Improved total sensitivity estimation for multiple receive coils in MRI using ratios of first-order statistics

Abstract

Object

Spatial variation in the sensitivity profiles of receive coils in MRI leads to spatially dependent scaling of the signal amplitude across an image. In practice, total sensitivity of the coil array is either calibrated or corrected directly by comparison to a uniform sensitivity image, fitting of coil profiles, or indirectly by constraining the reconstructed image or coil profiles. In the absence of these corrections, popular coil summation strategies are often designed to maximize the signal-to-noise ratio or optimize under-sampled encoding but not necessarily estimate the value of the signal unscaled by the coil spatial sensitivity.

Materials and Methods

We use ratios of first-order statistics to approach the unscaled value of the signal at any position. Motivated by the assumption that the coil array is a sample from much larger number of possible coils, we present two approaches to scale the mean signal in all coils: (1) an argument for use of the mode of the normalized signals, and (2) using a one-dimensional analog derive an approximate expression for scaling with the ratio of the square-of-the-mean to the mean-of-the-squares. We test these approaches with simulation where idealized coil elements are arrayed around an object, and on directly acquired data with an 8-channel coil array on a uniform 13C phantom, and on Hyperpolarized 13C pyruvate brain MRI.

Results

We show improved image uniformity using the ratios of first order statistics compared to a simple homomorphic filter, noting that these approaches are more sensitive to noise.

Discussion

We present simple methods for correcting the spatial variation in sensitivity profiles in the context of a coil array. These methods can be used as an initial or adjunct step in data post-processing.

Introduction

In multi-channel MRI, the sensitivity profiles of individual coil elements are expected to follow the spatial distribution of the Biot-Savart Law, given the reciprocity principle, noting the contribution from additional design parameters specific to each coil array. The sensitivity profile spatially scales the signal amplitude across an image. While the relative sensitivity of each coil element can be derived, estimation of the unscaled signal amplitude at any position equivalently requires estimation of the total sensitivity.

This issue is relevant for current approaches to ¹³C hyperpolarized (HP) MRI. Because of the inherently lower sensitivity of ¹³C MRI compared to ¹H MRI, early human studies have made use of small surface receive coils placed close to the anatomy of interest. These arrays of surface coils allow for flexibility in imaging different structures, but also inherently have reduced sensitivity at increasing distances from the coils, leading to spatial scaling across an image; namely the structures deep to the surface where the coils are placed demonstrate decreased signal amplitude.

Direct methods of estimating the total sensitivity include acquisition of a uniform sensitivity image with a volume coil or modeling coil geometry [1, 2]. However, for applications such as ¹³C hyperpolarized MRI, acquisition of a uniform field of view image may be challenging. Due to technical challenges, large ¹³C volume (body) coils are typically designed as transmit-only coils. Alternatively, modeling the profiles of the receive coils would require some estimate of the effective coil geometry, possibly simplified, but would ideally be done at the time of imaging if the coil array is flexible and not easily re-approximated with a phantom, which cannot be easily placed adjacent to deeper structures. We note that for hyperpolarized applications, the estimate of the relative coil-specific variances is, in general, possible from time points with low signal (i.e., after decay of the hyperpolarized signal), assuming negligible effects of multiplicative noise.

Indirect methods fall into the general field of bias-field estimation and include constraining the reconstructed data/image, for example, using sparsity in some domain such as enforcing smoothly varying individual coil sensitivity maps [3-5]. Hence, most reconstruction methods either focus on simplicity/robustness or maximizing the signal-to-noise ratio. For example, the simplest “brute-force” method is to sum the squares of the signal in each coil. This not only has the advantage of removing phase artifacts but also suffers from decreased noise averaging. Other well-known strategies that result in a high signal-to-noise ratio include singular value decomposition and estimation of the principle eigenvector of the covariance matrix [6].

In this work, we reason that the relative coil sensitivity profiles at a given position are non-random samples from some distribution (e.g., a beta distribution), and consider the limits of a very large number of coils. We then suggest the use of the mode of this distribution rather than the mean (which is more sensitive to outliers), to achieve a "typical” or “representative” value and scale the mean by the mode-to-mean ratio to improve noise characteristics. Subsequently, we consider a simple one-dimension analog, and after considerable simplification, suggest the use of the square-of-the-mean to mean-of-squares ratio. We think that this approach may be useful for ¹³C hyperpolarized MRI or for insensitive or transient signals from other nuclei that require multi-coil systems with freely movable/surface coil elements, without the need for pre-calibration.

Theory motivation

We assume that the observed signal follows a Gaussian distribution, where in any given voxel, for a given coil element, c, the observed signal B_c, is distributed around the unscaled value, U, scaled by the position-dependent coil-specific sensitivity S_c, and coil-specific variance, σ_c. We allow that the unscaled value is itself a relative amplitude with scaling (e.g. U ∈ (−1, 1)), with appropriate scaling for the variances, so to avoid confusion with the intrinsic signal-to-noise ratio [7] as follows:

$$P(B_c∕S_c\mid U,{\sigma }_c^2∕S_c^2)=\frac{\text{exp}\frac{-(B_c∕S_c-U{)}^2}{2{\sigma }_c^2∕S_c^2}}{{\sigma }_c∕S_c\sqrt{2\pi }}$$ (1)

If coil variances and sensitivities are known, differentiating the log of the probability gives the maximum-likelihood estimator of the unscaled value, U, as stated in [8]:

$$U_{MLE}=\frac{{\sum }_{c=1}^M{B}_c{S}_c∕{\sigma }_c^2}{{\sum }_{c=1}^M{S}_c^2∕{\sigma }_c^2}$$ (2)

The relative positional information in a flexible coil array with multiple elements can be used to obtain relative sensitivities, which are functions of coil-specific positions, Fig. 1.

Fig. 1.

Relative positional information of the field-of-view (FOV). The coil sensitivities (S₁, S₂, S₃, …) are functions of the relative positions and geometries of the coils (p₁, p₂, p₃ ), which scale the observed signal in each coil (B₁, B₂, B₃, …) for a voxel at position r with unscaled value signal U

For a voxel r, the relative sensitivities, R, can be obtained by normalizing the signal from each coil, B_i, to the sum-of-squares, simplified by assuming equal coil-specific variances as folows:

$$R^2=\frac{[B_1^2,B_2^2,\dots B_M^2]}{{\sum }_i{B}_i^2}=\frac{[S_1^2,S_2^2,\dots S_M^2]}{{\sum }_i{S}_i^2}=\frac{S^2}{\Vert S{\Vert }_2^2}$$ (3)

$$R=\frac{S}{\Vert S{\Vert }_2}$$ (4)

This can be extended to the spatial profiles of sensitivities over the field of view, for example with additional low-rank constraints to implicitly arrive at an estimate of the total sensitivity, ∥S∥₂ [9, 10]. Alternatively, it is possible to constrain the coil-specific spatial profiles to the functional form of the Biot-Savart Law, provided some model for their geometry, potentially simplified [2]. In this model, the sensitivities are spatial functions of the coil geometry and position, as path integrals of the infinitesimal tangent to a coil element, dI′, and distance from the coil, r_p. In these approaches, the constraint to the total sensitivity, ∥S∥₂, is a function of the coil model(s).

$$S_p=\oint \frac{d{I}^{{}^{\prime }}\times {\widehat{r}}_p}{r_p^2}$$ (5)

First-order statistics

At a given point r, the normalized sample of sensitivities, R, is bounded on the interval (0, 1). In general, this is a non-random sample due to geometric constraints in coil placement. This suggests exploring the mode of coil-specific signals, $\widehat{\theta }$, as an estimate for the unscaled signal intensity, U, to balance the lower sensitivities at longer distances with the higher sensitivities at closer distances and arrive at a “typical” or “characteristic” sensitivity. In a sense, the mode is less sensitive to outliers, which are the fewer coils very close/very far from a spatial point, demonstrating higher/lower sensitivity. Similarly, it favors the majority of coils when they are equidistant.

$$\widehat{U}\approx \widehat{\theta }\left(\mid B_i\mid \right)$$ (6)

The problem with simply using the mode, however, is that it disfavors the high signal-to-noise ratio of coils close to spatial point. Because the mode increases the effect of noise relative to the mean, we consider scaling the mean by the mode of the mean-normalized signals, and approximate this scaling by the mode of R (non-negative dimensionless quantities that are invariant to the unscaled value signal):

$$\widehat{U}\approx \widehat{\theta }\left(\mid B_i\mid \right)\approx \widehat{\mu }\left(B_i\right)\widehat{\theta }\left(\frac{\mid B_i\mid }{\widehat{\mu }(B_i)}\right)\approx \widehat{\mu }\left(B_i\right)\widehat{\theta }\left(\frac{\mid B_i\mid }{\Vert B\Vert }\right)=\widehat{\mu }\left(B_i\right)\widehat{\theta }(R)$$ (7)

Further, we also consider a robust mode estimator, namely Bickel’s half-range mode with bootstrapping, HRMB [11, 12]. This in effect captures properties of the mode and some noise reduction in using the mean as follows:

$$\widehat{U}\approx \widehat{\mu }\left(B_i\right)HRMB(R)$$ (8)

Ratios of first-order statistics

We consider a simplified one-dimensional analog of the sensitivity S_p,r at position r, in an idealized single-point coil at position p ≠ r, with some positive scalar sensitivity α, and a spatial decay exponent x. The observed signal B_p,r, is then the unscaled value U_r, scaled by the sensitivity as follows:

$$S_{p,r}=\frac{\alpha }{(p-r{)}^x}$$ (9)

$$B_{p,r}=U_r{S}_{p,r}$$ (10)

In the limit of increasing number of coils uniformly distributed on some interval (0, c), we estimate the expectation value for the observed signal from all the coils as the integral over the interval, for positions r > c, as follows:

$$E\left(B_r\right)=U_{r}E(S_r)\approx \frac{1}{c}{\int }_0^c{U}_r{S}_{p,r}\approx \frac{U_{r}E(\alpha )}{c(x-1)}\left((r-c{)}^{1-x}-r^{1-x}\right)$$ (11)

By analogy to R in the preceding, we then consider the ratio of first-order statistics $R_B^2=E(B{)}^2∕E(B^2)$ for a decay exponent x = 2 (given the Biot–Savart law), and show that it is a function of the expectation of the sensitivities, E(S_r), independent of the unscaled value, U_r:

$$R_B^2\approx \frac{E(\alpha )∕E({\alpha }^2)}{E(S_r)\frac{c^2}{3E(\alpha )}+1}$$ (12)

We can consider correcting the sensitivity scaling of the mean E(S_r), using $R_B^2$ and a tunable global scalar parameter γ arising from geometric considerations, whereas$R_B^2\to \gamma$, the mean sensitivity approaches zero:

$$E(S_r)\propto \frac{\gamma -R_B^2}{R_B^2}$$ (13)

$$\frac{1}{E(S_r)}\propto \frac{E(B_r{)}^2}{\gamma E(B_r^2)-E(B_r{)}^2}$$ (14)

Using the mean estimate as the expectation value, $E(.)=\widehat{\mu }(.)$, we arrive at an expression for the unscaled value, U_r as function of γ as follows:

$$U_r(\gamma )=\frac{E(B_r)}{E(S_r)}\approx \widehat{\mu }\left(B_r\right)\left\{\frac{\widehat{\mu }{\left(B_r\right)}^2}{\gamma \widehat{\mu }\left(B_r^2\right)-\widehat{\mu }{\left(B_r\right)}^2}\right\}$$ (15)

To avoid the potential discontinuity at $R_B^2=\gamma$, we consider the even simpler (although obviously biased in this model) approximate scaling by $R_B^2$ as follows:

$$U_r\frac{E(S_r)}{E\left(S_r\right)+1}\approx \widehat{\mu }\left(B_r\right)\left\{\frac{\widehat{\mu }{\left(B_r\right)}^2}{\widehat{\mu }\left(B_r^2\right)}\right\}=\widehat{\mu }\left(B_r\right)R_B^2$$ (16)

$$U_r\approx \widehat{\mu }\left(B_r\right)R_B^2$$ (17)

As previously, we can improve the noise characteristics by bootstrapping and subsequent spatial filtering, WB:

$$U_r\approx \widehat{\mu }\left(B_r\right)WB\left\{R_B^2\right\}$$ (18)

Methods

Data acquisition

All data were acquired on a 3 T wide-bore MRI (GE Healthcare). Hyperpolarized brain MRI was performed as in [13]. Patients were recruited under an institutional review board-approved protocol (IRB 14–205, PI: Keshari) at Memorial Sloan Kettering Cancer Center (MSKCC). An investigational drug acknowledgement was granted for HP [1 – ¹³C] pyruvate (IND #11259470, PI: Keshari). Written, informed consent was documented. The study was performed in accordance with the Declaration of Helsinki, Belmont Report, U.S. Common Rule guidelines, and the International Ethical Guidelines for Biomedical Research Involving Human Subjects. All information was accessed, stored, and published in keeping with HIPAA protections. Briefly, dynamic nuclear polarization was performed with a 5.0 T SPINlab hyperpolarizer (GE Healthcare). Sterile fluid paths, containing [1 – ¹³C] pyruvic acid (14.2 mol/L), trityl-OX063 radical (15 mmol/L), sterile water for injection (USP), and a neutralizing base solution, were prepared under cGMP conditions. Following hyperpolarization and quality control, 0.43 mL/kg of 250 mmol/L HP pyruvate was injected intravenously at 5 mL/s, followed by a 20-mL saline flush. Phantom data were acquired using a human head-shaped phantom containing ethylene glycol. Phantom and HP data were acquired with a 2D ¹³C EPSI sequence, using a large volume excitation ¹³C coil (GE Healthcare) and 2 surface receive coil arrays, each composed of 4 × 1 coil elements (GE Healthcare, 8 coils total). The EPSI spectral width was 579 Hz, 13 lead points, 16 read points, 11 fly-back points,16 phase encode points, FOV of 20 cm, 1.5 cm slice thickness, 58 dynamic points, 4.3 s repetition-time (TR), ~ 0.5 ms TE (understood as the location of echo center for the read points), 15 or 90 degree flip angle (HP or thermal phantom).

Data processing

All data processing was performed in MATLAB (Math-works, Inc.). Post-processing included Kaiser-window apodization, zero-filling, Fourier transformation, spatial coil-specific first-order phasing, and histogram-based baseline correction. Means were calculated from phased absorptivemode data.

Homomorphic filtering was performed by subtracting a low-pass filtered image in the log-transform domain [5]. Low-pass filtering (LPF) was performed with a Gaussian image filter (sigma = 10).

$$I_{\text{filtered}}=\text{Re}\{\text{exp}(\log (I)-\text{LPF}(\text{Re}\{\log (I)\}))\}$$ (19)

The mode was calculated using Bickel’s half-range method with boot-strapping (HRMB). Bootstrapping (1000 trials) was performed for simulated data (HRMB(R), $R_B^2$) and all acquired data, including comparison to the mean. For acquired data, local spatial smoothing/denoising was performed using a Wiener filter.

Results

The theoretic results were investigated using a simplified multi-coil simulation. The proposed methods to scale the mean of the signal for each coil demonstrate improved uniformity over the field-of-view, Fig. 2. Compared to a standard bias-field correction method, homomorphic filtering, the residual bias maps from the proposed methods are more uniform. Notably, scaling the mean by the robust mode of R (HRMB(R)) slightly overcompensates at the center of the field-of-view (Fig. 2O, Q), where coils have similarly low sensitivity. Scaling the mean by the square-of-mean to mean-of-squares ratio, $R_B^2$, improves on this (Fig. 2S, U).

Fig. 2.

Simulation of sensitivity profile using first-order statistics. A 2D Phantom image (64 × 64). B Phantom image embedded in circle (blue) on which coil elements are uniformly arrayed at 3 × radius of the phantom FOV. Sensitivity profiles were calculated as r⁻² decay from an idealized point-coil, and 0.2 spatial scaling for the FOV. Gaussian noise was added to achive noise standard deviation of 0.1 normalized to the signal over the FOV. Simulations were with 8 coils (C, D, G, H, K, L, O, P, S, T) or 1000 coils (E, F, I, J, M, N, Q, R, U, V). (C, E, G, I, K, M, O, Q, S, U) Performance using the various methods (recon). (D, F, H, J, L, N, P, R, T, V) Corresponding residual multiplicative bias map spatial uniformity (res. bias) relative to the original phantom image, thresholded to signal, normalized and effectively scaled (0.3–0.8). (C–F) Sum-of-squares across all coil elements showing decreased sensitivity in the center (RMSD = 0.0102). (G–J) Homomorphic filtering performance (RMSD = 0.0103). (K–N) Mean signal across all coil elements showing decreased sensitivity in the center (RMSD = 0.095). (O–R) Mean scaled by HRMB(R) showing subtle star-shaped artifact for 8 coils and slight over-correction for 1000 coils (P, R: RMSD = 0.095,0.144). (S–V) Mean scaled by $R_B^2$, showing relatively uniform signal intensity, although this is probably countered by slightly reduced SNR (RMSD = 0.085)

Application to acquired ¹³C phantom data and ¹³C hyperpolarized brain MRI

Sensitivity correction of multiple coils was tested for hyperpolarized ¹³C MRI. This represents a unique situation where for various reasons separate coil bias images are not easily acquired; coil-specific variances, however, are easily acquired. For both ¹³C phantom images and hyperpolarized ¹³C pyruvate MRI, scaling of the mean by first-order statistics improves spatial uniformity as compared to use of the mean, Fig. 3. Specifically, signal intensity in the phantom images is improved at the center of the image; because the phantom is uniform, the improved uniformity in the scaled images is consistent with improvement (Fig. 3D, E compared to A-C). By the analogy the scaled ¹³C hyperpolarized brain data show improvement in the center of the image (Fig. 3, I, J compared to F-H). As with the simulation, scaling of the mean by the robust mode of R, (HRMB(R)), probably overcorrects the center of the image, and the improvement in spatial uniformity of the proposed methods is a trade-off with slightly reduced SNR. In this acquired data example, the two approaches (HRMB(R) and $R_B^2$) show relative similarity, likely due to their similar formulation.

Fig. 3.

¹³C phantom and HP human brain data acquired using 8 coils showing improved uniformity using first-order statistics. (A, B, C, D, E) ethylene glycol head phantom, single slice EPSI. (F, G, H, I, J) ¹³C HP pyruvate human brain MRI, single axial slice pyruvate map showing relatively higher signal in the cortical regions and superior sagittal sinus.(C,H) Schematic position of 8 receive coils (2 arrays of 4 channels) indicated by linked blue circles. (A,F) Sum-of-squares of real spectral integrals. (B, G) Homomorphic filter of the mean of real spectral integrals (bootstrapped 1000 trials). (C, H) Mean of real spectral integrals (bootstrapped 1000 trials). (D, I) Mean scaled by HRMB(R) (bootstrapped 1000 trials). (E, J) Mean scaled by $R_B^2$, square-of-mean to mean-of-squares ratio. Bootstrapping (1000 trials) and spatial Weiner filtering were applied, and presented images were normalized to (mean + 2 × standard deviation). (D, E, I, J) show improved uniformity of the center of the images compared to the mean (C, H—green arrow on H vs. J)

Discussion

In this work we explore simple methods for total sensitivity signal estimation in multi-coil systems in Magnetic Resonance Imaging using simple statistics on the signals detected by individual coils. Our approach is motivated by considering the signal in multiple coils at a given point as a sample from a larger number of potential coils. We then try to use simple methods to estimate the total sensitivity at that point and spatially scale the image.

First, we discuss the utility of the mode to balance the higher sensitivity of voxels close to a given a coil element with those far from the coil element, in the sense that the few voxels close to a coil element can be considered outliers. Subsequently, we consider a one-dimensional analog and scale the mean by a ratio of squares of the relative sensitivities (ratio of the square-of-the-mean to the mean-of-the-squares). We note that this ratio is also invariant to the underlying signal and allows for spatial filtering to reduce noise contamination.

The major limitations of these methods are that they tend to amplify noise slightly more than simply using mean, can introduce possible artifacts, and are not a complete solution. Nevertheless, the major benefits are that they offer a simple solution with reasonable performance.

For hyperpolarized ¹³C MRI with multi-channel coils, these methods have important implications for calibrating sensitivity profiles when separate coil bias images cannot be easily acquired to correct the relative decreased signal in “deeper” structures farther from the coils, noting that differences in the excitation profile and other sources of bias will also modulate the signal. The methods presented in this work provide a simple and fast method to correct images for spatially dependent coil sensitivity variation across an image, in the context of a coil array, if care is taken to reduce the effects of noise. We believe that they can be used as an initial or adjunct correction step, allowing for more advanced methods in the subsequent processing pipeline.

Future extensions to this work are multifold and include robust statistical estimation of bias with physical motivations and varying degrees of knowledge of coil geometries. Our approach using a one-dimensional analog is a step toward this goal.