To the Editor: Brillouin microscopy is an optical method to map the mechanical properties of materials1,2,3,4,5. The first Brillouin micrographs with cellular resolution were reported in Nature Methods in 20154 and inspired a growing number of applications in biomechanics and biophotonics. In these publications, Brillouin measurements were considered a proxy for stiffness. In contrast, we show here in hydrogels that water content dominates Brillouin signals in hydrated materials.
Brillouin microscopy relies on the phenomenon of Brillouin scattering, whereby photons exchange energy with thermally driven acoustic waves, or phonons, leading to a frequency shift ωb between the incident and scattered light, given by
where ρ and n are the density and refractive index of the material, and λ and θ are the in vacuo wavelength and angle between the incident and scattered wave vectors. M is the longitudinal elastic modulus and represents the compressibility of a material or, specifically, the mechanical stress necessary to compress or expand it in one direction without lateral strain.
Despite the fact that equation (1) depends on M, recent reports have interpreted Brillouin micrographs in terms of Young’s modulus E for cells and tissues3,4,5. E represents the stiffness of a material, that is, how resistive it is to deformation, and describes the stress necessary to compress or extend a material in one direction while allowing lateral strain. As biological materials are composed mostly of water, which is relatively incompressible, M is several orders of magnitude greater than E. Further, Brillouin scattering is sensitive to gigahertz frequencies where, owing to viscoelastic effects, mechanical properties may diverge from those at biologically relevant strain rates, which typically occur over seconds. Nonetheless, empirical correlations between M and E for cells4, hydrogels3,4 and other biological tissues3 have suggested that variations in M, as measured by Brillouin scattering, reflect variations in E.
We set out to examine the relationship between Brillouin measurements and Young’s modulus, accounting for the potential influence of water content ε, which can affect both M and E in hydrated materials. We used hydrogels as a simplified model of biological materials because both contain fluid interspersed within a flexible solid network that provides elasticity.
As the molecular weight of polyethylene oxide (PEO) increases, hydrogels change from a dilute suspension with zero Young’s modulus to a semi-dilute entangled network with finite E. When the molar concentration is decreased in proportion to the increase in molecular weight, water content ε can be fixed while E increases. Thus, we could vary E independently of ε while measuring M using a custom Brillouin microscope5 (Supplementary Methods). With increasing molecular weight, E measured by rheometry increased for a given ε. However, M was unaffected by the change in molecular weight, but decreased with ε (Fig. 1a and Supplementary Note). Thus, for PEO hydrogels, changes in E were uncorrelated with changes in M when controlling for water content (Fig. 1b and Supplementary Note).
To understand the relationship between M and ε, we considered a biphasic model where the aggregate compressibility is equal to the sum of the individual fluid and solid compressibilities weighted by their respective volume fractions6,
where Mf and Ms are the longitudinal elastic moduli (inverse compressibilities) of the fluid and solid, respectively. Equation (2) captures the relationship between M and ε for PEO (Fig. 1a and Supplementary Note).
Previously reported correlations between M and E were based on polyacrylamide (PA) hydrogels3,4, which swell over time (Supplementary Fig. 1). Using different concentrations of bis-acrylamide to vary PA stiffness, we measured M, E and ε during swelling (Supplementary Figs. 2–4). All values of M collapsed onto a single relationship with ε, consistent with equation (2), despite differences in E or the extent of swelling (Fig. 1c and Supplementary Note).
During swelling of PA, E measured by unconfined compression decreased according to
where Q = (1 – ε0)/(1 – ε) is the swelling ratio and E0 is the estimated Young’s modulus immediately after gelation.
There was no clear relationship between M and E when considering all data. However, equations (2) and (3) predicted how M changed versus E for individual hydrogels as ε increased from ε0 = 0.9 to 1. These predicted trajectories (shown by solid curves in Fig. 1d and the Supplementary Note) explain how a correlation may arise owing to a mutual dependence on ε without an explicit relationship between M and E.
In conclusion, Brillouin measurements appear insensitive to Young’s modulus after accounting for the influence of water content in PEO and PA hydrogels. This suggests that Brillouin measurements applied to biological tissues may be similarly affected by water content, although differences in hydration between tissues and the role of bound water may have additional effects. This work cautions against the straightforward application of Brillouin microscopy, or Brillouin scattering in general, as a form of optical bioelastography and motivates further research into the factors influencing the Brillouin frequency shift in biological materials.
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Datasets generated and analyzed during the current study are available upon reasonable request.
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Acknowledgements
This work was supported by NIH grants EY022359 (D.R.O.) and EY019696 (D.R.O.), a PhD studentship from the Ministry of Education, Republic of China (P.-J.W.) and the Imperial College Junior Research Fellowship (I.V.K.). We thank C. Song (Imperial College London) for help in acquiring the Brillouin measurements.
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P.-J.W., I.V.K., C.P., P.T., J.W.R., J.M.S., I.E.D. and D.R.O. planned the study. P.-J.W. and I.K. conducted experiments. All authors participated in and contributed to data analysis. D.R.O. and P.-J.W. wrote the manuscript. All authors contributed to editing and revision of the manuscript.
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Integrated supplementary information
Supplementary Figure 1 Polyacrylamide hydrogel swelling.
a, Hydrogel mass m increased during swelling, with larger swelling observed for lower cross-linker concentrations. b, The swelling ratio Q increased during swelling, as calculated by equation S6. Data are taken from the studies shown in Fig. 1c,d. Each data point represents an individual hydrogel made from the same stock solution. Data from four hydrogels are shown here. Curves show exponential fits.
Supplementary Figure 2 Setup of the Brillouin microscope and associated frequency analysis.
a, Schematic of the Brillouin microscope. Laser light is directed into an inverted confocal microscope. Backscattered light is collected and filtered by an interferometer to reduce the intensity of the Rayleigh peak by up to 40 dB. The filtered signal is passes through a VIPA to separate spectral components that are detected by an sCMOS camera. b, Pixel locations in the spectrum are converted into frequency (Supplementary Methods) to identify the Brillouin frequency shift \(\omega _b\) after the peaks are fitted by a Lorentzian function, where \(\omega _b = \left( {{\mathrm{FSR}} - \Delta f} \right)/2\). FSR, full spectral range. \(f\left( {x_i} \right)\) represents the frequency at pixel location \(x_i\), as needed for equation S8. Similar results were obtained for each individual Brillouin measurement, 50 of which were acquired at each location, averaging over three locations per hydrogel.
Supplementary Figure 3 Representative measurement of the Young’s modulus of a PEO hydrogel by rheometry.
a, The viscoelastic storage modulus (G′) as a function of oscillatory strain magnitude. b, The viscoelastic loss modulus (G″) as a function of frequency. Young’s modulus E was calculated as E=3 G′. For this sample, the molecular weight was 8 MDa with ε = 1.5%. Similar results were obtained for each of the 2–3 hydrogel samples per condition.
Supplementary Figure 4 Representative measurement of the Young’s modulus of a PA hydrogel by uniaxial unconfined compression.
The stress-strain data were used to calculate Young’s modulus as the slope of the linear regression to a full cycle (blue line). The initial bis-acrylamide concentration of this sample was 0.06%, measured after 12 h of swelling. One similar compression measurement was done per hydrogel per time point.
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Supplementary Figures 1–4, Supplementary Methods and Supplementary Note
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Wu, PJ., Kabakova, I.V., Ruberti, J.W. et al. Water content, not stiffness, dominates Brillouin spectroscopy measurements in hydrated materials. Nat Methods 15, 561–562 (2018). https://doi.org/10.1038/s41592-018-0076-1
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DOI: https://doi.org/10.1038/s41592-018-0076-1
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