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Abstract
High-throughput label-free measurements of the optical and mechanical properties of single microparticles play an important role in biological research, drug development, and related large population assays. However, mechanical detection techniques that rely on the density contrast of a particle with respect to its environment cannot sense neutrally bouyant particles. On the other hand, neutrally buoyant particles may still have a high compressibility contrast with respect to their environment, opening a new window to their detection and analysis. Here we present a label-free high-throughput approach for measuring the compressibility (bulk modulus) of freely flowing microparticles by means of resonant measurements in an opto-mechano-fluidic resonator.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The mechanical properties of individual cells (e.g. density, Young’s modulus, and compressibility) are important in determining how they interact with their environment. Variations of mechanical properties are also known to correlate with specific disease states including anemia [1], malaria [2], and cancers [2], and can influence cell differentiation [3]. While optical flow cytometry is an extremely powerful tool for single cell analysis and sorting [4,5], an equivalent high-throughput single cell mechanical assay is not yet available. Knowledge of mechanical parameters of single microparticles or cells could enable new discoveries and aid the development of next generation diagnostics.
Traditionally, measurement of mechanical properties of individual microparticles requires the application of forces, upon which the mechanical responses (e.g. deformation) can be quantified to determine the properties of interest. Such forces can be applied through direct contact techniques like AFM deformational probing [7,8], optical tweezing of adhered beads [9], micropipette aspiration [10], and mechanical resonator loading [11–13]. These methods, however, are inherently slow since the analyte particle must be temporarily immobilized. In contrast, flow-through type sensors can offer much higher throughput measurements, for instance, single cell mass can be rapidly measured by flowing it through an internal channel within a mechanical resonator [14]. Such methods, however, have so far relied on the density contrast of a particle with its environment and therefore cannot detect neutrally buoyant particles. The solution to detecting neutrally buoyant particles lies in their compressibility contrast against their environment, which can open a window to detection. Recently, Hartono et. al. demonstrated [15] a non-contact compressibility measurement for single cells using the acoustic radiation force and cell trajectories. However, the direct measurement of single microparticle compressibility with high throughput has not yet been achieved.
2. Methods
In this work, we report an acoustic-based high-throughput technique for measuring the compressibility of single particles without physical contact, using an opto-mechano-fluidic resonator (OMFR) [16,17]. The OMFRs are fabricated from fused silica capillary preforms (Polymicro Technologies TSP-700850) by means of linear drawing under local heating provided by a CO2 laser. The diameter of each OMFR can be locally controlled by modulating the heating laser power, thereby forming localized micro-bottle resonators along the length of the microcapillary. These resonators can be used as cavity optomechanical sensors [18–22] that support ultrahigh-Q optical whispering gallery modes (confined to the outer silica shell) that are coupled to co-localized mechanical modes. As a result, fluid analytes can be flowed within the OMFR without influencing the optics at the outer surface. Phonons occupying certain radially symmetric mechanical breathing modes permeate the entire cross-section of the capillary, including the fluid, casting a near-perfect net for measuring particles flowing inside (Fig. 1(b)). All analyte particles within the sample must therefore transit and perturb such a mechanical mode (Fig. 1(c)), which in turn perturbs the optical readout due to the optomechanical coupling (Fig. 1(d)) [6]. The optical probing of the OMFR in this work is performed with the assistance of a tapered optical fiber waveguide that is evanescently coupled to the optical modes of the OMFR.
Fig. 1 (a) False colored SEM image of an opto-mechano-fluidic resonator (OMFR). (b) A fluid-shell hybrid breathing mechanical (phonon) mode in an OMFR. (c) Particles of density contrast Δρ and compressibility contrast Δκ change the vibrational mode shape as indicated by the slight broken symmetry. (d) The thermal mechanical fluctuations of the OMFR mode can be measured optically [6] via a single-point tapered fiber measurement. Analysis of this spectrum conveys information on the particle.
In previous work, optomechanical sensing of fluid density and speed of sound [16,17], fluid viscosity [23], and flowing particles [6, 24] have been demonstrated using this platform. A recent report [24] demonstrated an OMFR-based optomechanical particle detection rate that can potentially exceed 50,000 particles-per-second, without any binding, labeling, or reliance on random diffusion. It has also been shown [6, 25] that the size, position, density, and compressibility of the flowing particles, and even the vibrational modeshape of the OMFR, all influence such measurements. The contributions of these individual parameters can be modeled using a Helmholtz equation [6], resulting in the a linearized model in the limit of small perturbations as follows:
where , . Δ f = f2 − f1 is the frequency difference between the perturbed situation i.e. with a particle present (denoted by subscript “2”) and the unperturbed situation (denoted by subscript “1”). The subscript “s” denotes the solid particle such that κs (Vs) represents the particle compressibility (volume). The subscript “ℓ” denotes the liquid core of the OMFR for which κℓ (ρℓ) is the compressibility (density) of the fluid. For the OMFR mechanical mode under consideration Wsp (Wsk) is the acoustic potential (kinetic) energy in the fluid, but evaluated only over the volume displaced by the particle. Wℓp (Wℓk) is the acoustic potential (kinetic) energy in the entire liquid core including the particle. The operation 〈·〉 denotes the time average over the period of oscillation T as . Therefore, A and B represent sensitivity factors of the OMFR to the added particle and are both spatially dependent on the basis of the mechanical mode shape. The parameter A is in particular responsible for the sensitivity to particle compressibility.The above model is derived using the Helmholtz equation with the consideration of the fluid portion of the resonator cavity alone, but does not incorporate the acoustic pressure modification by the particle, i.e. the scattering effect. To more accurately predict the frequency shift, we must incorporate both the pressure field modification by the particle and the energy change of the resonator shell into our model. We thus derive an improved “energy balance” perturbation model in the Supplement (below), in which we leverage energy conservation, including all energy components of the resonant motion. Using this model, we will be able to quantify how the change of each parameter (particle size, material properties, acoustic field etc.) contributes to the resonant frequency change for the OMFR mechanical mode. Moreover, we will be able to explicitly model the contribution from the particle properties and from the acoustic scattering effect. The final form obtained for the perturbed frequency through this revised model is
where ,, and Wwp (Wwk) is the elastic strain (kinetic) energy associated with the resonator shell or wall “w”. Here, the parameter C provides the sensitivity to the compressibility contrast between the particle and the surrounding carrier liquid. As we can see, the frequency perturbation prediction equations derived from the energy balance (Eq. (2)) and the Helmholtz equation (Eq. (1)) have very similar form. Essentially, the frequency perturbation occurs due to the modification of system potential energy through compressibility contrast and the modification of system kinetic energy through density contrast. The energy term in the denominator is the only difference between the two methods. However, the effect of this modification in the frequency perturbation calculations is small (for resonators with thin shell) as shown later in Fig. 2(b).Fig. 2 Theoretical predictions vs experimental measurements. (a) The sensitivity variables A, B, C, and D are required for making a prediction of the mechanical frequency shift using the models presented in Eq. 1 and Eq. 2. These coefficients can be obtained from finite element simulation of the OMFR device. The results are fitted to third-order polynomial curves for interpolation. (b) The simulated coefficients A, B, C, and D are then used to predict the frequency shifts anticipated due to 6 um spherical silica and polystyrene particles as described in the text. Experimentally measured frequency shifts of the mechanical mode follow the trend predicted by theory. The particle position is subject to both the image processing fitting error (shown here by the error bar) and a roughly 2 um error (not shown in figure) that exists in determining the central axis of the OMFR [6].
3. Results and discussion
In order to verify the theoretical model for sensing compressibility, we performed a series of experiments on 6 μm polystyrene particles (Corpuscular C-PS-6.0) and 6 μm silica particles (Corpuscular 100235-10). The OMFR that we used has approximate maximum outer diameter of 55 μm and estimated wall thickness of 4.5 μm. The device supports a 34 MHz breathing vibrational mode as well as many ultrahigh-Q optical whispering gallery modes near 1550 nm wavelength. This breathing vibrational mode of the OMFR mode can be measured optically [6] via light scattering with optical coupling provided through a single-point tapered fiber measurement. Analysis of this spectrum with spectrum analyzer conveys information on the particle. More details on the experimental setup have been described in [6] (Fig. 1(d)).
The OMFR is vertically oriented, so that particles can flow through the sensing region by gravity. The concentration of the particle suspension is kept very low for this study to ensure that only one particle is transiting the sensing region at a time. Since the radial position of a particle during a transit through the OMFR is not fixed or known, each transit is confirmed visually, and the particle’s radial location is triangulated with the assistance of two microscopes. The error bar associated with the particle location is determined by image fitting error as described previously [6]. The material properties of the particles under study are summarized in Table 1, where cP (cS) is the P-wave (S-wave) acoustic velocity. Example measurements of the optomechanically detected mechanical mode perturbations generated by each of these particles are shown in Fig. 2(b). The sensitivity to particle compressibility, A, dominates the sensitivity to particle density, B, when the particle is near the pressure antinode (OMFR center axis), resulting in positive mechanical frequency shift. On the other hand, when the particle is near a pressure node (near the OMFR wall), B dominates A and therefore a negative mechanical frequency shift is observed.
Table 1. Material properties [26]
To compare the results between the Helmholtz method and the energy balance method, we first performed a finite-element simulation (Fig. 1(b)) of the unloaded resonator using the method described in [27]. This simulation is used to numerically estimate the A, B, C, and D parameters as a function of particle location (Fig. 2(a)). A third-order polynomial curve fit is generated for each parameter. These fits are then used to predict the frequency perturbation using Eq. (2) and Eq. (1), which are also plotted in Fig. 2(b). While the predictions from either model can be used to qualitatively describe the experimental results, there is significant discrepancy between the predictions and the experimental measurements – for example, the prediction overestimates frequency shift when the particles are close to the center axis of the OMFR and underestimates frequency shift when particles are away from the center axis. This shared trend could be produced simply by a mischaracterization of the system, for instance, by a poor estimation of the device geometry leading to a mismatch between simulated and actual mode shapes. It is not possible to determine the precise OMFR device geometry since the inner device walls are hidden from view.
Presently, we attempt an empirical correction to the model of the OMFR system using a single parameter N, by utilizing the known properties of one particle type for calibration. This is introduced as a pre-factor into Eq. (2) as follows:
This simplistic empirical correction is chosen to demonstrate that using one type of particle to calibrate the system is viable. Other empirical methods are also possible and could be applied in a similar way.Before proceeding, we illustrate a constraint inherent to this approach. When the test particle has high stiffness with respect to the ambient fluid, i.e. the compressibility κs is very small (κs ≪ κl), the relative influence of particle compressibility on the mechanical frequency is not significant since . This makes it challenging to extract the particle compressibility with good accuracy, because a small uncertainty associated with the frequency shift measurement or the density contrast can result in a large relative error in the estimation of κs, due to ill-conditioning of the numerical fitting. To illustrate this limitation, we attempt to find the compressibility of silica (which has much lower compressibility than water), using 6 μm silica particles themselves as the calibration. Using the silica particle test data, we first perform a least-square curve fit to Eq. (3) with N as an unknown parameter and all the other parameters as known (Fig. 3(a)). To support this, we use the material properties from Table 1 and N = (0.812 ± 0.182) is obtained. For all curve fits, we cite the 95% confidence interval as the uncertainty for the extracted values, which include N and the material compressibility shown below. Next, we assume that the compressibility of silica is the unknown variable and use N = 0.812 as the known calibration parameter. We again perform a curve fitting of Eq. (3) to the silica experimental data. The compressibility for silica extracted by this fitting is (1.7 ± 9.1)×10−11 Pa−1. The impractically large error bar indicates that we simply cannot trust this extracted value.
Fig. 3 (a) Least-square curve fitting for Eq. (3) is applied to the silica data to obtain N = 0.812. The material properties are listed in Table 1. The prediction with N = 1 is given as a comparison. (b) The obtained N is now used for least-squares curve fitting to the polystyrene data with the compressibility of polystyrene as the unknown variable. Fitting to the experimental data extracts the compressibility of polystyrene as 2.5 × 10−10 Pa−1, which is only 3% different from the data in Table 1. The predictions with compressibility from Table 1, with and without scaling factor N, are given as comparisons.
On the other hand, this system is fairly capable of extracting the compressibility of softer (low κs) particles. Once again, we use the calibration N = 0.812 as obtained above as the known parameter. We now fit Eq. (3) to the polystyrene particle experimental data setting the compressibility of polystyrene as the unknown parameter (Fig. 3(b)), from which we estimate the compressibility of polystyrene as (2.5 ± 0.3)×10−10 Pa−1. The error of this fitted value compared with data provided in Table 1 is only 3%. As a comparison, if we perform a curve fit for Eq. (3) again but with N = 1 (i.e. no calibration), the compressibility for polystyrene is extracted at (2.9 ± 0.2)×10−10 Pa−1, which is a 18% error compared with data provided in Table 1. We can also see this improvement from the difference between the experimental data and predicted value: the root mean square error (RMSE) value for polystyrene data against the predicted value from Eq. (3) when N=1 is 5.4 kHz, while same the RMSE value when N=0.812 is improved to 2.8 kHz. We thus conclude that this single-parameter calibration can improve the compressibility prediction by an appreciable factor.
The optomechanical technique that we have demonstrated in this work permits the measurement of compressibility for single particles in fluid media, with potentially high throughput [24], without contact or labeling [6]. The approach is particularly powerful for measurements on soft particles, or biological particles, that have relatively large compressibility (low κs) and for which label-free analysis of large population statistics is a critical need.
4. Supplement: energy balance perturbation model
Here we derive a perturbational model for contributions of particle compressibility, density, size, and position on the OMFR mechanical frequency. For a harmonic oscillator, the average kinetic energy per oscillation cycle must equal to the average potential energy per oscillation cycle. For the OMFR, when there is no particle in the resonator, this relation can be expressed as:
where Wwp (Wwk) is the elastic strain (kinetic) energy associated with the resonator shell or wall “w”. We can then express this unperturbed situation as follows: where ρw (Vw) is the shell material density (shell volume), Vℓ is the resonator’s fluid core volume, and Ωi is the frequency of the vibrational mode. The subscript “1” denotes the unperturbed case. By assuming the oscillation to be time harmonic, i.e. the elastic displacement of the shell is U⃗i(r, t) = u⃗i(r) cos(Ωit) and the pressure field Pi(r, t) = pi(r) cos(Ωit), we can further simplify Eq. (5) to: We now place a particle of volume Vs inside the resonator liquid volume at some fixed location and rewrite this energy balance as: For added simplicity, we further assume that the elastic displacement field of the shell does not change when the particle is added, i.e. u⃗1 = u⃗2 = u⃗, and thus 〈Wwp1〉 = 〈Wwp2〉. We can then subtract Eq. (7) from Eq. (6) and obtain: where we have introduced notations: , , , , and . Since the frequency perturbation is assumed to be small compared with the resonance frequency, we have: , and , where ΔΩ = Ω2 − Ω1. We then obtain the fractional frequency perturbation of the mechanical mode as: The terms T1 − T2 and G1 − G2 represent the liquid volume excluding the particle, and are thus non-zero due to the scattering effect. The effect is subtle but still observable in the FEM simulation in Fig. 1(b) and (c). In theory, by simulating both the perturbed and unperturbed situations, we can use Eq. (9) to estimate the frequency perturbation quite precisely. However, the computational model has to be accurate enough to get a good estimation of all the difference terms in the numerator of Eq. (9), which is challenging and computationally expensive. Nonetheless, if we can theoretically calculate the scattering field in the future, we should be able to get a much better prediction. Presently, we further simplify Eq. (9) by neglecting the scattering effects such that that p1 = p2 = p, then T1 = T2, G1 = G2, t1 = t2 = t, and g1 = g2 = g. We then obtain: Finally, by defining Ω2 = 2π f2 and Ω1 = 2π f1, and treating for the denominator since the particle is much smaller compared to the mode volume, we obtain the perturbed resonance frequency where ,. The geometry and mode shape based parameter C provides the sensitivity to the compressibility contrast between the particle and the surrounding carrier liquid, while D provides sensitivity to the density contrast.Funding
National Science Foundation (NSF) (ECCS-1509391, ECCS-1408539)
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