The Effects of Spatial Sampling Choices on MR Temperature Measurements

Abstract

The purpose of this paper is to quantify the effects that spatial sampling parameters have on the accuracy of MR temperature measurements during high intensity focused ultrasound (HIFU) treatments. Spatial resolution and position of the sampling grid were considered using experimental and simulated data for two different types of HIFU heating trajectories (a single point and a 4 mm circle) with maximum measured temperature and thermal dose volume as the metrics. It is demonstrated that measurement accuracy is related to the curvature of the temperature distribution, where regions with larger spatial second derivatives require higher resolution. The location of the sampling grid relative temperature distribution has a significant effect on the measured values. When imaging at 1.0 × 1.0 × 3.0 mm resolution, the measured values for maximum temperature and volume dosed to 240 CEM or greater varied by 17% and 33% respectively for the single point heating case, and by 5% and 18% respectively for the 4 mm circle heating case. Accurate measurement of the maximum temperature required imaging at 1.0 × 1.0 × 3.0 mm resolution for the single point heating case and 2.0 × 2.0 × 5.0 mm resolution for the 4 mm circle heating case.

Introduction

Magnetic Resonance temperature measurements have been shown to play a crucial role in the planning, monitoring, control, and assessment of thermal therapies for non-invasive treatment of tumors (1–6). Because these treatments are non-invasive, it is critical that clinicians be able to predict the heat deposition pattern, monitor the treatment in real-time, determine when a region of tissue has been treated to satisfaction, and retrospectively assess the amount of damage done to the treated area. Magnetic Resonance Imaging (MRI) can play an important role in each of these stages due to its ability to measure temperature changes through the temperature dependence of the proton resonance frequency (7–8).

Investigators have used MR temperature measurements to obtain estimates for tissue thermal and acoustic properties which, in turn, can be used in the patient treatment planning process (9–11). MR temperature imaging is also widely used during thermal therapies to monitor in real-time where the thermal energy is being deposited ensuring that the targeted area is receiving treatment and that normal tissue is not being damaged (1,11–14). As part of this monitoring, many investigators use either maximum temperature or accumulated thermal dose as the criterion for determining when a region of targeted tissue has been sufficiently treated. Total accumulated thermal dose can be calculated retrospectively after a treatment is complete and used as part of the histological analysis to compare predicted tissue necrosis with actual necrosis (15). For all of these reasons, it is vitally important that MR temperature measurements be accurate, reliable, and repeatable.

MR temperature maps are necessarily a discrete representation of a physical quantity that is continuously varying in both space and time. The MR sampling scheme may affect the discrete temperature values measured. Due to averaging effects, different choices for the sampling grid location, effective voxel size, and scan time can lead to different measurements of the same underlying temperature distribution. These differences could have a large impact on how a thermal therapy procedure is planned, carried out, and assessed. This is especially true for high intensity focused ultrasound (HIFU) treatments, where the spatially non-uniform deposition of energy into a small volume creates large temperature gradients with a focal spot that has dimensions on the order of an MR voxel.

This paper investigates and quantifies the extent to which sampling choices affect MR temperature measurements. While temporal averaging does affect temperature accuracy, this paper only considers the effects due to variations in the spatial sampling scheme, specifically the spatial resolution and image sample grid location. Experimental and simulation results are presented quantifying the effects that variations in the sampling scheme have on measuring the maximum temperature and total accumulated thermal dose.

Methods

Theory

For HIFU heating applications with a small focal spot, the temperature increase will vary spatially over the volume of a voxel. The discrete temperature measured within a voxel is therefore an average of the continuous temperature over the entire voxel. In MRI, the signal sensitivity is not constant throughout the voxel, but rather is determined by a sinc-like voxel sensitivity function (VSF) (16–19). In one dimension, the VSF is given by:

$$q(x)=\frac{1}{N_X}·\frac{\text{sin}\left(\pi \frac{x}{\mathrm{\Delta }x}\right)}{\text{sin}\left(\pi \frac{x}{N_x\mathrm{\Delta }x}\right)}$$ [1]

where N_x is the number of voxels and Δx is the voxel dimension. The three-dimensional VSF would be the product of q(x), q(y) and q(z). The signal sensitivity is maximal at the center of the voxel volume, but drops off quickly away from the center. For a three-dimensional isotropic voxel, the sensitivity is 64% of the maximum at the sides of the voxel, 41% at the edges, and only 26% at the corners.

An MR image is the discrete Fourier transform of discrete samples of the continuous spatial frequency space (k-space) of the object, which allows the image grid properties to be chosen retrospectively. The location of the sampling grid can be manipulated by applying a linear phase offset to the k-space data. For 2-D MR data, the sampling grid can only be shifted in-plane, while for 3-D MR data the sampling grid can be shifted in all three dimensions. The spacing of the image grid points can be manipulated by zero-filling the k-space data to a larger matrix size. Zero-filled interpolation (ZFI) will create a denser grid of overlapping VSFs, reducing the distance between sampling points in the image domain. It must be emphasized that ZFI does not alter the size or shape of the VSF and therefore has no effect on the spatial resolution of the original data. However, the more tightly packed VSFs will reduce the amount of area with low signal sensitivity and allow for better capture of image details that are smaller than the VSF’s dimensions. In theory, ZFI can achieve arbitrary placement of the sampling grid, but this would require creating an arbitrarily large image matrix.

The extent to which a temperature distribution is accurately measured depends on how the temperatures are changing spatially over the volume of a voxel. Consider the example in Figure 1, where a simple one-dimensional temperature distribution consisting of a linear ramp up and down, changing 20 °C over 10 mm, is taken as truth. The 2 mm wide voxel shown on the left side of Figure 1 would measure a value that is an average of the continuous temperatures over the entire voxel (white dot). Using ZFI, the voxel spacing can be made arbitrarily small in order to obtain measurements that are closer together than the size of the voxel; in this example the voxels are 0.25 mm apart. It can be seen that these finely spaced voxels will accurately measure the temperature distribution, no matter how steep the gradient, as long as the temperature is changing linearly in space. The errors occur when the temperature distribution within a voxel has a non-zero second derivative, as seen in the 2 mm wide voxel centered over the peak. The linear temperature changes are measured accurately, while the peak temperature, where the curvature is greatest, is underestimated by almost 2 °C.

Figure 1.

Simple simulation demonstrating temperature measurement accuracy. If the changes in space are linear, a 2 mm voxel can accurately measure the temperature distribution (left voxel and white dot). Zero-filled interpolation can be used for arbitrarily small voxel spacing. For regions with non-zero second derivative, the 2 mm voxels cannot accurately measure the temperature distribution (central voxel and white dot).

Experiments

HIFU heating experiments were performed on a tissue-mimicking agar phantom (20). The heating was induced using an MRI-compatible 256-element phased-array ultrasound system (Imasonics, Inc. and Image Guided Therapy, Bordeaux, France) with the transducer operating at a frequency of 1.0 MHz with a 14.5 cm aperture, 13 cm radius of curvature and focal zone full width at half maximum (FWHM) dimensions of 1.8 × 1.8 × 10.2 mm. A bath of de-ionized and de-gassed water coupled the ultrasound transducer to the agar phantom. Two different heating trajectories were carried out: 1) a single shot (70 W for 28 seconds) and 2) a 4 mm radius circle (130 W, 8 points, 50 ms per point, 60 total seconds). Each trajectory was run twice under identical circumstances, imaged first at 1.0 × 1.0 × 3.0 mm resolution, and then at 2.0 × 2.0 × 3.0 mm resolution. All imaging was performed on a Siemens TIM Trio 3T MRI scanner (Siemens Medical Solution, Erlangen, Germany). The MR imaging parameters used for temperature monitoring were as follows:

• Higher Resolution: 2-D spoiled gradient echo, TR/TE = 50/9 msec, 3 slices, 1.0 × 1.0 × 3.0 mm resolution, 256×80 mm FOV, 4.0 seconds/scan

• Lower Resolution: 2-D spoiled gradient echo, TR/TE = 50/9 msec, 3 slices, 2.0 × 2.0 × 3.0 mm resolution, 256×160 mm FOV, 4.0 seconds/scan

The imaging slices for all runs were positioned over the center of the focal zone in a coronal plane perpendicular to the path of the ultrasound beam. Temperature maps from the hottest time frames of each heating run are shown in Figures 2A, 2B, 2D, and 2E.

Figure 2.

Temperature maps from four experimental HIFU heatings and the corresponding simulated heatings. The simulations were carried out at 0.1 mm isotropic resolution.

To investigate voxel grid location effects, linear phases were applied across the readout and phase-encode directions of the original k-space data. Combinations of linear phases ranging in total variation from 0 to 2π shifted the location of the sampling grid by sub-voxel increments in the two directions. During investigation of spatial resolution effects, all k-space data sets were zero-filled to give 0.25 × 0.25 × 3.0 mm voxel spacing. This small voxel spacing mitigated the effects of the sampling grid location. After modifications to the k-space data, temperature measurements were calculated using the standard proton resonance frequency (PRF) method (7).

Simulations

Ultrasound beam propagation software was used to model the power deposition of the experimental phased-array ultrasound transducer at 0.1 × 0.1 × 0.1 mm spatial resolution (21). The FWHM dimensions for the simulated ultrasound power deposition were the same as in the experimental data. Using this power deposition matrix and the Pennes bioheat equation (22), simulated temperature maps with 0.1 mm isotropic spatial resolution and 1 second temporal resolution were created for the same two heating trajectories as used for the experimental data. All thermal and acoustic parameters used in the simulation process were assumed to be constant and are summarized in Table 1. Complex MR image data was created from the simulated temperature maps by converting the temperature into phase (7) and assuming constant image magnitude over the region of interest. These complex images were then transformed into k-space using a three-dimensional FFT. Images of the simulated temperature maps are shown in Figure 2C and 2F.

Table 1.

Thermal and acoustic parameters used for numerical simulation of temperature maps.

Property	Value
Thermal conductivity (W/m*K)	0.47
Specific heat (J/kg*°C)	4186
Density (kg/m3)	1000
Perfusion	0
Speed of sound (m/s)	1540
Attenuation (Np/(cm*MHz)	0.054

The spatial resolution of the simulated temperature maps was changed by manipulating the k-space data. In order to best mimic the image spatial resolution achieved during MR imaging, the k-space data was truncated from N_i × N_j × N_k voxels to M_i × M_j × M_k voxels (M < N), with the peak of k-space kept at the center of the matrix. Transforming the truncated k-space matrix back into image space, the new spatial resolution in the i, j, and k directions would be:

$$\mathrm{\Delta }{x}_{i,j,k}=\frac{N_{i,j,k}}{M_{i,j,k}}·0.1\mathit{\text{mm}}$$ [2]

In this way, the spatial resolution of the temperature maps could be set to any value ranging from 0.1 mm to 10.0 mm in each of the three dimensions. The k-space matrix was zero-filled to its original size, giving 0.1 × 0.1 × 0.1 mm voxel spacing in the temperature maps.

Analysis

To quantify the effects that changes to the spatial sampling scheme had on the measured temperatures, three metrics were considered. The first was the maximum measured temperature change over the course of the HIFU heating. The second and third metrics were the volume of the sample that received a total thermal dose of 30 CEM or greater (V_D30) and the volume with a total thermal dose of 240 CEM or greater (V_D240). The total thermal dose was calculated using the Sapareto and Dewey equation (23).

Results

Experiments

The FWHM dimensions of the temperature distributions at the hottest time frame were 2.9 × 2.9 × 12.5 mm for the single point case and 11.3 × 11.3 × 16.0 mm for the 4 mm circle case. The standard deviation of the temperature measurements were 0.16 °C (Figure 2A), 0.45°C (2B), 0.18°C (2D), and 0.48°C (2E). Results are shown in Figure 3 demonstrating how the location of the sampling grid can affect the measured temperatures. Both temperature maps are for the single point heating case with 2.0 × 2.0 × 3.0 mm resolution at the time frame of maximum temperature. In Figure 3A, with the sampling grid placed such that the hotspot peak is centered within one voxel, the maximum temperature is measured to be 26.3 °C and only that central voxel is measured to have a thermal dose greater than 240 CEM. In Figure 3B, where the sampling grid has been shifted such that the peak of the temperature distribution is evenly split among four voxels, the maximum temperature is measured to be only 18.5 °C but each of those four voxels is measured to receive a thermal dose greater than 240 CEM. Table 2 summarizes the range of possible measured values for maximum temperature, V_D240, and V_D30 based on the sampling grid location for all four experimental heating runs.

Figure 3.

The same single point, 2×2×3 mm resolution temperature data, reconstructed with two different sampling grid placements. When the temperature peak is centered within one voxel, the maximum temperature is measured to be 26.3°C (A), while splitting the peak between voxels gives 18.5°C (B).

Table 2.

Experimental results. Effects of sampling grid location on maximum measured temperature and thermal dose volume.

	Max Temp Range (°C)	VD240 Range (mm3)	VD30 Range (mm3)
Single shot, 1×1×3 mm	27.4 – 33.1	42 – 63	81 – 102
Single shot, 2×2×3 mm	18.5 – 26.3	36 – 96	60 – 144
4mm circle, 1×1×3 mm	16.5 – 17.4	108 – 132	582 – 612
4mm circle, 2×2×3 mm	16.2 – 16.8	96 – 156	564 – 672

Results summarizing the effects of different spatial resolution on the maximum measured temperatures and measured thermal dose volumes for the four different experimental heating runs are presented in Table 3. Figure 4A shows plots of the temperature cross section through the focal zone at the hottest time frame for the two single point heating runs. The different resolution temperature measurements match well at the sides of the focal zone where the temperature gradients are steep but linear. However, large discrepancies occur at the peak of the focal zone, where the second derivative is maximal, leading to measurements of the maximum temperature that differ by more than 6 °C. Figure 4B shows the same cross section temperature plots for the 4 mm circle heating runs. In this case, the second derivative is never as large as it is in the single point heating and the different resolution temperature measurements match well over the entire extent of the heated region.

Table 3.

Experimental results. Effect of spatial resolution on maximum measured temperature and thermal dose volume.

	Max Temp (°C)	VD240 (mm3)	VD30 (mm3)
Single shot, 1×1×3 mm	32.9	52.3	91.5
Single shot, 2×2×3 mm	26.3	55.5	95.1
4mm circle, 1×1×3 mm	17.4	118	600
4mm circle, 2×2×3 mm	16.7	129	615

Figure 4.

Experimental data temperature line plots through the center of the focal zone, perpendicular to the path of the ultrasound beam. Plots from the single point heating imaged at 1×1×3 mm and 2×2×3 mm are shown in A, and the corresponding plots from the 4 mm circle heating are shown in B.

Simulations

The simulated temperature maps were reconstructed to have spatial resolutions of 1.0 × 1.0 × 3.0 mm and 2.0 × 2.0 × 3.0 mm with 0.1 mm isotropic voxel spacing. Figure 5 shows plots of these two resolutions for both simulated heating trajectories corresponding to the plots of experimental data shown in Figure 4. As with the experimental results, image resolution affects temperature measurement accuracy the most where the spatial second derivative is large. The maximum measured temperatures differ by 4.9 °C for the single point heating case but only by 0.2 °C for the 4 mm circle heating case. The simulated 2.0 × 2.0 × 3.0 mm data shown in Figure 5A and B were used to calculate the temperature error as a function of the spatial second derivative. The results are shown in Figure 6 where the strong correlation between error and the second derivative can be seen.

Figure 5.

Simulated data temperature line plots through the center of the focal zone, perpendicular to the path of the ultrasound beam. Plots from the single point heating imaged at 1×1×3 mm and 2×2×3 mm are shown in A, and the corresponding plots from the 4 mm circle heating are shown in B.

Figure 6.

The simulated 2×2×3 mm data from Figures 6A and B were used to calculate the temperature error as a function of the spatial second derivative.

More general results for how spatial resolution affects the maximum measured temperature for the case of the simulated single point heating are shown in Figure 7. The maximum temperature as a function of in-plane resolution (for a 3mm thick slice) is plotted in Figure 7A with the 0.1 mm isotropic maximum temperature of 33.0°C shown for reference (gray line). Figure 7B shows the maximum measured temperature as a function of slice thickness where four different scenarios were considered: 1.0 × 1.0 mm and 2.0 × 2.0 mm in-plane resolution combined with the imaging slice oriented either perpendicular or parallel to the path of the ultrasound beam. Small Gibbs ringing artifacts were present in the reconstructed temperatures, causing slight undulations in some of the maximum temperature curves.

Figure 7.

A) Simulation results showing the maximum measured temperature as a function of in-plane resolution for t a 3 mm thick slice oriented perpendicular to the path of the beam. B) Simulation results showing the maximum measured temperature as a function of slice thickness for four different cases: 1×1 mm or 2×2 mm in-plane resolution combined with the slice oriented perpendicular or parallel to the path of the beam.

Discussion

In this paper we have investigated the dependence of MR temperature imaging measurements on the relationship between the MRI sampling choices and detail in the temperature distribution to be measured. We have shown through simulation and experiments that sampling grid location and image spatial resolution each have a significant effect on how the underlying temperature distribution is measured, including critical metrics such as maximum temperature, maximum thermal dose, and the volume of tissue dosed to a certain level. Because of the symmetry of the MRI VSF, the errors due to sampling effects are greatest when the temperature distribution has large curvature over the dimension of a voxel.

The results demonstrate that the location and spacing of the image spatial sampling grid is critical to achieving accurate, repeatable measurements of a HIFU induced temperature distribution. For the case of single point heating, Table 2 indicates large variations in the measured maximum temperature and thermal dose volumes could occur depending on where the sampling grid is placed, even when sampling at the higher resolution of 1.0 × 1.0 × 3.0 mm. Variations in the maximum measured temperature values for the case of 4 mm circle heating are much less, although the measured thermal dose volumes still vary between 5% (1×1×3 mm, V_D30) and 38% (2×2×3 mm, V_D240).

These variations in the measured temperature and thermal dose values due to grid placement have important implications for HIFU applications where multiple single point sonications will be delivered at different locations or times, such as repeatability studies or multi-point tumor ablation treatments. In such cases, it is inevitable that the peak of the temperature distribution will fall into different positions relative to the image sampling grid, leading to different measured temperatures even if the true underlying distribution is the same. These errors will create larger variations than may actually be present in repeatability studies and effect important monitoring criteria in ablation treatments. For tumor treatments, sub-optimal grid placement will always underestimate the true maximum temperature or peak thermal dose and therefore these errors will cause tissues to be over-treated, not under-treated. While over-treating a region of tissue that needs to be necrosed is not harmful, there are still adverse effects. The longer heating time will increase thermal build up in tissue in the near field of the ultrasound beam. The increase in energy deposition in normal tissue will require longer cooling times between sonications to ensure no tissue damage occurs. Additionally, thermal dose must be measured accurately for histological studies that correlate predicted tissue necrosis with actual tissue necrosis.

While ZFI cannot overcome measurement inaccuracy due to sampling at insufficient resolution, it can create smaller voxel spacing to mitigate the measurement discrepancies due to varying grid placement. The simulations and experiments both show that ZFI does improve measurement accuracy at the edges of the focal zone where the temperature changes are linear. Thus, it can be expected that ZFI will improve the consistency and accuracy of the important dose metrics V_D240 and V_D30. For the case of single point heating sampled at 2.0 × 2.0 × 3.0 mm resolution, ZFI to 0.25 × 0.25 mm in-plane spacing reduced the variation in measurements of maximum temperature, V_D240, and V_D30 from 30%, 63%, and 58%, respectively, to 0.5%, 4.0%, and 3.0%, respectively. In most instances, zero-filling to 0.5 × 0.5 mm in-plane voxel spacing was found to be sufficient, with the only penalty being a small computation burden and additional memory usage.

The MRI spatial resolution necessary to accurately capture the non-uniform temperature distributions created by HIFU heating is highly dependent on the shape of the temperature distribution. Regions of a temperature distribution that change linearly in space can be accurately measured at low resolution, even if the slope is very steep. It is regions where the second derivative of the temperature distribution is large that require high spatial resolution for accurate measurement. The curvature of a HIFU-induced temperature distribution will depend on many factors including the shape and size of the focal zone, the heating trajectory, and the ultrasound power deposition and duration.

The single point heating case presented in this paper is an example of a HIFU sonication that will create a temperature distribution with large curvature due to the small focal zone size, fixed trajectory and relatively large applied power. For this particular case, the simulation results in Figure 7 indicate that imaging at 1.0 × 1.0 × 4.0 mm spatial resolution or higher with the slices oriented perpendicular to beam path is necessary for accurately capturing the peak temperature information. The elongated shape of the focal zone creates less curvature in the direction of the beam, allowing for somewhat thick slices to be used. The 4 mm circle heating case is an example of a trajectory that will create a temperature distribution with less curvature. Similar simulation results indicate that the peak temperature for this case can be measured accurately to within less than 1.0 °C by using voxels as large as 2.0 × 2.0 × 5.0 mm.

Both the experimental and simulation results indicate that imaging at insufficient spatial resolution will lead to significant errors in the maximum measured temperature. Errors can arise from low in-plane resolution, overly thick slices, or slices oriented in the wrong direction. For the single point heating case, there is a 6.6°C drop off in the maximum measured temperature between the 1.0 × 1.0 × 3.0 mm temperatures, which simulation suggest are accurate, and the 2.0 × 2.0 × 3.0 mm temperatures. It is important to note the tissue volume dosed to a certain threshold was not similarly affected for the specific cases presented. As Table 3 indicates, there was little variation in V_D240 and V_D30 between the two different resolution temperature maps. This is due to the fact that low resolution images can accurately capture temperature changes that are linear in space. Figure 4 shows that the two temperature measurements match quite closely at the edges of the heated region, which defines the extent of the volume that received a certain thermal dose.

High spatial resolution is only one consideration for acquiring accurate temperature maps. Other error sources exist that could effect the accuracy of the temperature measurements that this study has not considered, including: inaccurate estimation of the baseline temperature where each 1°C offset error will affect the thermal dose calculation by a factor of two; inaccuracies in the chemical shift coefficient where reported values in different tissue types range from −0.007 to −0.011 ppm/°C (15); resolution effects due to T2* decay during segmented echo-planar imaging; temporal averaging; and low signal-to-noise ratio (SNR). In MR imaging, there is always a trade-off between spatial resolution, temporal resolution, and SNR. While we have demonstrated that high spatial resolution scans are necessary for accurately measuring a HIFU induced temperature distribution, it may be that the long scan times that come with high spatial resolution cause similar or worse errors or that the decrease in SNR causes the temperature maps to be excessively noisy. Fortunately, there are a number of steps that can be taken to minimize the error due to spatial averaging without increasing scan time. Increasing the number of samples in the read-out direction does not add significant (if any) scan time and therefore adequate spatial resolution can always be achieved in at least one direction with the caveat that image noise will be increased. Orienting the slices such that the thickest imaging dimension is aligned with the longest dimension of the HIFU focal zone is also important. Finally, zero-filling the k-space in all encoded directions does not add scan time and should be computationally efficient enough to be done in real-time.