Faculty 03 Center for Industrial Mathematics Mathematics/Computer Science Compressed Sensing in Imaging Mass Spectrometry Andreas Bartels 1 Joint work with ulk 1 , Dennis Trede 1 , 2 , Theodore Alexandrov 1 , 2 , 3 and Peter Maaß 1 , 2 Patrick D¨ 1 Center for Industrial Mathematics, University of Bremen, Bremen, Germany 2 Steinbeis Innovation Center SCiLS (Scientific Computing in Life Sciences), Bremen, Germany 3 MALDI Imaging Lab, University of Bremen, Bremen, Germany Copenhagen 26.-28. March 2014 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion 1
Faculty 03 Center for Industrial Mathematics Mathematics/Computer Science Outline 1 Imaging mass spectrometry (IMS) 2 Compressed sensing in IMS 3 Numerics: Implementation & Results 4 Conclusion Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion 2
Faculty 03 Center for Industrial Mathematics Mathematics/Computer Science 1 Imaging mass spectrometry (IMS) 2 Compressed sensing in IMS 3 Numerics: Implementation & Results 4 Conclusion Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion 3
Faculty 03 Center for Industrial Mathematics Mathematics/Computer Science Mass spectrometry (MS) Technique of analytical chemistry that identifies the elemental com- position of a chemical sample based on mass-to-charge ratio of charged particles. What is it used for? drug development detect/identify the use of drugs of abuse (dopings) in athlets identification of explosives and analysis of explosives in postblast residues ( puffer machine ) study the interaction of two (or more) bacterial cultures detection of disease biomarkers determination of proteins, peptides, metabolites and . . . Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion 4
Soft Laser Desorption + A breakthrough for the laser desorption method in its application to large biomolecules was reported at a symposium in Osaka in 1987, when Koichi Tanaka at the Shimadzu Corp. in Kyoto presented results of a mass spectrometric analysis of an intact protein. In two publica- tions and lectures in 1987-1988, Tanaka presented ionisation of proteins such as chymotrypsi- nogen (25,717 Da), carboxypeptidase-A (34,472 Da) and cytochrome c (12,384 Da) [12-14]. The missing link to make laser desorption work for large macromolecules was a proper com- bination of laser energy and wavelength with the absorbance and heat transfer properties of a chemical/physical matrix plus the molecular structure of the analytes in this matrix. Tanaka showed that gaseous macromolecular ions could be formed using a low-energy (nitrogen) laser, 2+ matrix could be used to volatilise small analyte molecules, but were without initial success for + + sample in matrix + + 2+ 2+ + + large molecules. M. Karas and F. Hillenkamp in Münster. In 1985, these scientists showed that an absorbing + work for small but polar molecules, like amino acids. This approach was further developed by Figure 2. The soft laser desorption process. + 2+ + m/z Rel. Intensity 60000 50000 40000 30000 10000 20000 + + 4 (13) Soft laser desorption (SLD) During the 1980s several groups tried to solve the volatilisation/ionisation problem of mass spectrometry using laser light as an energy source. By focusing a light beam onto a small spot of a liquid or solid sample, one hoped to be able to vaporise a small part of the sample and still avoid chemical degradation. V.S. Letokhov in Moscow demonstrated that the method could + Faculty 03 Center for Industrial Mathematics Mathematics/Computer Science MS methods Matrix-Assisted Laser Desorption/Ionization (MALDI) Secondary Ion Mass Spectrometry (SIMS) Desorption Electrospray Ionization (DESI) . . . 1 sample is cut and mounted on glass slide Sample 2 matrix solution is applied (acid crystalisation) 3 laser desorption of ’spots’ (grid ∼ 20 µ m – 200 µ m) TOF – Time-of-flight 4 computer aided analysis Laser of m / z -slices [Markides et al. ’02] Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion 5
Faculty 03 Center for Industrial Mathematics Mathematics/Computer Science Matrix-assisted laser desorption/ionization B) A) optical image intensity n y a r t c e p s 4,000 n x 5 mm 5,000 6,000 7,000 8,000 9,000 m / z [Alexandrov et al. ’11] Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion 6
Faculty 03 Center for Industrial Mathematics Mathematics/Computer Science Matrix-assisted laser desorption/ionization B) A) optical image intensity n y a r t c e p s 4,000 n x 5 mm 5,000 6,000 7,000 8,000 9,000 m / z [Alexandrov et al. ’11] Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion 6
Faculty 03 Center for Industrial Mathematics Mathematics/Computer Science Matrix-assisted laser desorption/ionization B) A) optical image intensity n y a r t c e p s 4,000 n x 5 mm 5,000 6,000 7,000 8,000 9,000 m / z [Alexandrov et al. ’11] Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion 6
Faculty 03 Center for Industrial Mathematics Mathematics/Computer Science Matrix-assisted laser desorption/ionization B) A) optical image C) m / z 4,966 intensity n y a r t c e p s 4,000 n x 5 mm D) m / z 6,717 5,000 6,000 7,000 m / z 4,966 8,000 9,000 m / z 6,717 m / z [Alexandrov et al. ’11] Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion 6
Faculty 03 Center for Industrial Mathematics Mathematics/Computer Science Matrix-assisted laser desorption/ionization B) A) optical image C) m / z 4,966 intensity n y a r t c e p s 4,000 n x 5 mm D) m / z 6,717 5,000 6,000 7,000 m / z 4,966 8,000 9,000 m / z 6,717 m / z [Alexandrov et al. ’11] n x × n y × c = ⇒ IMS data: Hyperspectral data X ∈ R ( m / z -spectra and -images) + Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion 6
Faculty 03 Center for Industrial Mathematics Mathematics/Computer Science The information disaster – data overflow Data X ∈ R n x × n y × c typically contains + n x · n y = 10 , 000 − 100 , 000 pixels c = 10 , 000 − 100 , 000 m / z -values 10 8 − 10 10 values, altogether Write X ∈ R n × c , n = n x · n y . + (General) Questions: y x � How to interpret the data? � What is the main information? � How to compress the data? � Where to compress the data? m / z Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion 7
Faculty 03 Center for Industrial Mathematics Mathematics/Computer Science Compression perspectives Mass spectrometry data X ∈ R n × c is typically large! + Nonnegative matrix factorization X ≈ MS , where M ∈ R n × ρ and S ∈ R ρ × c with ρ ≪ min { n , c } . + + min M , S α Θ 1 ( M ) + β Θ 2 ( S ) s.t. � X − MS � F ≤ ε � M – pseudo m / z -images, S – pseudo spectra Compressed Sensing Y = Φ X ∈ R m × c , + where Φ ∈ R m × n , m ≪ n . + min α Θ 1 ( X ) + β Θ 2 ( X ) s.t. � Y − Φ X � F ≤ ε � X Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion 8
Faculty 03 Center for Industrial Mathematics Mathematics/Computer Science 1 Imaging mass spectrometry (IMS) 2 Compressed sensing in IMS 3 Numerics: Implementation & Results 4 Conclusion Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion 9
Faculty 03 Center for Industrial Mathematics Mathematics/Computer Science Compressed Sensing in IMS Problems: MALDI measurements require several hours in time Data interpretation on full data Example: Rat brain dataset ∼ 5 hours Idea: Make use of compressed sensing with the knowledge of sparse m / z -spectra ( ℓ 1 minimization) and sparse m / z -images (TV minimization) Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion 10
Faculty 03 Center for Industrial Mathematics Mathematics/Computer Science Compressed Sensing in IMS Problems: MALDI measurements require several hours in time Data interpretation on full data Example: Rat brain dataset ∼ 5 hours Idea: Make use of compressed sensing with the knowledge of sparse m / z -spectra ( ℓ 1 minimization) and sparse m / z -images (TV minimization) A.B., P. D¨ ulk, D. Trede, T. Alexandrov and P. Maaß, � ”Compressed Sensing in Imaging Mass Spectrometry”, Inverse Problems , 29 (12), 125015 (24pp), 2013. Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion 10
Faculty 03 Center for Industrial Mathematics Mathematics/Computer Science CS-IMS model - The data IMS data is a hyperspectral data cube consisting of n x · n y ( m / z -)spectra of length c (number of channels), whereas n x and n y are the number of pixels in each coordinate direction. Thus, X ∈ R n x × n y × c . + Concatenating each m / z -image as a vector the data X becomes X ∈ R n × c , + where n := n x · n y . Each column corresponds to one m / z -image Each row to one m / z -spectrum. Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion 11
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