Estimating Particle Non-sphericity from the Fourier Spectrum of Its - - PowerPoint PPT Presentation

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

10 20 30 40 50 60 70 500 1000 1500 2000 2500 3000 Intensity Scattering angle, degree experiment Mie theory

?

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Spectrum of light-scattering pattern

10 20 30 40 50 60 2000 4000 6000 8000 Intensity Scattering angle, degree 10 20 30 40 50 60 70 500 1000 1500 2000 Spectrum amplitude Frequency 10 20 30 40 50 60 70

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• 3
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1 2 3 4 Spectrum phase Frequency

𝑇 𝑤 = ෠ ℱ 𝐽(𝜄)

Romanov et al. J. Quant. Spectrosc. Radiat. Transfer 200, 280–294 (2017).

Fourier-based operator

෠ ℱ 3

Spectrum parameters

10 20 30 40 50 60 70 500 1000 1500 2000 Spectrum amplitude Frequency

𝐵0 – Zero-frequency amplitude 𝑀 – peak position 𝐵𝑞 – peak amplitude 𝜒 – peak phase

10 20 30 40 50 60 70

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1 2 3 4 Spectrum phase Frequency

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2 4 6 8 10 12 14 1.35 1.40 1.45 1.50 1.55 Refractive index Size, m

𝜇 = 660 𝑜𝑛 𝑜𝑛𝑓𝑒𝑗𝑣𝑛 = 1.331

Spectrum parameters

10 20 30 40 50 60 70 500 1000 1500 2000 Spectrum amplitude Frequency

𝐵0 – Zero-frequency amplitude 𝑀 – peak position 𝐵𝑞 – peak amplitude 𝜒 – peak phase

10 20 30 40 50 60 70

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• 3
• 2
• 1

1 2 3 4 Spectrum phase Frequency

The goal:

𝑔: 𝑇 𝑤 → 𝑄 𝑕: 𝑄 → 𝜁 - non-sphericity 𝑄 = 𝑕(𝜁, 𝑏, 𝛽, 𝑜, … )

Weak dependency

𝜁 ≅ 𝑕−1 (𝑄)

5

1 ) 1 ( 1 1     m x m

Rayleigh–Gans-Debye approximation

Changes in main spectral peak = 𝐺(𝑏𝛿)

𝛿 ≝ 𝜁2 − 1 (squared eccentricity)

𝑏

𝛽 – orientation Incident light direction

𝑐

𝑦

𝜁 = 𝑏 𝑐 (aspect ratio) Changes in phase 𝛽 < 40° Changes in amplitude 𝛽 > 40° 6

Algorithm for non-sphericity estimation

* Romanov et al. J. Quant. Spectrosc. Radiat. Transfer 200, 280–294 (2017).

𝑄 = 𝑎 𝜉 − 𝑎0 𝜉

w ≝ න 𝜉 𝑎 𝜉 − 𝑎0 𝜉

𝑒𝑤

Complex shape of the peak

LSP Spectrum Spectral method* LSP (Mie) Spectrum 𝑎 𝜉 𝑎0 𝜉 𝑎 𝜉 − 𝑎0 𝜉 Spectral parameter 𝑄 = 𝑎 𝜉 − 𝑎0 𝜉

w

(𝑦sp, 𝑜sp)

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Weighted norm of complex spectral peak

𝑎 𝜉 𝑎0 𝜉 𝑎 𝜉 − 𝑎0 𝜉

w ≝ න 𝜉 𝑎 𝜉 − 𝑎0 𝜉

𝑒𝑤

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• 10

10 20 0.0 0.2 0.4 0.6 0.8 1.0 Spectrum amplitude Frequency Non-sphere Sphere

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10 20

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1 Spectrum phase Frequency Non-sphere Sphere

Weight

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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

5 10 15 20 25 30 5 10 15 20 25 30 35 40 Spectral parameter P

xsp

20 40 60 80 100 120 140 50 100 150 200 250 300 Spectral parameter xsp

Classifying using the spectral parameter 𝑄

Spheres

Linear part

Non-spheres

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Estimating non-sphericity

5 10 15 20 25 30 5 10 15 20 25 30 35 40 Spectral parameter P xsp min max

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Experiment with milk fat globules

25 50 75 100 125 150 175 200 225 250 50 100 150 200 250 300 350 400

Count Spectral parameter P

5 10 15 20 25 30 35 40 45 50 20 40 60 80 100 120 140 160

Count Spectral parameter P

Konokhova et al., International Dairy Journal/ Vol. 39, p. 316-323, 2014

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10 20 30 40 50 60 70 80 90 100 50 100 150 200 250 300

Count Spectral parameter P

Spheroids Spheres

Experiment with milk fat globules

Spheres

Konokhova et al., International Dairy Journal/ Vol. 39, p. 316-323, 2014

F-test

Agreement ~95%

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Add lysis buffer The spherization process

Red blood cells lysis

Experiment with RBC’s spherization

2000 4000 6000 8000 10000 12000 50 100 150 200 250 300 350 400 Spectral parameter P RBC's index Chernyshova et al. J. Theor. Biol./ Vol. 393, p. 194-202, 2016

𝜁𝑓𝑔𝑔 𝑛𝑗𝑜 = 1.04

2000 4000 6000 8000 10000 12000 1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.16 1.18 1.20 Effective aspect ratio sp RBC's index

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Effective aspect ratio of red blood cell

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1 2 3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Z(r) r Fully spherized RBCs Spheroid  = 1.04 Sphere

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

• Оpen question about the final form of RBCs in the spherization

process 17

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Thank you!

email: a.v.romanov94@gmail.com

Norm of the complex spectrum peak shape

𝑎 𝜉 𝑎 𝜉 − 𝑎0 𝜉

w ≝ න 𝜉 𝑎 𝜉 − 𝑎0 𝜉

𝑒𝑤

Weight

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10 20 30 40 50 60 70 80 90

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2 4 6 8 10 Spectrum amplitude, 10

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Frequency Real part Imaginary part Amplitude 64.4 L= Ap

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10 20

• 1.0
• 0.5

0.0 0.5 1.0 Spectrum amplitude / Ap Frequency  L Re Z(v) Im Z(v) Amplitude Z(v)