PALMA for DOSY Analysis


Theory

A polydisperse sample has to be analysed with a distribution $X(D)$ of diffusion coefficients: $$ I(q) = \int_{D_{\min}}^{D_{\max}} X(D) exp( -D \Delta q^2 ) dD $$ Determining the distribution $X(D)$ from $I(q)$ requires to solve the Laplace inversion of the $q^2$ dependence of $I(q)$.

Given that the experiment was performed over a series of $M$ values of $q$ and measured as a series of intensities $y_m$ for one given chemical shift value. The problem stated by the equation above can be discretised: $$ y_m = \sum_{n=1}^N x_n exp( -D_n \Delta q_m^2 ) $$ with $D_n$ sampling the $D_{\min}$ to $D_{\max}$ axis in an exponential manner.

As this expression is linear in $x_n$, it can be rewritten as follows: $$ Y = \pmb{H} {X} $$ where $Y = \{ y_m, 1 \leq m \leq M \}$ is the experimental series, $X=\{ x_n, 1 \leq n \leq N \}$ is the reconstructed distribution, and $\pmb{H}$ is a $M \times N$ matrix with values $\pmb{H}_{m,n} = exp( -D_n \Delta q_m^2 )$.

A general approach to solving this linear equation is to generate a solution $\hat{X}$ which solves the following constrained optimisation problem: $$ \text{minimize}_{X \in \mathbb{R}^N}\,{\Psi(X)} \quad \text{subject to}\quad \| \pmb{H} X - Y \| \le \eta $$ where $\eta$ is an estimate of the expected quality of the fit, based on an estimate of the experimental noise.

In order to favour both smooth and sparse shapes in the estimated signal, we use a regularisation defined as follows: $$ \Psi(X) = \lambda \text{ent}(X,a) + (1 - \lambda) \ell_1(X) $$ where $\text{ent}{(X,a)}$ is given by $$ \text{ent}(X,a) = \begin{cases} \sum_{n=1}^N \frac{x_n}{a} \log \left(\frac{x_n}{a}\right) & \mbox{if} \, \,x_n > 0 \\ 0 & \mbox{if} \, \, x_n = 0\\ +\infty & \mbox{elsewhere}, \end{cases} $$ corresponds to the MaxEnt regularisation with a flat prior $a>0$.

$\ell_{1}(X)$ is the $\ell_1$ norm of the vector $X$: $$ \ell_{1}(X) = \sum_{n=1}^N |x_n| $$ and $\lambda \in [0,1]$ allows to control the balance between the sparsity prior and the entropy prior.

Algorithm

This problem is solved by using a modified Douglas-Rachtford approach, using the proximal operator of the regularization function $\Psi$. The details of this algorithm are outside the scope of this presentation, and can be found in the publication.

The implementation requires the following parameters :

$\lambda$
the weight between $\ell_1$ and MaxEnt regularisation. typically chosen here to be $\lambda=0.1$ which gives the best general results, but modified in the special cases
$\eta$
the noise corrupting the experimental decay Estimated from a polynomial fit of the decay
$a$
the prior using in the Entropy definition, chosen here as $y_o \approx \sum_n{X_n}$

Program

The program takes the dataset provided in the zip file, analyse the acquisition the PULPROG and the parameters to determine the diffusion scale. Then using the difflist file as the list of the intensities of PFG, the analysis is done.

The program scans all the columns of the 2D, and process only columns corresponding to signal with a minimum SNR. Other columns are just set to zero. The program is naturally parallelized by submitting to each processor a subset of the columns to be processed.

The program is provided as a plugin to the spike spectroscopy program.

Spike can be downloaded on bitbucket, HERE. At this date, the devel branch should be used.

The python code of PALMA on which the plugin is based is HERE