The impulse response is then used to estimate the input function to the gradient amplifier, xp(t), that is required to achieve a desired MS-275 cell line output gradient

shape, yd(t). equation(5) xp(t)=F-1F[yd(t)]F[h(t)]. Eq. (5) describes a deconvolution of the impulse response from the desired shape. Deconvolution is prone to amplifying noise as it involves division in the frequency domain. To overcome this, Goora et al. [22] model their measured gradient using a polynomial. Here, we use a Gaussian filter, with a standard deviation of 200,000 Hz, in the Fourier domain to suppress the noise. The resulting shape is then applied to the MRI system and the output gradient shape is measured using the method of Duyn et al. [32]. An oscilloscope is used to measure the approximate timing of the r.f. and gradient pulses. The time at which the r.f. pulse is applied is adjusted until the two pulses end at the same time. This timing is later optimized experimentally as there is a slight delay of approximately 20 μs between the input from the gradient amplifier and the actual

applied gradient. The optimization is performed using the slice measurement technique that will be described in Section 3.2.3. The k-space data for UTE is acquired on a non-Cartesian grid. The NUFFT algorithm of Fessler and Sutton [29] is used to perform the re-gridding and subsequent fast Fourier-transformation of the k-space data points. The k-space trajectory is measured using selleck kinase inhibitor the technique of Duyn et al. [32] as accurate image reconstruction requires

precise knowledge of the trajectory. The images are reconstructed using the total variation based regularization method as described in Benning et al. [33]. A brief description of the approach used here is equation(6) uα,β∈argminu12||Fu-f||22+α||∇u||2,1+β||u||1which is a Tikhonov-type reconstruction with total variation prior and, in the context of under sampled MRI, is often referred to as CS [3]. Here u   denotes the spin density image, F   the non-uniform fast Fourier transform (NUFFT) operator, ∇ a forward finite difference discretization of the gradient operator, ||u  ||1 the one-norm, ||∇u  ||2,1 the one-norm applied mafosfamide to (∇xu)2+(∇yu)2, α the regularization parameter for the gradient term, β the regularization parameter for the image, f the measured k-space data and uα,β the image to be recovered. Lustig et al. [17] used a similar method (with ||∇u||1,1 instead of ||∇u||2,1) as an optimization method as it had been shown to successfully overcome the blurring and ringing artifacts present in the zero-filled reconstruction. The regularization parameters, α and β, are chosen heuristically. For robust methods used to select regularization parameters see Benning et al. [33]. The slice selection profile was measured using the one dimensional imaging sequence shown in Fig. 3. The sequence uses a frequency encoded acquisition applied in the same direction as the slice selection.