Fast Integral-Equation Method for Simulating High-Field Radio Frequency Coil Arrays in Magnetic Resonance Imaging

Abstract

A fast full-wave numerical approach was developed for simulating high-field multi-channel radio-frequency (RF) receive coil arrays in magnetic resonance imaging (MRI). To improve the efficiency, the impedance matrix was compressed by a multilevel Adaptive Cross Approximation (ACA) method. Furthermore, careful organization of multiple coil simulations was applied so that the impedance matrix associated with biological subjects is constructed and pre-conditioned only once. Numerical examples demonstrate the efficacy of the proposed approach for RF coil simulations.

1. Introduction

Multi-channel RF receive arrays have become the method of choice for signal reception in high-field (≥ 3 Tesla) MRI (Roemer et al 1990, Pruessmann et al 1999). The signal-to-noise ratio (SNR) is improved by using small elements of higher sensitivity. The coverage of a large coil is retained by optimally combining the sensitivity profile of small coils. The performance of RF coil arrays depends on both coil/subject and inter-element electromagnetic interactions (electric and magnetic coupling). Coil/subject interactions determine the tuning and the sensitivity profile of each coil element. Coil coupling determines the noise correlation and therefore the combined sensitivity map.

Full-wave simulations are effective to study these electromagnetic interactions (Jin 1999). On the one hand, they enable performance improvement for a general population when designing coils. On the other hand, subject-specific simulations are often desired when interpreting high-field images. Thus one needs fast simulation speed to examine a large number of different coils with a fixed generic subject model, or to simulate a fixed coil array with a subject-specific model.

In the past, several full-wave methods have been applied to RF coil simulations. These include the finite-difference time-domain (FDTD) method (Taflove and Hagness 2005, Zhao et al 2002), the finite-element method (FEM)(Silvester and Ferrari 1996, Petropoulos et al 1993), and the integral-equation (IE) methods (Chew et al 2000, Wang and Duyn 2006). The FDTD is simple to implement but lacks geometric modeling accuracy of RF coils. The FEM is superior at geometric modeling, but is computationally expensive and requires non-trivial three-dimensional (3D) unstructured mesh generation (George 1991). The IE method has comparable geometrical modeling capability as the FEM. It is more accurate (no dispersion errors as in the FDTD and the FEM) and does not require high-quality 3D meshes. Hybrid approaches that combine the advantages of different methods have also been proposed (Wang and Duyn 2008). However, to the authors’ best knowledge, no methods have been developed specifically for improving the speed of simulating RF coil arrays.

In this article, we present a fast full-wave simulation strategy based on the surface integral-equation (SIE) method (Chew et al 2000, Wang and Duyn 2006). Efficiency is improved by applying two approaches. First of all, a multilevel ACA method was applied to compress the impedance matrix (Bebendorf 2000, Zhao et al 2005). This results in less floating-point operations (FLOPs) when iteratively solving a linear system of equations (Saad 2003). Secondly, simulations of multiple coil elements were organized so that the impedance matrix representing biological subjects is constructed only once. This reduces the CPU time because matrix fill-in often takes longer than iterative solving the matrix. Results of a 7 Tesla 32-channel array demonstrates a 6.6-fold speed-up with 1.68% decrease of numerical accuracy.

2. Surface Integral-Equation Method

The SIE method in this article applies the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulation to model dielectric bodies and the Electrical Field Integral Equation (EFIE) to model conductors (Chew et al 2000, Wang and Duyn 2006). For simplicity, let us consider one dielectric body. The computational domain is divided into interior $\left({ϵ}_i,{\mu }_i\right)$ and exterior regions $\left({ϵ}_e,{\mu }_e\right)$. In each region, the total field is the superimposition of the incident and the scattering fields.

$${\overrightarrow{E}}_{(i,e)}={\overrightarrow{E}}_{(i,e)}^{inc}+{\overrightarrow{E}}_{(i,e)}^s$$ (1)

$${\overrightarrow{H}}_{(i,e)}={\overrightarrow{H}}_{(i,e)}^{inc}+{\overrightarrow{H}}_{(i,e)}^s$$ (2)

The scattering field generated by surface electric current $(\overrightarrow{J})$ and magnetic current $(\overrightarrow{M})$ can be written as

$${\overrightarrow{E}}_{(i,e)}^s=\frac{1}{{ϵ}_{(i,e)}}{ℒ}_{(i,e)}(\overrightarrow{J})-{𝒦}_{(i,e)}(\overrightarrow{M})$$ (3)

$${\overrightarrow{H}}_{(i,e)}^s={𝒦}_{(i,e)}(\overrightarrow{J})+\frac{1}{{\mu }_{(i,e)}}{ℒ}_{(i,e)}(\overrightarrow{M})$$ (4)

According to surface equivalent theorem (Jackson 1999), ${\overrightarrow{E}}_i^s$ and ${\overrightarrow{H}}_i^s$ are equal to those generated by electric and magnetic currents on the interior phantom surface, i.e., ${\overrightarrow{J}}_i^d$ and ${\overrightarrow{M}}_i^d$, in an infinite homogeneous region filled with ${ϵ}_i$ and ${\mu }_i$. The same applies to ${\overrightarrow{E}}_e^s$ and ${\overrightarrow{H}}_e^s$, which are equal to those generated by electric and magnetic currents on the exterior phantom surface, i.e., ${\overrightarrow{J}}_e^d$ and ${\overrightarrow{M}}_e^d$, and conduction currents ${\overrightarrow{J}}^c$ on coil surface in an infinite homogeneous region filled with ${ϵ}_e$ and ${\mu }_e$. In each medium, operators $ℒ$ and $𝒦$ are written as

$${ℒ}_{(i,e)}(\overrightarrow{F})=\frac{\left(\nabla \nabla \cdot +k_{(i,e)}^2\right){\int }_{S^{\prime }}{G}_{(i,e)}(\overrightarrow{r},{\overrightarrow{r}}^{\prime })\overrightarrow{F}d{S}^{\prime }}{j\omega }$$ (5)

$${𝒦}_{(i,e)}(\overrightarrow{F})=\nabla \times \underset{S^{\prime }}{\int }{G}_{(i,e)}(\overrightarrow{r},{\overrightarrow{r}}^{\prime })\overrightarrow{F}d{S}^{\prime }$$ (6)

where $k_{(i,e)}=\omega \sqrt{{\mu }_{(i,e)}{ϵ}_{(i,e)}},\overrightarrow{F}$ can be either $\overrightarrow{J}$ or $\overrightarrow{M}$, and $G\left(\overrightarrow{r},{\overrightarrow{r}}^{\prime }\right)$ is Green’s function

$$G_{(i,e)}(\overrightarrow{r},{\overrightarrow{r}}^{\prime })=\frac{e^{-j{k}_{(i,e)}|\overrightarrow{r}-{\overrightarrow{r}}^{\prime }|}}{4\pi |\overrightarrow{r}-{\overrightarrow{r}}^{\prime }|}$$ (7)

In the above, $j=\sqrt{-1}$ and we adopted the $e^{j\omega t}$ convention.

In the PMCHWT formulation, the interior field is coupled to the the exterior field by enforcing field continuity on phantom surfaces, i.e., ${\overrightarrow{J}}_i^d=-{\overrightarrow{J}}_e^d$ and ${\overrightarrow{M}}_i^d=-{\overrightarrow{M}}_e^d$. The coupled problem is solved by applying Galerkin’s method with a set of basis and testing functions. This yields the following so-called impedance matrix

$$\left[\begin{array}{ccc}{Z}_{11}& Z_{12}& Z_{13}\\ -Z_{12}^T& Z_{22}& Z_{23}\\ Z_{13}^T& -Z_{23}^T& Z_{33}\end{array}\right]\left[\begin{array}{c}{\overrightarrow{J}}^d\\ {\overrightarrow{M}}^d\\ {\overrightarrow{J}}^c\end{array}\right]=-\left[\begin{array}{c}{\overrightarrow{b}}_1\\ {\overrightarrow{b}}_2\\ {\overrightarrow{b}}_3\end{array}\right]$$ (8)

where ${\overrightarrow{J}}^d$ and ${\overrightarrow{M}}^d$ are unknown electric and magnetic currents on phantom surface respectively. Each sub-matrix is given by

$$Z_{11,ij}={\overrightarrow{\alpha }}_i\cdot \left[\frac{1}{{ϵ}_i}{ℒ}_i({\overrightarrow{\alpha }}_j)+\frac{1}{{ϵ}_e}{ℒ}_e({\overrightarrow{\alpha }}_j)\right]$$ (9)

$$Z_{12,ij}=-{\overrightarrow{\alpha }}_i\cdot [{𝒦}_i({\overrightarrow{\alpha }}_j)+{𝒦}_e({\overrightarrow{\alpha }}_j)]$$ (10)

$$Z_{13,ij}={\overrightarrow{\alpha }}_i\cdot \frac{1}{{ϵ}_e}{ℒ}_e({\overrightarrow{\alpha }}_j)$$ (11)

$$Z_{22,ij}={\overrightarrow{\alpha }}_i\cdot \left[\frac{1}{{\mu }_i}{ℒ}_i({\overrightarrow{\alpha }}_j)+\frac{1}{{\mu }_e}{ℒ}_e({\overrightarrow{\alpha }}_j)\right]$$ (12)

$$Z_{23,ij}={\overrightarrow{\alpha }}_i\cdot {𝒦}_e({\overrightarrow{\alpha }}_j)$$ (13)

$$Z_{33,ij}={\overrightarrow{\alpha }}_i\cdot \frac{1}{{ϵ}_e}{ℒ}_e({\overrightarrow{\alpha }}_j)$$ (14)

One of the advantages of the SIE method is that only surfaces are meshed. Compared to volume methods such as the FEM, this results in much fewer unknowns to be solved. However, the impedance matrix in Eq. (8) is dense because it represents mutual interaction between currents. The computational costs of storing and inverting dense matrices are typically one polynomial order higher than sparse matrices in the FEM (Golub and Van Loan 1996, Saad 2003). To reduce memory cost, one approach is to apply symmetry. Although the global impedance matrix is asymmetric, a certain part of the impedance matrix is either symmetric or skew-symmetric as denoted in Eq. (8). Furthermore, $\left[Z_{11}\right],\left[Z_{12}\right]$ and $\left[Z_{22}\right]$ are all symmetric sub-blocks. In most RF coil simulations, the impedance matrix is mainly associated with biological subjects. In this scenario, about $3/8$ of the entire impedance matrix needs actual storage.

3. Fast Simulation Scheme

3.1. The Multilevel ACA Method

The multilevel ACA method is a way to reduce both memory and CPU time in iterative matrix solvers (Bebendorf 2000, Zhao et al 2005). In order to understand the principle, we first take a look at an example of the so-called outer-product format (Golub and Van Loan 1996)

$$\left[\begin{array}{cccc}{A}_{11}& A_{12}& \dots & A_{1N}\\ A_{21}& & \dots & ⋮\\ ⋮& ⋮& \ddots & ⋮\\ A_{N1}& \dots & \dots & A_{NN}\end{array}\right]=\left[\begin{array}{c}{\overrightarrow{u}}_1\\ ⋮\\ {\overrightarrow{u}}_N\end{array}\right][{\overrightarrow{v}}_1,\dots ,{\overrightarrow{v}}_N]$$ (15)

The N-by-N matrix $[A]$ can be written exactly as the product of a N-by-1 matrix (or a vector) $\overrightarrow{u}$ and a 1-by-N matrix (or the transpose of a vector) $\overrightarrow{v}$. This equivalent format favors the computation of matrix-vector products, e.g., $[A]\cdot \overrightarrow{x}$ where $\overrightarrow{x}$ is a N-by-1 vector, which are the dominant operation in an iterative matrix solver (Saad 2003). Using the conventional matrix format of $[A]$, it requires $N^2$ storage units and $N^2$ FLOPs for one matrix-vector multiplication. On the contrary, the equivalent outer-product format only requires $2N$ storage units and one vector inner product and one scalar-vector product, a total of $3N$ FLOPs for one matrix-vector multiplication. When $N$ is large, the outer-product format reduces computational costs by one polynomial order. For this reason, the outer-product format can be viewed as matrix compression.

The rank of the matrix $[A]$ in Eq.(15) is one according to definition. In general, a matrix can be written as the sum of $r\le N$ sets of outer products, i.e.,

$$[A]=\sum _{i=1}^r{\overrightarrow{u}}_i{\overrightarrow{v}}_i^T=[U]{[V]}^T$$ (16)

To determine the rank $r$ of $[A]$, one can apply the singular value decomposition (SVD), column-pivoted QR or column-pivoted LU (Golub and Van Loan 1996). The actual computational efficiency of matrix compression via outer-products depends on rank $r$. Outer-products are advantageous in matrix-vector multiplications if $r<N/2$.

The ACA method is an on-the-fly approach to compress the impedance matrix of the SIE method via outer-products. Before introducing the ACA, one needs to make sure that the impedance matrix can be compressed. According to Eq. (7), the magnitude of Green’s function varies by $1/r$. The spatial profile is smooth in far-field and can be interpolated by low-order polynomials. This interpolatability indicates a degenerated degree of linear independence in the part of the impedance matrix that represents far-field mutual interactions (far-field blocks). As rank $r$ serves as a measure of the degree of linear independence, far-field blocks are rank-deficient and can be compressed via outer-products. Or in other words, compression should only be selectively applied to far-field blocks (Rokhlin 1985).

In the ACA method, identification of far-field interactions starts from dividing a computational domain into multiple sub-domains. This is illustrated in Fig. 1. If the radii of the circumscribed spheres of two sub-domains and the geometrical center-to-center distance $d_g$ satisfy the following criteria (Bebendorf 2000)

$$d_g\ge q\cdot (r_1+r_2)$$ (17)

where $0.5\le q\le 2.0$ is the so-called admissible ratio, the mutual interactions between the two sub-domains are determined to be weak and the associated sub-matrix should be compressed.

Figure 1.

The 32-channel receive coil array (a) and the sub-domains of the decomposed head model (b).

Matrix compression can take place once the entire impedance matrix is known. Alternatively, a more CPU-efficient approach can be derived from the column-pivoted Crout algorithm. The Crout algorithm is a variant of the LU decomposition but with a specific memory accessing sequence. It decomposes a matrix on-the-fly and has been used in out-of-the-core matrix decomposition (Duff et al 1986). Upon finishing the LU decomposition, one matrix is factorized into the product of two matrices, which resembles the format in Eq. (16). Meanwhile, column-pivoted LU can be used to determine matrix ranks (Golub and Van Loan 1996). Thus column-pivoted Crout is a rank-revealing on-the-fly matrix factorization method that we need.

The ACA method is essentially the column-pivoted Crout algorithm without actually performing pivoting^‡. The detail of the ACA algorithm is listed here for completeness (Bebendorf 2000)

The ACA algorithm

$k=1,\mathit{and}{I}_k=1$

$\mathit{while}(\mathrm{\Delta }\ge \tau )$

$\left\{R\left(I_k,:\right)=A\left(I_k,:\right)-\sum _{p=1}^{k-1}{\overrightarrow{u}}_p\left(I_k\right){\overrightarrow{v}}_p^T\right.;$

$J_k:=\left|R\left(I_k,J_k\right)\right|=\underset{j}{\mathrm{m}\mathrm{a}\mathrm{x}}\left|R\left(I_k,j\right)\right|;$

${\overrightarrow{v}}_k=R\left(I_k,:\right)/R\left(I_k,J_k\right);$

${\overrightarrow{u}}_k=A\left(:,J_k\right)-\sum _{p=1}^{k-1}{\overrightarrow{v}}_p\left(J_k\right){\overrightarrow{u}}_p;$

$I_{k+1}:=\left|{\overrightarrow{u}}_k\left(I_{k+1}\right)\right|=\underset{i}{\mathrm{m}\mathrm{a}\mathrm{x}}\left|{\overrightarrow{u}}_k(i)\right|;$

$\parallel \tilde{A}{\parallel }_F^2=\parallel \tilde{A}{\parallel }_F^2+2\sum _{p=1}^{k-1}\left({\overrightarrow{u}}_p\cdot {\overrightarrow{u}}_k^T\right)\left({\overrightarrow{v}}_p\cdot {\overrightarrow{v}}_k^T\right)+{∥{\overrightarrow{u}}_k∥}_2^2{∥{\overrightarrow{v}}_k∥}_2^2$

$\mathrm{\Delta }=\frac{{∥{\overrightarrow{u}}_k∥}_2{∥{\overrightarrow{v}}_k∥}_2}{\parallel \tilde{A}{\parallel }_F}$

$k=k+1;\}$

In the context of electromagnetic simulations, ACA can be physically interpreted as an alternating and iterative way of finding the most significant transmitters and receivers in two groups of antennas (Zhao et al 2005). The strongest receiver is first selected after examining the interactions between an arbitrary transmitter and all receivers. By reciprocity, the strongest receiver then transmits and the strongest transmitter is determined by examining the interactions. Then the next strongest receiver and transmitters are found similarly and the algorithm proceeds.

According to antenna theory, the minimum rank of far-field interactions is two, which correspond to the two orthogonal linear polarizations in far-field (Heldring et al 2007). The ACA determines the actual rank by examining the convergency of the Frobenius norm $\parallel A{\parallel }_F$ (Golub and Van Loan 1996),

$$\parallel A{\parallel }_F=\sum _{i=1}^r{\sigma }_i^2$$ (18)

where ${\sigma }_i$ denotes a singular value. If the relative change of Frobenius norm converges to a prescribed tolerance $\tau$, matrix compression stops. Thus $\tau$ indicates the amount of uncertainty introduced by the ACA to the impedance matrix. As shown in (Wang and Duyn 2010), $\tau$ also indicates the additional solution error introduced by matrix compression. Thus it provides a practical tool to control simulation errors.

The compressed $\left[Z_{11}\right]$ (in Eq. (8)) of the impedance matrix associated with the human head (in Fig. 1) is illustrated in Fig. 2, where dark blocks represent incompressible interactions and light-colored blocks represent compressed ones with the associated rank denoted inside. Instead of compressing far-field interactions between individual sub-domains, one can combine sub-domains into larger groups and reduce the total number of compression required. This is a multilevel scheme and its detail can be found in (Wang and Duyn 2010). The resulting rank map of this multilevel ACA is also shown in Fig. 2, where compression on a larger scale can be identified clearly.

Figure 2.

Rank map of the single-level ACA method (a) and the multilevel ACA method (b).

3.2. Multi-Coil Simulations

When simulating different RF coil elements, the phantom does not change its shape and constitution. Only coils and coil/subject interactions change. Thus we can generate matrix blocks $\left[Z_{11}\right],\left[Z_{12}\right]$ and $\left[Z_{22}\right]$ (in Eq. (8)) that are associate with subjects once and apply them repeatedly. When simulating a single element, the CPU time of matrix fill-in is comparable or even longer than that of solving the matrix iteratively (see examples below). Therefore, this scheme reduces the CPU time of multiple coil simulations by a factor of two or more in practice.

4. Numerical Example

We simulated a 32-channel receive coil array for human brain proton imaging at 7.0 Tesla (300 MHz). This array was designed in cooperation with, and built by, Nova Medical Inc. (Wilmington, MA, USA). It comprises 32 small loop coils modeled by quadrilateral elements as shown in Fig. 1. On average, between 20 to 30 roof-top basis functions were applied to model each element (Chew et al 2000). The head model was adapted from the specific anthropomorphic mannequin (SAM) model, which was published by the IEEE standard 1528-2003, by scaling it in the two transversal directions according to the size of the actual subject being scanned. The model has a relative permittivity of ${ϵ}_r=43$ and a conductivity of $\sigma =0.41\mathrm{S}/\mathrm{m}$. These values correspond to those of the white matter at 300 MHz (Gabriel 1996).

Both the combined SNR map and the geometric noise amplification factor (g-Factor) were simulated (Pruessmann et al 1999). This required 32 simulations to calculate the $B_1^{+}$ profiles and the noise correlation matrix (Wang and Duyn 2006). The generalized minimal residual method (GMRES) method with a successive over relaxation (SOR) pre-conditioner was applied (Saad 2003). In order to construct the pre-conditioner, a sparse representation of the impedance matrix was extracted at first by retaining all elements in a row that are larger than 0.5% of the largest element of that row. The stopping criteria of the GMRES method was 0.001. In the ACA method, $\tau$ was 0.01. The computer program was developed in $\mathrm{C}++$ with double precision numbers. Simulations were performed on a 3.0 GHz AMD Opteron 256 processor.

4.1. Evaluation

The proposed approach was evaluated at first by comparing its results with the conventional approach without matrix compression (Wang and Duyn 2006). The SNR map was calculated by combining the simulated $B_1^{+}$ profiles with a noise correlation matrix obtained by adding coil noise to simulated noise correlation. The amount of noise added corresponded to a measured 2:1 unloaded-to-loaded Q ratio. Note that if one knows the detail of the electronics on coils, accurate simulation of $Q$ factors can be performed (Wang et al 2007).

Table I lists the computational costs of simulating one element by using different approaches. Both matrix fill-in and iterative solution time were reduced when applying the multilevel ACA method. Meanwhile, memory consumption is reduced by a factor of 2.6. Note that matrix symmetry as mentioned in Section 2 was applied in the conventional approach as well. Otherwise the memory cost of the conventional approach will be 2.67 times higher.

Table 1.

Comparison of computational costs, where $t_1$ is fill-in time, $t_2$ is pre-conditioning time and $t_3$ is the total iterative solution time.

	$t_1(\mathbf{s})$	$t_2(\mathbf{s})$	$t_3(\mathbf{s})$	Memory (MB)
Conventional approach	14.8	1.7	16	296
Single-level ACA	10	2.7	8.1	138
Multi-level ACA	8.6	2.5	6	111

On the other hand, matrix pre-conditioning time increased by a factor of 1.5. This is because each row of the impedance matrix needs to be reconstructed in the multilevel ACA method for pre-conditioning. Since the impedance matrix of the phantom is generated and pre-conditioned just once, this additional CPU overhead is expected to have an insignificant impact.

The simulated combined SNR maps are compared in Fig. 3. As can be seen, the results are nearly identical. Accuracy was further evaluated by using a 2-norm error defined by

$$ϵ=\sum \frac{{\Vert SN{R}_{ACA}-SN{R}_{\mathit{conventional}}\Vert }_2}{{\Vert SN{R}_{\mathit{conventional}}\Vert }_2}$$ (19)

where the sum was taken over all pixels. As shown in Table II, the relative error introduced by the multilevel ACA method is 1.68%. This corresponds well to the ACA tolerance of $\tau =0.01$, which indicates 1% uncertainty in the compressed impedance matrix.

Figure 3.

Combined SNR maps computed by conventional approach (a) and the proposed approach (b).

Table 2.

Comparison of the total simulation time of 32 elements and the relative error.

	Total CPU (s)	$ϵ$
Conventional approach	1270	(N/A)
Single-level ACA	238	1.61%
Multi-level ACA	192	1.68%

Table II also compares the total CPU time of simulating all 32 coil elements. The multilevel ACA finished in 192 seconds, or a 6.6-fold speed-up. Note that different coil elements may require different number of iterations of the GMRES solver (between 20 to 39 in this example). Typically, if an element is closer to the phantom, the coil/subject interaction is stronger and more iterations are expected.

We also mention that compared to the FDTD method, the conventional approach is already two orders of magnitude faster if similar modeling accuracy is required (Wang and Duyn 2006). The inefficiency of the FDTD is mainly due to its inferior geometry modeling capability of RF coils.

4.2. Experiment Verification

Experiments were performed on a General Electric (Waukesha, WI, USA) 7.0 Tesla whole body scanner. Axial multi-slice data were acquired using a gradient-echo sequence with a $256\times 192{\mathrm{m}\mathrm{m}}^2$ field-of-view (FOV) and 2-mm spatial resolution. The repetition time (TR) is 3.0 s and the echo time (TE) is 9.6 ms. The thickness and spacing of 128 axial slices were 1 mm. Image reconstruction and computation of the noise amplification factor (g-Factor) maps (for rate 2, 3 and 4 in the left-right direction) were performed as described in (deZwart et al 2002). The brain mask used to compute the coil sensitivity maps and the g-Factor maps were extrapolated results over 4 mm. Furthermore, the FOV was truncated to $230\times 156{\mathrm{m}\mathrm{m}}^2$ to narrowly encompass the head.

Results on a single axial slice 8-cm below the top of the head was used to compare simulation and measured results. Since g-Factor maps are sensitive to masks and the definition of the FOV, the brain mask from the acquired data was used to compute the simulated g-Factor maps as well. FOV of the simulated data was also reduced to the same $230\times 156{\mathrm{m}\mathrm{m}}^2$. The simulated and measured g-Factor maps at acceleration rate four are shown in Figs. 4, where same gray scale ranging from 1.0 to 3.0 was applied. A close match of simulation and measured results is observed. Further comparison of the average g-Factors results on the same slice are listed in Table III, where it shows a maximum difference of about 5.5%. This discrepancy may result from the approximate head model, residual coil mutual coupling in experiment and other experiential conditions not considered in simulations.

Figure 4.

The simulated combined SNR map and the masking of simulation results with the brain profile (a), the simulated (b) and the measured (c) g-Factor maps at acceleration rate four.

Table 3.

Comparison of average g-Factors at different acceleration rates.

Acceleration rate	2	3	4
Simulated g-Factor	1.01	1.14	1.73
Measured g-Factor	1.02	1.20	1.83

5. Conclusions and Discussions

We presented a full-wave approach for fast simulations of high-field MRI receive coil arrays. In order to improve efficiency, the impedance matrix was compressed by applying a multilevel ACA method. Furthermore, multiple coil simulations were organized so that the impedance matrix associated with biological subjects is constructed only once. Numerical examples demonstrated the validity and the effectiveness of the proposed approach. It was shown that on a single CPU, 32 full-wave simulations of a human head coil array finished within a few minutes. As demonstrated separately in (Wang et al 2010), this method provides online simulation results with realistic subject models generated by an in-house image processing program.

Though examples in this article show that a single-medium phantom yields good agreement between simulation and experimental results of receive coil arrays, inhomogeneous phantoms are useful in certain applications, especially in the prediction of the Specific Absorption Rate (SAR) of transmit coils. Incorporating tissue inhomogeneity can be accomplished in several ways. For instance, the surface of each organ can be meshed and the proposed SIE method can be carried out straightforwardly. Alternatively, different types of tissues can be modeled by finite-difference or finite-element methods, and the proposed SIE method provides a boundary condition such as the one in the finite-element boundary-integral (FEBI) method (Chew et al 2000). The first approach has been implemented and verified by the electromagnetic scattering of a double-layer sphere. We plan to extend the proposed method to simulate inhomogeneous phantoms in the future.

As shown in (Wang and Duyn 2010), the computational complexity of the multilevel ACA method applied in this article scales by $𝒪\left(N^{1.28}\mathit{log}N\right)$. Several approaches can be applied to further improve simulation speed when the number of unknowns becomes very large, e.g., modeling the whole body. For instance, advanced matrix solvers such as the one proposed in (Lucente et al 2008) can be applied to reduce matrix inversion time. On the other hand, the application of multi-processor, multi-core or Graphic Processing Unit (GPU) is expected to reduce simulation time significantly. As demonstrated in (Lezar and Davidson 2010), the application of GPUs can increase the speed by an order of magnitude in finite-element simulations. Since the ACA method divides computational domains into packets, it is parallelization-friendly. We focus on algorithm development in this article and the hardware implementation on different parallel platforms will be studied in the future.

Finally, we noticed the actual number of iterations depends on the relative position of RF coils with respect to phantom. Thus developing robust matrix inversion techniques, which are less sensitive to the aforementioned factor and other potential performance-degrading factors, such as coil tuning and the Q-factor, is important when considering a broad spectrum of RF coils. This topic will ensue in our future research as well.