Modified Kelvin-Voigt fractional derivative model for viscoelasticity measurement in optical coherence elastography

Chenming Yang; Zhongliang Li; Nan Nan; Teng Liu; Yaoli Luo; Xiangzhao Wang

doi:10.3788/COL202523.011701

Chinese Optics Letters, Volume. 23, Issue 1, 011701(2025)

Modified Kelvin-Voigt fractional derivative model for viscoelasticity measurement in optical coherence elastography

Chenming Yang^1,2, Zhongliang Li^1,2、*, Nan Nan^1、**, Teng Liu^1,2, Yaoli Luo^1,3, and Xiangzhao Wang⁴

¹Advanced Light Source and System R&D Center, Department of Advanced Optical and Microelectronic Equipment, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China

²Center of Materials Science and Optoelectronics Engineering, University of Chinese Academy of Sciences, Beijing 100049, China

³School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China

⁴State Key Laboratory of Extreme Photonics and Instrumentation, College of Optical Science and Engineering, Zhejiang University, Hangzhou 310027, China

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Fig. 1. Schematic of the OCE system.

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Fig. 2. Storage and loss moduli of gelatin samples with four different concentrations obtained in the rheometer test, and the curve fitting results of the KV, KVFD, and mKVFD models. The shaded areas indicate 95% confidence intervals.

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Fig. 3. Shear viscosity of gelatin phantoms with four concentrations and the fitting curves.

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Fig. 4. Fitting results of the high-frequency storage and loss moduli of gelatin phantoms using the KV, KVFD, and mKVFD models. The shaded areas indicate 95% confidence intervals.

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Fig. 5. Fitting results of the complex shear modulus of PDMS using the mKVFD model. The shaded areas indicate 95% confidence intervals.

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Table 1. The Fitting Parameters of the Models in the Rheometer Test

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Table 1. The Fitting Parameters of the Models in the Rheometer Test

Gelatin concentration	KV model				KVFD model					mKVFD model
Gelatin concentration	µ₀/kPa	RMSE/kPa	η₀/Pa·s	R²	µ₀/kPa	RMSE/kPa	µ_a/Pa·s^a	a	R²	µ₀/kPa	RMSE/kPa	µ₂/Pa·s^a₂	a₂	R²
10%	4.89 ± 0.28	0.016	2.07 ± 0.19	−4.8	4.67 ± 0.02	0.044	113 ± 17	0.23 ± 0.01	0.95	4.89 ± 0.28	0.016	40.6 ± 4.9	0.23 ± 0.01	0.95
12%	7.26 ± 0.29	0.046	2.64 ± 0.15	−5.1	6.88 ± 0.03	0.110	204 ± 67	0.22 ± 0.05	0.86	7.26 ± 0.29	0.046	65.8 ± 8.3	0.22 ± 0.05	0.86
14%	9.87 ± 0.12	0.071	3.44 ± 0.09	−4.7	9.28 ± 0.02	0.076	294 ± 39	0.22 ± 0.02	0.91	9.87 ± 0.12	0.071	99.8 ± 5.4	0.22 ± 0.02	0.91
16%	13.0 ± 0.6	0.092	4.58 ± 0.01	−5.9	12.10 ± 0.10	0.200	472 ± 68	0.20 ± 0.02	0.85	13.00 ± 0.60	0.092	146 ± 10	0.20 ± 0.02	0.85

Table 2. The Fitting Parameters of the Models in the OCE Test

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Table 2. The Fitting Parameters of the Models in the OCE Test

Gelatin concentration	KV model				KVFD model					mKVFD model
Gelatin concentration	µ₀/kPa	RMSE/kPa	η₀/Pa·s	R²	µ₀/kPa	RMSE/kPa	µ_a/Pa·s^a	a	R²	µ₁/Pa·s^a₁	a₁	RMSE/kPa	µ₂/Pa·s^a₂	a₂	R²
10%	5.43 ± 0.04	1.04	0.14 ± 0.01	−0.32	2.10 ± 0.04	0.391	85.8 ± 22.0	0.40 ± 0.02	0.92	131 ± 1	0.39 ± 0.01	0.134	50.4 ± 11.0	0.40 ± 0.02	0.92
12%	8.83 ± 0.01	1.91	0.23 ± 0.01	0.12	4.37 ± 0.01	0.956	73.7 ± 6.4	0.46 ± 0.01	0.93	152 ± 4	0.42 ± 0.01	0.392	48.4 ± 3.5	0.46 ± 0.01	0.93
14%	13.0 ± 0.1	3.60	0.21 ± 0.01	0.38	6.73 ± 0.74	1.94	70.5 ± 34.2	0.48 ± 0.04	0.91	103 ± 11	0.48 ± 0.01	0.710	48.1 ± 12.0	0.48 ± 0.04	0.91
16%	16.3 ± 0.1	4.91	0.32 ± 0.01	0.62	8.88 ± 0.03	2.70	43.1 ± 5.7	0.56 ± 0.01	0.94	76.1 ± 6.2	0.53 ± 0.01	0.843	32.9 ± 3.9	0.56 ± 0.01	0.94

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Chenming Yang, Zhongliang Li, Nan Nan, Teng Liu, Yaoli Luo, Xiangzhao Wang, "Modified Kelvin-Voigt fractional derivative model for viscoelasticity measurement in optical coherence elastography," Chin. Opt. Lett. 23, 011701 (2025)

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