Synopsis

We present here a simple and efficient algorithm for designing of diffusion gradient profiles allowing one to implement prescribed theoretical properties. This algorithm generalizes the well-known sine and cosine decomposition and can handle various experimental constraints. Relying on recent diffusion MRI theoretical advances, we apply it to the challenging problem of estimating the surface-to-volume ratios in anisotropic media.

Introduction

Methods

We consider a spin-echo experiment with an applied diffusion encoding gradient $\boldsymbol{g}(t)$ from $t=0$ to the echo time $t=T$. The main idea of the algorithm is to choose a family of functions $\left(f_1(t),\ldots, f_k(t)\right)$ (for example, (co)sines, polynomials, ...) and write the three-dimensional gradient $\boldsymbol{g}(t)$ as a linear combination of these basis functions
$$\begin{pmatrix}g_x(t)\\g_y(t)\\g_z(t)\end{pmatrix}=\mathcal{X}\begin{pmatrix}f_1(t)\\\vdots\\f_k(t)\end{pmatrix}\;,$$
where $\mathcal{X}$ is the $3\times k$ weights matrix that we seek to optimize. This is a generalization of the classical sine and cosine decompositions. Some constraints on the gradient profile may be satisfied by a suitable choice of the basis functions $\left(f_1(t),\ldots, f_k(t)\right)$. For example, one can ensure the smoothness of the gradient profile by choosing a family of smooth functions such as polynomials. In the same way, the refocusing condition $\int_0^T \boldsymbol{g}(t)\,\mathrm{d}t=\boldsymbol{0}$ can be achieved by choosing zero-mean functions, for example, (co)sines with an integer number of periods. It is also possible to add some constraints as a part of the optimization process (see below some examples).

It was recently derived in Ref⁴ that one can estimate the surface-to-volume ratio of an anisotropic medium by using gradient sequences that satisfy a particular ``isotropy’’ matrix condition, $\mathcal{F}^{(3)}\propto I$, where
$$\mathcal{F}^{(m)}=-\frac{\gamma^2}{2}\int_0^T\int_0^T\boldsymbol{g}(t_1)\boldsymbol{g}(t_2)\lvert t_2 - t_1 \rvert^{m/2} \,\mathrm{d}t_1 \,\mathrm{d}t_2$$
(with $m=3$), and $I$ is the $3\times 3$ unit matrix. We used the tensor product notation: if $\boldsymbol{a}$ and $\boldsymbol{b}$ are vectors, then $\boldsymbol{a}\otimes\boldsymbol{b}$ is a matrix with components $\left(\boldsymbol{a}\otimes\boldsymbol{b}\right)_{ij}=a_ib_j$. Note that original isotropic diffusion weighting sequences<sup>3</sup> satisfy a different condition, namely $\mathcal{F}^{(2)}\propto I$ (one can indeed show that $\mathcal{F}^{(2)}$ is actually the $b$-matrix, with $\mathrm{Tr}\left(\mathcal{F}^{(2)}\right)=b$. We thus search for a 3D gradient profile $\boldsymbol{g}(t)$ that makes the $\mathcal{F}^{(3)}$ matrix ``isotropic''. Moreover, one can incorporate other constraints such as vanishing $\mathcal{F}^{(4)}$ matrix for a more accurate estimation of the surface-to-volume ratio.

From a numerical point of view, the computation of the $\mathcal{F}^{(m)}$ matrices involves a tradeoff between precision and speed. Improving the speed of computations is especially important in the context of optimization algorithms, which usually require numerous iterations. By pre-computing the $k\times k$ matrices $\mathcal{A}^{(m)}$:
$$\mathcal{A}^{(m)}_{i,j}=-\frac{\gamma^2}{2}\int_0^T\int_0^T f_i(t_1)f_j(t_2)\lvert t_2 - t_1 \rvert^{m/2} we \,\mathrm{d}t_1 \,\mathrm{d}t_2\;,$$
we can compute the $\mathcal{F}^{(m)}$ matrices directly from $\mathcal{X}$ by $\mathcal{F}^{(m)}=\mathcal{X}\mathcal{A}^{(m)}\mathcal{X}{}^T$. The initial computation of the $\mathcal{A}^{(m)}$ matrices is the most time-consuming step but it has to be done only once. The consequent computations of $\mathcal{F}^{(m)}$ involve just matrix multiplications whose size is given by the size of the family $\left(f_1(t),\ldots,f_k(t)\right)$, independent of the numerical sampling of the time interval $[0,T]$ that controls the accuracy of the computations. A similar matrix representation is applicable to any linear or bilinear form of the gradient profile. Common examples include: imposing zeros of the gradient; flow-compensation or motion artifact suppression conditions; controlling heat generation of the sequence; computing the $b$-value.

Results

Conclusion

Acknowledgements

References