Calidad de la Evidencia

Signal localisation methods introduced earlier in this chapter rely upon field gradients to encode position, and read out data sequentially using a single receive probe. In particular, phase encoding constrains the image acquisition rate, as each phase encoded point requires a separate echo acquisition.

An alternative encoding scheme is to use multiple receive probes to acquire data simultaneously, each probe providing a different component of spatial information. The simplest implementation of this approach is to use two receive probes, placed sufficiently far apart that they do not interact, si- multaneously imaging two regions of the body [30]. A more challenging approach is to image a single region using multiple probes with overlapping fields-of-view.

Methods for combining images from multiple probe elements for uniform sensitivity or uniform noise were derived by Roemer et al [31]. They also show that, where no probe sensitivity information is available, the pixel- by-pixel root-sum-of-squares (RSS) method is optimal. If a is a vector containing the pixel values detected by each probe element (the superscript T indicates the transpose)

a= [a1 a2 · · · ane]

the pixel magnitude V in the reconstructed image is

v =√aH_Ψ−1_{a (2.60)}

where the superscript H indicates the conjugate transpose and Ψ is the ne× ne receiver noise matrix, which is discussed in more detail in section 3.10.4. Assuming the probe elements are non-interacting, Ψ is the identity matrix.

Rather than increasing the SNR, parallel methods can also be used to ac- celerate image acquisition. There are two approaches: massively parallel and partially parallel methods. Massively parallel methods use array probes with approximately the same number of probe elements as data points to entirely replace phase encoding and acquiring a complete image from a sin- gle echo signal [32, 33]. Partially parallel methods use a smaller number of receive elements in combination with phase encoding, to reduce the number of required acquisitions [34]. Massively parallel methods have not yet seen wide use, but partially parallel methods have become a standard technique in regular clinical use.

Initial reconstruction methods, while showing promise in simulation, were not stable enough for clinical application [35]. The first clinically applicable method was SMASH (SiMultaneous Acquisition of Spatial Harmonics) [36]. SMASH uses a probe array with a series of elements in the phase encoding direction. The elements are designed such that channels can be combined to produce different spatial harmonics in sensitivity in the phase encoding direction. These spatial harmonics reproduce the same effect in the acquired data as phase encoding gradients. Hence several phase encoding steps may be reconstructed from a single, parallel, acquisition. A generalisation of SMASH has since been demonstrated, allowing arbitrary probe configura- tions designed to optimise SNR rather than produce spatial harmonics [37].

Shortly after SMASH, the SENSE (SENSitivity Encoding) method was in- troduced [38]. While SMASH performs the reconstruction in k-space, the SENSE reconstruction is done in image space. During a SENSE acquisition, k-space is undersampled in the phase encoded direction, causing Nyquist aliasing in image space (fig. 2.24). Data is simultaneously acquired by multi- ple probe elements, each with a different sensitivity profile. Using knowledge

Figure 2.24: Undersampling the image in k-space causing Nyquist aliasing, folding the edges of the image into the centre, superimposing pixels.

of the sensitivity profiles, aliased images may be unfolded and combined to produce a single, fully sampled image. Undersampling each channel reduces the number of phase encoding steps, reducing the overall acquisition time.

The SENSE algorithm is applicable to an arbitrary k-space trajectory, and does not place restrictions on the probe configuration (although probe de- signs may be optimised for SENSE acquisition, as discussed in section 3.10.4). In the general case, SENSE reconstruction involves the inversion of a large matrix, requiring considerable computing resources. Cartesian reconstruc- tion is described in this section, which considerably simplifies the recon- struction, as each aliased pixel is unfolded separately.

Data is acquired using an array probe with ne elements. Data from each channel is Fourier transformed to produce an image in the normal manner. However, the phase encoded direction is undersampled such that np pixels are superimposed in each subimage. For each pixel ρ, a vector is constructed from the aliased data for each probe element

a= [a1 a2 · · · ane]

T _(2.61)

Central to the reconstruction is the sensitivity matrix S. Complex element sensitivities at np superimposed pixels form an ne× np sensitivity matrix

where γ is the element number, ρ is the superimposed pixel, rρis the location of pixel ρ and sγ is the sensitivity of element γ.

Folded pixel values a may be calculated from the separate pixel values v using the sensitivity matrix S

a= Sv (2.63)

However, we wish to perform the reconstruction in the reverse direction, generating an unfolded image from folded subimages. This is done using the unfolding matrix U , which is the pseudo-inverse of S [39]

U = SHΨ−1S
−1

SHΨ−1 (2.64)

where SH _{indicates the complex conjugate transpose of S and Ψ is the} ne × ne receiver noise matrix, which describes noise correlation between receive channels. The unfolded pixels values v for the originally aliased locations are then

v = U a (2.65)

SENSE reconstruction relies upon highly accurate sensitivity maps, which are acquired by taking a set of reference images. A full image is acquired using a volume probe (usually the same probe that is used for excitation) and a full image for each element of the receive array. Noise is removed from the reference images by polynomial fitting. The array images are then divided by the volume reference to give the sensitivity maps [38].

Undersampling an image inevitably reduces the SNR. The SNR of a SENSE reconstructed image is SNRSENSE_ρ = SNR full ρ gρ √ R (2.66)

where SNRfull_ρ is the signal to noise ratio of the full image and R is the reduction factor. The extra factor gρ is a geometry factor, which accounts for noise amplification in the reconstructed image due to poor conditioning of the sensitivity matrix S. This occurs where probe elements have similar

sensitivities to aliased pixels, making the pixels difficult to separate. It is defined as gρ= q (SH_Ψ−1_S)−1 ρ,ρ(SHΨ−1S)ρ,ρ>1 (2.67) Increasing the reduction factor R increases the number of pixels superim- posed in the folded images, which tends to increase the geometry factor. In practice the maximum usable reduction factor is limited by the g-factor before the ultimate limit, the number of receive elements ne, is reached.