An Internal Reference Model–Based PRF TemperatureMapping Method With Cramer-Rao Lower Bound NoisePerformance Analysis

Cheng Li,1 Xinyi Pan,1 Kui Ying,1* Qiang Zhang,2 Jing An,2 Dehe Weng,2 Wen Qin,3 andKuncheng Li3

The conventional phase difference method for MR thermometrysuffers from disturbances caused by the presence of lipid pro-tons, motion-induced error, and field drift. A signal model ispresented with multi-echo gradient echo (GRE) sequence usinga fat signal as an internal reference to overcome these prob-lems. The internal reference signal model is fit to the water andfat signals by the extended Prony algorithm and the Levenberg-Marquardt algorithm to estimate the chemical shifts betweenwater and fat which contain temperature information. A noiseanalysis of the signal model was conducted using the Cramer-Rao lower bound to evaluate the noise performance of variousalgorithms, the effects of imaging parameters, and the influ-ence of the water:fat signal ratio in a sample on the temperatureestimate. Comparison of the calculated temperature map andthermocouple temperature measurements shows that the max-imum temperature estimation error is 0.614°C, with a standarddeviation of 0.06°C, confirming the feasibility of this model-based temperature mapping method. The influence of samplewater:fat signal ratio on the accuracy of the temperature esti-mate is evaluated in a water-fat mixed phantom experimentwith an optimal ratio of approximately 0.66:1. Magn ResonMed 62:1251–1260, 2009. © 2009 Wiley-Liss, Inc.

Key words: MRI; thermometry; water proton resonance fre-quency; PRF; fat; multigradient echo sequence; Crame-Raolower bound

Quantitative MRI thermometry has become an attractivemethod to noninvasively monitor the evolution of tissuetemperatures and to guide tumor ablation based on localthermal therapy, such as high-intensity focused ultra-sound surgery (1,2). Temperature imaging based on thetemperature dependence of the water proton resonancefrequency (PRF) (3,4) is currently the main choice formany applications, especially for high magnetic fieldsabove 1 T, because there exists a linear relationship be-tween PRF and temperature that is independent of thephysiologic effects of hyperthermia (5). Typically, a tem-perature map from PRF is generated using phase-differ-

ence images from a single long gradient echo sequence(4,6). Despite many advantages, the conventional phase-difference method still has several problems that may in-fluence the precision of the quantitative temperature map-ping, such as the presence of lipid protons (7,8), inter- andintrascan motion (9), and frequency drift due to an unsta-ble magnetic field (10). The presence of lipids modifies thephase difference because the lipid resonance frequenciesare not temperature dependent, which leads to tempera-ture estimation errors. Fat-suppressed slice-selective exci-tation or spectral-spatial pulses (7,11) have been proposedby several research groups to overcome this problem.However, these techniques are sensitive to field inhomo-geneities and typically yield nonuniform suppression overthe field of view (12,13).

The motion artifacts and field drift associated with thetraditional phase-difference approach have been reducedusing an internal frequency reference that has a tempera-ture-insensitive resonance. Magnetic resonance spectro-scopic imaging (14) can trace the changes in the waterresonance frequency, using a non–temperature-dependentcomponent as an internal reference. Thus, magnetic reso-nance spectroscopic imaging has great potential to reducethe error from these sources. However, the long acquisitiontime (typically on the order of a couple of minutes) andpoor spatial resolution of magnetic resonance spectro-scopic imaging limit its applications, especially for real-time temperature monitoring, although the spatiotemporalresolution problem is mitigated by applying fast imagingmethods such as echo-planar spectroscopic imaging (15)and line scan echo-planar spectroscopic imaging (16,17).Some research groups (18) have developed an approachcalled chemical shift selective phase mapping to exploitthe complementary advantages of both the phase-differ-ence and spectroscopic imaging methods. However, thesuboptimal fat-water separation from chemical shift selec-tive phase mapping causes a temperature estimation errorthat is proportional to the amplitude ratio of the desiredand “to-be-suppressed” signal. More recently, an im-proved method that used a triple spin-gradient echo se-quence was developed to reduce the temperature error, butit still cannot eliminate the parasitic phase term arisingfrom the suboptimal fat-water separation (19).

All of the quantitative MR thermometry approaches ex-pect optimization of the imaging parameters to maximizethe temperature signal-to-noise ratio (SNRT) since the ac-curacy of the quantitative temperature mapping is an im-portant factor. Some investigators analyzed the SNRT de-pendence on the imaging parameters with a simple gradi-ent-echo type sequence and found that the optimal echo

1Engineering Physics, Tsinghua University, Beijing, People’s Republic ofChina2Siemens Mindit Magnetic Resonance Ltd., Shenzhen, Guangdong, People’sRepublic of China3Radiology, Xuanwu Hospital, Capital Medical University, Beijing, People’sRepublic of ChinaGrant sponsor: CSMRM; Grant number: N4-011.*Correspondence to: Kui Ying, PhD, Department of Engineering Physics,Tsinghua University, Haidian District, Beijing, 100084, PR China. E-mail:yingkui@mail.tsinghua.edu.cnReceived 16 July 2008; revised 30 April 2009; accepted 28 May 2009.DOI 10.1002/mrm.22121Published online 24 September 2009 in Wiley InterScience (www.interscience.wiley.com).

Magnetic Resonance in Medicine 62:1251–1260 (2009)

© 2009 Wiley-Liss, Inc. 1251

time (TE) is simply the transverse magnetization decayrate (T2*) of the tissue (20). This work presents a system-atic noise analysis of the PRF-based multi-echo gradientecho (GRE) sequence method (21,22), which is more com-plex than the single-echo GRE method. This noise analysismethod can be extended to other PRF-based methods, suchas the magnetic resonance spectroscopic imaging method.

This work presents a signal model for temperature map-ping using a multi-echo GRE sequence. The model-basedmethod exploits the advantages of both the phase differ-ence method, which has high spatiotemporal resolution,and the magnetic resonance spectroscopic imagingmethod, which utilizes lipid signals as the internal refer-ence. The new method does not require fat suppression,which may result in nonuniform fat-water separation. Thefat and water separation is achieved by solving the non-linear signal model with a noniterative frequency estima-tion–extended Prony algorithm and a nonlinear leastsquares Levenberg-Marquardt (LM) method. The tempera-ture map is calculated along with the fat and water images.Both Monte Carlo simulations and phantom experimentsare used to evaluate the method. The Monte Carlo simu-lations employ the Cramer-Rao lower bound (CRLB) toanalyze the noise performance of each algorithm (Pronyand LM) since the CRLB provides the minimum varianceof an unbiased estimator (23). The noise performance isevaluated to study the impact of the imaging parameters,such as the TEs, on the signal-to-noise ratio (SNR) of thetemperature map. The impact of the fat-water compositionon the noise performance is also studied because it isequally important, especially when the fat signal is used asan internal reference. Phantom experiments are conductedto provide experimentally verified results.

THEORY

Signal Model

The general model based on the multiecho GRE sequencedescribed here can be used for any number of species.However, for simplicity, assume that the received demod-ulated MR signal comes from the water and the fat, whichis a reasonable assumption in most temperature-mappingcases. When a multiecho GRE pulse sequence is used, thesignal within a voxel at TE tn can be represented as

s�tn� � �i�water,fat

�iej�ie��R*2i�j2�fi�tn � w�n� [1]

where �i is the magnitude, including factors such as spindensity, T1 and the imaging parameters, i is the initialphase, R2,i* is the spin-spin relaxation rate, and fi are theresonance frequencies for water and fat. These are un-known parameters that need to be determined. w(n) Is acomplex gaussian white noise that is independent identi-cally distributed, with a zero mean and variance 2 (24).

From the established relationship between the chemicalshift of water and fat and the temperature, the resonancefrequencies of the water and fat protons are:

fwater � ��B0�T � Tref� �

ffat � �B0�f�w � [2]

where T is the temperature, is the field inhomogeneity,�is the hydrogen proton gyromagnetic ratio, B0 is the mainfield strength, � is the temperature coefficient in ppm/oC,and �f�wis the chemical shift in ppm between fat and waterat the reference temperature, Tref.

The signal is acquired at various TEs at each tempera-ture point, represented by s0,s1,· · ·,sN�1, where N is thenumber of echoes with different TEs. The sample signalsare arranged as a vector �s0,s1,· · ·,sN�1�

T. Then, an unknownparameter vector p is defined with eight elements: p1

� �water, p2 � �water, p3 � R*2,water, p4 � fwater, p5 � �fat,p6 � �fat, p7 � R*2,fat, and p8 � ffat. These unknownparameters require at least eight independent measure-ments, which would be four echoes, since each echo has acomplex signal with real and imaginary parts. Since thenoise in the different samples is independent, the jointprobability density function of the samples can be given by

f(s;p) � � 12�2�N

exp��1

22 �n�0

N�1

(snR�s�n

R)2 � �snl � s�n

l �2� [3]

Where snR and sn

l are the real and imaginary parts of thesamples and s�n

R and s�nl are the expectation values of sn

R andsn

l , which are the real and imaginary parts of�

i�water,fat�iej�ie��R*2i�j2�fi�tn.

Temperature Estimation

The temperature can be estimated by first determining thelocation of the water and the fat resonance frequencies.The fast Fourier transform is a simple and computationallyefficient way to locate the water peak and the fat peak.However, there are several inherent performance limita-tions of the fast Fourier transform approach. The mostprominent limitation is the frequency resolution (25),which affects the temperature accuracy. The performanceof the fast Fourier transform approach has been improvedby using a complex Lorentzian curve-fitting method in thefrequency domain (14,15) or the z-domain (22). Here, wedid curve fitting in the time domain since the time domainanalysis is straightforward and simplifies the followingnoise analysis.

A criterion is needed to measure how well the estimatorfits the measured data before any analysis. Among themany criteria, the maximum-likelihood (ML) criterionmost closely meets the CRLB since the CRLB has a naturalconnection to ML, i.e., if there exists an unbiased estimatorthat achieves the CRLB, it will maximize the likelihood.The ML estimate of p, referred to as p̂, maximizes theprobability density function f (s;p) (see Eq. 3), which isequivalent to maximizing ln f (s;p), which is called thelikelihood function. In this sense, the ML estimator can beexpressed as

p̂ � arg maxp

ln f(s:p) [4]

By substituting the probability density function in Eq. 3into Eq. 4, the ML estimation becomes a nonlinear leastsquares problem:

1252 Li et al.

p̂ � arg minp �

n�0

N�1

�s�TEn� � �i�water,fat

�iej�ie��R*2,i�j2�fi�TEn�2

2

[5]

where

�p� � �n�0

N�1

�s�TEn� � �i�water,fat

�iej�ie��R*2,i*�j2�fi�TEn�2

2

is the cost function.The nonlinear least squares optimization problem can

be solved by the trust region LM algorithm (26). A criticalproblem in nonlinear least squares fitting is to select theinitial value so that the LM algorithm can converge to theglobal minimum. Thus, the choice of the initial valueplays a vital role in the iterative algorithm for nonlinearoptimization problems. To give insight into the effect ofthe initial value, the cost function is plotted as a functionof the water and fat frequencies in the absence of noise inFig. 1. As Fig. 1 shows, the cost function can have manylocal minima. If the initial value is selected close to one ofthe local minima, the iterative algorithm does not convergeto the global minimum. Therefore, a noniterative fre-quency estimate, the extended Prony algorithm, was usedto obtain the initial value. The Prony algorithm result isalways close to the global minimum when the data has arelatively high SNR (25). Therefore, the nearly global-min-imum solution generated by the Prony algorithm is used toprovide an initial value for the iterations. After the waterand fat resonance frequencies are extracted from the sig-nal, the chemical shift between the water and fat is calcu-lated and the temperature estimate is obtained using

�H2O-CH2�ppm� �fwater � ffat

�B0� �T � � [6]

where the temperature coefficient, �, and the intercept, �,are obtained by calibration procedure (15).

CRLB

Generally, a parameter (such as the temperature in thiscase) can be estimated by many estimators, which havedifferent accuracies. In practice, an unbiased estimator ispreferred, so the variance of the estimator, which is iden-tical to the mean square error (MSE), is the most importantand is often used as a measure of the estimation inaccu-racy. The variance of any unbiased estimator is lowerbounded by the CRLB, which provides the minimum vari-ance for a given data set, independent of the estimationalgorithm. Therefore, the CRLB can be used to optimizethe imaging parameters and to choose the estimation algo-rithm. The CRLB has been applied in MRI to estimate themagnitude, phase, and variance of the image (24), thediffusion coefficient (27), and the three-point decomposi-tion of water and fat (28). Here, the CRLB is used for thetemperature estimate.

The CRLB can be derived from the joint probabilitydensity function of the elements of the sample vector, S.Here the derivation of the Fisher information matrix isdescribed to explain how the temperature term is addedinto the CRLB. Specifically, the CRLB asserts that thecovariance matrix of any unbiased estimator P̂, which isdenoted by Cp̂, is always “larger” than the inverse of theFisher information matrix Fp. The bound is expressed as(23):

Cp̂ � Fp�1 [7]

which indicates that the matrix �Cp̂ � Fp�1� is positive

semidefinite. In particular, for a positive semidefinite ma-trix, the diagonal elements are all nonnegative, whichimplies that

p̂i

2 � �Fp�1�iii � 1,2,· · ·,8. [8]

where p̂i

2 is the variance of the estimator of the ith un-known parameter.

Assuming that the temperature coefficient is a knownparameter with zero uncertainty, the temperature estima-tor variance is bounded by

T̂2

�1

���B0�2��FP

�1�4,4 � �FP�1�8,8 � 2�FP

�1�4,8� [9]

More details of the derivation of the CRLB for the temper-ature estimator are given in the Appendix. Eq. 9 can beanalyzed to show that the lower bound depends on theimaging parameters (pulse repetition time, flip angle, andTEs), tissue parameters (spin density, water:fat ratio, T1,and T2*) and temperature.

FIG. 1. Cost function of ML estimate. The magnitude for both thewater and the fat is 1, the initial phase and R2* are all set to 0, theresonance frequencies of fat and water are 0 and 0.1 Hz, and thenumber of echoes � 8 with TE0 � 0 sec and �TE � 1 sec. Forsimplicity, the time parameters are all normalized. As displayed, thecost function has many local minima. If the initial value is selectedclose to one of the local minima, the iterative algorithm does notconverge to the global minimum.

Model-based PRF Temperature Mapping With Noise Analysis 1253

MATERIALS AND METHODS

Sequence

The MR temperature imaging was performed with a 3-TTrio scanner (Siemens Medical Systems, Erlangen, Ger-many). A multiple gradient-echo sequence was used thatsimultaneously encodes multiple TEs with the samephase-encoding gradient using the bipolar readout gradi-ent. The multiecho GRE sequence is time efficient and hasa larger effective spectroscopic bandwidth than the single-echo GRE with a long TE. The increased spectral band-width removes the phase wrapping caused by high tem-perature increase. More important, the multiecho se-quence has reduced motion sensitivity and multiplespectral component sensitivity (21).

Monte-Carlo Simulation

The CRLB tool provides a theoretical bound on the noiseperformance that may not be achieved by a practical esti-mator. Monte Carlo simulations were used to compare theProny and ML estimators with the CRLB. For the MonteCarlo simulations, the water and fat magnitudes were bothsimply set to 1, the initial water and fat phases were bothset to 0, the T2*s of water and fat were 85 ms and 15 ms, theresonance frequencies of the water and fat were 0 and–420 Hz (corresponding to the –3.35 ppm chemical shift in3 T), and the number of echoes was eight, with TE0 � 4 msand �TE � 3.5 ms. The signal was then generated as thesum of the two damped complex exponents from the waterand fat with these parameters. Then, an independent gaus-sian white noise with mean 0 and variance 2 was added toboth the real and imaginary parts of the signal to realizethe samples. The SNR was defined as

SNR � 10log10

12dB [10]

The SNR varied from 0 dB to 40 dB. Five thousand sam-ples were analyzed for each SNR by the extended Pronyalgorithm, and the ML estimation was performed by theLM algorithm. The simulations, including the generationof noisy signals and the implementations of the algo-rithms, were done on the Matlab platform (Mathworks,Natick, MA, USA). The estimation performance was eval-uated based on the MSE of the frequency difference be-tween the water and the fat. The MSE was compared withthe CRLB at each SNR point. The MSE of the estimator isdefined as

MSE �

�i�1

N

��f̂water,i � f̂fat,i� � �fwater � ffat��2

N[11]

where N ( � 5000) is the number of samples, f̂water,i and f̂fat,i

are the ith estimated resonance frequencies of water andfat, and fwater and ffat are the true resonance frequencies ofwater and fat.

Phantom Temperature Experiment

Phantom temperature experiments were performed todemonstrate the feasibility of the method. A 36% fat whip-ping cream mixed with agar was first boiled and thencooled to a solid jelly and later placed in a container filledwith boiled water and imaged on a Siemens Trio 3-T MRscanner with a single-channel wrist coil. A copper-con-stantan thermocouple probe (Physitemp Instruments, Inc.,Clifton, NJ) was used to record the temperature for calibra-tion, as well as verification of the model. An eight-echoGRE sequence with TE0 � 4.15 ms and �TE � 3.59 ms wasused for the continuous measurements. The TE0 and theTE spacing were empirically chosen to produce imageswith reasonable image SNR (40 dB). The other imagingparameters were pulse repetition time � 50 ms, BW �� 19.60kHz, flip angle � 25°, field of view � 100 �100 mm2, slice thickness � 5 mm, and data matrix � 128 �128. The heating rate was approximately 1.82°C/min.About 70 consecutive images (each with a scan time of 6.4sec) were acquired throughout the experiment and ana-lyzed offline with a PC using Matlab to calculate the tem-perature evolution. The calculated chemical shift, �H2O-CH2,between the water and the lipid (3 � 3-pixel averagingnear the probe) with the extended Prony algorithm and thecorresponding temperature recorded by the thermocoupleprobe for the first 25 measurements were fit to a linearregression to determine the slope and intercept of theregression line for the calibration process. The calibrationregion was selected to avoid the susceptibility artifactcaused by the thermocouple probe. Although the calibra-tion region was not exactly located at the probe position,the temperature difference between this region and theprobe was quite small because the temperature distribu-tion in the phantom is relatively uniform. The tempera-tures for each of the remaining measurements were thencalculated via the linear relationship defined by the regres-sion line between the temperature and the chemical shift(see Eq. 6).

Water:Fat Ratio Phantom Experiment

A water-fat phantom experiment similar to that by Reederet al. (29) was conducted to quantitatively examine theinfluence of the water and fat content on the temperatureestimate and to verify the results of the theoretical temper-ature variance analysis for various water:fat signal ratios.The phantom consisted of agar gel with pure water in thebottom layer of a container and lard rendered to separateout the impurities placed above the gel. The phantom wasthen scanned on a Siemens 3T at room temperature. Fig. 2shows the phantom imaged in the coronal plane using asingle-echo GRE pulse sequence with nonsuppression, wa-ter suppression, and fat suppression. An obliquely ori-ented slice was selected through the water-fat interface togenerate a continuous range of water:fat signal ratiosacross the slice. A series of 135 images was acquired witha single-channel wrist coil, along with an eight-echo GREsequence with the following image parameters: pulse rep-etition time � 50 ms, bandwidth � 256�256, flip angle �40°, field of view � 120 � 120 mm2, slice thickness �8 mm, and data matrix � 256 � 256. The initial TE and the

1254 Li et al.

echo spacing were TE0 � 4.15 ms and �TE � 4.04 ms. Thescan time of each image was 12.8 sec, and the total time forthe experiment was approximately 1 h. The temperaturewas first calculated using the ML estimator, and then theMSE of the temperature was determined. The mean of thecalculated water:fat signal ratio for the 135 measurementswas used as the true water:fat signal ratio since the MLestimator is unbiased. The MSE of the calculated temper-ature of each pixel was plotted with the water:fat signalratio of that pixel to investigate the influence of the water:fat ratio on the temperature estimate accuracy. The statis-tical result was also compared with the CRLB.

RESULTS

Fig. 3 compares the MSE results from the Monte Carlosimulations with the two algorithms along with the CRLB.The figure shows that the CRLB is achievable by the MLestimator when the SNR is relatively high (more than 15dB). However, when the SNR is low, the MSE of the MLestimator increases and is higher than the CRLB. The MSEof the extended Prony algorithm is generally higher thanboth the ML estimator and the CRLB, but asymptoticallyapproaches the CRLB as the imaging SNR increases.

Fig. 4a shows the phantom image from the first echo andthe thermocouple probe, indicated by an arrow. Fig. 4bshows a temperature map of the sample at the 20th timepoint when the heating was not entirely stable. The resultdemonstrates that the temperature distribution across thesample is captured reasonably well. A more detailed tem-perature profile is given in Fig. 4c for the line shown inFig. 4b through the sample. Fig. 4d shows the calibrationline �H2O-CH2 � � 0.01021T � 3.80284 of a 3 � 3-pixelregion near the probe (the calibration region is shown by arectangle in Fig. 4b), which had a correlation coefficient,r2, of 0.998. The calculated temperature coefficient is con-sistent with those reported previously (15,22). Fig. 4eshows the temperature evolution curves measured by thethermocouple and calculated from the data using both theProny and LM algorithms. No significant difference be-tween the results of the two algorithms is observed. Themaximum temperature estimation error is 0.614°C, and thestandard deviation is 0.06°C, which indicates very goodagreement between the calculated temperatures and thethermocouple measurements.

Fig. 5a-c shows the calculated water, fat, and recom-bined images acquired obliquely through the water-fat in-terface of the water-fat phantom. Fig. 5d displays one

example of a continuum of water:fat signal ratios acrossthe 128th line.

Fig. 6a shows the experimental MSEs of the calculatedtemperature from the water-fat phantom experiments as afunction of the water:fat signal ratio at each pixel (dots),with the CRLB theoretical result shown as a solid curve.The comparison indicates that the CRLB agrees well withthe experimental measurements. The calculated relativetemperatures of the 135 measurements for three typicalwater:fat ratios (10:1 (●); 4:1 (�); 0.66:1 (‚) are shown inFig. 6b. For these water:fat ratios, the best temperatureestimate is for the water:fat ratio of 0.66:1, as indicated bythe smallest standard deviation for the calculated temper-ature of only 0.20°C. As the water content or the fat contentincreases, the temperature estimate becomes less accurate.

Fig. 7 shows the chemical shift of water �H2O(E), fat�CH2(�), and their difference �H2O-CH2(�) for the 135 temper-ature measurements. The apparent temperature changesdemonstrate that the magnetic field drift has been mostlyremoved by using the fat as the internal reference duringthe long time temperature measurements. Since both thewater protons and lipid protons experience a similar res-onance frequency drift, their difference cancels the drifteffect.

DISCUSSION

The initial model for temperature mapping (30) was sug-gested by the IDEAL water-fat decomposition method (29)since the fat can be separated from the water in the IDEALmethod and has the potential to be used as an internalreference. For temperature mapping, the current methodadds the temperature-related component into the water-fatdecomposition model and also considers the T2* effectwhen multiple gradient echoes are acquired. The internal

FIG. 3. The MSE results of Monte Carlo simulations with the ex-tended Prony algorithm (dashed line), the ML estimation with the LMalgorithm (solid line), and the CRLB (dashed single-dot line). TheCRLB is achieved by the ML estimator with a relatively high SNR(more than 15 dB). When the SNR is low, the MSE of the MLestimator increases. The MSE of the extended Prony algorithm isgenerally higher than for the CRLB, but it is asymptotic to the CRLBas the imaging SNR increases.

FIG. 2. A mixed water-fat phantom imaged in the coronal planeusing the GRE pulse sequence with water suppression (a), fatsuppression (b), and nonsuppression (c). From this localized plane,an obliquely oriented slice through the fat-water interface generatesa continuous range of water:fat ratios across the slice.

Model-based PRF Temperature Mapping With Noise Analysis 1255

reference-based fitting of the water and fat signals with themultiecho sequence acquisition presented here not onlyremoves artifacts associated with the traditional phase dif-ference method but also provides higher spatiotemporal

resolution than spectroscopic imaging. These results showthat the spatial resolution is 0.78 mm � 0.78 mm and theimaging time is 6.4 sec. To further reduce the scan time,future work will focus on combining parallel imaging orother fast imaging techniques with this method. These testresults verify that the model provides absolute tempera-ture measurements as long as the calibration is accurate.

The CRLB tool is used to evaluate the noise effect of eachestimator for the temperature mapping. The ML estimatoris experimentally shown to be efficient, achieving theCRLB when the SNR is above a threshold of 15 dB. How-ever, at lower SNR, the MSE of the ML estimation is higherthan the CRLB because the LM algorithm converges to alocal rather than a global minimum for the ML criterion.This phenomenon can be termed the “threshold effect”and frequently plagues nonlinear estimates (23). The ex-tended Prony algorithm utilizes an important property thatthe damped exponential signal generated by the GRE pulsesequence can be described by linear equations so that theProny algorithm can change the minimization of the mod-ified cost function to a linear estimation problem. Thecomputational complexity (25) is then dramatically re-duced, but at the cost of a higher MSE estimate than withthe ML estimator-LM algorithm (see Fig. 3). However, be-cause of its computational simplicity, the Prony algorithmcan also be used as an alternative temperature estimatorfor the ML throughout the measurements. Phantom tem-perature experiments demonstrate that there is no signifi-cant difference between the results of the LM algorithmand the Prony algorithm when the SNR is relatively high(see Fig. 4), which is consistent with the Monte Carlosimulation result.

FIG. 4. Phantom temperature experiment results. a: Phantom image from the first echo. The thermocouple probe is indicated by the arrow.b: Temperature map of the sample at the 20th time point. The rectangle shows the calibration region. c: Temperature profile of a line throughthe sample for the line shown in (b). d: Linear regression between the temperature and the chemical shift (r2 � 0.998). e: Temperatureevolution curves measured by thermocouple (‚) and calculated from our method using both the extended Prony (�) and LM (E) algorithms.No significant difference between the results of the LM and extended Prony algorithms was observed. The maximum temperatureestimation error was 0.614°C and standard deviation was 0.06°C.

FIG. 5. Water:fat ratio phantom experimental results with calculatedwater (a), fat (b), and recombined (c) images of the oblique slicethrough the water-fat interface of the mixed water-fat phantom withmultiecho GRE. A plot of the water:fat signal ratio through the 128thline of the water-fat phantom image displayed in (d) shows a con-tinuous range of water:fat signal ratios.

1256 Li et al.

Besides the dependence on the temperature estimationalgorithms, the SNRT is also dependent on the sequenceand tissue parameters. The analysis of the influence ofthese parameters on SNRT is also based on the CRLB. Theeffects of the sequence parameters were analyzed for asimple, ideal case, assuming that water is the only species,with no field drift, susceptibility effect, motion, or otheradverse effects. In this case, SNRT can be expressed simplyas:

SNRT���n�0

N�1

�water2 TEn

2e�2R*2,waterTEn [12]

where �water is the signal intensity dependent on the tissueparameters, such as the water proton density T1, and theimaging parameters, such as pulse repetition time and theflip angle. N is the number of echoes, and TEn is the nthTE. When N � 1, Eq. 12 reduces to Eq. 4 or Eq. 5 in Chunget al. (20). Although single-echo GRE has an analyticalsolution for the optimal TE, which is equivalent to T2*,

this is not the case for multiecho GRE. Therefore, theoptimization of the TEs in multiecho GRE requires numer-ical analysis. Moreover, the optimal TE determinationwould involve readout bandwidth issues since shorter TEspacing requires higher readout bandwidth. Because ofthese difficulties, global TE optimization is beyond thescope of this work. Nonetheless, the optimal initial TE(TE0) can still be determined when the readout bandwidthor TE spacing is fixed. The normalized SNRT is plotted asa function of TE0 in Fig. 8 for T2* � 20 ms, �TE � 4 ms andone, two, four, and eight echoes. Note that the SNRT isnormalized with respect to the maximum SNRT for thesame number of echoes. From Fig. 8, the optimal TE0 forthe different numbers of echoes is TE0,single-echo �20 ms, TE0,two-echo � 18.20 ms, TE0,four-echo � 15.00 ms, andTE0,eight-echo � 10.06 ms.

All internal reference methods rely on the presence ofthe reference. Therefore, the amount of the reference con-tained in the sample influences the accuracy of the param-eter estimate. For this case, the fat content could have asignificant effect on the temperature estimate accuracy.The CRLB is used to theoretically predict the dependenceof the temperature estimate on the water:fat signal ratio,with experiments conducted to verify the prediction. In-tuitively, the optimal water:fat ratio is 1:1 if the relaxationtime is not considered. However, since the T2* of fat isshorter than that of water, the fat signal decays faster thanthe water signal. Therefore, the fat content should behigher than the water content to compensate for the signalloss. In the experiments, the average fat T2* was 16.6 ms,and the average water T2* was 95.0 ms. Both the experi-mental results and the CRLB indicate that the optimalwater:fat signal ratio is approximately 0.66:1. Generally,the internal reference–based methods work effectively fora range of water:fat ratios. If fat is the only species so that

FIG. 7. Field drift effects on each chemical shift and their difference.Apparent temperature changes of water �H2O (E), fat �CH2 (�), andtheir difference �H2O-CH2(‚) for the 135 temperature measurements.The chemical shifts have been converted to temperatures with thetemperature coefficient 0.01 ppm/oC. A field drift of about0.047 ppm/h for both the water and the fat resonance frequencieswas observed, but not for the difference between the water and thefat frequencies.

FIG. 6. Comparison of CRLB theoretical result (solid curve) and theexperimental MSEs of the calculated temperature from the water-fatphantom experiments as a function of the water:fat ratio at eachpixel (dots) displayed in (a) shows that the CRLB agrees very wellwith the experimental measurements. The calculated relative tem-perature of the 135 measurements with three typical water:fat ratios(10:1 (●); 4:1 (�); 0.66:1 (‚) in (b) shows that the temperatureestimation accuracy depends on water:fat ratios. The smallest stan-dard deviation for the calculated temperature with a water:fat ra-tio � 0.66:1 is only 0.20°C, which is the optimal water:fat ratio in theexperiments.

Model-based PRF Temperature Mapping With Noise Analysis 1257

the ratio is out of the effective range, all the PRF-basedmethods fail. In this case, other MR parameters sensitive totemperature should be utilized to measure the tempera-ture, such as T1 (31). Another tissue parameter that influ-ences the temperature estimate accuracy is the T2* of bothwater and fat. An evident demonstration of the T2* effecton the SNRT can be seen in Eq. 12. As T2* decreases (or R2*increases), the SNRT also decreases and, hence, the tem-perature estimation error increases. The effect of T2* canalso be understood from a spectral perspective. When T2*decreases in the time domain, the line width of the signalpeak broadens in the corresponding frequency domain,which makes the resonance frequency determination lessaccurate. A similar conclusion has been reached for thespectroscopic methods (10,32).

As stated in the introduction, the phase-differencemethod suffers from several adverse factors that cause thewater resonance frequency shift that are unrelated to thetemperature change. This method offers a solution thataddresses these problems. First, since the water and fat areseparated, the intravoxel signal interference from the fat iseliminated. Second, by using the resonance frequency oflipid proton as an internal reference in this method, thestatic magnetic field drift is also removed, as demonstratedin these experiments. The water-fat mixed phantom exper-iments showed that the field drift was about 0.047 ppm/h(apparent temperature drift of 4.7°C/h) for both the waterand fat resonance frequencies, but not for the frequencydifference between the water and fat, since the field driftinfluence was removed by subtraction. The field drift prob-lem is severe with long heating and low fields. Therefore,it is desirable to eliminate the field drift effect. Third, asreported by others, using fat as an internal reference cansignificantly reduce the sensitivity of the temperature es-timation to interscan motion artifacts within inhomoge-

neous fields (15,18,19). More experimental and theoreticalwork is needed to evaluate this anticipated advantage.

Another method to address the problems of interscanmotion and field drift is called referenceless thermometry,or self-referenced thermometry, which extrapolates thebackground phase in the treatment area from the un-wrapped background phase in the surrounding unheatedregion and estimates the temperature from every individ-ual image itself, without a preheating reference scan (33).However, this technique is limited by the requirement ofhaving enough surrounding, unheated, and relatively ho-mogeneous tissue with which to accurately estimate thebackground phase and the need to account for TE-depen-dent phase discontinuities between the water and fat re-gions (34). Our method is free of these limits. However, thecurrent method will not work when no fat signal ispresent. The comparison between these two methodsshows that the complementary advantages of these twoapproaches can be exploited by combining them. For in-stance, the frequency in the nonfat pixels can be extrapo-lated from the fat-containing pixels, as the referencelessthermometry method does. More work needs to be done tofurther investigate and utilize the benefits of combiningthese two methods.

CONCLUSIONS

A signal model using a multiecho GRE sequence was de-veloped for PRF thermometry using the fat signal as aninternal reference. The internal reference signal model isfit to the water and fat signals using the extended Pronyalgorithm and the nonlinear LM algorithm to estimate thechemical shifts between the water and fat that contain thetemperature information. The results from the phantomtemperature experiments demonstrate the feasibility of themodel-based temperature measurements. More important,the theoretical CRLB can be used to analyze the noiseperformance of each estimator. The noise performance ofthe combination of the Prony and LM algorithms used inthis work reaches the CRLB, which provides the minimumvariance of the unbiased estimator, indicating that thismethod is an efficient temperature estimation method. Inaddition, the CRLB is used to construct a general SNRexpression to quantitatively evaluate the impact of se-quence parameters on the SNRT. The CRLB is also used totheoretically analyze the quantitative influence of the sam-ple water and fat content on the temperature estimate. Thetheoretical predictions are verified by phantom experi-ments. The model-based temperature mapping methodwill be an efficient method for temperature measurementsin clinical applications, and the noise analysis based onthe CRLB provides a guideline for clinical applications ofMR thermometry.

ACKNOWLEDGMENTS

The authors thank Prof. Kim Butts Pauly (Stanford Univer-sity) and Prof. Zhi-Pei Liang (UIUC) for valuable com-ments and the anonymous reviewers for their helpful com-ments and suggestions.

FIG. 8. Normalized SNRT as a function of TE0 of GRE for variousnumbers of echoes. The other parameters are �TE � 4 ms, T2* �20 ms. The SNRT is normalized with respect to the maximumSNRT for the same number of echoes. The optimal TE0s for thedifferent numbers of echoes are TE0,single-echo � 20 ms,TE0,two-echo � 18.20 ms, TE0,four-echo � 15.00 ms, and TE0,eight-

echo � 10.06 ms.

1258 Li et al.

APPENDIX

Derivation of the Elements of the Fisher Information Matrix

The joint probability density function of the sample vectorS is given by Eq. [3]:

f�s;p� � � 12�2�N

exp��1

22 �n�0

N�1

�snR � s�n

R�2 � �sn1 � s�n

1�2� [A.1]

Then, the log of the joint probability density function(likelihood function) is:

ln f(s;p) � N ln� 12�2� �

122 �

n�0

N�1

��snR � s�n

R�2 � �sn1 � s�n

1�2�

[A.2]

The first-order partial derivative with respect to pi is:

�ln f(s;p)�pi

�12 �

n�0

N�1��snR � s�n

R��s�n

R

�pi� �sn

1 � s�n1�

�s�n1

�pi� [A.3]

The second-order partial derivative with respect topi and pj is:

�2ln f(s;p)�pi�pj

��

�pj��ln f(s;p)

�pi� �

12 �

n�0

N�1���s�n

R

�pi

�s�nR

�pj�

�s�n1

�pi

�s�n1

�pj

� �snR � s�n

R��2s�n

R

�pi�pj� �sn

1 � s�n1�

�2s�n1

�pi�pj� [A.4]

The elements of the Fisher information matrix are relatedto the expectation of

�FP�ii � � E��2ln f(s;p)�pi�pj

� �12 �

n�0

N�1��s�nR

�pi

�s�nR

�pj�

�s�n1

�pi

�s�n1

�pj� [A.5]

The expectation of the third and fourth term in [A.4] is 0since E�sn

R� � s�nR,E�sn

1� � s�n1.

Because of the symmetry of the Fisher information ma-trix, we need only calculate the upper triangle of thematrix. We will also use the symmetry in the derivation ofthe CRLB for the temperature.

Derivation of the CRLB for the Temperature

The temperature can be written as a linear function of thefrequency difference between the water and the fat:

T �fwater � ffat

��B0� c [A.6]

where c is a constant value resulting from the temperaturecalibration.

The Jacobian matrix of the linear transformation is

�T�p

� � �T��water

,�T

��water,

�T�R*2,water

,�T

�fwater,

�T��fat

,�T

��fat,

�T�R*2,fat

,�T�ffat

��

1��B0

[0,0,0,1,0,0,0,�1] [A.7]

The CRLB for the temperature is:

T̂2

��T�p

Fp�1��T

�p�T

�1

���B0�2��Fp

�1�4,4 � �Fp�1�8,8 � �Fp

�1�4,8

� �Fp�1�8,4� �

1���B0�

2��Fp�1�4,4 � �Fp

�1�8,8 � 2�Fp�1�4,8� [A.8]

The second equation exploits the symmetry of the Fisherinformation matrix, �FP

�1�4,8 � �FP�1�8,4.

REFERENCES

1. Wlodarczyk W, Hentschel M, Wust P, Noeske R, Hosten N, RinnebergH, Felix R. Comparison of four magnetic resonance methods for map-ping small temperature changes. Phys Med Biol 1999;44:607–624.

2. Quesson B, de Zwart JA, Moonen CTW. Magnetic resonance tempera-ture imaging for guidance of thermotherapy. J Magn Reson Imaging2000;12:525–533.

3. Hindman JC. Proton resonance shift of water in the gas and liquidstates. J Chem Phys 1966;44:4582–4592.

4. IshiharaY, Calderon A, Watanabe H, Okamoto K, Suzuki Y, Kuroda K,Suzuki Y. A precise and fast temperature mapping using water protonchemical shift. Magn Reson Med 1995;34:814–823.

5. Peters RD, Hinks RS, Henkelman RM. Ex vivo tissue-type indepen-dence in proton-resonance frequency shift MR thermometry. MagnReson Med 1998;40:454–459.

6. De Poorter J, De Wagter C, De Deene Y, Thomsen C, Stahlberg F, AchtenE. Noninvasive MRI thermometry with the proton resonance frequency(PRF) method: in vivo results in human muscle. Magn Reson Med1995;33:74–81.

7. de Zwart JA, Vimeux FC, Delalande C, Canioni P, Moonen CTW. Fastlipid-suppressed MR temperature mapping with echo-shifted gradient-echo imaging and spectral-spatial excitation. Magn Reson Med 1999;42:53–59.

8. Hynynen K, Pomeroy O, Smith DN, Huber PE, McDannold NJ, Ketten-bach J, Baum J, Singer S, Jolesz FA. MR imaging-guided focused ultra-sound surgery of fibroadenomas in the breast: a feasibility study. Radi-ology 2001;219:176–185.

9. Bolan PJ, Henry P-G, Baker EH, Meisamy S, Garwood M. Measurementand correction of respiration-induced Bo variations in breast 1H MRS at4 tesla. Magn Reson Med 2004;52:1239–1245.

10. De Poorter J, De Wagter C, De Deene Y, Thomsen C, Stahlberg F, AchtenE. The proton-resonance-frequency-shift method compared with mo-lecular diffusion for quantitative measurement of two-dimensionaltime-dependent temperature distribution in a phantom. J Magn ResonB 1994;103:234–241.

11. Weidensteiner C, Quesson B, Caire-Gana B, Kerioui N, Rullier A, Tril-laud H, Moonen CTW. Real-time MR temperature mapping of rabbitliver in vivo during thermal ablation. Magn Reson Med 2003;50:322–330.

12. Bernstein MA, King KF, Zhou XJ. Handbook of MRI pulse sequences.Boston: Elsevier; 2004. 857–858.

13. Reeder SB, Wen Z, Yu H, Pineda AR, Gold GE, Markl M, Pelc NJ.Multicoil Dixon chemical species separation with an iterative leastsquares estimation method. Magn Reson Med 2004;51:35–45.

14. Kuroda K, Suzuki Y, Ishihara Y, Okamoto K, Suzuki Y. Temperaturemapping using water proton chemical shift obtained with 3D-MRSI:feasibility in vitro. Magn Reson Med 1996;35:20–29.

15. Kuroda K, Mulkern RV, Oshio K, Panych LP, Moriya T, Okuda S,Hynynen K, Jolesz FA. Temperature mapping using the water protonchemical shift: self-referenced method with echo-planar spectroscopicimaging. Magn Reson Med 2000;44:220–225.

Model-based PRF Temperature Mapping With Noise Analysis 1259

16. McDannold N, Hynynen K, Oshio K, Mulkern RV. Temperature mon-itoring with line scan echo planar spectroscopic imaging. Med Phys2001;28:346–355.

17. McDannold N, Barnes AS, Rybicki FJ, Oshio K, Chen NK, Hynynen K,Mulkern RV. Temperature mapping considerations in the breast withline scan echo planar spectroscopic imaging. Magn Reson Med 2007;58:1117–1123.

18. Kuroda K, Oshio K, Chung AH, Hynynen K, Jolesz FA. Temperaturemapping using the water proton chemical shift: a chemical shift selec-tive phase mapping method. Magn Reson Med 1997;38:845–851.

19. Shmatukha AV, Harvey PR, Bakker CJG. Correction of proton resonancefrequency shift temperature maps for magnetic field disturbances usingfat signal. J Magn Reson Imaging 2007;25:579–587.

20. Chung AH, Hynynen K, Colucci V, Oshio K, Cline HE, Jolesz FA.Optimization of spoiled gradient-echo phase imaging for in vivo local-ization of a focused ultrasound beam. Magn Reson Med 1996;36:745–752.

21. Mulkern RV, Panych LP, McDannold N, Jolesz FA, Hynynen K. Tissuetemperature monitoring with multiple gradient echo imaging se-quences. J Magn Reson Imag 1998;8:493–502.

22. Taylor BA, Hwang KP, Elliott AM, Shetty A, Hazle JD, Stafford RJ.Dynamic chemical shift imaging for image-guided thermal therapy:analysis of feasibility and potential. Med Phys 2008;53:793–803.

23. Kay SM. Modern spectral estimation: theory and applications. Engle-wood Cliffs, NJ: Prentice Hall; 1993. 407–443.

24. Sijbers J, den Dekker AJ. Maximum likelihood estimation of signalamplitude and noise variance from MR data. Magn Reson Med 2004;51:586–594.

25. Kay SM, Maple SL. Spectrum analysis—a modern perspective. In:Proceedings of IEEE. 1981;69:1380–1419.

26. Nocedal J, Wright SJ. Numerical optimization. New York: Springer;1999. 262–264.

27. Brihuega-Moreno O, Heese FP, Hall LD. Optimization of diffusionmeasurements using Cramer-Rao lower bound theory and its applica-tion to articular cartilage. Magn Reson Med 2003;50:1069–1076.

28. Pineda AR, Reeder SB, Wen Z, Pelc NJ. Cramer-Rao bounds for three-point decomposition of water and fat. Magn Reson Med 2005;54:625–635.

29. Reeder SB, Pineda AR, Wen Z, Shimakawa A, Yu H, Brittain JH, GoldGE, Beaulieu CH, Pelc NJ. Iterative decomposition of water and fat withecho asymmetry and least-squares estimation (IDEAL) with fast-spinecho imaging. Magn Reson Med 2005;54:636–644.

30. Li C, Pan XY, Ying K, Zhang Q. PRF thermometry using iterativedecomposition of water and fat. In: Proceedings of the 15th AnnualMeeting of ISMRM, Berlin, Germany, 2007. p 3372.

31. Hynynen K, McDannold N, Mulkern RV, Jolesz FA. Temperature mon-itoring in fat with MRI. Magn Reson Med 2000;43:901–904.

32. Childs C, Hiltunen Y, Vidyasagar R, Kauppinen RA. Determination ofregional brain temperature using proton magnetic resonance spectros-copy to assess brain-body temperature differences in healthy humansubjects. Magn Reson Med 2007;57:59–66.

33. Rieke V, Vigen KK, Sommer G, Daniel BL, Pauly JM, Butts K. Refer-enceless PRF shift thermometry. Magn Reson Med 2004;51:1223–1231.

34. Rieke V, Kinsey AM, Ross AB, Nau WH, Diederich CJ, Sommer G, PaulyKB. Referenceless MR thermometry for monitoring thermal ablation inthe prostate. IEEE Trans Med Imaging 2007;26:813–821.

1260 Li et al.