A key advantage of rational polynomials as a quotient of two polynomials is their automatically built-in polar representation. Rational polynomials are thus the most suitable for describing functions with peaks, such as those in magnetic resonance spectra (MRS). The Padé approximant is the most important of these rational polynomials because of its uniqueness for the power series expansion of the given function. Non-parametric analysis through the fast Padé transform (FPT) is a convenient initial step for processing MRS time signals, since it can be carried out once the expansion coefficients of the polynomials are generated from the time signal, without polynomial rooting. We applied the FPT to synthesized MRS time signals similar to those encoded in vitro from breast cancer. Padé-based non-parametric envelopes generated with and without spectra partitioning are studied. Comparisons of these total shape spectra with the related Padé component spectra were made. Phosphocholine (PC) and phosphoethanolamine (PE), separated by a mere 0.001 parts per million of chemical shift, were resolved in the non-parametric partitioned envelopes. However, in the non-parametric FPT without partitioning, a single composite smooth Lorentzian peak (PC $$+$$ + PE) was generated in envelopes, with no indication whatsoever that two resonances were present. Subsequent parametric analysis (quantification) by the FPT confirmed that PC completely underlies the much more abundant PE. This problem, chosen to illustrate the usefulness in the non-parametric partitioned envelopes, has clinical implications. Namely, PC is a cancer biomarker which thus far was not identified with in vivo MRS using envelopes from the conventional Fourier-based (single-polynomial) processing.
Journal of Mathematical Chemistry – Springer Journals
Published: Jul 6, 2017
It’s your single place to instantly
discover and read the research
that matters to you.
Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.
All for just $49/month
Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly
Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.
Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.
Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.
All the latest content is available, no embargo periods.
“Hi guys, I cannot tell you how much I love this resource. Incredible. I really believe you've hit the nail on the head with this site in regards to solving the research-purchase issue.”Daniel C.
“Whoa! It’s like Spotify but for academic articles.”@Phil_Robichaud
“I must say, @deepdyve is a fabulous solution to the independent researcher's problem of #access to #information.”@deepthiw
“My last article couldn't be possible without the platform @deepdyve that makes journal papers cheaper.”@JoseServera