The shellshad variable sizes, but a constant thickness of one i.u.

The higher-precision temperature-increase maps in theharmonic regions were obtained by replacing each voxelvalue with the SMV value over an entire sphere of a certainradius centered at the voxel. The larger the radius, themore voxels were used for the averaging, and the better thetemperature-increase map precision. However, the radiusshould be small enough for the sphere to be entirely main-tained within the 3D space of the harmonic region for theanalysis. Noise levels were obtained for both raw and SMVtemperature-increase maps by fitting a 2D temperature-increase map with polynomial functions and then takingthe difference of the fit and the original map (8).

RESULTS

RF heating caused the sample to heat nonuniformly, witha maximum temperature rise of ;1.7 K. Figure 1 shows the

FIG. 1. Comparison of the rawtemperature increase map withthe L-maps. a: Typical tempera-ture increase map slices in threeorthogonal directions (sagittal,coronal, and transverse). Highertemperature increases are ob-served at locations closer to theRF coil, lying in the coronal planeunderneath the cylindrical sam-ple. All images are scaled with anidentical window size ([21, 2] K).The x, y, z axes are also indicated.b: The L-maps shown with thesame window size ([21, 1] K). Formost regions, the L-maps are uni-formly close to zero with standarddeviations smaller than that ofraw maps. The original imageshad non-isotropic voxel sizes. Inorder for the images to reflectproportionally the relative physi-cal sizes of the phantom, sagittaland transverse images are en-larged by five times along the zdimension.

Precision Improvement for Temperature Mapping 679

raw temperature-increase map and corresponding L-mapfor three typical orthogonal slices (sagittal, coronal, andtransverse) in the phantom. As we can see, the effectiveregion of temperature increase after masking is quite nar-row along the y axis of the phantom. The means and SDsof both the temperature-increase map and the L-map forfive typical regions of interests (ROIs) in the effective tem-perature change region are listed in Table 1. Throughoutmost of the effective temperature-increase region, the L-map was quite uniform (close to zero) with much lessdeviation than the temperature-increase map in the corre-sponding region. The means of the L-map in typical ROIswere all less than 2 3 10–3 K, with SDs smaller than thetemperature-increase map noise (0.06 K). This meant thatthe temperature distribution in these regions satisfied theLaplace equation within the noise level. The temperaturedistribution tended to deviate from the Laplace equation atthe boundaries of the effective temperature-increase re-gion.

Figure 2 demonstrates the mean value property of thetemperature distribution for typical points in harmonicregions. The SMVs over spherical shells of various sizesare plotted against radii. As long as the whole shells werecontained in the harmonic regions, SMVs remained con-stant within the noise level as the radius varied up to10 i.u., in agreement with the mean value property ofharmonic functions. In the figure, the initial SMV values atsmall radii showed relatively larger fluctuation comparedto the SMV at large radii, in agreement with the increasingprecision improvement by SMV as the radius increases.

The surface plots of the raw temperature-increase map,L-map, and high-precision SMV temperature-increasemaps for one typical 2D ROI are displayed in Fig. 3. TheL-map was quite uniformly close to zero, with small fluc-tuations. The SMV maps had significantly higher precisionthan the raw map. The temperature-increase map fromSMV over an entire sphere (r 5 5 i.u.) had a noise level of0.009 K, compared to 0.06 K in the raw map. The signifi-cant noise reduction is further visually demonstrated inFig. 4, comparing the raw and SMV temperature-increaseimages for another ROI.

DISCUSSION AND CONCLUSIONS

We have proposed that the temperature distributionwithin a homogeneous agarose phantom is a harmonicfunction during the MRI scans both before and after a longperiod of high-power heating. This is assuming a quasi-steady-state temperature distribution, homogeneous ther-

mal diffusivity (k), and no heat sources and sinks in the gelphantom. For real data, this harmonic distribution is dem-onstrated throughout most of the effective heating regionby the L-maps which have close-to-zero mean values andSDs comparable to or less than the raw data noise level.We applied the SMV method and obtained a sixfold pre-cision improvement for the temperature-increase map-ping. The SMV method is based on the mean value prop-erty of harmonic functions, with this property being con-firmed by the data in Fig. 2.

The SMV method is a special smoothing method, differ-ent from conventional smoothing methods over a rectan-gular template. Figure 5 shows a typical line plot of SMVtemperatures (obtained over entire spheres of a 5-i.u. ra-dius) that is substantially different from a line plot oftemperatures at the corresponding positions, after the rawdata were smoothed with a rectangular template of com-parable size (11 3 11 pixel) in the x-y plane. The differencebetween the SMV and the smoothing results is 0.04 60.05 K, which is much larger than the SMV map noisepreviously shown. The basis of the SMV method is theharmonic function mean value property, which is an in-trinsic characteristic of the temperature distribution understeady state (Eq. [2]). Therefore, this method maintains thetrue value of the measurement, unlike conventionalsmoothing, which is an approximation.

In this work, the sizes of harmonic regions or the spheresfor SMV are limited by the narrow size of the effectivetemperature-increase region along the y axis. Therefore,the magnitude of the temperature precision improvementis restrained. When larger sizes of harmonic regions arepossibly achievable with an RF volume coil, the precisionof temperature mapping can be improved such that verysmall temperature changes can be detected that are notdetectable with conventional MR methods. The high-fieldmapping precision of 10–5;–6 ppm achieved with SMVprocessing (8) corresponds to a temperature mapping sen-sitivity of 10–3;–4°C for water proton signals. This is asensitivity improvement of three to four orders of magni-tude over the regular MR temperature-increase mappingbased on proton frequency shift. With more precise tem-perature change measurements, previously unobtainablesubtle temperature changes caused by mild heating can bemapped to a better extent.

Table 1Comparison of the Means and the Standard Deviations of Boththe Raw Temperature Map and the L-Map for Typical Three-Dimensional ROIs Within the Effective Temperature IncreaseRegion in the Phantom

ROI size (voxel) Raw T map (K) L-map (K)

a: 11 3 16 3 15 0.63 6 0.16 1.1 3 1023 6 0.034b: 31 3 16 3 11 1.30 6 0.33 22.9 3 1024 6 0.033c: 11 3 11 3 16 0.71 6 0.23 1.1 3 1023 6 0.034d: 21 3 11 3 11 1.02 6 0.30 1.3 3 1023 6 0.045e: 21 3 11 3 11 0.83 6 0.24 25.1 3 1024 6 0.039

FIG. 2. Validation of the mean value property. SMV values overspherical shells are plotted against the radii of the shells for severaltypical points (1, { and h) of interest in the harmonic regions. Allthe shells have a thickness of 1 i.u.

680 Li et al.

The validity of harmonic function distribution (Eq. [2])may not be true everywhere in the phantom, presumablydue to noise and other factors, such as localized heatdeposition, gel inhomogeneity, and artifacts due to theair-sample susceptibility difference. For instance, we no-ticed large nonzero deviations in the L-maps near wherethe sample contacts the RF coil. Therefore, we needed touse the L-maps to define only the harmonic regions beforewe applied the SMV method. As long as the L-maps aresufficiently close to zero, the field can be treated as aharmonic function and the SMV method should work. Weare currently in the process of developing a quantitativestandard for the application of the SMV method over ageneral field distribution.

Another factor that may violate the harmonic functiondistribution and generate errors for precision temperaturemapping is heat dissipation during the probe scan. Withonly a small RF heating-induced temperature difference(,2°) between the phantom and the environment, the tem-perature change due to heat dissipation within the 31-sacquisition time of the probe image was negligible, accord-ing to our observation. In the case of more rapid tempera-ture changes due to large initial temperature increases, wecan adopt faster sequences, such as echo planar and spiralscans, which can acquire a 3D data set in a few seconds.

This kind of quick imaging would minimize the error dueto the temperature change during the probe scan.

MR phase signal is subject to the influence of severalfactors, such as static field inhomogeneity, air-sample sus-ceptibility difference, local eddy currents, and nonlineargradients. However, these effects are mostly canceled out(8) because the temperature-increase maps are obtainedfrom MR phase difference maps. The image distortion orthe displacement of a voxel position caused by the abovefactors is assumed to be negligible in the central area of theeffective temperature-increase region, and can be reducedby choosing stronger gradients.

Future work will include an experimental justification forusing the SMV method to improve the estimates of temper-ature. The SMV results can be compared with temperaturevalues directly measured by other precision instruments,such as inserted thermocouples or infrared thermometers.

We can also test this SMV methodology in vivo. Intheory, the SMV method is limited by the condition ofhomogeneous thermal diffusivity in the ROI. Generallyspeaking, this condition is not valid in vivo. However, ithas been shown that the SMV method can be applied forimproving the precision of magnetic field mapping forcertain regions in muscles (8) and brain tissues (unpub-lished data) in vivo. This suggests that within the image

FIG. 4. Comparison of the temperature increase maps from rawdata (a) and SMV (b) (r 5 5 i.u.) for a ROI (36 3 7 pixels) denoted bythe small white rectangular marker in the coronal phase image inFig. 1. Both images are scaled with an identical window size [0.2,0.8] K, and enlarged for better visualization (five-fold more verticallythan horizontally).

FIG. 3. Temperature increase maps (in K) from a typical ROI (x 5 80:200, y 5 100, z 5 20:30, 121 3 11 pixels, denoted by the largerectangular marker in the coronal phase image in Fig. 1. a: The raw temperature increase map with a noise level of 0.06 K. b: The L-map.c: The SMV (r 5 5 i.u.) temperature increase map with a noise level 0.009 K.

FIG. 5. Comparison of the temperature increase line profiles (x 580:200, y 5 100, z 5 23) of the raw data, after smoothing (dashedline), and after SMV (r 5 5 i.u.) processing (smooth solid line). Thesmoothing was applied to the two-dimensional temperature imageslice at z 5 23 with a rectangular template of 11 3 11 pixels.

Precision Improvement for Temperature Mapping 681

noise level the intrinsic physiological inhomogeneity incommon types of tissues is not so severe as to violate theworking hypothesis of harmonic functions. Although theremay be significant differences between the thermal diffu-sivities of tissue, fat, and bones, presumably Eq. [2] canstill be valid within an individual type of tissue if thethermal diffusivity is sufficiently uniform. Therefore, itmay be possible to apply the SMV method for improvingthe precision of temperature mapping in the tissue.

However, some caution must be exercised for the appli-cation of this method in evaluating temperature-increasemaps during thermal therapies with laser or focus ultra-sound. In these interventions, the focal energy dispositionmay generate a strong heat source in the ROI. The muchlarger temperature difference (tens of degrees compared to,2° in this study) may make it difficult to achieve thequasi-steady state, because the heat dissipation into sur-rounding environment would increase. All of these factorscan make the temperature field deviate from the harmonicfunction distribution.

In conclusion, the SMV method has been applied toimprove the temperature-mapping precision in a phantom.The improvement in temperature precision is importantbecause it more accurately measures heating effects andcan presumably better evaluate thermal therapies.

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