Magnetic Resonance Perfusion Quantification using QR-based Deconvolution

Phaneendra Kumar Yalavarthy^{1}, Kasireddy Viswanatha Reddy^{1}, and Junki Lee^{2}

In
DSC-MRI, a mathematical description in the matrix form that is necessary to
describe the complex relationship between signal intensity and contrast agent
concentration by taking into account the physical contrast mechanisms, has the
form $$C_t = \mathbf{C_a}R
+ e,$$ where $$$C_t$$$ is the vector samples tissue concentration $$$C(t_1, t_2, ..,t_n)^T$$$ and $$$R$$$ (known as the tissue residue
function from which the quantitative perfusion maps are obtained) is given by
$$$(R(0), …R(t_n))^T$$$, where $$$n$$$ is the number of time points and
$$$T$$$ denotes the transpose operation with $$$e$$$ representing the noise vector. The
entries of $$$\mathbf{C_a}$$$ (dimension:
$$$n\times n$$$) are in the circulant form formed by the arterial input function. The
matrix $$$\mathbf{C_a}$$$ is nearly
singular (highly ill-conditioned) in nature. The e represents the additive
noise in the MR signal. To obtain R one
needs to minimize the following objective function^{1}: $$\Omega = || \mathbf{C_a}R - C_t||^2
+ \lambda^2 ||R||^2.$$ Traditionally this is minimized either using
Oscillatory-limited SVD (oSVD)^{2} or Frequency-Domain deconvolution (FDD)^{3}.

The
proposed method performs
QR decomposition of $$$\mathbf{C_a}$$$ by
Lanczos bidiagonalization as given in Ref. [4]. With these bidiagonalizations the problem of
deconvolution decomposes into the iterative Least Squares QR. It turns $$\mathbf{C_a}R - C_t = U_{k+1}(B_kr_k
- β_0e_1); R = \mathbf{V_k}r_k.$$
Here $$$\mathbf{B}$$$
represents the lower bidiagonal matrix (dimension $$$k+1\times k$$$), with $$$(\alpha_1,…,\alpha_k)$$$
on the main diagonal and $$$(\beta_1,…, \beta_k)$$$ in the lower sub-diagonal), $$$\beta_0$$$ is the $$$L_2$$$-norm of the $$$C_t$$$, $$$\mathbf{U}$$$
and $$$\mathbf{V}$$$ represent the left and right
orthogonal Lanczos matrices, respectively. The unit vector of dimension $$$k\times 1$$$ is
represented by $$$e_k$$$ (=1 at the k^{th} row and 0 elsewhere).
The dimensions of $$$\mathbf{U_k}$$$ and $$$\mathbf{V_k}$$$ are $$$(n\times k)$$$ and $$$(n\times k)$$$,
with $$$k$$$ representing the number of iterations the bidiagonalization is
performed. Equivalently, it converts the minimization function into $$\Omega = || \mathbf{B_k}r_k - \beta_0e_1||^2
+ \lambda^2 ||r_k||^2.$$ This results in
update equation $$r_k = (\mathbf{B_k}^T\mathbf{B_k} + \lambda^2\mathbf{I})^{-1} β_0\mathbf{B_k}^Te_1); R = \mathbf{V_k}r_k.$$ Note that $$$k \ll n$$$ leading to dimensionality reduction of the original
problem (i.e. lesser number of linear system of equations to be solved)^{4}.

The $$$k$$$
is chosen based on the oscillation index of the deconvolved signal similar to
oSVD approach^{2}, giving it the name Oscillatory-limited QR method (oQR).

1. Baird A. E. et al. J. Cereb. Blood Flow Metab. 1998; 18: 583–609.

2. Wu O. et al. Magn. Reson. Med. 2003; 50:164–174.

3. Straka M. et al. J. Magn. Reson. Imaging; 2010; 32:1024–1037.

4. Shaw C. B. et. al. J. Biomed. Opt. 2013; 18:080501.

5. Kohsuke K. et al. Radiology. 2013; 267:201–211.

Figure-1:
Perfusion
map (MTT) obtained using the real stroke patient with diagnosis of right Middle
Cerebral Artery (MCA) stroke (region indicated with red arrow) DSC-MRI (1.5
Tesla) data using the standard methods of deconvolution (oSVD and FDD) along
with proposed method (oQR). Out of total 12 slices, here maps corresponding to
slice numbers 7 and 8 are presented.

Table-1: Computed Root Mean Square Error (RMSE) for
perfusion maps estimated using digital phantom data for each slice. The type
here represents the utilized residue function (R(t)).

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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