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) Dževad Belkić & Karen Belkić Karolinska Institute, Stockholm, Sweden

Approximation and extrapolation of convergent and divergent sequences and series CIRM Luminy - September 28, 2009 -- October 2, 2009

Outline

Direct Applications of the

FPT for MRS cancer data

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

Key advantages of Magnetic Resonance Based-Diagnostics

• Superior contrast among

tissues compared to computerized tomography (CT)

• Submillimeter resolution

(1 order of magnitude superior to CT)

• Multi-planar capabilities
• No ionizing

No ionizing radiation radiation

?

kx ky

) , ( ~

y x k

k

• Spatial localization via inverse 2D

Fourier transform of the k space Momentum representation (k space)

• f the measured data

kx ky x y

) , ( y x

• Physics

Mathematics INTERDISCIPLINARITY: PHYSICS, MATHEMATICS & MEDICINE 2D FFT 2D IFFT

MRI versus MRS

• Images of the human

body produced with Magnetic Resonance Imaging (MRI) primarily depict the distribution of water and fatty acid chains

• Magnetic Resonance can

also be used to obtain information about many

• ther chemical

constituents ⇒ Magnetic Magnetic Resonance Resonance Spectroscopy (MRS) Spectroscopy (MRS)

) 1 (

• +

=

L P

:

P

• :

L

• the actual resonant frequency of a given

constituent δ : the screening constant/chemical shift (dimensionless units) (δ << 1) Larmor Frequency

Proton MRS in Brain Tumour Diagnostics: Low Grade Astrocytoma vs. Non-malignant Lesion

MRS spectrum from a small size tumour in a human brain Cho Cr NAA

4 3 2 1 Chemical Shift (ppm)

MRS spectrum from a small size tumour in a human brain Cho Cr NAA

4 3 2 1 Chemical Shift (ppm)

From: Dr. Erik Akkerman, Academic Medical Centre Amsterdam

MRS spectrum from a small size tumour in a human brain Cho Cr NAA

4 3 2 1 Chemical Shift (ppm)

Cho Cr NAA

4 3 2 1 Chemical Shift (ppm)

Normal MR Spectrum Brain Low Grade Astrocytoma

• It is remarkable that

thus far MRS has made gigantic strides in cancer diagnostics by relying merely upon a handful of metabolites or their ratios.

• This restricted metabolite

window stems directly from the limitations of the conventional data analysis based upon the FFT and the accompanying post- processing via fitting and

• ther, related

phenomenological phenomenological approaches.

Belkić Dž, Belkić 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

time. sampling and (epoch) n time acquisitio total length, signal the is 1),

• 0,1,2,3

( where points grid Fourier

• n the
• nly

defined is spectrum FFT The . epoch by the determined is /

• 2

separation minimal whose s, frequencie angular assigned

• pre

with ) F(

min

= = = … = ± = =

• N

T T N N k T k T T k

k k k

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

• f in vivo MRS

Belkić Dž, Belkić K, Signal Processing in Magnetic Resonance Spectroscopy with Biomedical Applications, Taylor & Francis Publishing, 2009. Belkić Dž, Quantum Mechanical Signal Processing and Spectral Analysis, Institute of Physics Publishing, Bristol, 2004. Belkić K, Molecular Imaging through Magnetic Resonance for Clinical Oncology, Cambridge International Science Publishing, Cambridge, 2004.

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

) ( ) ( ) ( ) (

, 1 , 1 1 1 1 ± ± ± ± = ± ± ± ± ± ±

• =
• Q

k P k K k K K K K

z z z z q p z Q z P

The FPT represents the input spectrum as a polynomial quotient:

) ( ) ( 1 ) (

1 1 1 1 ± ± ± ±

• =
• =
• z

Q z P z c N z F

K K n N n n

variable) (harmonic signal, time

• i

n

e z c

• =
• (Canonical form)

) ( , ) (

, ,

= =

± ± ± ± Q k K P k K

z Q z P

(Characteristic equation)

Belkić Dž, Phys Med Biol 51, 6483-6512 (2006),

• =

± ± ± ± ± ± ± ±

• =

K k Q k k K K

z z z d z Q z P

1 , 1 1 1 1

) ( ) (

(Partial fractions)

) ( ) (

, ' , Q k K Q k K k

z Q z P d

± ± ± ± =

(Amplitudes)

) ( ) (

1 1 1 ' ± ± ± ± ±

= z Q dz d z Q

K K

• =
• ±

± ± ± ± ± ±

• =

K k k k Q k Q k P k Q k K K k

z z z z q p d

1 ' ' , ' , , ' ,

) ( ) (

k Q k K k k n

z d c ) (

, 1 ± = ±

• =

Inversion of the spectrum

Belkić Dž, Adv Quantum Chem 56, 95-179 (2009).

• P

k Q k

i P k i Q k

e z e z

, ,

, ,

,

• =

=

• 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.

Belkić Dž, 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. 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.

Belkić Dž, Phys Med Biol 51, 6483-6512 (2006). Belkić Dž, Adv Quantum Chem 56, 95-179 (2009) .

P k Q k

z z Q P

, ,

: / =

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. The computation is carried out by gradually and systematically increasing the degree of the Padé polynomials.

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

• f Froissart doublets.

Belkić Dž, Phys Med Biol 51, 6483-512 (2006). Belkić Dž, Adv Quantum Chem 56, 95-179 (2009).

Froissart doublets (pole-zero cancellations) in the frequency range from 0 to 6 ppm. Froissart doublets: Amplitude = 0 The absolute value of the amplitudes of genuine resonances ≠ 0.

) ( , ,

FPT by the genuine are that those from separated totally are doublets Froissart The

+

=

P k Q k

z z P/Q

SIGNAL- NOISE SEPARATION (SNS)

Belkić Dž, Phys Med Biol 51, 6483-512 (2006).

SNS: Frequency SNS: Amplitude

PADÉ

S N N S

The number of spurious resonances is always several times greater than the true metabolites. Here, e.g. of 128 resonances, only 25 were genuine. 103 or over 80% were spurious.

) (

FPT +

PADÉ SPECTRA OCCIPITAL GREY MATTER COMPONENT ENVELOPE

Belkić Dž, Phys Med Biol 51, 2633-70 (2006).

The Noiseless Case

Belkić Dž, Belkić K, J Math Chem 45, 563-97 (2009).

) (

FPT +

SEPARATION OF NOISE from PHYSICAL SIGNAL A solution of one of the most difficult problems hampering wider implementation of MRS in clinical oncology

Breast cancer, fibroadenoma and normal breast tissue*, demonstrating the ability to unequivocally resolve and precisely quantify extremely closely lying resonances, including phosphocholine, a marker of malignant transformation of the breast.

Direct Applications of the fast Padé transform for Oncology—to MRS data as encoded in vitro from:

K total = 750 K spurious = 741 K genuine = 9

Belkić Dž, Belkić K, J Math Chem 45, 790-818 (2009). *Gribbestad IS, et al. Anticancer Res, 19, 1737 – 1746 (1999).

Metabolite Concentration Map for Data from Breast Cancer Reconstructed via the FPT

Prior to Convergence NP = 1000: Phosphocholine Undetected, Phosphoethanolamine

• verestimated

At Convergence: NP = 1500: [Phosphocholine] correctly assessed Past Convergence:

NP = 2000 & thereafter

Completely stable

For data from benign and malignant ovarian fluid*, the FPT dramatically improved SNR

and highly accurate determination of key metabolite concentrations for identifying

• varian cancer

K total = 32 K spurious= 20 K genuine = 12

Belkić Dž, Belkić K, J Math Chem 43, 395-425 (2008). *Boss EA et al., NMR Biomed, 13, 297 – 305 (2000).

Belkić K, Nucl Instrum Methods Phys Res A 580, 874-80 (2007), Belkić Dž, Belkić K, J Math Chem 43, 395-425 (2008).

Benign Ovarian Cyst Malignant Ovarian Cyst FOURIER SPECTRA PADÉ SPECTRA

Normal glandular and stromal prostate and prostate cancer*, with unequivocal resolution and exact quantification of numerous overlapping resonances including multiplets of metabolites that distinguish cancerous prostate from normal tissue from these 2 sites.

Belkić Dž, Belkić K, J Math Chem 45, 819-58 (2009) .

K total = 350 K spurious= 323 K genuine = 27

*Swanson MG, et al. Magn Reson Med, 55, 1257 – 1264,2006.

We have also applied the FPT to time

signals encoded in vivo* via MRS at 7T from the brain of a healthy volunteer. Convergence rates of FFT and FPT are compared, using truncated and full signal-lengths N/M (M = 1 – 32). For M > 1 FFT employs N - N/M zeros for completion to N=2048, but FPT does not.

*Tkáč I, et al., Magn Reson Med 46, 451 – 456 (2001)

Belkić Dž, Nucl Instr Meth Phys Res A, 525, 366 – 371 (2004).

Resolution is much better in

the FPT than the FFT at any truncation M > 1. Marked improvement is seen at N/16 = 128, where the FPT resolves some 5 metabolites, which appear in the FFT as broad overlapping bumps. At N/M (8 ≤ M ≤ 32) heights and widths of the major metabolites are predicted much more accurately by the FPT. At N/2 = 1028 the FPT is in complete agreement (up to random noise) with both FFT and FPT computed at full signal length N = 2048.

This steady

convergence of the FPT sharply contrasts with most non-linear estimators that wildly oscillate before eventually stabilizing and

• nly perhaps

converging.

Belkić Dž, Nucl Instr Meth Phys Res A, 525, 366 – 371 (2004). Belkić Dž, Belkić K, Signal Processing in Magnetic Resonance Spectroscopy with Biomedical Applications, Taylor & Francis Publishing, 2009.

Conclusions / Perspectives

We have only begun to tap the potential of MRS for cancer diagnostics. Advances in signal processing based

• n the fundamental theory of

quantum mechanics will be vital for further progress.

With the fast Padé transform (FPT), pole-zero cancellation (Froissart doublets) can be used to unequivocally disentangle true from spurious resonances for MR time signals corrupted by noise. This offers the unique solution to one of the most difficult problems that has hampered wider implementation of MRS in clinical oncology. Moreover, our signal-noise separation is general, as it applies to any time signal.

CONCLUSIONS

This work is supported by King Gustaf V Jubileums Fund for Cancer Research and the Swedish Cancer Society