Estimating Particle Non-sphericity from the Fourier Spectrum of Its Light-scattering Pattern Andrey Romanov Laboratory of Cytometry and Biokinetics Voevodsky Institute of Chemical Kinetics and Combustion SB RAS
Introduction • Measuring light scattering by a single particle is a powerful method for its characterization • Most methods rely on the assumption of a particle shape model • Deviations from the proposed shape model cause many problems • Consider the simplest case – the deviation from a homogeneous sphere 3000 experiment Mie theory 2500 2000 ? Intensity 1500 1000 500 0 10 20 30 40 50 60 70 2 Scattering angle, degree
Spectrum of light-scattering 2000 pattern Spectrum amplitude 1500 𝑇 𝑤 = ℱ 𝐽(𝜄) 1000 500 Fourier-based operator 0 0 10 20 30 40 50 60 70 Frequency ℱ 4 8000 3 2 6000 Spectrum phase Intensity 1 4000 0 -1 2000 -2 -3 0 10 20 30 40 50 60 -4 Scattering angle, degree 0 10 20 30 40 50 60 70 3 Frequency Romanov et al. J. Quant. Spectrosc. Radiat. Transfer 200 , 280 – 294 (2017).
Spectrum parameters 𝐵 0 – Zero-frequency amplitude 2000 𝑀 – peak position Spectrum amplitude 1500 𝐵 𝑞 – peak amplitude 1000 𝜒 – peak phase 1.55 500 𝜇 = 660 𝑜𝑛 𝑜 𝑛𝑓𝑒𝑗𝑣𝑛 = 1.331 0 1.50 0 10 20 30 40 50 60 70 Frequency Refractive index 4 3 1.45 2 Spectrum phase 1 1.40 0 -1 1.35 -2 -3 0 2 4 6 8 10 12 14 -4 4 0 10 20 30 40 50 60 70 Size, m Frequency
Spectrum parameters 𝐵 0 – Zero-frequency amplitude 2000 𝑀 – peak position Spectrum amplitude 1500 𝐵 𝑞 – peak amplitude 1000 𝜒 – peak phase The goal: 500 0 𝑔: 𝑇 𝑤 → 𝑄 0 10 20 30 40 50 60 70 Frequency 4 : 𝑄 → 𝜁 - non-sphericity 3 2 𝑄 = (𝜁, 𝑏, 𝛽, 𝑜, … ) Spectrum phase 1 0 -1 -2 Weak dependency 𝜁 ≅ −1 (𝑄) -3 -4 5 0 10 20 30 40 50 60 70 Frequency
Rayleigh – Gans-Debye approximation 1 1 m ( 1 ) 1 x m 𝑐 𝛽 – orientation Incident light direction 𝑏 𝑦 Changes in main spectral peak = 𝐺(𝑏𝛿) 𝛿 ≝ 𝜁 2 − 1 (squared eccentricity) 𝛽 < 40° Changes in phase 𝜁 = 𝑏 Changes in amplitude 𝛽 > 40° 𝑐 (aspect ratio) 6
Algorithm for non-sphericity estimation LSP 𝑎 𝜉 Spectrum Spectral parameter 𝑎 𝜉 − 𝑎 0 𝜉 𝑄 = 𝑎 𝜉 − 𝑎 0 𝜉 Complex shape of the w peak 𝑎 0 𝜉 Spectral method* (𝑦 sp , 𝑜 sp ) LSP (Mie) Spectrum 𝑄 = 𝑎 𝜉 − 𝑎 0 𝜉 w ≝ න 𝜉 𝑎 𝜉 − 𝑎 0 𝜉 𝑒𝑤 7 * Romanov et al. J. Quant. Spectrosc. Radiat. Transfer 200 , 280 – 294 (2017).
Weighted norm of complex spectral peak Non-sphere 1.0 1 𝑎 𝜉 Sphere 0.8 Spectrum amplitude 0 Spectrum phase 0.6 -1 0.4 -2 Non-sphere 0.2 Sphere 𝑎 0 𝜉 -3 0.0 -20 -10 0 10 20 -20 -10 0 10 20 Weight Frequency Frequency 𝑎 𝜉 − 𝑎 0 𝜉 w ≝ න 𝜉 𝑎 𝜉 − 𝑎 0 𝜉 𝑒𝑤 8
Spectral parameter 𝑄 Simulated spheroids in the range of milk fat globules 𝑏 ∈ 3, 6 𝜈𝑛 𝑜 ∈ 1.44, 1.49 𝜁 ∈ 1, 1.4 𝛽 ∈ 0°, 90° 𝑏 ∈ 60, 140 𝑛 ∈ 1.07, 1.11 9
Classifying using the spectral parameter 𝑄 Non-spheres 40 300 35 250 30 Spectral parameter P Spectral parameter 200 25 20 150 15 100 10 50 5 0 0 0 20 40 60 80 100 120 140 0 5 10 15 20 25 30 x sp x sp Spheres Linear part 10
Estimating non-sphericity 40 35 30 Spectral parameter P 25 20 15 10 5 0 0 5 10 15 20 25 30 x sp min 11 max
Experiment with milk fat globules 400 160 350 140 300 120 250 100 Count Count 200 80 150 60 100 40 50 20 0 0 0 5 10 15 20 25 30 35 40 45 50 0 25 50 75 100 125 150 175 200 225 250 Spectral parameter P Spectral parameter P 12 Konokhova et al., International Dairy Journal/ Vol. 39, p. 316-323, 2014
Experiment with milk fat globules 300 250 Spheroids Spheres 200 Count 150 100 50 F-test 0 0 10 20 30 40 50 60 70 80 90 100 Spheres Spectral parameter P Agreement ~ 95% 13 Konokhova et al., International Dairy Journal/ Vol. 39, p. 316-323, 2014
Red blood cells lysis Add lysis buffer The spherization process 14
Experiment with RBC’s spherization 400 1.20 350 1.18 1.16 300 Effective aspect ratio sp Spectral parameter P 1.14 250 1.12 200 1.10 1.08 150 1.06 100 1.04 50 1.02 1.00 0 0 2000 4000 6000 8000 10000 12000 0 2000 4000 6000 8000 10000 12000 RBC's index RBC's index 𝜁 𝑓𝑔𝑔 𝑛𝑗𝑜 = 1.0 4 15 Chernyshova et al. J. Theor. Biol./ Vol. 393, p. 194-202, 2016
Effective aspect ratio of red blood cell 3.5 3.0 2.5 2.0 Z(r) Fully spherized RBCs 1.5 Spheroid = 1.04 Sphere 1.0 0.5 0.0 -3 -2 -1 0 1 2 3 r 16
Conclusion • New method to estimate an effective aspect ratio (close to 1) of single particles using the Fourier spectrum of its LSP • Based on the weighted complex deviation of the spectral peak from that for the equivalent sphere • The applicability range includes large milk fat globules, red blood cells, and other biological objects • Qualitative agreement (and superior sensitivity) with other methods on the real experimental data • О pen question about the final form of RBCs in the spherization process 17
Thank you ! email: a.v.romanov94@gmail.com 18
Norm of the complex spectrum peak shape L = 64.4 10 1.0 8 Spectrum amplitude / A p 6 3 Spectrum amplitude, 10 0.5 A p 4 2 0.0 0 -2 -0.5 -4 𝑎 𝜉 Re Z( v ) Real part -6 Im Z( v ) Imaginary part -1.0 -8 Amplitude Z( v ) Amplitude -10 -20 -10 0 10 20 0 10 20 30 40 50 60 70 80 90 Frequency L Frequency Weight 𝑎 𝜉 − 𝑎 0 𝜉 w ≝ න 𝜉 𝑎 𝜉 − 𝑎 0 𝜉 𝑒𝑤 19
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