Estimation of the mass density of biological matter from refractive index measurements

Abstract

The quantification of physical properties of biological matter gives rise to novel ways of understanding functional mechanisms. One of the basic biophysical properties is the mass density (MD). It affects the dynamics in sub-cellular compartments and plays a major role in defining the opto-acoustical properties of cells and tissues. As such, the MD can be connected to the refractive index (RI) via the well known Lorentz-Lorenz relation, which takes into account the polarizability of matter. However, computing the MD based on RI measurements poses a challenge, as it requires detailed knowledge of the biochemical composition of the sample. Here we propose a methodology on how to account for assumptions about the biochemical composition of the sample and respective RI measurements. To this aim, we employ the Biot mixing rule of RIs alongside the assumption of volume additivity to find an approximate relation of MD and RI. We use Monte-Carlo simulations and Gaussian propagation of uncertainty to obtain approximate analytical solutions for the respective uncertainties of MD and RI. We validate this approach by applying it to a set of well-characterized complex mixtures given by bovine milk and intralipid emulsion and employ it to estimate the MD of living zebrafish (Danio rerio) larvae trunk tissue. Our results illustrate the importance of implementing this methodology not only for MD estimations but for many other related biophysical problems, such as mechanical measurements using Brillouin microscopy and transient optical coherence elastography.

Figure 1

Correlative distributions $\mathcal{P}\left({\alpha }_{\mathrm{p}}^{\mathrm{B}},{\theta }_{\mathrm{p}}\right)$ of the RI increment ${\alpha }_{\mathrm{p}}^{\mathrm{B}}$ and PSV ${\theta }_{\mathrm{p}}$ for the human proteome (blue; 82,127 proteins included) and the zebrafish proteome (red; 46,517 proteins included). The solid and dashed lines indicate the 68% and 95% confidence contours, respectively.

Figure 2

2D visual interpretation of a voxel with volume $v_v=N_0{v}_0$ consisting of $N_0$ voxelinos with volumes $v_0$. Each number stands for one type of voxelino; 1 corresponds to the solvent voxelinos and $2-5$ correspond to the solute voxelinos, e.g., four different proteins. Each voxelino is characterized by its PSV ${\theta }_i$, refraction per gram $R_i$ and has a corresponding mass of $m_i=v_0/{\theta }_i$. This illustrative depiction is based on (47).

Figure 3

Results of MC simulations and according analytical solutions for the mixture of human proteins and the neutral lipid triolein in water. (A) Relative deviations of the MD ρ, the RI n, the effective RI increment ${\alpha }_{\text{eff}}$, and PSV ${\theta }_{\text{eff}}$ in dependence of the number of voxelinos per voxel $N_0$ obtained from MC simulations (symbols) and analytical solutions (dashed lines) for different mean relative lipid volume fractions ${\overline{x}}_{\text{lip}}$, associated deviations $\Delta x_{\text{lip}}$, and relative deviations of the number of the solute mass $\Delta m_{\mathrm{s}}/{\overline{m}}_{\mathrm{s}}$. The MC simulations were performed for a mean water volume fraction of ${\overline{\phi }}_1=0.9$ and $N_{\mathrm{v}}={10}^3$. (B) MD ρ in dependence of the RI contrast $\delta n$ for different mean relative lipid volume fractions ${\overline{x}}_{\text{lip}}$ and mean water volume fractions (${\overline{\phi }}_1=0.1$, ${\overline{\phi }}_1=0.3$, ${\overline{\phi }}_1=0.5$,  ${\overline{\phi }}_1=0.7$, ${\overline{\phi }}_1=0.9$) for $\Delta x_{\text{lip}}=0$, $N_0={10}^3$ and $N_{\mathrm{v}}={10}^2$. The dashed lines indicate the analytical solutions of Eq. 15. (C) Correlative distribution of the MD ρ and the RI contrast $\delta n$ (the solid and dashed lines indicate the 68% and 95% confidence contours) with the corresponding normalized marginal probability density distributions $\hat{\mathcal{M}}\left(\rho \right)$ and $\hat{\mathcal{M}}\left(\delta n\right)$, respectively (the solid line represents the median, the dash-dotted and dashed lines indicate the 68% and 95% CIs), for the exact relative lipid volume fraction ${\overline{x}}_{\text{lip}}=0.5$ (purple) and the relative lipid volume fraction following a truncated normal distribution $x_{\text{lip}}\sim \mathcal{T}\left({\overline{x}}_{\text{lip}}=0.5,\Delta x_{\text{lip}}=0.1\right)$ (cyan; see main text). The MC simulations were performed for a water volume fraction following a truncated normal distribution ${\phi }_1\sim \mathcal{T}\left({\overline{\phi }}_1=0.9,\Delta {\phi }_1=0.1\right)$, $N_0={10}^5$ and $N_{\mathrm{v}}={10}^3$.

Figure 4

Results of the concentration-dependent measurements of the MD and RI of bovine SM and 20% IL in water as well as the theoretical predictions based on the chemical composition using Eq. 15. (A) RI contrast $\delta n$ in dependence of the solute concentration $c_{\mathrm{s}}$ of SM (△, $N=5$ technical repetitions, mean $\pm$ SD) and IL (□, $N=5$ technical repetitions, mean $\pm$ SD) with the respective fits of Eq. 8 (solid lines) and fit residuals. (B) MD ρ in dependence of the solute concentration $c_{\mathrm{s}}$ of SM (△, $N=5$ technical repetitions, mean $\pm$ SD) and IL (□, $N=5$ technical repetitions, mean $\pm$ SD) with the respective fits of Eq. 6 (solid lines) and fit residuals. (C) Effective RI increment ${\alpha }_{\text{eff}}$ and PSV ${\theta }_{\text{eff}}$ of various substances that compose SM and IL, as well as the measured and predicted values for SM and IL. (D) MD ρ in dependence of the RI contrast $\delta n$ for different concentrations of SM in water and IL. The symbols represent measured values ($N=5$ technical repetitions, mean $\pm$ SD) and the predicted values using the Biot mixing rule (Eq. 7) for $N_0={10}^4$ and $N_{\mathrm{v}}={10}^3$. The dashed lines indicate Eq. 15 for the predicted values of the effective RI increment and PSV.

Figure 5

Results of MC simulations and according analytical solutions for the trunk tissue of larval zebrafish. Predicted correlative distributions of the MD ρ and the RI contrast $\delta n$ of larval zebrafish trunk tissue with the corresponding normalized marginal probability density distributions $\hat{\mathcal{M}}\left(\rho \right)$ and $\hat{\mathcal{M}}\left(\delta n\right)$, respectively (the solid line represents the median, the dash-dotted and dashed lines indicate the 68% and 95% CIs). The ellipsoids indicate the predictions using Eq. 9 (blue) and the Lorentz-Lorenz mixing rule of RIs Eq. S8 (orange). The green band indicates the RI measurement of (21). The MC simulations were performed for $N_0={10}^5$ and $N_{\mathrm{v}}={10}^3$.