Model Predictive Filtering MR Thermometry: effects of model inaccuracies, k-space reduction factor, and temperature increase rate

Abstract

Purpose

Evaluate effects of model parameter inaccuracies (thermal conductivity, k, and ultrasound power deposition density, Q), k-space reduction factor (R), and rate of temperature increase (Ṫ) in a thermal model-based reconstruction for MR-thermometry during focused-ultrasound heating.

Methods

Simulations and ex-vivo experiments were performed to investigate the accuracy of the thermal model and the model predictive filtering (MPF) algorithm for varying R and Ṫ, and their sensitivity to errors in k and Q. Ex-vivo data was acquired with a segmented EPI pulse sequence to achieve large field-of-view (192x162x96mm) 4D temperature maps with high spatio-temporal resolution (1.5x1.5x2.0mm, 1.7s).

Results

In the simulations, 50% errors in k and Q resulted in maximum temperature root mean square errors (RMSE) of 6°C for model only and 3°C for MPF. Using recently developed methods, estimates of k and Q were accurate to within 3%. The RMSE between MPF and true temperature increased with R and Ṫ. In the ex-vivo study the RMSE remained below 0.7°C for R ranging from 4–12 and Ṫ of 0.28–0.75°C/s.

Conclusion

Errors in MPF temperatures occur due to errors in k and Q. These MPF temperature errors increase with increase in R and Ṫ, but are smaller than those obtained using the thermal model alone.

Introduction

Magnetic resonance (MR) imaging is commonly used for monitoring thermal therapies such as high intensity focused ultrasound (HIFU) (1–5). It has gained popularity because of its excellent soft tissue contrast and ability to accurately and non-invasively measure temperature change in a variety of tissues. Of the many MR parameters that are sensitive to temperature changes and have been used for MR temperature imaging (MRTI) (6–10), the gold standard is the proton resonance frequency (PRF) shift method (11,12). It relies on temperature dependent shifts in the water proton resonance frequency, and temperature maps are calculated by scaling the phase difference between consecutively acquired phase images.

One general challenge in MRTI is achieving accurate and precise temperature measurements with adequate spatial and temporal resolution over a sufficiently large field-of-view (FOV). For HIFU high spatio-temporal resolution is needed to accurately measure the focal zone heating and avoid temperature-averaging effects (both spatially and temporally), and large FOV is needed to monitor inadvertent near- and far-field tissue heating (13,14).

In MRTI, increased data acquisition speed, which can be traded for higher spatial resolution and/or larger FOV (usually more slices), has been achieved through steady state free precession (15,16), single-shot echo planar imaging (EPI) (17–20) and segmented-EPI (17,21–25) pulse sequences. As in MRI in general, k-space subsampling in combination with dedicated reconstruction methods are also widely used for faster imaging. Parallel reconstruction methods utilize the different sensitivities of multiple receiver coils to reconstruct subsampled k-space measurements (26–31). Recently, reconstruction of subsampled MRTI data with various iterative, compressed-sensing-like approaches have shown promising results (25,32–35). Such iterative approaches lead to long reconstruction times that hinder real-time availability of the temperature maps. A third approach for reconstructing temperature maps from subsampled data is utilizing a thermal model such as the Pennes bioheat transfer equation (PBTE) (36) to estimate missing data in k-space (37) or in image space (18,38,39). These methods are non-iterative and inherently better suited for real-time reconstruction. In general, temperature measurement accuracy in model-based approaches will depend on the accuracy of the model parameters.

The purpose of the present study was to investigate the effect of k-space reduction factor (R), rate of temperature increase (Ṫ - total temperature rise divided by total heating time), and model parameter inaccuracies (in ultrasound (US) power deposition density, Q, and in thermal conductivity, k) on a previously described model-based MRTI reconstruction algorithm, called model predictive filtering (MPF) (37). Simulation and ex-vivo porcine muscle studies using HIFU tissue heating were performed to assess the accuracy and robustness of the thermal model and the MPF algorithm. The methods were evaluated by calculating the root-mean-square-error (RMSE) of the temperature maps and by investigating what volume of tissue reached different levels of thermal dose.

Methods

Model predictive filtering

A full description of the MPF algorithm is given in the original paper (37), and only a brief description as background is given here. The first time frame (n) needs to be fully sampled so an artifact free complex image can be reconstructed. All subsequent time frames can be subsampled with arbitrary subsampling factor and sampling scheme. To reconstruct time frame (n+1) a thermal model (see below) is used to forward predict temperature map T_n to T_n+1 and then convert it into a phase map using the PRF equation:

$${\varphi }_{n+1}={\varphi }_n+\gamma B_0\alpha TE(T_{n+1}-T_n)$$ (1)

(Φ_n=phase at time frame n, γ=gyromagnetic ratio, B₀=magnetic field strength, α=PRF coefficient assumed to be −0.01 ppm/°C, TE=echo time). The phase map is combined with the magnitude image from time frame (n), creating a complex image that is transformed into k-space using fast Fourier transform (FFT). In k-space the data acquired for time frame (n+1) replaces corresponding predicted data before transforming back into image space and finally calculating an updated temperature map for time frame (n+1).

As a thermal model the MPF algorithm uses PBTE:

$$\rho C_t\frac{\partial T}{\partial t}=k{\nabla }^{2}T-W{C}_b(T-T_{\mathit{blood}})+Q$$ (2)

(ρ=tissue density [kg/m³], C_t/C_b=specific heat of tissue/blood [J/kg/°C], T/T_blood=tissue/arterial blood temperature [°C], k=thermal conductivity [W/m/°C], W=Pennes perfusion parameter [kg/m³/s], Q=power deposition density [W/m³]), although any thermal model can theoretically be used. Since only ex-vivo experiments were performed W=0 was used throughout this work.

The original MPF work utilized 2D and 3D GRE MR pulse sequences and the methods of Cheng and Plewes (40) and Roemer et al. (41) to estimate k and Q. In this work we used recently developed and improved methods by Dillon et al. (42,43) to estimate k and Q by fitting a 2D radial Gaussian analytical solution to a pre-treatment low-power heating. A segmented-EPI pulse sequence was also used for faster data acquisition.

Simulation studies

Two sets of simulations were performed (1. Model-only and 2. MPF, both performed with and without determining model parameters using the methods of Dillon et al. (42,43)) to assess the effect of R, Ṫ, and model parameter inaccuracies on reconstructed temperature maps. All simulations used 3D Q-patterns modeled from a 256-element phased array US transducer (Imasonic, Besançon, France) with 2x2x8 mm focal spot full-width-half-max and 0.5 mm isotropic voxels, assuming speed of sound 1540 m/s, and attenuation 0.054 Np/cm/MHz using the hybrid angular spectrum method (44). The Q-pattern, together with literature values for ρ (1000 kg/m³) (45), C_t (2760 J/kg/°C) (46), and k (0.40 W/m/°C) (43), were used in PBTE to create 3D temperature maps. All simulations mimicked the ex-vivo experiments described below as closely as possible in terms of MR- and US-parameters used and were performed with three k-space reduction factors (R=4, 6, and 12) using the same subsampling scheme as for the ex-vivo experiments (see Supporting Figure S1 and below), three US heating durations (21, 42, and 63s), and two US powers (low/high resulting in 9/18°C temperature increase). The high-power sonications for the three durations resulted in Ṫ of 0.86, 0.43, and 0.29 °C/s. The purpose and methods (see Appendix A for detailed descriptions) of the simulations were:

1a) Model-only accuracy (MOacc)

To assess how Ṫ and errors in model parameters affect temperature predictions from the thermal model, errors in k and Q magnitude (from −50% to +50%) were introduced in simulated high-power sonications and temperatures were compared to truth.

1b) Model-only accuracy with parameter estimation (MOacc-param)

k and Q were determined from a simulated low-power sonication and then used in PBTE to predict high-power model only temperatures (by linearly scaling Q). The simulation was repeated 25 times for each of three sonication durations for a total of 75 repetitions, each with a different noise realization (zero mean Gaussian noise with standard deviation corresponding to the ex-vivo experiment) for the low-power sonication.

2a) MPF accuracy (MPFacc)

To assess how R, Ṫ, and errors in model parameters affect temperature reconstructions using the MPF algorithm, a temporal sequence of subsampled k-spaces obtained with known (true) k and Q values were reconstructed with erroneous k and Q values (from −50% to +50%) using the MPF algorithm. The simulation was repeated for all three values of R and all three sonication durations.

2b) MPF accuracy with parameter estimation (MPFacc-param)

To assess the errors that occur in the MPF reconstruction when using model parameters determined from low-power sonications, k and Q were first determined from simulated noisy low-power sonications and then used in combination with noisy and subsampled k-space data obtained during simulated high-power sonications (by linearly scaling Q) to predict temperatures. The simulation was repeated 25 times for each of three sonication durations and three values of R for a total of 225 repetitions, each with different noise realizations for both low-power sonication and subsampled k-space.

For all simulations, temperatures obtained with the erroneous or estimated k and Q values are compared against truth obtained using known (true) values. Two different metrics were used to evaluate temperature accuracy. To determine local focal spot error the mean and STD of the peak temperature voxel were plotted as a function of time, and the RMSE of this voxel was calculated. As a more global error, the RMSE was calculated over all voxels that experienced significant temperature increase, here defined as >30% of the peak temperature truth voxel.

Ex-vivo HIFU experiments

All imaging was performed on a 3T MRI scanner (TIM Trio, Siemens Medical Solutions, Erlangen, Germany), and all HIFU heating was performed using an MR-compatible phased array US transducer (256 elements, 1 MHz frequency, 13 cm radius of curvature, 2x2x8 mm focal spot FWHM, Imasonic, Besançon, France, and Image Guided Therapy, Pessac, France). Two in-house built 2-channel receive-only RF-coils (47) were used for signal detection. All data processing was performed using Matlab (R2013a, MathWorks, Natick, MA).

Experiments were performed on three separate samples of porcine muscle. For each sample, five sets of HIFU experiments were performed, where each set consisted of three sonications:

• #1Small FOV low-power sonication to determine k and Q;
• #2Small FOV high-power sonication to determine a fully sampled estimate of truth; and
• #3Large FOV high-power sonication that was sampled with a k-space sampling order designed to allow retrospective subsampling (see next paragraph and Supporting Figure S1) with R=4, 6, and 12 for reconstruction with the MPF algorithm.

For #1 and #2 above, each k-space was fully sampled, i.e. no subsampling was performed for these datasets. The 15 sonications in each of the three tissue samples were performed at 14/28 W (low/high-power, respectively) for 21 s in the first sample, 14/28 W for 42 s in the second sample, and 12/24 W for 63 s in the third sample. The goal was to perform low-power sonications with temperature increases of 8–10 °C for model parameter determination (#1 above), and high-power sonications with temperature increases of 16–20 °C to determine truth (#2 above) and for MPF reconstructions (#3 above). The starting temperature of the samples was approximately 20 °C so limiting temperature increase to 16–20 °C ensured minimal thermal dose accumulation and allowed repeatable sonications within each sample. A 10-minute cooling period was applied between each sonication. Except for the number of slices and the slab thickness, the same MR parameters were used in all experiments: TR/TE=36/11 ms, readout bandwidth (BW)=752 Hz/pixel, echotrain length (ETL)=9, voxel size=1.5x1.5x2.0 mm³, matrix=128x108x(12/48), FOV=192x161x(24/96 mm), where 12 slices (24 mm) were used for the small FOV, and 48 slices (96 mm) were used for the large FOV (Supporting Table S1). The acquisition time was 5.2 s for the small FOV and 20.8 s for the large FOV (before subsampling). For R=6 and 12 the acquisition time per time frame was less than 5.2 s, so only time frames coinciding with truth time frames were used for plotting and error calculations. The flip angles used, which gave the maximum signal-to-noise-ratio for each tissue sample, were 35°, 25°, and 25°, respectively. Fat saturation by a frequency specific off-resonance RF-pulse was applied in all sonications (48). All data was zero-filled interpolated (ZFI) to 0.5 mm isotropic voxel spacing to mitigate partial volume effects and ensure accurate tissue properties determination (13,49).

The 3D segmented-EPI pulse sequence used throughout this study samples a set of k_y phase encodes for a k_z slice encode with the first echo train. The same set of k_y phase encodes is then sampled for remaining k_z slice encodes, before changing to the next set of k_y phase encodes and repeating them for all k_z, and so on until k-space is fully sampled. This sampling pattern enables retrospective down-sampling with arbitrary R without discarding any data, by dividing each fully sampled 3D k-space into multiple subsampled 3D k-spaces (Supporting Figure S1).

For each of the 15 sets of sonications the methods of Dillon (42,43) and the low-power sonication (#1 above) were used to estimate k and Q. The magnitude of Q was then linearly scaled to correspond to the high-power sonications, and model-only and MPF reconstructions were performed. Both reconstructions were evaluated against the fully sampled truth (#2 above). For model-only the estimated k and (scaled) Q were used in the thermal model (equation 2) - these ex-vivo results correspond to simulation 1b) “MOacc-param”. For MPF the estimated parameters were used together with the subsampled k-space data from the large FOV high-power sonications (#3 above) in the MPF algorithm. These ex-vivo results correspond to simulation 2b) “MPFacc-param”.

The same accuracy metrics used in the simulation study were used for the ex-vivo experiments. In addition MPF was also compared to reconstructing the subsampled data with a simple sliding window (SW) reconstruction, and a previously described compressed-sensing-like temporally constrained reconstruction (TCR) method (25,50). For all three reconstruction methods, the 15 high-power large FOV datasets (#3 above) were reconstructed with R=4, 6, and 12, corresponding to acquisition times of 5.2 s, 3.5 s, and 1.7 s. To further compare the relative performance of the three reconstruction methods, the RMSE of all voxels with temperature increase between 30–60%, 60–90%, and >90% of the hottest truth voxel were calculated and compared. The method of Sapareto and Dewey (51) was used to calculate the thermal dose cumulative equivalent minutes at 43 degrees (CEM43) for the three reconstruction methods. Tissue volumes that reached levels of 30, 100, and 240 CEM43 were compared to corresponding volume obtained using the fully sampled truth. To make meaningful dose volume comparisons and avoid effects due to different end temperatures (Table 1), the dose calculations were made assuming that an end temperature of 58 °C was reached in all cases.

Table 1.

Maximum temperature increase and Ṫ achieved for low- and high-power simulation and ex-vivo heatings for all three durations.

		Simulation Low-power	Simulation High-power	Ex-vivo Low-power	Ex-vivo High-power
21 s	Max ΔT (°C)	9	18	8.3±0.62	15.8±0.98
	Ṫ (°C/s)	0.43	0.86	0.40	0.75
42 s	Max ΔT (°C)	9	18	10.4±0.30	19.3±0.37
	Ṫ (°C/s)	0.21	0.43	0.25	0.46
63 s	Max ΔT (°C)	9	18	9.4±0.72	17.3±0.89
	Ṫ (°C/s)	0.14	0.29	0.15	0.28

One-way analysis of variance (ANOVA) tests, with follow-up multiple comparison of means tests using Bonferroni adjustments, with a level of significance α=0.05 were performed to investigate if the different reconstruction methods resulted in significantly different RMSE and thermal dose values.

The temperature STD was investigated by calculating the mean of the STD through time of 40x40x20 voxels in an un-heated region of each porcine sample. This was done for all three durations for the fully sampled truth as well as for all sonication durations and Rs for all three reconstruction methods.

To investigate the possibility of using a tabular lookup value for k, all 15 large FOV high-power datasets were further reconstructed with the MPF algorithm using the average value of k from the 15 low-power sonications in the three different porcine samples. A multiple comparison of means ANOVA test (α=0.05) was also performed to investigate statistically significant differences.

Results

Figure 1 shows the RMSE of all voxels receiving significant heating for the “MOacc”- (a–c) and “MPFacc”-simulations (d–i). Arrows in b) and e) indicate that errors in k/Q of −9%/+9% result in 1 °C RMSE (black contour) for “MOacc”, whereas for “MPFacc” substantially larger errors of −17%/+21% result in 1 °C RMSE.

Figure 1.

Global RMSE of all voxels with temperature increases greater than 30% of the maximum true temperature. a)–c) “MOacc”-simulations for 21s, 42s, and 63s, d)–i) “MPFacc”-simulations with R=4 for 21s, 42s, and 63s, and for 42s with R=4, 6, and 12. Errors in Q (vertical axis) and k (horizontal axis) vary from −50% to +50%. Red/Black contours indicates RMSE of 0.5/1.0°C. Arrows in b) and e) show that k/Q errors of −9%/+9% and −17%/+21% result in RMSE of 1°C for “MOacc” and “MPFacc”, respectively. Please note the difference in scale between a)–c) and d)–i).

Mean and STD of the hottest voxel vs. time for simulated (mean±STD of 25 sonications) and ex-vivo (mean±STD of 5 sonications) “MOacc-param”-experiments, for all sonication durations, are shown in Figure 2. All simulations underestimate the temperature increase, whereas for the ex-vivo data both over- and underestimations can be observed. For the simulations the RMSE of the hottest voxel was 0.28±0.01°C, 0.32±0.02°C, and 0.33±0.02°C for the three durations, and corresponding values for the ex-vivo experiment were 0.46±0.11°C, 0.63±0.16°C, and 0.64±0.23°C.

Figure 2.

Temperature rise of hottest voxel vs. time for a) simulated and b) ex-vivo “MOacc-param”, both for 21s, 42s, and 63s heatings. The RMSE shows a slight increase for longer heating durations (i.e. slower heatings) in both simulations and ex-vivo experiments.

In figures 3 and 4 mean and STD of the temperature increase of the hottest voxel vs. time for simulated and ex-vivo “MPFacc-param”-experiments are shown. For the simulation studies (Figure 3) all sonication durations and R can be seen to perform similarly, whereas larger differences are observed for ex-vivo experiments (Figure 4). The ex-vivo MPF reconstructions are compared to SW and TCR reconstructions, as well as to fully sampled truth. In general MPF performs better than TCR, which in turn performs better than SW. The differences are most noticeable for higher values of R (rightmost column), and close to sharp temporal gradients, i.e. when the ultrasound is turned on/off.

Figure 3.

Mean and STD temperature rise of the hottest voxel vs. time for “MPFacc-param”-simulations. a) for R=4, b) for R=6, and c) for R=12, all three for 21s, 42s, and 63s heating duration.

Figure 4.

Mean and STD temperature rise of the hottest voxel vs. time for ex-vivo “MPFacc-param” experiments. a)–c) 21s heating duration, d)–f) 42s heating duration, and g)–i) 63s heating duration, all for R=4, 6, and 12. MPF is compared to truth, SW and TCR. Note that for R=6 and 12, which have higher temporal resolution than the truth, only time frames that coincide with the truth are plotted.

The global RMSEs for the simulation and ex-vivo data in Figures 2–4 are shown in Figure 5 as mean and standard error of the mean (SEM). For both simulated and ex-vivo MPF data (Figure 5b) the RMSE decreases for slower heating rates as well as for lower Rs, and this is also the case for ex-vivo TCR and SW data. In Figures 6a)–c) the mean and SEM of the RMSE of all voxels with temperature increase between 30–60%, between 60–90%, and >90% of the hottest truth voxel, respectively, are shown for the ex-vivo data. In agreement with Figures 4–5 it can again be seen that MPF always performs better than SW, is always similar or better than TCR, and the improvements are much more significant for the hottest voxels. Asterisks in figures 5 and 6 indicate that the RMSE is statistically significantly different as investigated with the multiple comparison of means ANOVA test.

Figure 5.

Mean and SEM of RMSE of all voxels with significant temperature rise for a) “MOacc-param” for simulated and ex-vivo data, and b) “MPFacc-param” for simulated and ex-vivo data, the latter reconstructed with MPF, TCR and SW. For the ex-vivo data in b), the asterisks (*) indicate that the RMSE is significantly different than the RMSE for the other reconstruction methods as evaluated by the multiple comparison of means ANOVA test.

Figure 6.

Mean and SEM for RMSE of all voxels with temperature rise a) between 30% and 60%, b) between 60% and 90%, and c) above 90% of the hottest truth voxel. Asterisks indicate that the RMSE is statistically significantly different (as tested by the multiple comparison of means ANOVA test) than the RMSE for the other reconstruction methods.

As shown in Figure 7, SW is significantly less accurate than MPF and TCR in terms of dosed tissue volume, with worse accuracy for faster heating rates and higher dose levels (i.e. larger errors for 240 than for 30 CEM43). Overall MPF is most accurate and dosed volumes are within 5% for all R and Ṫ of 240 CEM43, within 8% for 100 CEM43, and within 11% for 30 CEM43. The difference in accuracy between MPF and TCR is smaller in terms of dose than in temperature RMSE, and MPF is only significantly more accurate in 4 of the combinations of R and Ṫ compared.

Figure 7.

Mean and SEM of tissue volume reaching a) 30, b) 100, and c) 240 CEM43 compared to true volume. Asterisks indicate that the dosed volume is statistically significantly different (as tested by the ANOVA test) than the dosed volume for the other reconstruction methods.

Figure 8 shows three orthogonal views of the spatial error of the focal spot for the hottest time frame for the three reconstruction methods and truth for one of the 42 s and R=12 ex-vivo sonications, and in Supporting Figure S2 corresponding temperature maps are shown. Both Figures indicate that the MPF data agrees best with truth, although with a slight temperature overestimation at the peak of the focal zone rather than an underestimation, which is the case for TCR and SW.

Figure 8.

Three orthogonal views of spatial error for the hottest time frame for a) MPF, b) TCR, and c) SW, all for one of the 42s ex-vivo runs with R=12. Subfigure d) shows the truth run for comparison.

The maximum temperature increase and Ṫ for all experiments are summarized in table 1.

Both simulations and ex-vivo experiments provided estimates of k and Q for comparison with true or previously published values. From the 300 simulated low-power sonications k and Q were overestimated compared to true values by 3%±0.9% and 1%±2.5%, respectively. k estimated from the 15 ex-vivo low-power sonications in the 3 tissue samples was 0.41±0.03W/m/°C (individual values shown in Supporting Table S2). For comparison, the value 0.40W/m/°C for porcine muscle was obtained with standard invasive methods (43).

The effect of the range of measured values of k on the RMSE for the ex-vivo experiment was very small. The largest difference in global RMSE between reconstructing the 15 large FOV high-power datasets with the individual values of k (Supporting Table S2), and the average value of 0.41, was 0.02°C (data not shown). The ANOVA tests further showed no statistically significant differences, suggesting that a tabular value for k could have been used without significant loss of measurement accuracy.

The mean and STD of the temperature STD through time of an un-heated region in each porcine sample are shown in Supporting Table S3. All three reconstruction methods can be seen to experience lower STDs than the fully sampled data, with MPF being slightly lower than TCR, which in turn is lower than SW.

Reconstruction time for one time frame of a zero-filled 3D volume in a non-optimized Matlab implementation of the MPF algorithm on a standard laptop (2.5GHz Intel Core i7 processor and 8GB RAM, MacBook Pro, Apple, Cupertino, CA) was 1.13 s, i.e. faster than the shortest acquisition time (1.7s for R=12) in the present study.

Discussion and conclusion

This paper has investigated the performance of a previously described thermal model-based reconstruction method for MR thermometry data, in terms of effect of errors in model parameters, k-space reduction factor, and rate of temperature increase. Simulations and HIFU experiments in ex-vivo porcine muscle show that MPF is less sensitive to errors in model parameters than the thermal model alone, and that increased accuracy is achieved for lower k-space reduction factors and lower rates of temperature increase. Utilizing a 3D segmented EPI pulse sequence and k-space subsampling for increased data acquisition speed, and recently described and improved methods for model parameter determination, MPF enables accurate (RMSE<0.7°C) and real-time reconstructions (~1s) of high-resolution (1.5x1.5x2.0 mm), large FOV (192x162x96 mm) 3D MRTI. Results also suggest that tabular values for k can be used without loss in measurement accuracy.

Simulation studies (Figure 1) show that both the thermal model and MPF are more robust to errors in model parameters if both k and Q are either over- or underestimated (in the simulation studies both parameters were slightly overestimated), in which case their effects counteract each other minimizing the errors. As can be expected, substantially reduced sensitivity to errors in model parameters is gained if subsampled k-space data is incorporated in the reconstruction algorithm as in MPF. The errors in k and Q that result in a given RMSE in the “MPFacc”-simulations are approximately double those in the “MOacc”-simulations. This is highlighted by the considerably larger red/black contours in Figures 1d)–i) than in 1a)–c). The slope of the contours (i.e. the contours are closer to being horizontal than vertical) also suggests that the MPF reconstruction is more sensitive to relative errors in Q than in k. The “MPFacc”-simulations further show that 1) for a given R, better accuracy is achieved for lower Ṫ, and 2) for a given sonication duration, better accuracy is to be expected for lower values of R.

The “MPFacc”-simulations and ex-vivo “MPFacc-param”-data both show increased accuracy for lower Ṫ and R, and so do TCR and SW reconstructions of the ex-vivo data. Slower heating results in smaller changes occurring from time frame to time frame, which simplifies model prediction for MPF and minimizes temporal smoothing-effects for TCR and SW. Although higher values of R utilize the acquired k-space data in the reconstruction algorithm closer in time to where it was actually acquired, errors increase in MPF because the erroneous model values carry more weight. The cause for the error increase for TCR and SW requires more investigation.

The ex-vivo MPF data (Figures 4–6) further show that MPF can accurately (RMSE<0.7°C) monitor sonications with R=4–12 and Ṫ=0.28–0.75°C/s, and that MPF generally performs better than TCR and SW. For areas experiencing lower temperature increases (Figure 6a) there is no statistically significant difference between MPF and TCR for any of the R and Ṫ values investigated. For these areas SW also performs as well as MPF and TCR for low values of R and Ṫ, suggesting that for low to moderate R and Ṫ these off-focus areas can be reconstructed with simple reconstruction methods without loss in measurement accuracy. For regions experiencing higher temperature increase (Figures 6b–c), the RMSE for SW is significantly larger than for MPF and TCR for all cases, highlighting that, for these areas, more advanced reconstruction methods are really needed. For these higher temperatures, it is also clear that MPF is significantly more accurate than TCR in most cases, especially for low values of Ṫ and for high R. The higher accuracy and smaller errors over the focal spot for MPF is also evident in the spatial temperature and error distributions (Figure 8 and Supporting Figure S2).

In Supporting Figure S2 all three reconstruction methods can further be seen to experience less background (i.e. off-focal) temperature variation compared to fully sampled truth, in agreement with the STD through time data (Supporting Table S3). For MPF all predicted data is inherently noise-less which reduces the STD through time. In TCR the temporal constraint tends to smooth out variations in time, and for SW the same k-space lines are used in multiple consecutive time frames, which also results in lower STD through time.

Larger temperature increases than the 16–19 °C reached in this study can be achieved by increasing the sonication duration or by using higher power. Increasing the sonication duration will have a small effect on Ṫ, and it can be assumed that it will not affect the temperature measurement accuracy. Using a higher power, which would minimize treatment time, will lead to higher Ṫ and hence decrease accuracy as discussed above. We note that higher temperatures can also lead to lower signal magnitude over the focal spot region, due to T₁ increase with temperature (6). Although a decreased signal might introduce a problem for the MPF algorithm, no such problems were observed in the current study. This may become an issue for tissues with already long T₁ relaxation times and grant further studies.

The thermal dose evaluation shows that MPF temperatures can also accurately predict treated volume (dose>240 CEM43), being within 5% of the true volume for all cases of R and Ṫ (Figure 7c). The fractional differences in calculated dose between MPF and TCR are smaller than the corresponding fractional difference in temperature RMSE. As was the case for temperature RMSE, SW temperatures can be used to estimate dosed volume without loss in accuracy for slow heatings, but performs significantly worse for higher Ṫ.

The simulation results indicate that accurate estimates for k and Q can be achieved with the methods used throughout this work (42,43), with the 300 simulated low-power sonications resulting in slightly overestimated values (103% for k, and 101% for Q). For the ex-vivo experiments, the mean value of k determined from the 15 low-power datasets (0.41 W/m/°C) agrees well with previously published values for porcine tissue obtained with invasive measurements using a thermal properties analyzer. The spread in the value of k between the three different pieces of porcine muscle, and its effect on the temperature RMSE, further indicate that a tabular lookup value for k can possibly be used without decreasing the accuracy of the MPF reconstructions. There is no direct way to assess the accuracy of the determined Q in the ex-vivo experiments. However, the fact that both k (compared to gold standard invasive method) and the reconstructed temperature maps (compared to fully sampled truth) were accurate gives strong evidence that Q, just as in the simulation study, was accurately determined as well.

One of the main advantages of model-based reconstruction approaches (like MPF) compared to iterative reconstruction approaches (like TCR) is the decreased computational burden, allowing real-time reconstruction. The reconstruction time of 1.13 s per 3D volume presented here is faster than the shortest acquisition time (t_acq=1.7s for R=12). This and the fact that the code has not yet been optimized for fast and parallel reconstruction, suggests that real-time availability of large FOV 3D temperature maps is achievable with MPF. If multiple receive coils are used, these can be reconstructed in parallel without an increase in reconstruction time.

Both local and global RMSEs for simulated and ex-vivo “MOacc-param”-studies (Figures 2 and 5a) show, contrary to corresponding MPF results, larger errors for slower heating rates. Since these are model-only reconstructions, any errors caused by erroneous k and Q estimates will propagate in time, resulting in the larger errors observed for longer heating durations. All three simulated predictions (Figures 2a and 5a) are underestimated, in agreement with the larger observed overestimation in k (103%) than in Q (101%). From corresponding ex-vivo data (Figures 2b and 5a) the 21 s sonications agree well with truth, while the 42 and 63 s sonications demonstrate over- and underestimations, respectively. While nothing can be deduced about Q in this case, the observed over-/underestimations are consistent with observed under-/overestimations of k (0.39 and 0.43 W/m/°C, respectively, Supporting Table S2), compared to the previously reported value for porcine muscle.

Since the tissue used throughout this study was relatively homogenous, it was assumed that ρ, C, and k were spatially constant, and only Q was estimated in 3D. For inhomogeneous tissue, it can be hypothesized that Q can still be accurately determined in 3D since it is based on 3D temperatures. If inhomogeneous tissues can be accurately segmented, tabular values for ρ, C, and k for the different tissue types can be used, but further studies are needed to investigate the effect on MPF accuracy. More inhomogeneous tissues will also lead to susceptibility artifacts, but these are likely to be small as long as significant temperature increase does not occur near tissue-air interfaces. Since only non-perfused tissue was investigated W was set to zero in all reconstructions. For in-vivo applications the effects of blood perfusion will need to be taken into account, and can be determined with previously described methods (41,52–56). Lastly, MPF in the presence of motion was described in the original MPF paper (37), but was beyond the scope of this manuscript. In-vivo motion might be a concern depending on the organ targeted.

Supplementary Material

Supp MaterialS1

Supporting Figure S1. Retrospective k-space down-sampling. Schematic figure of k_y-k_z (phase encode – slice encode) plane of k-space for one fully sampled, and the first three retrospectively subsampled time frames for R=6. Solid lines represent sampled data. If two echo trains are left in each slice (k_z) in each time frame, R=6 (108/(2*9), see Supporting Table S1) and t_acq of 3.5 s is achieved. Similarly, by leaving 1 or 3 echo trains R=12 (108/(1*9)) or R=4 (108/(3*9)), with t_acq of 1.7s and 5.2s, can be achieved. The k_x read out direction is always fully sampled.

Supporting Figure S2. Three orthogonal views of temperature distributions for the hottest time frame of a) MPF, b) TCR, c) SW, and d) Truth, all for one of the 42s ex-vivo runs with R=12.

Supporting Table S1. MR Parameters used for the ex-vivo experiments. TR – Repetition time, TE – Echo time, BW – Bandwidth, ETL – Echo train length, t_acq - acquisition time, FOV- field-of-view, ES – Echo spacing. *Note: the acquisition time of 20.8 s for #3 is to sample the full k-space matrix, and with R=4, 6, and 12 acquisition times of 5.2, 3.5, and 1.7 s, respectively, are achieved. It should further be noted that the way k-space is sampled allows for retrospective down-sampling with arbitrary R without discarding any data, as seen in Supporting Figure S1.

Supporting Table S2. Thermal conductivity for the 15 low-power ex-vivo heatings [W/m/°C].

Supporting Table S3. Mean and STD of the STD of temperature through time for an un-heated region in the three porcine samples. Note that all the “truth” heatings are fully sampled, and the same “truths” are used for all values of R within each heating duration.