Multiple-Input Multiple-Output (MIMO) MRI: Combining Parallel Excitation and Parallel Reception for Enhanced Imaging

Abstract

Magnetic resonance imaging (MRI) plays a critical role in visualizing the structure and functions of the human body. In order to accelerate imaging time and improve image quality, radio-frequency (RF) coil receive arrays are commonly employed to acquire the magnetic resonance (MR) signal. Similarly, multiple transmit coils have been shown to accelerate and refine RF excitation. In this work, we investigate the optimization of total imaging time and image accuracy when considering both the transmit and receive coil arrays; we term this strategy multiple-input multiple-output (MIMO) MRI. Our RF pulse design method is modeled by minimizing the excitation errors while simultaneously maximizing the signal-to-noise ratio (SNR) of the reconstructed MR image. It further allows a key tradeoff between the two optimizers. Additionally, multiple acceleration factors, varying numbers of receive coils used, maximum excitation error tolerance, and different excitation patterns are simulated and analyzed in this model. For a given excitation pattern, our method is shown to improve the SNR by 18–130% under certain acceleration schemes, as compared to conventional parallel transmission methods, while simultaneously controlling the excitation error within a desired scope (NRMSE ≤ 0.12).

I. INTRODUCTION

Over the last 30 years, clinical applications of magnetic resonance imaging (MRI) systems have become widespread, and research on multi-coil MRI has already yielded commercial deployment in clinical settings. The most popular multi-coil techniques used for MR image reception are forms of parallel imaging [1]. Parallel imaging is a robust method used to accelerate the acquisition of MRI data and has resulted in many new MRI applications [2]-[4]. Other research has addressed MRI systems using multiple independent transmitter coils under the theory of parallel transmission (pTX) [5]-[7]. Transmission of radio-frequency (RF) pulses with transmit coil arrays has been shown to reduce RF energy deposition, facilitating compliance with specific absorption rate (SAR) safety limits [8], [9].

Multi-coil RF pulse design enables the acceleration of multidimensional excitation pulses by making use of multiple independently transmit coils [10]. To date, the majority of pulse designs assume operation in the small-tip-angle regime [11]. Pioneering pulse design methods in small-tip-angle regime were introduced by Katscher et al. [5], Zhu [9], and Grissom et al. [12]. Katscher et al. [5] proposed Transmit SENSE, which introduced parallel transmission with arbitrarily shaped transmit coils. Transmit SENSE can be applied to general cases of three dimensional imaging and arbitrary k-space trajectories. In contrast, the method used by Zhu [9] uses spatially selective excitation, allowing for both multidimensional excitation acceleration and SAR control during parallel excitation.

Despite extensive research on multi-coil transmission and/or reception, there have been only minor advancements in multi-coil processing at both sides of the scanner. Transceive multi-coil design for both transmission and reception has been the focus of several papers [13]−[16]. These studies suggest that proper transceiver coil design can effectively enable transmit and/or receive techniques while maintaining a high signal-to-noise ratio (SNR). Other research has proposed a mean-squared error (MSE)-based excitation pattern design for parallel transmit and receive MR image reconstruction [17].

In this paper, we address multi-coil RF pulse designs using multiple transceivers. We refer to this technique as multiple-input multiple-output (MIMO) MRI. This research contributes to the investigation of transmit-side multi-coil RF pulse design by jointly considering the operation of multi-coil transmitters and receivers.

Specifically, we propose an alternative RF pulse design method for MIMO MRI, similar to the spatial domain method introduced by Grissom, et al. [12], [18]. Our method is specifically designed for the scenario of multiple transceive coils using an SNR cost function. Traditional multi-coil RF design approaches allow for spatially-varying excitation error weighting, such as region of interest (ROI) specification, and also have adaptability to the main magnetic field inhomogeneity. This MIMO MRI model provides a theoretical method to estimate and maximize the SNR at the receive side, while maintaining the excitation error in a controllable/manageable range. This paper outlines the theoretical framework of the MIMO MRI model, using numerical simulations for verification. Preliminary research on this method using an echo planar imaging trajectory was presented in an IEEE conference proceeding [19].

This paper is organized as follows: Section II reviews the fundamental MRI theory and introduces the MIMO MRI model setup; Section III outlines the novel RF pulse design method; several computer simulations are employed to verify this theoretical study in Section IV; discussions and conclusions are presented in Sections V and VI.

II. THEORY

We consider an MRI system with a transceive coil array, where a number of coil elements are used for signal transmission, N_t, and another number of elements are used for signal reception, N_r. We are explicitly interested in systems optimized for the case where min(N_t, N_r) > 1, but results still apply when min(N_t, N_r) = 1. All derivations in this section are made using the small-tip-angle approximation.

A. Parallel Excitation with Multi-Coils

Parallel excitation uses multiple coils with each transmitting a different RF pulse to excite spins over a certain bandwidth, yielding visible MR images at the receive side. During the spin excitation, the multi-dimensional RF pulse follows a certain k-space trajectory to enable imaging.

MR signal intensities are proportional to the transverse magnetization and dependent on the user-defined excitation parameters, e.g., repetition time (T_R), echo time (T_E), and slice selection. Each coil transmits a complex RF waveform with the p^th coil sending b_{1, p}(t). These RF pulses generate the transverse magnetization M(r) (e.g., $\mathit{r}={[x,y]}^{\top }$ in 2D imaging), as determined from the Bloch equation. The transverse magnetization can be expressed as

$$M(\mathit{r})=\sum _{p=1}^{N_t}{S}_p(\mathit{r})M_p(\mathit{r}),$$ (1)

where S_p(r) is the transmit sensitivity map of the p^th transmit coil at position r, and M_p(r) is the transverse plane magnetization produced by the p^th transmit waveform.

Similiar to methods outlined in [11] and [12],

$$M_p(\mathit{r})=i\text{γ}{M}_0{\int }_0^T{b}_{1,p}(t)e^{i\text{γ}\mathrm{\Delta }{B}_0(\mathit{r})(T-t)}{e}^{i\mathit{r}\cdot \mathit{k}(t)}dt,$$ (2)

where γ is the gyromagnetic ratio, M₀ is the magnetization at the equilibrium state, T is the pulse length, k(t) represents the two-dimensional k-space trajectory $\left({[k_x(t),k_y(t)]}^{\top }\right)$, and $e^{i\text{γ}\mathrm{\Delta }{B}_0(\mathit{r})(T-t)}$ represents the phase accrued due to the static field deviation ΔB₀(r). We normally assume the static field is uniform over the magnetization domain.

Similar to approaches in [12] and [20], we discretize the problem. We assume that all position vectors, r, are restricted to a finite set, $\mathcal{R}$ = {r₁,...,r_R}, and time values are represented by t ∈ {t₁, t₂,…,t_K}, where t_k = t₁ + (k − 1)Δt for a sample time, Δt. Using these assumptions, we define

$${\mathit{m}}_p={\left[M_p\left({\mathit{r}}_1\right)M_p\left({\mathit{r}}_2\right)\cdots M_p\left({\mathit{r}}_R\right)\right]}^{\top }$$ (3)

$$\text{=}\mathit{A}{\mathit{b}}_{\text{1},p},$$ (4)

where $A\in {ℂ}^{R\times K}$ has entry (j, k) defined as

$$a_{jk}=i\text{γ}{M}_0\mathrm{\Delta }t{e}^{i\text{γ}\mathrm{\Delta }{B}_0\left({\mathit{r}}_j\right)\left(T-t_k\right)}{e}^{i{\mathit{r}}_j\cdot \mathit{k}\left(t_k\right)},$$ (5)

and ${\mathit{b}}_{1,p}={\left[b_{1,p}\left(t_1\right)\cdots b_{1,p}\left(t_K\right)\right]}^{\top }$ . The transverse magnetization ($\mathit{m}\in {ℂ}^{R\times 1}$) can then be expressed as

$$\mathit{m}=\sum _{p=1}^{N_t}{\tilde{\mathit{S}}}_p{\mathit{m}}_p=\sum _{p=1}^{N_t}{\tilde{\mathit{S}}}_p\mathit{A}{\mathit{b}}_{1,p}={\tilde{\mathit{S}}}_A\mathcal{B},$$ (6)

where ${\tilde{\mathit{S}}}_p=diag\left({\mathit{S}}_p\left({\mathit{r}}_1\right),{\mathit{S}}_p\left({\mathit{r}}_2\right),\dots {\mathit{S}}_p\left({\mathit{r}}_{\mathcal{R}}\right)\right),{\tilde{\mathit{S}}}_A=\left[\begin{array}{lll}{\tilde{\mathit{S}}}_1\mathit{A}\hfill & \cdots \hfill & {\tilde{\mathit{S}}}_{N_t}\mathit{A}\hfill \end{array}\right]$.

Using this discrete version, we can analyze the transverse magnetization by optimizing the matrix RF pulse

$$\mathcal{B}=\left[\begin{array}{c}{\mathit{b}}_{1,1}\\ ⋮\\ {\mathit{b}}_{1,N_t}\end{array}\right].$$ (7)

Thus, the problem lies in the construction of the RF pulses in both space and time.

Previously, researchers have approached the design of $\mathcal{B}$ to excite a given pattern m_ref [21], or to mitigate undersampling artifacts [5]. To meet a reference magnetization, the problem becomes one of solving the optimization (‘H’ is the conjugate transpose operator)

$${\mathcal{B}}_{ref}=\underset{\mathcal{B}}{\mathrm{argmin}}\left({\Vert {\tilde{\mathit{S}}}_A\mathcal{B}-{\mathit{m}}_{ref}\Vert }^2+\alpha {\mathcal{B}}^{\text{H}}\mathcal{B}\right).$$ (8)

Integrated RF power is controlled via a Tikhonov regularization term α$\mathcal{B}$^H$\mathcal{B}$, where α is the tuning parameter. This magnetization pattern matching problem can be solved either by the pseudoinverse or the conjugate gradient (CG) method. Once $\mathcal{B}$_ref is obtained, all designed RF pulses can be obtained.

B. Parallel Reception with Multi-Coils

Parallel reception in an MRI system allows data to be separately and simultaneously acquired from multiple coils. Fast parallel reception methods are referenced as parallel imaging techniques, such as SiMultaneous Acquisition of Spatial Harmonics (SMASH) [2], SENSitivity-Encoding (SENSE) [3], and GeneRalized Auto-calibrating Partially Parallel Acquisition (GRAPPA) [4].

The ideal image-combining procedure in the receive coil array is shown in Fig. 1. This image combination method is similar to the sum-of-squares (SoS) method [22]. However, the SoS method usually causes the reconstructed image to appear dark at locations further away from the receive coils [23]. Hence, receiver weights are applied to avoid non-uniformity of the image intensity. In addition, each individual channel is digitized so channel combinations can be performed digitally. Various image reconstruction algorithms can be applied to complete the image reconstruction; here, we used SENSE to obtain the electromotive force (emf) output, V.

Fig. 1:

Block diagram of a phased array receiver coil. N_r receive coils with outputs phase shifted and summed through a set of transformers. η_q is the digitized receiver weight of q^th receive coil. V is the complex electromotive force (emf) of the NMR signals received after applying the receiver weights.

Often in the literature, $B_1^{-}$ fields are used as the receive sensitivity maps for SENSE reconstruction, but the framework presented herein distinguishes the difference between the $B_1^{-}$ fields and the receive sensitivity maps. The $B_1^{-}$ fields should not only contain the coil patterns, but also the complex-valued RF waveform. The amplitude of the applied RF pulses contribute to the final strength of the $B_1^{-}$ field. For q^th receive coil, we assume the corresponding receive coil sensitivity map is $S_q^{\prime }(\mathit{r})$. We define the $B_1^{-}$ field, detected by the q^th transceive coil, as

$${\tilde{B}}_{1,q}^{-}(\mathit{r})=S_q^{\prime }(\mathit{r}){\int }_0^T{b}_{1,q}(t)e^{i\text{γ}\mathrm{\Delta }{B}_0(\mathit{r})(T-t)}dt.$$ (9)

The body/coil geometry and the electromagnetic parameters (e.g., permittivity and conductivity), and circular polarization components are considered in the receive sensitivity maps. The distinctive ‘twisting’ asymmetry of the $B_1^{+}$ and $B_1^{-}$ fields are usually observed at high magnetic field strengths [24]-[26], so we use sensitivity maps to account for the distinction. The time integral of b₁(t) is equivalent to a homogenous scaling factor to the $B_1^{-}$ field (assume uniform B₀ field), and it has no interference with image reconstruction [3].

Consider an axial plane 2D MRI, for which the z-component of the gradient waveforms is zero during the readout. If C_q (r) is defined as the 2D MR signal received from the q^th coil in the image domain, it can be represented by [27], [28]

$$C_q(\mathit{r})\approx {\delta }_{z}M(\mathit{r})e^{-T_E/T_2(\mathit{r})}{\tilde{B}}_{1,q}^{-}(\mathit{r})+{\mu }_q(\mathit{r}),$$ (10)

where M is the transverse plane magnetization generated by multiple transmit coils (see (1)), T₂ is the transverse relaxation time constant, δ_z is the slice thickness, μ_q (r) is the noise in q^th channel during the MR signal reception, and we assume μ_q(r) follows a Gaussian distribution.

After applying receiver weights {η_q, q ∈ [1, N_r]}, the effective MR output signal received from each receive channel is (slice thickness and the T₂ weighting coefficients are ignored for simplicity; ‘*’ is the conjugate operator)

$$C_{eff,q}(\mathit{r})={\eta }_q^{*}(\mathit{r})M(\mathit{r}){\tilde{B}}_{1,q}^{-}(\mathit{r})+{\eta }_q^{*}(\mathit{r}){\mu }_q(\mathit{r}).$$ (11)

If the MR signals obtained from the receive coils are {C_eff,q(r),q ∈ [1,N_r]}, then the complex receving voltage after the image combination, V(r), is given as

$$V(\mathit{r})=M(\mathit{r})\sum _{q=1}^{N_r}{\eta }_q^{*}(\mathit{r}){\tilde{B}}_{1,q}^{-}(\mathit{r})+\sum _{q=1}^{N_r}{\eta }_q^{*}(\mathit{r}){\mu }_q(\mathit{r}),$$ (12)

$$=V_{\mathit{\text{desired}}}(\mathit{r})+\sum _{q=1}^{N_r}{\eta }_q^{*}(\mathit{r}){\mu }_q(\mathit{r}).$$ (13)

Specifically, the combined noise is defined as the spatially-variant additive Gaussian noise. If for each location r, the thermal noise across all coils is $\mathit{\mu }(\mathit{r})\in {ℝ}^{N_r\times 1}$, then it should satisfy the following two conditions:

(a) $E[\mathit{\mu }(\mathit{r})]=0$,

(b) $E\left[\mathit{\mu }(\mathit{r}){\mathit{\mu }}^{\text{H}}(\mathit{r})\right]=\mathbf{\Sigma }$, where Σ is the covariance matrix.

The noiseless MRI receive voltage can be described in the matrix form:

$${\mathit{V}}_{desired}(\mathit{r})=\mathit{m}(\mathit{r}){\mathit{\eta }}^{\text{H}}(\mathit{r}){\tilde{\mathit{B}}}_1^{-}(\mathit{r}).$$ (14)

III. METHODS

To date, researchers have implemented parallel excitation and parallel reception in the same MRI clinical machine. However, the optimal use of multi-coil MRI systems remains unknown. This novel MIMO MRI model aims to address this gap.

A. SNR Optimization Method

When using the SENSE method in the MIMO MRI model, two types of noise remain problematic during MRI acquistion: channel noise and thermal fluctuation noise.

There are two types of channel noises that commonly affect the reconstructed images: noise in k-space sample values and noise in the sensitivity data. Sensitivity-related noise can be reduced to a negligible level by smoothing, and the sample data related noise can be minimized by utilizing different optimization methods, e.g. Parallel magnetic resonance imaging with Adaptive Radius (PARS) method [29]. The MIMO MRI model works to minimize channel and thermal noise, as well as maximize the SNR after image combination.

In the MIMO MRI model, we aim to match a desired magnetization pattern while obtaining the maximized SNR of the MRI receive signal. We use N_t transmit coils for RF pulse transmission, and N_r receive coils for image reception. However, we assume that N_r = N_t in this RF pulse design procedure, and we use the coil selection matrix to select the corresponding receive coils for image reconstruction.

The SNR after the image combination can be defined as the ratio of the absolute value of nosisless MRI signal voltage to the standard deviation of noise voltage,

$$SNR(\mathit{r})=\frac{\left|V_{desired}(\mathit{r})\right|}{\sqrt{{\mathrm{\Sigma }}_{total}(\mathit{r})}}.$$ (15)

V_desired contains the complex receive voltage of the selected ROI. For each voxel, the NMR signal voltage is defined as the emf induced from that specific location, which is defined in (14). The covariance matrix Σ_total at each location is defined as

$${\mathrm{\Sigma }}_{total}(\mathit{r})=\sum _{j=1}^{N_r}\sum _{k=1}^{N_r}{\eta }_j^{*}(\mathit{r}){\eta }_k(\mathit{r}){\mathrm{\Sigma }}_{jk}={\mathit{\eta }}^{\text{H}}(\mathit{r})\mathbf{\Sigma }\mathit{\eta }(\mathit{r}),$$ (16)

where Σ is the covariance matrix of μ(r), η_j is the receiver weight of j^th receive coil. Σ_jk represents the correlated (j ≠ k) and uncorrelated (j = k) thermal noise between coils j and k. We assume Σ_jk is the noise covariance matrix induced by additive Gaussian noises in the simulation.

For simplicity, we ignore the constants which remain unchanged from voxel to voxel, and we further define the coil SNR, SNR_c, as

$$\left|SN{R}_c(\mathit{r})\right|=\frac{\left|M(\mathit{r}){\sum }_{q=1}^{N_r}{\eta }_q^{*}(\mathit{r}){\tilde{B}}_{1,q}^{-}(\mathit{r})\right|}{\sqrt{{\sum }_{j=1}^{N_r}{\sum }_{k=1}^{N_r}{\eta }_j^{*}(\mathit{r}){\eta }_k(\mathit{r}){\mathrm{\Sigma }}_{jk}}},$$ (17)

where ${\tilde{B}}_{1,q}^{-}(\mathit{r})$ is defined in (9).

The goal of this MIMO MRI model is to achieve the desired excitation pattern, i.e., match the induced transverse magnetization m (see (6)) to the desired excitation pattern m_ref (see (8)), while simultaneously maximizing the SNR_c in (17) to achieve high SNR in the receive signal. SNR_c is a useful metric for a RF pulse design algorithm since it preserves image quality of the reconstructed MR images and further provides the relative SNR of each voxel in the image. However, it is not convenient to apply the SNR_c matrix within the optimization, as it does not provide a single value to optimize. To combat this problem, we define the global SNR (gSNR) as the mean value of $SN{R}_c^2$in a specific ROI. The definition of gSNR will be given later in this section.

The efficient RF pulse design algorithm in this paper matches the magnetization pattern m_ref and the gSNR term simultaneously. In the optimization, we first take the square of the SNR_c so that (17) becomes

$$SN{R}_c^2(\mathit{r})=\frac{|M(\mathit{r}){|}^2{\sum }_{j=1}^{N_r}{\sum }_{k=1}^{N_r}{\eta }_j^{*}(\mathit{r}){\eta }_k(\mathit{r}){\tilde{B}}_{1,j}^{-}(\mathit{r}){\tilde{B}}_{1,k}^{-}{(\mathit{r})}^{\text{H}}}{{\sum }_{j=1}^{N_r}{\sum }_{k=1}^{N_r}{\eta }_j^{*}(\mathit{r}){\eta }_k(\mathit{r}){\mathrm{\Sigma }}_{jk}}.$$ (18)

Second, we discretize the term $SN{R}_c^2$ to a $R$ × 1 vector $\mathit{S}\mathit{N}{\mathit{R}}_c^2$, which is given as

$$\mathit{S}\mathit{N}{\mathit{R}}_c={\left[\mathit{S}\mathit{N}{\mathit{R}}_c\left({\mathit{r}}_1\right)\mathit{S}\mathit{N}{\mathit{R}}_c\left({\mathit{r}}_2\right)\cdots \mathit{S}\mathit{N}{\mathit{R}}_c\left({\mathit{r}}_R\right)\right]}^{\top }.$$ (19)

For each voxel in the selected ROI, its corresponding SNR is given as

$${\left|\mathit{S}\mathit{N}{\mathit{R}}_c(\mathit{r})\right|}^2=\frac{|\mathit{m}(\mathit{r}){|}^2\mathit{\eta }{(\mathit{r})}^{\text{H}}{\tilde{\mathit{B}}}_1^{-}(\mathit{r}){\tilde{\mathit{B}}}_1^{-}{(\mathit{r})}^{\text{H}}\mathit{\eta }(\mathit{r})}{\mathit{\eta }{(\mathit{r})}^{\text{H}}\mathbf{\Sigma }\mathit{\eta }(\mathit{r})}.$$ (20)

Additionally, ${\tilde{\mathit{B}}}_1^{¯}$ (r) is defined as a column vector that contains the receive fields at a specific position r from all receive coils (from (9)), given by

$${\tilde{\mathit{B}}}_1^{-}(\mathit{r})={\left[\begin{array}{llll}{\mathit{A}}_1^{\prime }(\mathit{r}){\mathit{b}}_{1,1}\hfill & {\mathit{A}}_2^{\prime }(\mathit{r}){\mathit{b}}_{1,2}\hfill & \cdots \hfill & {\mathit{A}}_{N_r}^{\prime }(\mathit{r}){\mathit{b}}_{1,N_r}\hfill \end{array}\right]}^{\top },$$ (21)

where the system matrix ${\mathit{A}}_q^{\prime }\left({\mathit{r}}_j\right)\in {ℂ}^{1\times K}$ has k entries

$${\mathit{A}}_q^{\prime }\left({\mathit{r}}_j,k\right)=S_q^{\prime }\left({\mathit{r}}_j\right)e^{i\text{γ}\mathrm{\Delta }{B}_0\left({\mathit{r}}_j\right)\left(T-t_k\right)}.$$ (22)

We assume N_r = N_t in this scenario, then (20) can be written as

$${\left|\mathit{S}\mathit{N}{\mathit{R}}_c(\mathit{r})\right|}^2=\frac{|\mathit{m}(\mathit{r}){|}^2\mathit{\eta }{(\mathit{r})}^{\text{H}}\mathit{E}\mathit{D}(\mathit{r})\mathcal{B}{\mathcal{B}}^{\text{H}}\mathit{D}{(\mathit{r})}^{\text{H}}{\mathit{E}}^{\text{H}}\mathit{\eta }(\mathit{r})}{\mathit{\eta }{(\mathit{r})}^{\text{H}}\mathbf{\Sigma }\mathit{\eta }(\mathit{r})},$$ (23)

where $\mathit{E}\in {ℝ}^{N_t\times N_t}$ is the receive coil selection matrix, and $\mathit{D}(\mathit{r})\in {ℂ}^{N_r\times \left(N_{r}K\right)}$ is defined as $\mathit{D}(\mathit{r})=diag\left({\mathit{A}}_q^{\prime }(\mathit{r})\right)$.

The expression of SNR_c² is a ratio of quadratic Hermitian forms, one that commonly arises in optimization problems. We can determine the optimal SNR by finding the point when the gradient of ${\left|SN{R}_c\right|}^2$ with respect to η(r) is zero (N_r equations). Solving for η yields optimum receiver weights. Using the methods in [30] and [31], the optimum receive weights for each voxel are given by

$$\mathit{\eta }(\mathit{r})={\mathbf{\Sigma }}^{-1}{\tilde{\mathit{B}}}_1^{-}(\mathit{r})={\mathbf{\Sigma }}^{-1}\mathit{E}\mathit{D}(\mathit{r})\mathcal{B}.$$ (24)

We can substitute the expression for η(r) from (24) into (20), and the equation becomes

$${\left|\mathit{S}\mathit{N}{\mathit{R}}_c(\mathit{r})\right|}^2=|\mathit{m}(\mathit{r}){|}^2\left|{\tilde{\mathit{B}}}_1^{-}{(\mathit{r})}^{\text{H}}{\mathbf{\Sigma }}^{-1}{\tilde{\mathit{B}}}_1^{-}(\mathit{r})\right|$$ (25)

$$=|\mathit{m}(\mathit{r}){|}^2\left|{\mathcal{B}}^{\text{H}}\mathit{D}{(\mathit{r})}^{\text{H}}{\mathit{E}}^{\text{H}}{\mathbf{\Sigma }}^{-1}\mathit{E}\mathit{D}(\mathit{r})\mathcal{B}\right|.$$ (26)

The gSNR is defined as the averaged value of the $SN{R}_c^2$ matrix in a specific ROI:

$$gSNR(\mathcal{B})=\frac{1}{\mathcal{R}}\left|{\mathcal{B}}^{\text{H}}\sum _r{\left|\mathit{m}(\mathit{r})\right|}^2\mathit{D}{(\mathit{r})}^{\text{H}}{\mathit{E}}^{\text{H}}{\mathbf{\Sigma }}^{-1}\mathit{E}\mathit{D}(\mathit{r})\mathcal{B}\right|.$$ (27)

Note that m ≈ m_ref because the simulated pattern should be very close to the target pattern after the excitation. Therefore, (27) is in quadratic form if we assume m = m_ref. In this case, the dominant eigenvector ($\mathcal{B}$_eig) of its system matrix can be used as RF pulse samples to maximize gSNR.

Recall that RF pulse samples, $\mathcal{B}$_ref (see (8)), achieve the desired excitation pattern in MRI parallel excitation, and $\mathcal{B}$_eig obtained from (27) maximizes the gSNR of the output signal in MRI parallel reception. Thus, a gradient descent algorithm is constructed to optimize a real β coefficent correlated with $\mathcal{B}$_eig and $\mathcal{B}$_ref, to update the resulting RF pulse samples, $\mathcal{B}$_opt. The finalized RF pulse samples, $\mathcal{B}$_opt, can maximize gSNR while controlling the excitation error and the RF power ($ϵ$_max is the maximum excitation error allowed; P is the maximum power constraint).

Note that we checked $\mathcal{B}$_eig (and scale back if necessary) to see if it satisfies the power constraint in (30) before the optimization starts. This is because $\mathcal{B}$_eig with any constant scale factor is also an eigenvector with the same eigenvalue regarding to the same Hermitian matrix. This scaling method will help controlling the RF power contraints.

$${\mathcal{B}}_{opt}=\underset{\mathcal{B},\beta }{\mathrm{argmax}}gSNR(\mathcal{B}),$$ (28)

     where $\mathcal{B}=\beta {\mathcal{B}}_{ref}+\sqrt{1-{\beta }^2}{\mathcal{B}}_{eig},|\beta |\le 1$;

$$\text{subject to}\frac{\Vert {\tilde{\mathit{S}}}_A\mathcal{B}-{\mathit{m}}_{ref}{\Vert }^2}{{\Vert {\mathit{m}}_{ref}\Vert }^2}-{ϵ}_{max}\le 0,$$ (29)

$${\mathcal{B}}^{\text{H}}\mathcal{B}-P\le 0.$$ (30)

The maximum excitation error is given by the Normalized Root-Mean-Square Error (NRMSE) [18] between the desired pattern and the simulated transverse plane magnetization pattern, and is given in the following equation:

$$\text{NRMSE}=\frac{{\Vert {\mathit{m}}_{ref}-{\mathit{m}}_{sim}\Vert }^2}{{\Vert {\mathit{m}}_{ref}\Vert }^2}.$$ (31)

The Karush-Kuhn-Tucker (KKT) conditions can be used to verify its optimality. We examined the lagrange dual problem of (28) and the duality gap. The resulting $\mathcal{B}$_opt might be sub-optimal, but it can substantially improve the resulting SNR.

B. Acceleration Factor in MIMO MRI model

Acceleration factor (AF) is defined as the maximum factor for which the imaging time is reduced with acceptable loss in SNR. We define AF_e as the AF in the parallel excitation stage of the MIMO MRI model.

In reality, acceleration in parallel excitation is achieved via undersampling, resulting in a reduction of the excitation field- of-view (FOV) for each individual coil’s excitation pattern. The practical AF can be defined as the ratio of the desired excitation FOV (FOV_f) to the accelerated excitation FOV (FOV_a) of a given k-space trajectory per transmit coil [9], [12], [18], which can be expressed as

$$A{F}_e=\frac{FO{V}_f}{FO{V}_a}.$$ (32)

In the MIMO MRI model, there is no acceleration during the MR image reception. Instead, we experimented with different numbers of receive elements (N_r) and investigated the effect of multi-coil reception in MRI.

IV. SIMULATIONS AND RESULTS

We performed Bloch simulations to test the new RF pulse design method. For the excitation, simulations were performed over a 64 × 64 transverse grid covering a 24 × 24 cm² region, assuming that all simulations were carried out by an eight-element active rung transceive array [10]. Transceive sensitivity patterns were obtained using finite-difference timedomain simulation of the array at 3.0 T in [10] and [18].

N_t = 8 transmit coils were used to reduce the parallel excitation-associated B₁ field inhomogeneity, and N_r ≤ 8 coils were used for parallel reception. The corresponding magnitudes of $B_1^{+}$ patterns from the eight-channel parallel excitation are shown in Fig. 2. The transmit and receive sensitivity maps are extracted from the given $B_1^{+}$ patterns [26]. The receive field of each q^th coil $\left({\tilde{B}}_{1,q}^{-}\right)$ , defined in (9), was calculated based on the designed RF pulses and the receive sensitivity maps.

Fig. 2:

Magnitude of $B_1^{+}$ patterns used in simulations of eight-channel parallel excitation, as measured on a spherical phantom. The circular region indicates the ROI mask (radius = 10 cm) has been applied.

In the simulation, we designed RF pulses to excite the desired slice in a sphere phantom. The RF pulses were designed not only to reduce erroneous excitation outside the desired region, but also to maximize the gSNR in the receive signal, V, when a specific excitation pattern was used. Different ${ϵ}_{max}$ values (0.05–0.15) were used for each simulation. During the image reconstruction, we assumed the noise across all receive coils to be μ(r), where μ(r) ~ $\mathcal{N}$(0, 0.5).

A. Acceleration Factor

The desired excitation pattern in this simulation, shown in Fig. 3(a), was a smoothed circular slice in a sphere phantom (radius ≈ 4.5 cm). The excitation pattern was convoluted with a Gaussian kernel to reduce the Gibbs ringing artifacts. Fig. 3(b) shows the transverse plane magnetization after exciting the plane with a (π/2) RF pulse. The applied ROI mask was a circular region shown in Fig. 2 (radius = 10 cm).

Fig. 3:

Desired excitation pattern and transverse plane magnetization used in the simulations. (a) the desired excitation pattern within given ROI (red dashed circle), which is a smoothed slice of a sphere phantom (radius ≈ 4.5 cm); (b) the transverse plane magnetization (M_xy), which is a smoothed circle block, whose peak was scaled to π/2, correponding to 90° tip angle.

We designed RF pulses with a single-shot, spiral-out excitation k-space trajectory. This k-space trajectory was used with the following GE Discovery MR750 3.0 T scanner parameters: 5 G/cm maximum gradient amplitude, 20 G/cm/ms maximum slew rate, and 4 μs sampling period [32]. To accelerate the excitation, we used the trajectory with variable sampling density, resulting in reduced FOV (FOV_a) for each individual transmit coil. The acceleration factor was defined as the ratio of the accelerated FOV and the desired FOV per transmit coil (given in (32)). The spiral trajectory was chosen for these simulations to obtain a spatial resolution of 0.75 cm for the case of no acceleration. In addition, the spatial resolution increased as sampling density decreased.

In this simulation, we investigated the excitation performance by applying different acceleration schemes during parallel excitation. Although these accelerations translate to undersampling the excitation k-space, aliasing effects were minimal since the optimization cost function and constraint functions limited the NRMSE. Furthermore, the ROI definition provided additional degrees of freedom for maintaining excitation accuracy. Different AF_e values were applied during the simulation to investigate the changes in gSNR. For AF_e = 6, the final gSNR improved by 18.36%, and the NRMSE between the desired magnetization pattern and the simulated pattern was 0.12. Overall, the improvement of gSNR decreased as AF_e increased.

As AF_e increased from 1 to 10, the pulse length T decreased from 9.3 ms (no reduction in each receive FOV) to 0.88 ms (10-fold reduction in each receive FOV). The left column in Fig. 4 shows the simulated receive voltage after image combination, V, produced by the designed RF pulses; the right column is the transverse plane magnetization after excitation. We aimed to maximize the gSNR of V, while maintaining a controlled erroneous excitation (NRMSE ≤ 0.12). The results in Fig. 4 show that gSNR decreased and that NRMSE was always less than 0.12 when AF_e was incremented from 2 to 6. However, the gSNR generated by the SNR-controlled RF design method improved compared to the multi-coil RF design method without SNR control.

Fig. 4:

The left column shows the simulated MRI receive voltage, V, produced by the designed RF pulses. The right column shows the transverse plane magnetization (M_xy) obtained through the exact Bloch simulation; excitation outside the cylinder is the erroneous excitation. (a) N_t = 8, N_r = 8, AF_e = 2, T = 4.7 ms, gSNR = 17.45 dB (improved 130.52%), NRMSE = 0.10; (b) Nt = 8, N_r = 8, AF_e =4, T = 2.3 ms, gSNR = 11.88 dB (improved 59.89%), NRMSE = 0.10; (c) N_t = 8, N_r = 8, AF_e = 6, T = 1.6 ms, gSNR = 9.09 dB (improved 18.36%), NRMSE = 0.12. The gSNR is shown to be substantially improved in the SNR-controlled method with the NRMSE constraint of 0.12.

Because the maximum excitation error tolerance was set with ${ϵ}_{max}$, some erroneous excitation outside the desired cylinder always remained. Erroneous excitation became apparent when AF_e ≥ 7 (over 85.7% reduction in individual k-space data), and corresponded to decreased efficiency of the SNR- controlled RF design method. The transverse magnetization in the right column of Fig. 4 suggests that there were minimal biases in the center of the excitation pattern.

Two different RF pulse design methods (i.e., with SNR control and without SNR control) were compared in Fig. 5 and Fig. 6. The RF pulse design method without SNR control was determined by minimizing the NRMSE between the desired magnetization pattern and the simulated magnetization pattern, which was the solution obtained from (8). The results illustrate that these two methods generated different SNRs during image reconstruction. When AF_e was increased, the gSNR generated by the SNR-controlled method was always higher than the method without SNR control; but the improvement decreased as AF_e increased. The gSNRs generated by these two methods are shown in Fig. 5, related to the RF transmission time (i.e., pulse length T). High accelerations led to shorter RF pulse lengths and lower SNRs. We observed that gSNR obtained in the SNR-controlled method decreased, although it was always greater than the gSNR obtained in the method with no SNR control. The multi-coil RF pulse design method without SNR control maintained the gSNR when T > 2 ms, but dropped when T ≤ 2 ms.

Fig. 5:

Two different gSNRs (gSNR_w/control, gSNR_{w/O control}) are plotted associated with the pulse length T. Varying AF_e (1–7.5) was applied in these simulations. N_t = 8, N_r = 8 elements were used, and the NRMSE ∈ [0.10,0.12] in all simulations. The gSNR obtained from the SNR- controlled method was improved when T > 2 ms.

Fig. 6:

gSNR in two different RF pulse design methods, with and without SNR control. Different numbers of receive elements N_r (1–8) were applied in these simulations. Other parameters were N_t = 8, AF_e = 2, T = 4.7 ms, NRMSE [0.08, 0.10] in all simulations. The gSNR increased when the number of receivers was increased in both methods, but the SNR-controlled method has greater improvement.

B. Receive Coil Array

In the second simulation, we investigated the gSNR associated with the receive coils. In Fig. 6, we observed that the gSNR increased when the number of receive elements was increased from 1 to 8. The gSNR in the SNR-controlled method increased faster than the method without SNR control, suggesting that the SNR-controlled method could substantially improve SNR when the number of receive coils was sufficient. All simulations were implemented when NRMSE ≤ ${ϵ}_{max}$. For the majority of cases, we optimized the RF pulses until NRMSE = ${ϵ}_{max}$ to fulfill the KKT conditions.

The gSNR of the simulated MRI receive voltage, V, was increased in both RF design methods (i.e., with SNR control and without SNR control). However, the SNR-controlled method increased the gSNR substantially compared to the method without SNR control, especially when N_r was small.

We also observed that the intensity values of the receive voltage, V, became more uniform when we used more receive coils during the image reception. In Fig. 7(a), we found that the output, V, appeared dark at locations further away from the used receive coils, but this phenomenon completely disappeared in Fig. 7(c), suggesting that more receive coil elements could reduce the intensity inhomogenity. There were also intensity variations when we used six receive coils for image reception (see Fig. 7(b)), but the variation became unnoticeable when eight receive coils were used. Fig. 7 illustrates that, when the number of receive elements was incremented from four to eight, the output, V, became more homogeneous, and the NRMSE between the simulated transverse magnetization and desired magnetization was equal to 0.1 for all simulations.

Fig. 7:

The simulated MRI receive voltage, V, using different receive coil elements, under three different conditions. The MRI output became more homogeneous as the number of MRI receivers increased, although the excitation accuracy remained the same for all three conditions. (a) N_t = 8, N_r = 4, AF_e = 2, T = 4.7 ms, gSNR = 15.40 dB (improved 203.75%), NRMSE = 0.10; (b) N_t = 8, N_r = 6, AF_e = 2, T = 4.7 ms, gSNR = 17.04 dB (improved 175.73%), NRMSE = 0.10; (c) N_t = 8, N_r = 8, AF_e = 2, T = 4.7 ms, gSNR = 17.45 dB (improved 130.52%), NRMSE = 0.10.

C. Maximum Excitation Error Tolerance

Fig. 8 plots the gSNR associated with ${ϵ}_{max}$ during the SNR optimation process. The desired excitation pattern in this simulation is shown in Fig. 3. An acceleration factor of 2 was chosen which corresponded to a 4.7 ms RF pulse in the MIMO MRI model.

Fig. 8:

gSNR associated with ${ϵ}_{max}$ changes. This simulation was implemented with the desired circular excitation pattern with radius ≈ 4.5cm, N_t = 8, N_r = 8, AF_e = 2, T = 4.7 ms. ${ϵ}_{max}$was increased from 0.03 to 0.3. Piecewise cubic interpolation method was applied to implement the regression. The gSNR growed rapidly with small ${ϵ}_{max}$, but the gains decreased and became more difficult as ${ϵ}_{max}$ increased. The gSNR would not increase to infinity.

We further applied an interpolation method to show the increasing trend of gSNR with respect to ${ϵ}_{max}$. The relationship trend in Fig. 8 shows that the gSNR started to logarithmically increase when ${ϵ}_{max}$ increased, suggesting that ${ϵ}_{max}$ played a pivot role in controlling gSNR values when it was small-valued. Furthermore, there was a tradeoff between the excitation errors and the resulting SNR. Note that ϵ_max can be used to optimize the excitation error tolerance and the SNR.

D. Different Excitation Pattern

Previous research showed that excitation patterns can be utilized to improve the receive performance in terms of g-factor [33], and this would further improve the performance in SNR. Fig. 9 shows results for a circular excitation pattern using various radius sizes. It can be concluded that gSNR increased when the size of the excitation region increased. There were minimal bias effects in the center of both transverse plane magnetizations, and it was observed that the resulting excitation patterns in both simulations remained homogenous in the excited region. The applied ROI mask helped reduce the B₁ field inhomogeneity when the excitation region was close to the ROI. Additionally, the iterative method was effective in suppressing the aliasing effect outside the cylinder because the effect was accounted for in the optimization cost function and the contraint functions.

Fig. 9:

Top row represents the desired circular excitation pattern with radius² = 20 cm², N_t = 8, N_r = 8, AF_e = 2, T = 4.7 ms, gSNR = 13.02 dB (improved 71.99%), NRMSE= 0.05; the second row represents the circular excitation pattern with radius² = 40 cm², N_t = 8, N_r = 8, AF_e =2, T = 4.7 ms, gSNR = 16.65 dB (improved 73.08%), NRMSE = 0.05. (a) and (d) are the desired excitation pattern; (b) and (e) are the simulated excitation pattern; (c) and (f) are the simulated MRI receive voltage V.

Fig. 10 illustrates the desired and simulated excitation patterns of the Shepp-Logan phantom, inverted to yield a larger region of appreciable magnetization. We applied an ROI mask to the edge of the desired excitation pattern. The SNR-controlled RF design method indicated higher excitation accuracy and improved gSNR compared to the RF design method without SNR control. This simulation illustrated that the new multi-coil RF design method with SNR control could be utilized for arbitrary excitation patterns.

Fig. 10:

Desired (a) and simulated (b) excited excitation pattern of the inverted Shepp-Logan phantom. The top row represents the M_xy magnetization, and the bottom row represents the M_z magnetization. N_t = 8, N_r=8, AF_e = 2, T = 4.7 ms, gSNR = 16.05 dB (improved 55.67%), NRMSE = 0.03. The red dashed ellipse indicates the applied ROI mask. The SNR-controlled method allowed for arbitrary excitation profile, and it indicated improved gSNR compared to the method without SNR control.

V. DISCUSSION

Together, the results confirm that MIMO MRI is a novel and useful theoretical framework for MRI parallel RF transmission and reception. Through computer simulations, we verified the RF pulse design strategy for this specific model. This MIMO MRI framework constructed a signal processing model between the transmitters and receivers of the MRI system by using both parallel excitation and parallel reception. This signal processing model is analogous to techniques developed for MIMO communication.

This framework was built based on the spatial domain method [12] and the Transmit SENSE method [5]. These methods focus on transmit coils and other parameters that could significantly affect parallel excitation. However, our results confirm that it is worthwhile to investigate the relationship between transmit and receive coils in an MRI system. Our proposed RF pulse design method used RF pulses to optimize several quadratic cost functions comprised of gSNR, weighted error terms, and Tikhonov regularization terms, while maintaining the excitation errors in a controllable scope. The Tikhonov regularization term was used to control the peak and integrated RF power. An ROI mask was incorporated in the model to instill high excitation accuracy at high acceleration factors since errors outside the ROI do not contribute to gSNR and NRMSE.

In an ideal experiment, the ROI can be determined during the RF design procedure with prescan images obtained from sensitivity mapping, such as ASSET or ARC in a GE MRI system. Furthermore, our method compensated for the B₁ field inhomogeneity since eight transmit channels were used for parallel excitation. When all eight receive channels were used, the simulated transverse plane magnetization showed uniform transverse plane magnetization, though it may have induced excitation errors outside the excited region. In addition to these advantages, our RF design method also allowed for SNR maximization, although there still existed a tradeoff between gSNR maximization and NRMSE minimization. These two cost functions must be balanced to achieve optimal system performance. When NRMSE = 0.12 and AF_e was increased from 2 to 6, the gSNR in the SNR-controlled method improved by 18–130% as compared to the gSNR obtained in the method without SNR control for a given circular excitation pattern (see Fig. 4).

Although most of the designed RF pulses were sub-optimal solutions (duality gap is not zero within KKT conditions), they still improved the gSNR tremendously and maintained excitation errors within the desired scope. Therefore, although these solutions were sub-optimal, they remained viable.

In the current study, the MIMO MRI model was described as a closed form that should be valid for all k-space trajectories and coil sensitivity profiles. This model worked successfully for tip angles ranging from 10° to 90°, although the results presented here only used a tip angle equal to 90°. However, this work cannot be applied on 180° refocusing pulses for spin echo sequences or inversion recovery sequences becasue small-tip-angle regime can lead to significant errors in excitation patterns. To overcome the limits in this regime, several approaches of large-tip-angle RF and gradient design have been proposed that apply or could apply to approximate the nonlinear Bloch equation [18], [34], [35]. The algorithm on large-tip-angle regime of MIMO MRI model needs to be researched on improving the inversion and refocusing at ultra-high fields.

VI. CONCLUSIONS

Previous researchers have studied multi-coil transmission techniques, but there has been minimal research on multi-coil transmission and reception in MRI systems. Here, we considered an MRI system with a transceive coil array, where a number of coil elements are used for transmission and another number were used for reception. We termed this framework the MIMO MRI model. We proposed a novel RF pulse design method not only to minimize the difference between the desired magnetization pattern and the simulated pattern, but also to maximize the SNR in the MRI receive signal. Multiple simulations have been verified using the new RF pulse design method, and we futher investigated the influence of acceleration factors, number of receive coil elements used, maximum excitation error tolerance, and different excitation patterns. We have demonstrated that our method improved the gSNR by 18–130% as compared to the gSNR obtained in the conventional pTX work (AF_e = 2 − 6), and simultaneously controlled the image difference between excitation error in a desired scope (NRMSE = 0.12). The simulated transverse plane magnetization was uniform using the new RF design method, and there were minimal bias effects within the transverse magnetization. Research on large-tip-angle simulations and different acceleration methods in parallel reception remain problems for future MIMO MRI investigation.