## Tuesday, 22 July 2008

### Bayesian DOSY: a New Approach to Diffusion Data Processing

Yesterday I blogged about basic concepts on DOSY NMR. From an experimental stand point, the pulse sequences are not very complex, being the Pulse Field Gradient Stimulated Echo (PFGSE) experiment proposed by Tanner one of the basic pulse sequences used to determine diffusion by NMR. This pulse sequence can be considered as a building block for a number of extended pulse sequences designed to minimize some sources of artefacts caused by thermal convection currents in the sample, background gradients, radiation damping, zero order coherences in strongly coupled spin systems, etc. Other effects such as J-modulation and Cross-Relaxation have also been considered.

The tricky part comes in the mathematical DOSY transformation. As I mentioned earlier, there exists different approaches, each of them with their own strengths and weaknesses. Today I would like to introduce a new method for DOSY transformation based on Bayesian Theory which has been implemented in our software Mnova as a result of our collaboration with Stan Sykora. We call this new algorithm BDT (Bayesian DOSY Transform)

A formal description of the Bayesian theory and its particular implementation for DOSY transformation is beyond this blog entry, but we are currently writing an article which will give all the details of the method as implemented in Mnova.
In short, this Bayesian approach assigns an a-priori probability (this is the key word in Bayesian context) to the elements in a space defined by the entities to be estimated. In this context, an entity is the pair (f,d) where f is frequency and d is a mono-diffusion coefficient. Actually, it should be (f,d,w) where w is a weight (intensity), but this weight is factored out by finding its optimal value (since the dependence is linear, this can be done explicitly). So we are actually finding the a-posteriori probability for the sentence there is a component - no matter how intense - at f that has such and such d. Next, a final normalization process modifies the probability still further (composite probability) by looking at the intensity in the spectrum at location f (this would be real probability if the spectrum were normalized to 1).

Stan Sykora will be presenting the mathematical background and physical insights necessary to understand this new method as well as some real life examples processed with Mnova in the GIDRM conference to be held in Bressanone (Brixen) next September.

So how does the algorithm perform? We first tested the algorithm by simulating with Matlab a diffusion experiment with 2 synthetic peaks with frequencies of 100 and 200 Hz and diffusion coefficients of 1 and 0.01 (dimensionless).

Application of BDT yields the following DOSY spectrum:

The synthetic DOSY spectrum was changed in order to include artificial noise and different degrees of peak overlap and diffusion distances. We’ve found the performance of the algorithm to be excellent in all the cases analyzed.

Next we implemented the algorithm in Mnova and we applied it with a sample of an aqueous solution of potassium N-methyl-N-oleoltaurate (a surfactant) with TSP at 23 C (The original Varian FID file has been obtained from the VARIAN NMR USER GROUP LIBRARY which was submitted by Brian Antalek as a sample for this DECRA algorithm). This is the original raw data after automatic Fourier Transform, phase and baseline correction in Mnova:

After applying BDT, we obtain a DOSY spectrum in which the three components are clearly well resolved in the diffusion dimension and the absolute values are in accordance with the expected values

Below you can see the DOSY spectrum of a mixture of Caffeine + 2-EthoxyEthanol + Water acquired in a Bruker instrument by my friend Andy Soper.

Main properties of the algorithm

Compared to other approaches, we believe that this Bayesian method implemented in Mnova appears extremely promising. It automatically avoids having exact, unnatural zeros anywhere in the resulting DOSY map since every point of the 2D map has a well defined value of statistical congruence with the data. Moreover, the BDT maps show ‘normal’ line widths in the f-direction, correctly positioned and resolved peaks in the d-direction and quantitatively correct horizontal and vertical projections – a combination difficult to achieve by any other means.
The approach can be easily extended to non-exponential cases arising from overlapping lines and, like all Bayesian methods, incorporate additional information available from other sources (a-priori knowledge). Likewise, it is possible to place a statistical premium on alignment of spectral peaks along horizontal lines in the [f,d] plot.

Do you want to try it out yourself?

The algorithm is readily available in the alpha stage in Mnova. From this post, I would like to offer this special version to anyone interested in trying the BDT algorithm with his own data sets. I will very much appreciate any feedback from you.
Just write to me at the email address below and I will give you the instructions on how to get the software.

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