Froissart Doublets: Unequivocal Signal-Noise Separation by Pad-Based - PowerPoint PPT Presentation

Froissart Doublets: Unequivocal Signal-Noise Separation by Padé-Based Quantifications of Time Signals Applied to Cancer Diagnostics through Magnetic Resonance Spectroscopy (MRS) D ž evad Belki ć & Karen Belki ć Karolinska Institute, Stockholm, Sweden Approximation and extrapolation of convergent and divergent sequences and series CIRM Luminy - September 28, 2009 -- October 2, 2009

Outline Brief Introductory Framework Limitations in the Current Applications of MRS related to Reliance upon Conventional Fast Fourier Transform (FFT) Approaches to Signal Processing The Padé Approximant : Its Fast Padé Transform (FPT) and Froissart Doublets Direct Applications of the FPT for MRS cancer data

Key advantages of Magnetic Resonance Based-Diagnostics • Superior contrast among tissues compared to computerized tomography (CT) • No ionizing No ionizing • radiation radiation • Submillimeter resolution (1 order of magnitude superior to CT) • Multi-planar capabilities

? k y k x

INTERDISCIPLINARITY: PHYSICS, MATHEMATICS & MEDICINE k y Momentum representation Physics (k space) of the measured data ~ ( k x k , ) � y 2D FFT k x y ( x , y ) � 2D IFFT Mathematics Spatial localization via inverse 2D Fourier transform of the k space x

MRI versus MRS • Magnetic Resonance can also be used to obtain • Images of the human information about many body produced with other chemical Magnetic Resonance constituents ⇒ Magnetic Magnetic Imaging (MRI) primarily Resonance Resonance depict the distribution of Spectroscopy (MRS) Spectroscopy (MRS) water and fatty acid chains

( 1 ) � = � + � P L : the actual resonant frequency of a given � P constituent δ : the screening constant/chemical shift (dimensionless units) ( δ << 1) : Larmor Frequency � L

Proton MRS in Brain Tumour Diagnostics: Low Grade Astrocytoma vs. Non-malignant Lesion MRS spectrum from a small size tumour in a human brain MRS spectrum from a small size tumour in a human brain Cho Cho NAA NAA Cr Cr 4 4 3 3 2 2 1 1 0 0 Chemical Shift (ppm) Chemical Shift (ppm) From: Dr. Erik Akkerman, Academic Medical Centre Amsterdam

Normal MR Spectrum Brain MRS spectrum from a small size tumour in a human brain Cho Cho Low Grade NAA NAA Astrocytoma Cr Cr 4 4 3 3 2 2 1 1 0 0 Chemical Shift (ppm) Chemical Shift (ppm)

• It is remarkable that thus far MRS has made gigantic strides • This restricted metabolite in cancer diagnostics window stems directly by relying merely from the limitations of the conventional data upon a handful of analysis based upon the metabolites or their FFT and the ratios. accompanying post- processing via fitting and other, related phenomenological phenomenological approaches. Belki ć D ž , Belki ć K, J Math Chem 42, 1-35 (2007).

Limitations in the Current Applications of MRS in Cancer Diagnostics • Lack of reliable quantification : Use of non-unique fitting algorithms ⇒ Reliance upon metabolite ratios & other semiquantitive or qualitative methods—lack of norms; difficult inter-center comparisons • Small Number of Observable Compounds Difficult to resolve overlapping resonances —although these may be the most diagnostically informative based upon in vitro studies • Resolution & signal/noise ratio (SNR)

Using the FFT: A spectrum is given as a single polynomial F( ) with pre - assigned angular frequencie s, � � k k whose minimal separation 2 � k / T � = min is determined by the epoch T . The FFT spectrum is defined only on the Fourier grid points k T � = ± k where ( k 0,1,2,3 N - 1), = … N is the signal length, T total acquisitio n time (epoch) = T N and sampling time. = � � =

Limitations of the FFT • Low resolution • Linearity: imports noise as intact from the measured time domain data to the analyzed frequency domain • Non-parametric: Supplies only a shape spectrum • Quantification by FITTING: NON-UNIQUE • Number of metabolites guessed prior to fitting This may ⇒ false peaks (over-fitting), as well as true metabolites being undetected (under-fitting)

How could mathematics play such a critical role in medical diagnostics? Because data encoded directly from patients by means of existing imaging techniques, e.g. CT, PET, as well as MRI and MRS are not amenable to direct interpretation, which therefore need mathematics via signal processing.

Recorded Data: Time Domain Abscissa in milliseconds (ms), ordinate is intensity in arbitrary units (au). Data Transformed into the Frequency Domain: Absorption total shape spectrum, abscissa in parts per million (ppm), indicating the frequency at which metabolites resonate : Fast Fourier Transform (FFT) Absorption total shape spectrum and underlying component spectrum. Provided by the fast Padé transform (FPT), but not by the FFT

Among the advanced signal processing methods, the fast Padé transform (FPT) is particularly suitable for improving the diagnostic yield of in vivo MRS Belki ć D ž , Quantum Mechanical Signal Processing and Spectral Analysis , Institute of Physics Publishing, Bristol, 2004. Belki ć K, Molecular Imaging through Magnetic Resonance for Clinical Oncology, Cambridge International Science Publishing, Cambridge, 2004 . Belki ć D ž , Belki ć K, Signal Processing in Magnetic Resonance Spectroscopy with Biomedical Applications, Taylor & Francis Publishing, 2009.

THE FAST PADÉ TRANSFORM A non-linear, rational polynomial as the Padé approximant to the exact spectrum of the encoded time signal points Extracts diagnostically important quantitative information, otherwise unavailable in full depth using conventional estimators for in vivo MRS

The FPT represents the input spectrum as a polynomial quotient: 1 1 N 1 P ± ( z ± ) � 1 n K F ( z � ) c z � � = � n 1 N Q ± ( z ± ) n 0 K = i � �� c time signal, z e (harmonic variable) � = n 1 1 ± ± ( z z ) P ± ( z ± ) p ± K � k , P K K � = (Canonical form) 1 1 ± ± ± ± ± Q ( z ) q ( z z ) � k 1 K K = k , Q P ± ( z ± ) 0 , Q ± ( z ± ) 0 (Characteristic equation) = = K k , P K k , Q Belki ć D ž , Phys Med Biol 51, 6483-6512 (2006),

1 1 P ± ( z ± ) K d ± z ± K k � (Partial fractions) = 1 1 Q ± ( z ± ) z ± z ± � k 1 K k , Q = P ± ( z ± ) d ± = ' 1 1 K k , Q ± ± ± ± Q ( z ) Q ( z ) d = (Amplitudes) K K k 1 ± ' dz Q ± ( z ) K k , Q ± ± ( z z ) ± K p � k , Q k ' , P K d ± � = k q ± ( z ± z ± ) � k ' 1 K = k , Q k ' , Q k k ' � K k ± ± c d ( z ) � = Inversion of the spectrum n k k , Q k 1 = i i � � � � � � z e , z e k , Q k , P = = k , Q k , P Belki ć D ž , Adv Quantum Chem 56, 95-179 (2009).

• The optimal mathematical model for the frequency spectrum of these time signals is prescribed quantum-mechanically quantum-mechanically , to be the ratio of two polynomials, i.e. the FPT. • Just as in the time domain where quantum mechanics predicts the form of the time signal as the sum of damped exponentials, by virtue to the time-frequency dual representation, • The same physics automatically prescribes that the frequency spectrum is given by the Padé quotient of two polynomials. • This is the origin of the unprecedented algorithmic success of the FPT, via its demonstrable, exact reconstructions . Belki ć D ž , Quantum Mechanical Signal Processing and Spectral Analysis , Institute of Physics Publishing, Bristol (2004)

Via the FPT representations, pole- zero cancellation (Froissart doublets) can be used to unequivocally distinguish true and spurious resonances . P / Q : z z = k , Q k , P This is demonstrated not only in the noise-free case, but also for MR time signals corrupted by noise at a level similar to realistic encoding conditions. Belki ć D ž , Phys Med Biol 51, 6483-6512 (2006). Belki ć D ž , Adv Quantum Chem 56, 95-179 (2009) .

The computation is carried out by gradually and systematically increasing the degree of the Padé polynomials. As this degree changes, the reconstructed spectra fluctuate until stabilization occurs. The polynomial degree at which the predetermined level of accuracy is achieved represents the sought exact number of resonances. The quotient form P / Q from the FPT leads to cancellation of all the terms in the Padé numerator P and denominator Q polynomials, when the computation is continued after the stabilized value of the order in the FPT has been attained.

Illustration of the concept of Froissart doublets for noise- corrupted time signals derived from realistic MRS data encoded at 1.5T Gauss-distributed zero-mean noise was added with a standard deviation σ = 0.00289 rms (root mean square) of the noiseless time signal (0.00289 is approximately 1.5% of the height of the weakest resonance in the spectrum (#13: aspartate at 2.855 ppm (parts per million), and this is considered realistic for encoded data, as well as being sufficient to illustrate the principles of Froissart doublets. Belki ć D ž , Phys Med Biol 51, 6483-512 (2006). Belki ć D ž , Adv Quantum Chem 56, 95-179 (2009).