Dynamic contrast-enhanced (DCE) MRI is a well-established technique for non-invasive prognosis of cancer. The Pharmacokinetic (PK) maps: $$$K^{trans}$$$, the flow of Contrast Agent (CA) from plasma to Extracellular Extra vascular Space (EES) and $$$K_{ep}$$$, the flow of CA from EES to plasma can be obtained through curve fitting of the Tofts model [1]. The changes in intensity during a DCE-MRI scan are predominantly in the low frequency range. Here, we demonstrate a method to obtain a feasible arbitrary k-space trajectory using the Active Contour (AC) technique and subsequently design gradient pulse sequences to traverse this path in k-space using convex optimization (cvx) to sample arbitrary k-space trajectories.

The AC technique [2] is a framework for delineating an object from a possibly noisy 2D image. It is a form of energy minimization, defined as $$$E_{snake}=\int_{0}^{1}E_{int}(V(s))+E_{image}(V(s))+ E_{con}(V(s)) ds $$$ where $$$E_{int}(V(s)) $$$ is the internal energy of the spline due to bending, $$$E_{internal}(V(s))$$$ is the image forces and $$$E_{internal}(V(s))$$$ is the external constraint force. This method can be used to obtain tweaked spiral like arbitrary k-space trajectory from a k-space mask. cvx [3] is used to solve the k-space trajectory to gradient waveform equation given by $$$k(t)=\frac{\gamma}{2\pi}\int_{0}^{T} g(t)dt $$$ [4], where k(t) is the k-space trajectory traversed at time t (mm-1), $$$\frac{\gamma}{2\pi}$$$ is the gyromagnetic ratio (42.56MHz/T), g(t) is the gradient amplitude at time t and T is the total time duration. A matrix for integration (A) is developed based on the trapezoidal rule and fed into the cvx to solve $$$\parallel(k-A\times g)\parallel$$$, where ‖ . ‖ represents the norm operator, under the constraints of maximum gradient amplitude, maximum slew rate and total time duration of the gradients thereby designing the optimal gradient waveform. All experiments were performed on six breast DCE data sets downloaded from quantitative imaging network collection [5]. The DCE-MRI acquisition parameters included TR/TE =6.2/2.9ms, temporal resolution = 18~20s, flip angle = $$$10^{o}$$$. The number of frames in each breast data set ranged from 28-32 slices, the CA used was Gd (HP-DO3A) [ProHance] IV injection (0.1mmol/kg at 2mL/s). The tumor Region Of Interest (ROI) was drawn for the two breast DCE data with 1x acceleration factor and the pixels within the tumor region were provided for curve fitting to obtain the PK maps. The k-space data was obtained for six data sets and k-space masks were generated for different acceleration factors (2x, 3x, 4x and 10x) for the respective images. Morphological operations were performed and mask with a distinct boundary was obtained. The AC technique was used to traverse across the mask from the boundary to its center and an arbitrary shaped k-space trajectory was obtained. The k-space trajectory was then verified to represent the mask. The number of points obtained on the k-space trajectory was subsampled to match the memory requirements of the computer. The cvx was used to solve the k-space trajectory to gradient waveform equation, under the constraints of maximum gradient height Gmax = 50mT/m, maximum slew rate SRmax = 100T/m/s and total time duration = 40ms by taking in the subsampled k-space trajectory and the integration matrix as inputs. The gradient waveforms were verified by obtaining k-space trajectory back by solving for gradient waveform to k-space equation analytically. The images were reconstructed from the verified mask by Fourier transform. Curve fitting for the Tofts model was performed to estimate the PK maps as shown in figure 3.Figure 1 depicts the tumor ROI drawn for the two breast DCE data and pixels within the tumor region given for curve fitting to estimate PK maps. Figure 2(a) represents the undersampled k-space mask for the acceleration factor of 10x, figure 2(b) represents the mask after performing morphological operations. Figure 3 represents the gradients obtained for 10x, confined to the added constraints. Figure 4 represents the $$$K^{trans}$$$ and Ve map for the two data sets obtained for the acceleration factors: 1x, 2x, 3x, 4x and 10x. $$$K^{trans}$$$ and Ve map for the two data sets obtained for the acceleration factors: 1x, 2x, 3x, 4x and 10x.The combination of active contour and cvx on DCE-MRI has been established retrospectively for the first time. Instead of looking at k-space specific to ROI, locations in k-space that are highly relevant to that ROI are arbitrary shaped. Active contour technique will efficiently sample the k-space according to the k-space shape, which results in better reconstruction. We can infer from the results that $$$K^{trans}$$$ and Ve maps for higher acceleration factors matches with the $$$K^{trans} $$$ and Ve maps of 1x.