The Ultimate Signal-to-Noise Ratio in Realistic Body Models

Abstract

Purpose

We compute the ultimate signal-to-noise ratio (uSNR) and G-factor (uGF) in a realistic head model from 0.5 to 21 Tesla.

Methods

We excite the head model and a uniform sphere with a large number of electric and magnetic dipoles placed at 3 cm from the object. The resulting electromagnetic fields are computed using an ultrafast volume integral solver, which are used as basis functions for the uSNR and uGF computations.

Results

Our generalized uSNR calculation shows good convergence in the sphere and the head and is in close agreement with the dyadic Green’s function approach in the uniform sphere. In both models, the uSNR versus B₀ trend was linear at shallow depths and supralinear at deeper locations. At equivalent positions, the rate of increase of the uSNR with B₀ was greater in the sphere than in the head model. The uGFs were lower in the realistic head than in the sphere for acceleration in the anterior-posterior direction, but similar for the left-right direction.

Conclusion

The uSNR and uGFs are computable in nonuniform body models and provide fundamental performance limits for human imaging with close-fitting MRI array coils.

INTRODUCTION

The ultimate intrinsic signal-to-noise ratio (uSNR) is defined as the maximum possible SNR achievable by any coil positioned outside of the object. The uSNR is computed for a specific object, field strength, and minimum distance between the coil and the object. One of the goals of uSNR computations is to compare existing coils to the best possible sensitivity, thus assessing the “room for improvement” of a specific array. Another goal is to assess the gain from varying a parameter, such as field strength or coil array coverage.

The uSNR concept was introduced in 1987 by Roemer and Edelstein (1) and was computed by several researchers in the quasi-static regime in infinite half-planes (2) and infinite cylinders (3) in the following years. In 1998, Ocali and Atalar published a seminal paper (4) in which they showed uSNR results for a finite uniform cylinder; a more realistic shape for the human body than the previously used semi-infinite shapes. Ocali and Atalar’s work described the fields providing the uSNR as a linear combination of plane wave basis fields spanning the space of solutions to Maxwell’s equations. They showed that, within the cylinder, the uSNR is bounded and that the central uSNR increases faster than linearly with B₀. A possible limitation of their work is that they did not prove that their basis set was complete. Schnell et al (5) then performed a similar computation using the dyadic Green’s function (DGF) formalism to generate the basis fields. They pointed out that DGFs allow modeling of a minimum distance between the “coil space” and the sample, providing a more realistic model of array coils. Moreover, given that in this approach both curl- and divergence-free surface current distributions are modeled explicitly, the resulting basis set is guaranteed to be complete (in other words, the uSNR can be approximated to an arbitrary accuracy by using a sufficiently large number of basis vectors).

Following work studied the limit of SENSitivity Encoding (SENSE) acceleration for simple shapes such as half-spaces (6), cylinders (7), and spheres (8) showing the existence of a SENSE acceleration rate threshold below which there is almost no ultimate G-factor (uGF) penalty and above which the uGF increases rapidly. Other studies have been published to establish how practically implementable coil arrays with increasing channel count may approach the uSNR (9–13). As expected from experimental work (14,15), increasing the number of coils allows the un-accelerated SNR to quickly reach its ultimate value at the center of the sphere or cylinder (eg, a 32-channel coil reaches ~95% of the uSNR value at the center of head-sized uniform spheres even for B₀ as high as 7 Tesla [T]). However, convergence to the uSNR slows close to the sphere surface and more and more coil elements are required at higher field strength. For accelerated imaging, the ultimate sensitivity is impacted by the unaccelerated uSNR, uGF, and acceleration rate. Because the center-of-object G-factor continues to improve as elements are added (at least up to 128 for head coils), the central SNR for accelerated images keeps improving as coils are added beyond 32 elements (16–20).

A more recent development is the utilization of ultimate current patterns (ie, current patterns yielding the uSNR) to intuit more sensitive coil designs (10,12). These studies used the DGF formalism in uniform spheres and cylinders to show that both curl-free (ie, electric dipoles) and divergence-free current distributions (ie, magnetic dipoles) are needed to reach the uSNR. This led to the proposal of novel coil geometries with both electric dipole antenna and loop elements (21–26).

Although previous studies have shed light on ultimate coil performance and the fundamental detection limits of MRI, their restriction to uniform spheres, cylinders, or infinite half-planes limits their comparability to modern array coils. In this work, we develop a generalized uSNR calculation applicable to realistic human body models. This is needed to assess current array designs, which are typically highly conformal to the body area being scanned and are not necessarily loaded properly by a half-space, sphere, or cylinder. This becomes increasingly important at high B₀ fields, where interactions between the radiofrequency (RF) coil and the body become stronger. Whereas the conductivity of the sphere or cylinder can be appropriately matched to the average of body tissues, the reduced symmetry of the human body compared to spheres or cylinders generally leads to a reduced “center brightening” effect (27–34) in the body compared to uniform phantoms. In addition to providing more realistic comparisons for arrays designed to tightly fit the body, uSNR and uGF maps in realistic human body models allow statements about these parameters, in particular, organs or suborgans, answering questions such as, “What is the maximum achievable SNR in the hippocampus? In the heart?”. In this work, we:

• Develop a methodology for computing the uSNR and uGF in realistic body models.

• Compare our approach with the DGF method in a uniform sphere to ensure continuity between the proposed method and previous uSNR studies.

• Assess the convergence and impact of key parameters of the method, such as the thickness of the dipole cloud and spatial resolution.

• Study the variation of the uSNR in the head as a function of field strength from 0.5 to 21T.

• Study the dependence of the uGF in the realistic head model as a function of field strength.

Given that uSNR and uGF maps are—by definition—ultimate limits dictated by Maxwell’s equations and the human anatomy, they only need to be calculated once per body model. Ideally, a distribution of body models would include those maps so that coil designers modeling a particular array design could compare it to the ultimate performance metrics in specific organs of interest. To this end, we distribute these maps for the Duke head model as well as all the source code required to compute them at http://ptx.martinos.org/index.php/Main_Page.

METHODS

Huygens’s Principle

The so-called surface equivalent current or Huygens’s principle states that any electromagnetic (EM) field distribution inside a closed volume devoid of sources can be generated by a current distribution flowing on the surface defining that volume (35). Thus, the EM fields created inside the human body by an external RF coil can be reproduced by some current density flowing on the outer surface of the body, which we refer to as the Huygens’s surface. The Huygen’s surface separates the “coil space” from the “body space.” In order to determine the ultimate SNR of close-fitting, body-contoured array coils, we choose the Huygens’s surface as the surface conformal to the outer shape of the body model and located at a distance D from it (Fig. 1b). The distance D defines the minimum distance allowed between the coils and the body.

FIG. 1.

(a) Cut through the Duke head model nonuniform permittivity distribution (3T). (b) 3D view of the Duke head model (solid red) and the associated Huygens surface (transparent green). The black space beyond the Huygens surface is the “coil space.” The minimum distance between the coil space and the head model is noted D and is equal to 3 cm in this work. (c) The Huygens equivalent current principle states that the EM field created by an arbitrary arrangement of external sources located at least at D from the head can be modeled by a unique continuous current distribution on the Huygens surface. (d) We discretize continuous current distributions on the Huygens surface using electric and magnetic dipoles. In this work, we place 263,862 dipoles at 43,977 distinct locations. There are six dipoles per location: three X, Y, and Z electric dipoles and three X, Y, and Z magnetic dipoles. (e) Example of an electric field created in the head by a random excitation of the dipole cloud.

The Huygens’s equivalent current is continuous (Fig. 1c), but we discretize it using a large number of magnetic and electric dipoles (Fig. 1d) placed at the center of cubic voxels arranged in a 3D rectangular grid. Note that the same grid is used for discretization of the Huygen’s surface and discretization of the body model. This is done because, in this case, computation of the incident fields can be done very quickly using fast Fourier transform (FFT)-accelerated electric and magnetic field volume integral operators (36,37). This common Cartesian grid forces us to approximate the shape of the Huygens’s surface by a “staircase” or piece-wise constant shape. As a result, in practice, the discretized Huygens’s surface has a nonzero thickness. We place six dipoles at each discretized location: three electric dipoles (orientated along X, Y, and Z) and, similarly, three magnetic dipoles.

Generation of the EM Fields Basis Set

We compute a basis-set of EM fields inside the body model in two steps. First, we compute a basis set of incident fields using the free-space DGFs (36,37) (also see Jin (38), chapter 2, and Jackson (39), chapters 9.2 and 9.3). Then, we compute the scattered component of the fields due to the presence of the body model using a fast volume integral solver. The scattering calculation constitutes the computational bottleneck of the method.

Choosing the Incident Field Basis Set

The basis fields used in the uSNR calculation can be chosen as any linear combination of the individual dipole fields. The simplest combination might be to consider each individual dipole’s field as a basis vector (with all other dipoles weighted by zeros). For neighboring dipoles, these basis vectors would be highly correlated. Therefore, the resulting basis set would need to compressed using, for example, singular value decomposition (SVD). In this work, we use a 3-mm isotropic discretization grid which, for the Duke model truncated below the neck (Fig. 1a), gives P = 193,542 nonzero body voxels, and 43,977 dipole locations (there are therefore N = 43,977 × 6 = 263,862 dipoles). The P × N matrix, which we call A, containing the electric fields of all dipoles, would require 1.2 TB of single-precision memory storage, which would render the SVD compression step impractical.

Instead, we seek a low-rank approximation of the electric field matrix A. To do this, we energize the entire dipole cloud using random magnitude and phases (Fig. 2, step 1). The hope is that these randomly generated dipole excitations are able to quickly explore the left subspace of A without having to actually form it. This approach is inspired by recent randomized matrix decomposition techniques that allow to quickly compute low-rank approximations of enormous datasets (40–42).

FIG. 2.

Flow chart of the computation of a single vector of the ultimate basis-set in the Duke head model. Step 1: Excitation of electric and magnetic dipoles on the Huygens surface using random amplitudes and phases (only a subset of the N = 263,862 dipoles are shown for visibility). Step 2: Incident electric and magnetic fields are computed using the analytical free-space Green’s function at the Larmor frequency. Step 3: The equivalent volume polarization current is computed in each voxel of the head model using the volume integral equation solver MARIE. This step is the computational bottleneck and is accelerated using a GPU. Step 4: The total electric and magnetic fields are obtained by applying the discretized integrodifferential operators K and N defined in Jin (38) and Jacson (39) to the equivalent current distribution and then summing the result with the incident fields.

Computation of total fields

In theory, any EM solver can be used to compute the scattered fields (ie, field distortions attributed to the presence of the body model) given the incident fields. However, most finite difference time domain (FDTD) (43–45) and finite element commercial (FEM) solvers (46–49) are not intended for such analysis and are likely too slow. Instead, we use a recently introduced ultrafast volume integral equation (VIE) solver, Magnetic Resonance Integral Equation (MARIE; https://github.com/thanospol/MARIE) (36). MARIE computes the average electric and magnetic field inside cubic voxels given any incident field illumination. This tool was specifically developed for MRI applications. The MRI targeted design (with sources external to a heterogeneous volume, combined with an efficient preconditioner and ultrafast fast matrix-vector products implemented using the FFT on the graphics processing unit [GPU]) provides MARIE with a significant speed advantage for this problem compared to FDTD and FEM (36).

uSNR and Ultimate SENSE G-Factors

We follow the method of Ocali et al (4) and Lattanzi et al (50) for computation of the uSNR from a set of basis EM fields. Below, we define the uSNR and ultimate G-factor (uGF-factor) metrics using the notations of Lattanzi et al (50). The intrinsic SNR per unit of square-root receiver bandwidth and unit sample volume is shown by Equation 1:

$$\mathit{SNR}(\mathbf{r})=\frac{\omega M_0{B}_{1-}(\mathbf{r})}{\sqrt{4k_{B}T{\displaystyle \underset{BM}{\int }\mathbf{\sigma }({\mathbf{r}}^{\prime })|\mathbf{E}({\mathbf{r}}^{\prime }){|}^{2}d{\mathbf{r}}^{\prime }}}}$$ [1]

where ∫_BM. denotes spatial integration over the body model, σ(r) is the conductivity at location r, and E (r) is the total electric field in the sample at r. This considers only sample losses and ignores instrumentation noise. The Larmor frequency, ω, and the steady-state magnetization, M₀, are both proportional to the field strength, B₀, therefore their product varies as $B_0^2$. The term B₁₋(r) is defined as $B_{1-}(\mathbf{r})={\sum }_{k=1}^K{x}_k{B}_{1-,k}(\mathbf{r})=\mathbf{B}{(r)}^T\mathbf{x}$ where B(r) and x are the coil sensitivities and combination weights stacked in column vectors, respectively (we assume that the basis-set contains K vectors). Similarly, the total electric field is the combination of the individual basis electric fields: $\mathbf{E}(\mathbf{r})={\sum }_{k=1}^K{x}_k{\mathbf{E}}_k(\mathbf{r})=\overline{\mathbf{E}}{(\mathbf{r})}^T\mathbf{x}$, where $\overline{\mathbf{E}}(\mathbf{r})$ is a Kx3 matrix of stacked electric fields at location r (three X, Y, and Z components per location). Replacing the continuous spatial integration by a sum over nonzero voxels and removing the constant terms in Equation 1 yields the following “condensed SNR metric” (Eq. 2):

$$\mathrm{\Psi }(\mathbf{r})=B_0^2\frac{\mathbf{B}{(\mathbf{r})}^T\mathbf{x}}{\sqrt{{\mathbf{x}}^H\mathbf{Qx}}}$$ [2]

where the loss matrix is given by $\mathbf{Q}=dV{\sum }_{i=1}^P\mathbf{\sigma }({\mathbf{r}}_i)\overline{\mathbf{E}}({\mathbf{r}}_i)\overline{\mathbf{E}}{({\mathbf{r}}_i)}^H$, P is the number of nonzero voxels, and dV is the voxel volume. The units of Ψ(r) are T³/W^1/2.

Maximization of Ψ(r) is equivalent to minimization of x^HQx subject to B (r)^Tx= C, where C is a constant (the final value of Ψ(r) is independent of the specific value of C). This quadratic programming optimization problem with a single linear constraint is convex and its solution is well known (see, eg, see Eq. 20 of Lattanzi et al (51) for an analytical expression).

Wiesinger et al showed that this computation of the unaccelerated SNR can be extended to SENSE reconstruction of regularly undersampled Cartesian MR data (8). Equation 2 can still be used in this case, but the K × 1 vector x must be replaced by a matrix X of size K × R, where R is the acceleration factor, and the single equality constraint must be replaced by multiple equality constraints of the form ${\mathbf{B}}_R^T\mathbf{X}=C{\mathbf{1}}_R$. Here, B_R is a matrix of size K × R containing the ${\mathrm{B}}_1^{-}$ sensitivities of the basis-set vectors at the R aliased locations. This set of equality constraints form the SENSE unfolding condition (52). Again, the problem of finding the optimal coil combination that unfolds the aliased data while maximizing SNR was solved in Pruessmann et al (52). In this reference, an analytical expression of the G-factors was also derived, which we do not repeat here. Pixel aliasing becomes less pronounced when the image field of view (FOV) is large compared to the object. Therefore, we study the “worst case G-factor” by cropping the FOV of all basis fields to enclose exactly the nonzero head voxels.

Uniform Sphere Comparisons

We validate our generalized uSNR computation methodology by comparison with the uSNR computed with the DGF formalism (5,12) in a uniform 15-cm-diameter sphere. We perform this comparison for field strengths 0.5, 1.0, 1.5, 3, 7, and 14T. We model frequency-dependent electrical properties matched to the average of the gray and white matter (ε_R = 227, 184, 141, 68, 64, and 56 and σ = 0.25, 0.30, 0.35, 0.50, 0.51, and 0.62 S/m at 0.5, 1.0, 1.5, 3, 7, and 14T, respectively (53)).

The DGF basis set was computed as explained in Lattanzi and Sodickson (12): The DGF for electric and magnetic dipoles were expanded on spherical harmonics functions with L_max = 35, which yielded 2(L_max + 1)² −2 = 2590 basis fields. Both curl- and divergence-free current distributions where modeled in the basis set. The minimal distance between the sphere and current patterns was set to D = 3 cm. For all field strengths, the DGF fields were exported on a 64 × 64 × 64 grid with 2.34-mm isotropic voxel size. The B₀ = 3T fields were also computed on a 96 × 96 × 96 (1.56-mm spatial resolution) and 128 × 128 × 128 (1.17-mm) voxel grids in order to assess the effect of spatial resolution on the agreement between DGF and the generalized uSNR calculation.

The 15-cm diameter sphere was also modeled with the generalized uSNR approach (same electrical properties, grid size, and resolution) using a dipole cloud with 2.5-mm thickness. The distance between the sphere and dipoles was also D = 3 cm, and the number of random excitations was K = 2,500. Both electric and magnetic dipoles were modeled. The B₀ = 3T simulation was performed on three grid sizes (64, 96, and 128 as explained in the paragraph before) and for four dipole cloud thicknesses (2.5, 7.5, 12.5, and 17.5mm). All grid-size simulations were performed using a 2.5-mm dipole cloud thickness whereas all dipole cloud thickness simulations were performed on the 64 × 64 × 64 voxel grid.

Additionally, we used the DGF method to calculate fields in an 18.5-cm diameter uniform sphere, which matches the cross-section area of the Duke model in the axial plane (cross-section ~267 cm², giving an equivalent spherical diameter of 18.5 cm). The dielectric properties of this uniform sphere where set to the average brain at each frequency.

Duke Simulations

We computed the uSNR and uG-factors in the Virtual Family “Duke” body model from the IT’IS Foundation (54) for field strengths 0.5, 1, 1.5, 3, 4.7, 7, 9.4, 11.7, 14, and 21T. All simulations were performed on a Windows server with 512 GB of RAM, 28 logical cores, and a Tesla K80 GPU. To limit computational requirements, we truncated the body model below the neck. Strictly speaking, dipoles located far from the head below the neck (eg, close to the chest or the feet) have a nonzero, but small, contribution to the SNR in the head—we assume, in this work, that it is negligible.

We exported the Duke head model on a uniform voxel grid with 3-mm isotropic resolution, resulting in P = 193,542 nonzero voxels (63 × 80 × 85 voxel grid). We modeled the frequency-dependent permittivity and conductivity of human tissues using the Gabriel database (53). The minimum distance between the dipoles and the body model was D = 3 cm. Note that we did not place any dipoles directly below the neck given that no coils can be placed there in practice (Fig. 1d). uGFs were computed for an axial slice passing through the middle of frontal lobe and for acceleration (R = 1, 2, 3, 4, 5, 6, and 7) in both the left-right (LR) and anterior-posterior (AP) directions. We also computed uGFs for 2D SENSE acceleration R = 4 × 5 (LR × AP directions) for the same axial slice and for R = 4 × 5 (AP × HF, head-foot) for a central sagittal slice.

RESULTS

Figure 3 shows uSNR convergence as basis vectors are added for the DGF and generalized uSNR method for field strengths varying from 0.5 to 14T in the 15-cm diameter uniform sphere. Supporting Figure S1 shows that convergence is slower for both methods in voxels close to the sphere surface, which is in agreement with previous studies. For DGF, convergence slows with increasing field strength at the “CENTER” and “INTERMEDIATE” voxel locations (no clear trends are visible for the “EDGE” location). For the generalized uSNR method, there is no clear convergence trend as a function of field strength for any location in the sphere, but, overall, the convergence is clearly slower than with DGF.

FIG. 3.

(a) Convergence of the uniform sphere ultimate SNR with the generalized uSNR approach as a function of field strength, number of vectors in the basis set, and position along the sphere diameter (center, intermediate, and edge locations). (b) Convergence of the uniform sphere uSNR with the DGF approach.

Figure 4 shows natural log plots of uSNR profiles though the 15-cm diameter sphere center. There is good overall agreement between DGF and the generalized uSNR method for all field strengths. Small differences between the two methods are better visualized on the linear scale plot of the ratio of the two methods (Fig. 5). Agreement is best at the center of the sphere (ratio = 1.016 for the highest resolution model) and worst near the edges of the sphere where the generalized method first underestimates the uSNR in edge voxels and then overestimates the uSNR just inside the sphere (ratio = 1.052 for the highest resolution model). These Figure 5 overestimation lobes just inside the sphere decrease in amplitude with increasing dipole cloud thickness (Fig. 5a). This increased accuracy is likely attributed to the greater number of dipoles in thick dipole clouds (the dipole count was 167,760; 522,528; 906,768; and 1,331,760 for the 2.5-, 7.5-, 12.5-, and 17.5-mm-thick dipole clouds, respectively). Figure 5b shows that the accuracy at the sphere edge also improved with increased model spatial resolution. Because all simulations of Figure 5b were performed for the smallest dipole cloud thickness (ΔD = 2.5mm), it is likely that these errors decrease even further with a thicker dipole cloud.

FIG. 4.

Ultimate SNR (natural log scale) computed in a uniform sphere (D = 15 cm, “brain-like” frequency-dependent permittivity and conductivity) with the DGF and generalized uSNR approaches for field strengths ranging from 0.5 to 14T.

FIG. 5.

(a) Impact of dipole cloud thickness on the agreement between the generalized uSNR method and DGF (B0 = 3T). We plot the ratio of the uSNR computed with the generalized method for different dipole cloud thicknesses ranging from 2.5 to 17.5mm to the uSNR computed with DGF. The minimum dipole-sphere distance is 3 cm for all cloud thicknesses simulations. (b) Impact of increasing the spatial resolution on the agreement between the generalized uSNR method and DGF. We plot the ratio of the ultimate SNR computed with the generalized uSNR and DGF methods for spatial resolution ranging from 2.34mm isotropic (64 × 64 × 64 grid) to 1.17mm (128 × 128 × 128 grid). The dipole cloud thickness is 2.5mm for all these simulations.

Supporting Figure S2 shows the uSNR convergence in the Duke head model for field strengths ranging from 0.5 to 21T. As in the uniform sphere, good convergence is achieved at all locations in the head and all field strengths using K = 2,500 basis fields. Figure 6 shows a more detailed comparison of the variation of the uSNR with field strength in both the Duke head model and in the 18.5-cm diameter uniform sphere. Given that it is somewhat difficult to compare the sphere and Duke’s head results, we plot the uSNR at four locations (POS1– 4) located at 1, 2, 3, and 9 cm away from the top edge of the object. Figure 6a and 6b (and their zoom Figure 6a′ and 6b′) show that the uSNR tends to be greater in Duke than in the sphere at POS2 and POS3. We point out that direct comparison of the uSNR plots in the sphere and Duke’s head at POS1 and POS2 is a bit precarious because the uSNR varies extremely quickly at these locations. Indeed, the uSNR varies 51%, 35%, 18%, and 3% per pixel at locations POS1, POS2, POS3, and POS4, respectively. Therefore, a 30% error between DGF and the generalized uSNR approach (see Fig. 5) is equivalent to a position uncertainty of 1.7 pixel, 1.2 pixels, 0.6 pixel and 0.1 pixel for POS1–4 in this order. In contrast, in-model comparisons are more robust: For POS2–4, the rate of increase of the uSNR with field strength is greater in the sphere than in Duke. We fit each curve of Figure 6 to the quadratic form $u\mathit{SNR}=\alpha B_0^2+\beta B_0+\delta$. All fits were excellent (R² > 0.99). For both the uniform sphere and Duke’s head and for all positions, the value of the coefficients α, β, and d are given in the Supporting Table S2, which may be useful to some readers to extract quantitative uSNR values from our simulations.

FIG. 6.

(a) uSNR (linear scale) as a function of position and field strength in the uniform sphere as computed with the DGF method. (a′) Same figure as (a), but with the y-axis scaled differently to zoom in on the inner positions. (b) Same thing as (a), but in the Duke head model, computed with the generalized uSNR approach. (b′) Same figure as (b), but with the y-axis scaled differently to zoom in on the inner positions. (c) Gray matter and white matter averages of the uSNR in Duke as a function of field strength. For figures (a), (a′), (b), and (b′), the positions #1, #2, #3, and #4 are located at 1, 2, 3, and 9 cm away from the top edge of the sphere/head. For all figures, the dashed black line show linear uSNR extrapolations at low field.

Computation times for the generalized uSNR approach in Duke’s head are shown in Supporting Table S1. Total solve time for the 2,500-vector basis set was 2.9, 1.6, 1.4, 1.3, 1, 1.4, 2, 2.6, 3.2, and 4.9 days for B0 = 0.5, 1, 1.5, 3, 4.7, 7, 9.4, 11.7, 14, and 21T, respectively. The total solve time for all basis sets was 22 days. The fact that the total computation time decreases as B₀ increased from 0.5 to 4.7T, but then increases for B₀ greater than 4.7T, is attributed to the fact that the MARIE solver convergence slows down with 1) increasing permittivity contrast (ratio of the highest to smallest permittivity values in the nonzero body voxels) and 2) increasing frequency. Long computation times at B₀ = 0.5 and B₀ = 1T are attributed to the fact that the permittivity of tissues increases very quickly at low frequencies. The MARIE solver convergence (BICGSTAB) was set to 10⁻⁸, which, we found, was very important to ensure good agreement between DGF and the general uSNR method.

The uSNR in the 18.5-cm diameter uniform sphere and Duke’s head show a similar trend: For voxels deeper than 1 cm in the object, the uSNR increases superlinearly with field strength. Thus, the uSNR map becomes more homogeneous at higher B₀ field. For example, the ratio of the uSNR at POS2 and POS4 (Fig. 6) decreases from 35.7 at 1T to 2.9 at 21T. This is confirmed visually in Figure 7, where the “central hole” in uSNR maps attributed to this “edge-to-center” ratio becomes less prominent at increasing field strengths. Figure 7 suggests that the potential for high-density Faraday detectors near the object’s surface seems to overwhelm the B₀ effect. In contrast, at the object center, the B₀ effect is very visible (it is supralinear, as shown in Fig. 6). Thus, in order to increase SNR at the edge, one should use more coils. At the center of the object, however, increasing B₀ is the main strategy.

FIG. 7.

(a) Axial ultimate SNR maps (natural log scale) in the head of Duke for field strengths ranging from 0.5 to 21 T. (b) Same thing in the sagittal view (Duke). (c) Same thing in the uniform sphere (DGF).

Figure 8 shows maximum ultimate SENSE G-factors (uGFs) for acceleration in a single direction up to R = 7 in both Duke and the 18.5-cm diameter uniform sphere. In the sphere, our results are consistent with Wiesinger et al (8) (see Fig. 8, FOV = 20 cm therein): At high acceleration factors, the maximum uGF increases slightly with increasing field strength for B₀≤7T and then decreases rapidly for B₀≥7T. The uGFs in Duke (LR) and the sphere are in rough agreement, but not in Duke (AP) and the sphere. This result is also visible in Figure 9, where, for B₀≤7T, the ultimate retained SNR for Duke (LR×AP) and Duke (AP×HF) is greater than in the uniform sphere. For B₀≥7T, this difference disappears, which is attributed to the fact that there is essentially no uGF penalty for R = 4×5 SENSE in the sphere and Duke’s head.

FIG. 8.

(a) First row: Maximum ultimate SENSE G-factors as a function of field strength (B0) and acceleration factor (R) in the AP direction in Duke. Second row: Interpolated acceleration factor R yielding an uGF = 1.1. (b) Same thing for SENSE acceleration in the LR direction in Duke. (c) Same thing in the uniform sphere (no distinction between the AP and LR directions in this case).

FIG. 9.

(a) Ultimate percent of retained SNR for transverse SENSE acceleration with R = 4 × 5 in Duke’s head for B0 ranging from 0.5 to 21T. (b) Same thing in the sagittal orientation. (c) Ultimate percent of retained SNR for SENSE acceleration with R = 4 × 5 in the 18.5-cm diameter uniform sphere.

Finally, Figure 10 shows the ultimate accelerated SNR (SENSE R = 4 × 5) as a function of field strength and position (POS1–4) in Duke’s head, normalized to the B₀=0.5T value. Although the absolute uSNR is smaller in the center of the head than at the edge, this graph shows that the rate of increase of the uSNR is much greater in the head’s center than at the edge. This is attributed to the fact that the unaccelerated SNR increases superlinearly with field strength in the center of the head whereas the uGF decreases more rapidly at the center than at the edge of the head.

FIG. 10.

Ultimate SNR at the center (POS4) and edge (POS1, POS3) of Duke’s head normalized to the B0 = 0.5T value (each curve is normalized to 1 at B0 = 1.5T). Solid lines with symbols are for the accelerated uSNR (SENSE acceleration R = 4×5 in the LR×AP direction), whereas the dash lines without symbols are for the unaccelerated uSNR. The numbers in parenthesis on the right of each curve indicate the uSNR ratio uSNR (21T)/uSNR (0.5T) for the accelerated uSNR (first number) and for the unaccelerated uSNR (second number).

DISCUSSION

Methodology

The methodology presented in this work is not the only way to compute a basis-set of EM fields in realistic body models. Step 1, especially, can be performed differently. For example, we initially attempted to compute (following the algorithm given in Rokhlin et al (41)) the randomized SVD (rSVD) of the A matrix holding the incident electric fields created by all the dipoles in the cloud. Very briefly, the rSVD algorithm consists in 1) computing a matrix Y of electric fields generated by K random excitations of the dipole cloud; 2) computing the QR decomposition of Y (Y = QR); 3) projecting the matrix A onto the subspace Q (B = Q * A); 4) computing the SVD of the KxN matrix B; and 5) finally multiplying Q with the left dominant subspace of B (obtained in the previous SVD step). Given that K (the number of random excitations) is much smaller than N (the number of dipoles), the QR and SVD steps (2 and 4) are feasible in practice.

We found that this approach was computationally slower than simply using random excitations of the dipole cloud without the SVD step (this corresponds to simply performing step 1 of the rSVD process). The main advantage of rSVD is to reduce the number of basis vectors needed to adequately span the solution space given that it yields an orthogonal basis. However, we found that the basis reduction savings did not make up for the extra computational steps required in this approach.

Another way to compute a basis set in the nonuniform head model would be to use a very thin dipole cloud of only 2,500 dipoles. However, using such a small number of dipoles is insufficient to cover the entire Huygens’s surface with a reasonable resolution: The 3-cm Huygens’s surface for Duke has an area of 2,660 cm². Assuming that we use 2,500 dipoles, this results in a coverage of 1.04 cm² per dipole and an average distance between neighboring dipoles of 10.15mm, which is large compared to the 3-mm voxel resolution. For deep voxels, this would likely not matter; however, this might bias the uSNR estimation near the edge of the head. In contrast, using random excitations of the entire dipole cloud ensures that, for all surface voxels, there are a large number of dipoles close to those locations, thus improving the chance of reaching the ultimate SNR at those locations.

Convergence and Computation Time

In the uniform sphere, we found that our generalized uSNR method converged more slowly than the DGF method, which is expected given that the DGF basis set is orthogonal. However, good convergence was achieved for both the sphere and Duke’s head at all locations and all field strengths when using 2,500 basis vectors. Supporting Figure S3 shows that, for all field strengths and positions in the head, the uSNR percent change per 100 basis vectors was smaller than 0.7% when using 2,500 basis fields. In fact, for the “CENTER” and “INTERMEDIATE” positions, the uSNR percent change was smaller than 0.04% per 100 additional basis vectors. Using 2,500 basis vectors yielded relatively long computation times, even when using GPU acceleration—the longest being 5 days for computation of the basis set at B₀ = 21T. As shown in Supporting Figure S4, we found that decreasing the dipole-object distance D resulted in slower convergence at the edge of the object. It is also apparent in this figure that decreasing D leads to an increase of the uSNR only at locations close to the body surface. For deeper locations, the uSNR computed with D = 3, 2, and 1 cm was essentially the same. Computational times for the generalized uSNR method are provided in Supporting Table S1.

We note that the computational requirements for a full body computation of the uSNR are far beyond the capability of our computer server, and some simplification of the body model was required. This is the reason why we modeled the Duke body model truncated below the neck, and we did not increase the spatial resolution of the voxel grid beyond 3mm. The main performance bottleneck that prevented us from simulating a larger voxel grid was the memory of the GPU card (6 GB in this work).

Validation and Accuracy of the Generalized uSNR Calculation

Our generalized uSNR approach was validated by comparison with the well-validated DGF computation of the uSNR in a uniform sphere for field strengths varying from 0.5 to 14T. We found that the agreement between the two methods improved with 1) increasing spatial resolution and 2) increasing thickness of the dipole cloud (and thus the total dipole count). For the highest resolution (1.26-mm isotropic), the agreement between the two calculations was found to be better than 5.2% for voxels at least 7.6mm deep inside the sphere and as low as 1.6% for center voxels. For voxels close to the surface, the disagreement was as high as 40%. It has been noted by other researchers that the uSNR calculation becomes unstable close to the object’s surface (4,8,12), therefore it is not surprising that the disagreement between our method and DGF is greatest for these edge pixels. This instability is attributed to at least two reasons. First, the electric and magnetic fields close to the object boundary vary rapidly compared to the inside of the object. As a consequence, more basis vectors are needed to capture uSNR variations at these locations. Even when a large number of basis vectors are used, inaccuracies may be present attributed to the incompleteness of the basis set at these locations, leading to errors in the uSNR estimation. A second problem is that the computation of optimal basis vector weights in the ultimate SNR and G-factor calculations is ill-posed at the edge of the body (4). This is related to the first problem: Given that more basis vectors are needed for convergence at the object’s boundary, basis fields overlap more, which increases the condition number of the basis-set. This causes the SNR weight optimization problem to become unstable and prone to numerical errors. Instability of the uSNR and uGF metrics at the object boundary affects any methods based on the projection of all possible EM fields onto a basis set: Indeed, this effect has also been noted with plane wave expansions (4) and the DGF method (8,12). A standard workaround is to disregard pixel locations too close to the surface of the object (4,8,12).

A major difference between the DGF and generalized uSNR methods is that the latter models a discretized version of the sphere, whereas the former models a perfect sphere. As the spatial resolution improves, the voxelized version of the sphere converges toward the perfect sphere. It is known, from the FDTD literature, that this “staircasing” approximation has a significant effect on electric field estimates (55,56). In addition, in our approach, the discretization grid used for the sample is also used for the Huygens’s surface given that the dipoles are placed at discretized locations and not at continuous positions along the true Huygens’s surface. In contrast, in the DGF approach, the current patterns are exactly placed at a distance D from the object so that the Huygens’s surface is also an exact sphere.

In theory, there should be no SNR advantage in placing additional dipoles beyond the Huygens’s surface given that the contribution of these “far dipoles” can be modeled by the dipoles close to the sphere. However, given that the generalized uSNR method approximates the true, continuous Huygens’s surface by its discretized version, in practice, additional dipoles located slightly further from the ones closest to the sample provide additional degrees of freedom. For example, a dipole placed one voxel away from a sets of dipoles closer to the sample may fall “in between” these dipoles as viewed from the center of the sphere.

uSNR and G-Factor Trends in Realistic Head Model

In the Duke head model, like in the sphere with matched cross-section and material properties, the uSNR was found to increase sublinearly with field strength for voxels close to the head’s surface (≤1 cm from skin). Deeper in the head (≥1 cm from skin), this trend becomes superlinear. We found, however, that the transition from the sublinear to the superlinear regime occurred at shallower depths in the head model than in the sphere. Moreover, at matched tissue depths, the uSNR was greater in the head than in the sphere. These differences are likely attributed to the fact that the sphere does not model accurately the shape of the head. For example, in the head, dipoles located on the left and right sides of the line defined by positions POS1–4 in Figure 6b are closer to these points than in the sphere. We point out that these conclusions are valid for the shallow locations POS3 and POS4, but that a direct comparison between the sphere and Duke results of Figure 6 is a little precarious. Indeed, the uSNR varies quickly at the edge of the head (51% per pixel at POS1, compared to 3% per pixel at POS4), so that a small error in positioning— which is expected when comparing such different objects as a sphere and Duke’s head—can have a large impact on the curves of Figure 6. Moreover, Figure 5 shows that the generalized uSNR approach tends to overestimate the uSNR at locations POS1 and POS2. In this model, comparisons such as the rate of increase of the uSNR as a function of field strength are more robust.

When using SENSE acceleration, the ultimate G-factor penalty decreases rapidly with increasing field strength for B₀ greater than 7T. This is true both for acceleration in the AP and LR direction (as well as HF, as shown in Fig. 9). For B₀ smaller than 7T, however, somewhat counterintuitively, the uGF increased slightly with field strength. Although this is in agreement with previous work by Wiesinger et al (8), this effect may not have been fully appreciated because it depends dramatically on the sphere diameter. For example, in Figure 8 of Wiesinger’s work (8), this effect is visible for the 10 and 20-cm diameter spheres, but not for the larger spheres, and therefore it was not clear whether it was true in the human head. Our results confirm that the uGF does indeed increases slightly in the human head for B₀ up to 7T. However, this effect is much less pronounced than the subsequent dramatic reduction in the uGF for very high B₀ fields. As in previous uGF calculations in uniform spheres (8), we found that there is an acceleration “cutoff” below which there is little G-factor penalty for SENSE acceleration. As shown in Figure 8 (bottom row), this acceleration cutoff increases with field strength: From R ≈ 4 (B₀ = 0.5T) to R ≈ 7 (B₀=21T) when accelerating in the AP direction and from R ≈ 3.5 (0.5T) to R ≈ 5 (21T) when accelerating in the LR direction.

Variability of the uSNR and uGFs in Realistic Body Models

In this work, we have studied the uSNR and uGF in the head of the Duke model only. We have compared these metrics to those obtained in a uniform sphere in order to assess the impact of modeling a realistic tissue distribution. Another way to assess the impact of the nonuniform tissue property distribution in the head is to simulate and compare different body models (eg, the different models of the Virtual Family). We hope to enable the computation of ultimate SNR and G metrics in all body models as they become available. Although the corresponding computation time is long, and may require improved computation platforms, this type of calculation has to be performed only once. Following the vast majority of EM simulations in body models, our calculation assumes isotropic electrical properties in the tissues, although it is known that these are best modeled using tensors (57,58) and that this can have an impact on SAR (59,60).

Supplementary Material

Supplemental Material

Fig. S1. Number of basis vectors required to achieve 95% of the ultimate SNR in the 15-cm uniform sphere for the generalized and the DGF approaches as a function of field strength and position along the sphere diameter.

Fig. S2. (a) Convergence of the ultimate SNR (natural log scale) in the Duke head model as a function of field strength, number of vectors in the basis-set, and position in the brain. (b) Number of basis vectors required to achieve 95% of the ultimate SNR value in Duke as a function of field strength.

Fig. S3. Percent uSNR change per 100 basis vectors in Duke’s head when using 2,500 basis vectors. The percent uSNR change is shown for all field strengths and three positions in the head.

Fig. S4. (a) Convergence of the generalized uSNR method (natural log scale) in the 15-cm diameter uniform sphere as a function of the number of basis vectors, position in the sphere, and the sphere-dipoles distance D (this calculation was performed for 2.34-mm isotropic voxel size, dipole cloud thickness of 2.5 mm, and B0 5 3T). (b) Number of basis vectors required to achieve 95% of the uSNR value in the uniform sphere as a function of the position in the sphere and the sphere-dipoles distance D.

Table S1. Computation time for the general uSNR approach in Duke’s head

Table S2. Quadratic fit of uSNR vs B0 trends shown in Figure 6.