DMRI has been established as a useful tool to obtain voxel-level information on tissue micro-structure. An important quantity measured in dMRI is the apparent diffusion coefficient (ADC), and it has been well established by in-vivo brain imaging experiments that the ADC depends significantly on the diffusion time [1,2]. We derive an explicit formula for the time-dependent ADC for the purpose of using it to estimate geometrical parameters.

Suppose $M(\mathbf{x},\Delta,\delta,\mathbf{u}, g)$ is the complex transverse magnetization in a voxel, where $\delta$ and $\Delta$ are the pulse duration and the delay between pulses of the PGSE sequence, $g$ is the magnitude of the diffusion-encoding magnetic field gradient, $\mathbf{u}$ the normalized gradient direction and, $\gamma$ the gyro-magnetic ratio, the time-dependent ADC is defined as

$$ADC(\Delta,\delta,\mathbf{u}):= \frac{1}{\gamma^2 \delta^2(\Delta-\delta/3)}\left.\frac{\partial}{\partial (g^2)} \left (\int_{\mathbf{x} } M(\mathbf{x},\Delta,\delta,\mathbf{u}, g)d\mathbf{x}\right)\right\vert_{g=0}.$$

A model for the time-dependent ADC was derived and validated in [3], in the regime of low biological cell membrane permeability (meaning water exchange between cells and extra-cellular space is neglected). The model requires the solution of a homogeneous diffusion equation with a time-dependent flux in each biological cell. Suppose the cell is denoted by $\Omega$, with surface area $|\partial \Omega|$ and volume $|\Omega|$, the model is

$$\frac{ADC^{ref}(\Delta,\delta,\mathbf{u})}{\sigma} =1-\frac{1}{\gamma^2\delta^2(\Delta-\frac{\delta}{3})} \int_0^{TE} F(t) p(t)dt,$$ where $\sigma$ is the intrinsic diffusion coefficient, TE is the echo time, and

$$p(t)=\frac{1}{|\Omega|}\int_{\Omega}\nabla \omega(\mathbf{x},t,\mathbf{u}) \cdot \mathbf{u} d\mathbf{x}.$$

We denote this quantity by $ADC^{ref}$ because it has been validated [3] as a good approximation of the ADC of the Bloch-Torrey Equation [4]. In the above formula, $\omega$ is the solution of

$$\frac{\partial}{\partial t} \omega(\mathbf{x},t,\mathbf{u}) - div\left(\sigma \nabla\omega(\mathbf{x},t,\mathbf{u})\right) = 0, \quad \mathbf{x}\in\Omega\\ \sigma\nabla\omega(\mathbf{x},t,\mathbf{u}) \cdot\nu=F\sigma\mathbf{u}\cdot\nu,\quad \mathbf{x}\in\partial \Omega \\ \omega(\mathbf{x},0,\mathbf{u}) = 0, \quad \mathbf{x}\in\Omega$$

where $\nu$ is external normal vector to $\Omega$.

In the form described above, $ADC^{ref}(\Delta,\delta,\mathbf{u})$ cannot be easily used to invert for model parameters. Thus, we proceed to obtain a simpler form of $ADC^{ref}(\Delta,\delta,\mathbf{u})$. During the first pulse we write $\omega$ as a single layer potential and obtain that

$$p(t)=\frac{4\sqrt{\sigma}}{3\sqrt{\pi}}\frac{A_\mathbf{u}}{|\Omega|}t^{3/2}+ O(t^2), \quad t\in[0,\delta],$$

where $A_\mathbf{u}:=\int_{\partial \Omega}(\mathbf{u} \cdot\nu)^2ds_\mathbf{x}$ is the projection of the surface area in the gradient direction. Between the pulses we use the eigenfunctions decomposition of $\omega$ to obtain

$$p(t)=\delta+\sum_{n=1}^{\infty} -\delta \left(a_n(\mathbf{u})\right)^2 \lambda_n e^{-\lambda_n\sigma(t-\delta)}, \quad t\in[\delta,\Delta],$$

where $\lambda_n$ are the Neumann eigenvalues associated to the Laplace operator, and $a_n(\mathbf{u}):=\int_{\Omega}\mathbf{u} \cdot \mathbf{x} \phi_n(\mathbf{x}) d\mathbf{x}$ is the first moment of the eigenfunction $\phi_n$ in the direction $\mathbf{u}$. In the second pulse we use a combination of the eigenfunctions expansion and the single layer potential.

If we integrate the ADC for all possible gradient directions $\mathbf{u}\in\mathbf{R}^{dim},\|\mathbf{u}\|=1$, we can obtain a new ADC formula that is independent of the orientation of the biological cells. Using $$\frac{1}{\int_{\mathbf{u}}d\mathbf{u}}\int_{\mathbf{u}} A_{\mathbf{u}} d\mathbf{u} = \frac{|\partial \Omega|}{dim},$$ and defining: $$k_n^2:=\frac{1}{|\Omega|}\frac{1}{\int_{\mathbf{u}}d\mathbf{u}}\int_{\mathbf{u}}\left(\int_{\Omega}\mathbf{u}\cdot\mathbf{x}\phi_n(\mathbf{x}) d\mathbf{x} \right)^2 d\mathbf{u},$$ we formulate the following explicit approximate formula for the time-dependent ADC:

$$\frac{ADC^{new}(\Delta,\delta)}{\sigma} = \frac{ \frac{\delta}{6}}{\Delta-\frac{\delta}{3}}-\frac{8\sigma^{1/2}\delta^{3/2}}{35\sqrt{\pi}(\Delta-\frac{\delta}{3})}\frac{E}{dim}-\sum_{n=1}^\infty \frac{-\delta k_n^2}{\sigma \delta^2(\Delta-\frac{\delta}{3})}\left(\delta+\frac{e^{-\lambda_n\sigma\Delta}\left(1-e^{\lambda_n\sigma\delta}\right)}{\lambda_n\sigma} \right),$$ $E = \frac{A}{|\Omega|}$ being the surface to volume ratio.

We validated our formula $ADC^{new}(\Delta,\delta)$ using computer simulations on a set of 2 ellipse-shaped cells $\Omega^1$ and $\Omega^2$ . See Fig 1. We solved the diffusion equation (using the PDEtoolbox of Matlab) for $\omega(\mathbf{x},t,\mathbf{u})$ in 18 gradient directions evenly distributed between 0 and 180 degrees and obtained $\frac{1}{18}\sum_{i=1}^{18}\frac{ADC^{ref}(\Delta,\delta,\mathbf{u}_i)}{\sigma}$, for fixed $\delta$ and 15 different values of $\Delta$ in both $\Omega^1$ and $\Omega^2$. We then averaged the ADC's over the two domains to get a total of 15 data points as inputs in our fitting procedure where we kept two terms in the infinite sum. In Fig 2, we show the 15 data points for $\delta=1.5$ms and the best fit using 5 parameters, $E, \lambda_1,k_1,\lambda_2, k_2$. We show the true and estimated parameters in the Table 1. We consider the true $\lambda_1, \lambda_2$ to be the first non-zero Neumann eigenvalue of $\Omega^1$ and $\Omega^2$, respectively, and $E$ to be the average surface to volume ratio of the two ellipses. For two choices, $\delta=0.5$ms, $\delta=1.5$ms, we obtained good estimates for the eigenvalues (relative errors less than 18%) and the first moments $k_1,\;k_2$ (errors between 0.5-16%). The surface to volume ratio $E$ can be estimated well using the smaller $\delta=0.5$ms (error 22%).

We have obtained an explicit formula for the time dependent ADC and, using the ADCs at multiple diffusion times and gradient directions, we estimated the surface to volume ratio and the eigenvalues as well as the first moment of the dominant eigen-functions. This information can potentially give us useful information on the tissue micro-structure, as they are intricately linked to the geometrical shapes of the biological cells in the imaged tissue.