## Abstract

Measuring the average refractive index (RI) of spherical objects, such as suspended cells, in quantitative phase imaging (QPI) requires a decoupling of RI and size from the QPI data. This has been commonly achieved by determining the object’s radius with geometrical approaches, neglecting light-scattering. Here, we present a novel QPI fitting algorithm that reliably uncouples the RI using Mie theory and a semi-analytical, corrected Rytov approach. We assess the range of validity of this algorithm *in silico* and experimentally investigate various objects (oil and protein droplets, microgel beads, cells) and noise conditions. In addition, we provide important practical cues for the analysis of spherical objects in QPI.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

## 1. Introduction

Quantitative phase imaging (QPI) is a collective term for interferometric techniques that quantify the phase retardation of otherwise transparent objects. The establishment of QPI as a swift tool in single-cell analysis has given rise to a broad spectrum of applications in cell biology. For instance, QPI has been used to quantify cell dry mass [1,2], cell culture quality [3], cell dynamics [4–7], bacterial infection [8], parasitic infection [9], cellular differentiation [10], or drug response [11]. An important quantity that can be measured with QPI is the cellular refractive index (RI). Besides characterizing cells based on RI and the associated local protein concentration [1,2,12], knowing the RI is also essential for related applications such as the optical stretcher to quantify optical forces [13–15], or for Brillouin microscopy to compute cell elasticity [16,17].

The accurate determination of the cellular RI with QPI is difficult, because the optical path difference (OPD) measured in QPI needs to be separated into integral RI and cell thickness. Thus, the integral RI can only be computed if the cell thickness is measured, which has been achieved using scanning probe microscopy [18, 19], confocal microscopy [20], deliberate variation of the OPD [4, 21], or spatial confinement [22, 23]. However, RI computation with such OPD approaches is based purely on geometric considerations, neglecting light-scattering. By taking the QPI analysis to the frequency domain using Fourier transform light scattering (FTLS), it is possible to determine the RI of strong spherical scatterers such as polystyrene beads [24, 25]. However, RI retrieval for comparatively weak scatterers such as cells has, to our knowledge, not been demonstrated with FTLS. A more general approach to this problem is the measurement of high-resolution 3D RI maps utilizing optical diffraction tomography (ODT) [26], which is used to address a multitude of biophysical and biomedical questions [9,27–29]. However, ODT depends on the acquisition of multiple phase images, which is accompanied by an elaborate experimental effort. Moreover, state-of-the-art ODT techniques approximate light propagation with the Rytov approximation [30,31], which can cause an underestimation of the intracellular RI for large RI gradients [32,33].

However, if a cell is spherical and sufficiently homogeneous in RI, then its average RI can be inferred from a single phase image. Several studies have exploited cell sphericity by computing the cell radius from the cell area visible in the phase image using OPD approaches [34–37]. However, a rigorous treatment of this problem by exactly modeling light propagation with Mie theory has not been presented so far due to the large computational effort required.

Here, we present a QPI phase fitting algorithm that reliably retrieves RI and size of spherical objects. We address the computational challenge with efficient implementations of the Mie- and Rytov-scattered fields. Additionally, we derive a correction factor for the Rytov approximation, resulting in accuracies similar to the results of Mie theory. Using *in silico* simulations, we compare our scattering approaches to OPD approaches for homogeneous spheres with RI values from 1.334 to 1.440. In the experimental part, we demonstrate 2D QPI phase fitting for several test targets and for various noise conditions. Our approach allows for a faster and better interpretation of QPI data for spherical objects and gives valuable insights into single-cell light-scattering.

## 2. Methods

#### 2.1. Modeling light-scattering by a sphere

The OPD approach resembles a highly simplified model for light-scattering, treating the propagation of light as a line integral through the sphere’s constant RI. To estimate the average RI of a cell from a single quantitative phase image with the OPD approach, the cell radius must be determined. The radius can be determined with a series of 1D phase profile fits [34], or by fitting a circle to a contour along the cell edge (found using e.g. the Canny edge detection algorithm) [38]. The former approach is equivalent to a direct fit of the projected RI of a sphere to the measured phase image (referred to as “OPD projection approach” in the remainder of the manuscript), while the latter approach only uses the contour data to determine the radius (referred to as “OPD edge-detection approach”). Note that the OPD edge-detection approach is somewhat erratic, because the contour found with an edge detection algorithm depends on image resolution. Both OPD approaches do not take into account diffraction and thus only describe optically thin objects correctly.

The Rytov approximation gives a better estimation of light propagation [39]. It is state-of-the-art in ODT applications, because it provides a simple linear model for diffraction and because it is sufficiently accurate for many cell types [32]. In general, to compute the Rytov approximation, a full 3D model of the specimen is required and its Fourier transform must be computed according to the Fourier diffraction theorem [26]. In case of a sphere, however, we reduced this problem to two dimensions by using the analytical solution of the Fourier transform of a sphere (see Appendix A.1 for a detailed description). Our semi-analytical approach enables a fast and accurate computation of the Rytov approximation for homogeneous spheres.

Mie theory yields an exact solution for the scattered field. Here, we used the software BHFIELD (versioned October 5th 2012, available at https://seafile.zfn.uni-bremen.de/f/2d6fc70841/?dl=1, accessed August 2017) [40, 41] and refocused the obtained fields to the plane at the sphere center using the Python package nrefocus (version 0.1.5, available at https://pypi.python.org/pypi/nrefocus). Generating the complete scattered field of a sphere using Mie theory is computationally expensive. Hence, we approximated the full 2D field by radially averaging two 1D fields, perpendicular and parallel to the polarization axis, resulting in an effectively unpolarized field. The 2D phase error made by this approach oscillates within a ±1% interval relative the the maximum optical path difference (data not shown) and can thus be neglected for the studies presented here.

A comparison of these scattering models with Mie theory as a reference is shown in Fig 1 for an exemplary homogeneous sphere. While the error made by the Rytov approximation is small, the error of the OPD projection approach scores high values, mostly due to the discontinuity in the gradient of the simulated phase image. To allow a comparison to the OPD edge-detection approach as well, the lower right quadrants of Figs. 1(a) and 1(b) show the OPD projection and resulting error based on RI and radius as obtained with the OPD edge-detection approach applied to the Mie data. The OPD edge-detection approach exhibits a large error which can be explained by an underestimation of the radius due to the fact that the contour found by the edge detection algorithm does not coincide with the lateral perimeter of the sphere. This example illustrates that the determination of the RI and the radius of spherical objects should be addressed with models that take into account diffraction (Mie theory, Rytov approximation), rather than plain OPD approaches (OPD projection, OPD edge-detection).

#### 2.2. Fitting scattering models in-silico

The determination of the RI and the radius of a homogeneous sphere from a single quantitative phase image using one of the scattering models introduced above requires a phase image fitting algorithm. Previous work on this topic employed a set of 1D OPD fits to the 2D phase image [34]. For Mie theory or the Rytov approximation, this 1D approach is inefficient, either because fitting requires an enormous number of field computations (Mie) or because each iteration of the 1D fit requires the computation of a 2D field (Rytov, our implementation). To take advantage of the performance-enhancing procedures described in the previous section (Mie and Rytov), an actual 2D phase image fit is, in fact, necessary.

Here, we propose an image fitting algorithm that we specifically designed for homogeneous spheres in QPI. Our algorithm iteratively fits RI, radius, and lateral position of a sphere as well as phase offset to a quantitative phase image. In contrast to commonly used fitting techniques that call the modeling function for every parameter set during fitting, our algorithm interpolates the modeled phase image in a predefined interval and thus requires less calls to the modeling function. This approach, which is described in detail in Appendix A.2, reliably retrieves RI, radius, lateral position, and phase offset for homogeneous, spherical phase objects.

### 2.2.1. Error of the scattering models

To assess the accuracy of our 2D fitting approach, we performed a series of Mie simulations and retrieved RI and radius using the light-scattering models described above. Figure 2 shows the relative errors in RI and radius made using the OPD edge-detection approach and the 2D phase fitting approach (OPD projection, Rytov approximation, Mie theory). The relative errors for RI (*n*) and radius (*r*) were computed using

*π*phase jump between two pixels) [37]. As a result, all approaches failed to determine the correct RI and radius in this region, which we also observed with the otherwise error-free 2D Mie fit (Figs. 2(h) and 2(f)). As discussed above, the OPD edge-detection approach underestimates the radius of the sphere and thus overestimates the RI, which is clearly visible in Figs. 2(a) (green) and 2(b) (brown). The 2D fit with the OPD projection (Figs. 2(c) and 2(d)) exhibited a noisy error signal, which can be attributed to the discontinuity in the gradient of the modeled phase. In comparison, the error of the 2D Rytov fit shown in Figs. 2(e) and 2(f) is smooth and has lower values than the OPD projection in both RI and radius for RI values below 1.39. The data suggest that the comparatively faster Rytov approximation can be preferred over Mie theory, if an error in the radius below 2% and an error in the RI below 0.1% is acceptable and if the imaged object has a radius above three wavelengths (3

*λ*) and an RI below 1.36.

### 2.2.2. Convergence with noise

In practice, quantitative phase images are subject to phase noise which can be described as a background pattern that is varying over the range of several pixels. Here, we modeled phase noise using 2D Perlin noise as implemented in the Python package noise (version 1.2.1, available at https://pypi.python.org/pypi/noise/). Figure 3 illustrates the convergence of the proposed fitting algorithm for various starting parameters and for realistic noise conditions ranging from 0% to 5% standard deviation measured relative to the maximum OPD. The data show that the algorithm converges to the same value after an average of seven iterations, independent of noise or initial conditions. Unexpectedly, the RI and radius obtained with the OPD edge-detection approach, labeled with a green triangle, is closer to the correct value. This is a coincidental result that can be explained by the fact that the OPD edge-detection approach is resolution-dependent, i.e. the resolution chosen lead to better results for the OPD edge-detection approach (see Figs. 2(a), 2(b)). Note that the initial conditions shown in Fig. 3(a) are rather extreme. In practice, we determined the initial guess for radius, RI, and position of the sphere with the OPD edge-detection approach. Thus, experimental phase noise does not affect the convergence of the proposed 2D phase image fitting algorithm.

#### 2.3. Systematic correction for the Rytov approximation

The errors in RI and radius made by the Rytov approximation shown in Figs. 2(e) and 2(f) become large for objects with RI values above 1.36. Thus, 2D fitting with the Rytov approximation would yield poor accuracies for many cell types with RI values reaching up to 1.40 and above. In terms of efficiency, this would be a drawback, because high accuracies could only be achieved by falling back to the computationally more expensive Mie theory. To address this issue, we derived a systematic correction for the Rytov approximation, making it possible to access large RI values with high accuracy. We found that our implementation of the Rytov approximation (see Appendix A.1) exhibits a systematic error that is independent of the image resolution used. As opposed to both OPD approaches, the Rytov approximation additionally exhibits a systematic error that is dependent only on the RI for object radii greater than three wavelengths. This allowed us to derive a systematic correction for the Rytov approximation (Rytov-SC), considerably reducing the error made. We derived the correction formulas for RI (*n*_{Ryt-SC}) and radius (*r*_{Ryt-SC}) by fitting a polynomial function to the Rytov error, yielding

*n*

_{med}is the RI of the medium and

*n*

_{Ryt}is the RI obtained using our 2D fitting algorithm and our implementation of the Rytov approximation. Note that we chose the variable

*x*such that the correction is independent of the RI of the medium

*n*

_{med}. As we used the error maps shown in Fig. 2 to derive Eqs. (3) and (4), the systematic correction is valid for spheres with RI

*n*

_{sph}and radius

*r*

_{sph}of at least

*λ*of the light used. This systematic correction extends the applicability of the Rytov approximation in 2D fitting to cells with high RI values (up to 1.44 and above), yielding high theoretical accuracies for RI (<0.1%) and radius (<1%).

## 3. Results

To compare the five approaches for the retrieval of RI and radius (OPD edge-detection, OPD projection fit, Rytov fit, Rytov-SC fit, Mie fit), we applied them to quantitative phase data of lipid droplets, microgel beads, and cells.

#### 3.1. 2D Mie fits to doplets, beads, and cells

Mie simulations in combination with our 2D fitting algorithm served as a benchmark for the other four approaches. Figure 4 shows a representative set of phase images, the corresponding Mie fits, and the fit residuals, which are discussed in the following.

Liquid droplets are ideal test samples for the investigation of scattering by spheres, because they are homogeneous and assume a spherical shape due to the surface tension at the droplet-medium interface. Figure 4(a) shows a quantitative phase image of a silicone oil droplet embedded in phosphate buffered saline (PBS). The oil droplet was produced by vortexing a two phase solution made of 1 mL of silicon oil (Sigma-Aldrich, 10cSt) and 10 mL of de-ionized water containing 2% w/v poly(ethylene glycol) monooleate (Sigma-Aldrich). The image was recorded with quadriwave lateral shearing interferometry (QLSI) [42] using a commercial QPI camera (SID4Bio, Phasics S.A.) attached to an inverted microscope (AxioObserver Z1, Zeiss) with a 40 × objective (NA 0.65, 421060-9900, Zeiss). The illumination wavelength was confined to an average of 647 nm using a bandpass filter (F37-647, 647/57, AHF analysentechnik). We separately measured the RI of the silicone oil using an Abbe refractometer (2WAJ, Arcarda), yielding a value of 1.402 that matches the value of 1.405 from the 2D Mie fit shown in Fig. 4(b). The resulting relative error of 0.04% is small which is reflected by residuals below 5% of the maximum OPD shown in Fig. 4(c). Figure 4(d) shows the quantitative phase image of a protein droplet that was recorded with the same setup as above, except for the bandpass filter. Here we used an average imaging wavelength of 550 nm for the RI analysis. The droplet consisted of the RNA-binding protein fused in sarcoma (FUS) and was embedded in a protein buffer that is described in detail in reference [43]. FUS has been linked to neurodegenerative diseases such as amyotrophic lateral sclerosis (ALS) (see e.g. [44,45]) which is correlated to a liquid to solid phase transition of the protein with time [43]. Here, we obtained an average RI of 1.435 for a liquid FUS protein droplet (Fig. 4(e)) using a protein buffer RI of 1.3465, measured with the Abbe refractometer named above. The fit residuals (Fig. 4(f)) are largely below 5%, indicating good agreement of theory and experiment. Note that the liquid droplets shown had comparatively high RI values and were imaged at low resolution, conditions that both were handled well by the proposed 2D fitting algorithm.

Microgel beads offer an additional, convenient possibility to test scattering by a sphere. Here, we used polyacrylamide (PAA) beads that were produced in a flow focusing microfluidic system described elsewhere [46,47]. We imaged two types of PAA beads with different density (named A and B for simplicity) using two different imaging setups. The microgel bead of type A was imaged with the QLSI setup used for the FUS droplet above with the addition of a telescope (*f*_{1}=−15 mm, LD2020-A and *f*_{2}=75 mm, LBF254-075-A, Thorlabs, Germany) to increase the magnification, and thus the sampling of the recorded phase image, by a factor of five. The microgel bead of type B was imaged with digital holographic microscopy (DHM). A detailed description of the DHM setup used can be found in reference [38]. A comparison of the resulting quantitative phase images in Figs. 4(g) and 4(k) shows that the DHM data have a higher background noise than the QLSI data (3.4% vs. 0.7%). The convergence of the 2D fitting algorithm is not affected by such noise magnitudes (see also Fig. 3) as can be seen in the fit residuals that are either below 5% (Fig. 4(j)) or reproduce the background phase noise magnitudes (Fig. 4(m)).

HL60 cells in suspension have a spherical shape. Even though the internal RI distribution of a cell is generally non-uniform, a 2D Mie fit can be used to estimate its average RI. Figures 4(n) and 4(r) show quantitative phase images of two HL60 cells, recorded with the DHM and QLSI setups described above; To match the wavelength of the laser used for DHM imaging (HeNe 633 nm, HNL050L-EC, Thorlabs), the illumination light of the QLSI setup was confined by a bandpass filter (BP 640/30, 488050-8001, Zeiss) in addition to filter F37-647 named above. Compared to liquid droplets or beads, the fit residuals shown in Figs. 4(q) and 4(f) exhibit heterogeneous structures that represent more and less dense regions within the cell. The similar RI values fitted to the representative cells, 1.3677 (QLSI) and 1.3673 (DHM), indicate consistency across the two different imaging setups and confirm resilience and accuracy of the 2D Mie fit with regard to phase noise, which is further discussed in the next section.

#### 3.2. Comparison of the scattering models

Each of the methods presented to determine the average RI of spherical objects has benefits and drawbacks. Mie theory yields the best theoretically possible result, but it is computationally expensive. The Rytov approximation is less expensive and, with a systematic correction (Eqs. (3) and (4)) it can achieve noteworthy accuracy. The OPD projection approach is faster than the Rytov approximation, but its accuracy is limited because it does not take into account diffraction. The same applies to the OPD edge-detection approach which is even faster but exhibits a resolution-dependent overestimation of the RI. To extend the theoretical insights obtained with the error maps shown in Fig. 2, a comparison of the scattering models based on four representative experimental data sets is shown in Fig. 5 and Table 1.

Figure 5(a) shows the RI values determined for a single FUS protein droplet from a time series recorded with the setup introduced in the previous section. Note that, due to the low resolution of the phase images (Fig. 4(d)), the RI values computed using the OPD edge-detection approach are severely overestimated and spread-out.

Figure 5(b) shows RI values for microgel beads of type A imaged with the corresponding QLSI setup described in the previous section (Fig. 4(g)). Here, the difference between the scattering models is not as large as for the protein droplet, because the resolution is higher and because the average RI of the beads is much closer to the surrounding medium (PBS, *n*_{PBS}=1.335).

Figures 5(c) and 5(d) show the RI values fitted to two different HL60 cell populations recorded with QLSI and DHM (see Figs. 4(n) and 4(r)). Note that each method produces consistent RI values across both imaging modalities (cell populations). A comparison of the average cell radii indicated slight differences between these two populations (6.5 μm QLSI and 6.9 μm DHM), which is supposedly caused by biological variation (data not shown).

In all examples shown, the systematically corrected Rytov approximation presents the best trade-off between accuracy and computational time. The high accuracy becomes most evident for the case of the FUS protein droplet which had a large RI and thus caused a prominent systematic error in the Rytov approximation. In case of the microgel beads, with RI values of about 1.34, the OPD projection approach and the non-corrected Rytov approximation still yield accurate results. However, at RI values that are observed in HL60 cells, the systematic correction of the Rytov approximation becomes important, producing RI values that are considerably closer to Mie theory than all other approaches. The total fitting time of the Rytov approximation is more than thirteen times shorter than the total fitting time of Mie simulations and less than three times longer than the fitting time of the OPD approach (e.g. for *N*=55 HL60 cells in Fig. 5(c): 17 minutes OPD projection, 48 minutes Rytov, 11 hours Mie) on a single CPU (Intel Core i7-4600U @2.10GHz, microcode version 0×21). Therefore, to accurately determine size and RI for spherical objects in QPI, we suggest the systematically corrected Rytov approximation in combination with a 2D fitting algorithm as presented here.

## 4. Discussion

#### 4.1. Summary

The present study provides an efficient method to accurately compute the RI and the radius for spherical objects from a single quantitative phase image. We investigated OPD approaches (OPD edge-detection, OPD projection), which were used in previous publications, and introduced efficient implementations of diffraction-based models (Mie theory, systematically corrected Rytov approximation) in combination with a 2D phase fitting algorithm, yielding improved accuracy and stability. Figure 6 illustrates the validity ranges of the models used (shaded regions) and provides orientation by indicating the exemplary samples shown in Fig. 4. We identified the systematically corrected Rytov approximation (Rytov-SC) as a feasible model for everyday-use in basic research, offering high accuracy at low computational costs over a broad range of sample sizes and RI values. Our 2D phase fitting approach can be applied to most biological cells (taking cell sphericity as given), is resilient to phase noise (see Fig. 3), and operates well even at low image resolution (see Figs. 4(a)–4(f)).

As a consequence, previous studies which employed the OPD approaches are likely subject to systematic errors. The OPD edge-detection approach overestimates the RI (e.g. [33,38] and possibly [37]). To illustrate this problem, a re-assessment of the RI of microgel beads presented in reference [33] reveals that the Rytov approximation (*n*_{hydro2D-Ryt}=1.354) is a better match for the RI determined using ODT (*n*_{ODT}=1.354) than the OPD edge-detection approach used in that study (*n*_{hydro2D-edge}=1.356). However, these systematic errors might not appear as prominent as shown in the present study when edge-detection algorithms are used that are able to correctly determine the cell boundary. The OPD projection approach slightly underestimates the actual RI, possibly affecting the studies presented in references [3,11,34]. In general however, we do not believe that our findings have any consequences regarding data interpretation of previous studies utilizing OPD approaches, because significant RI differences measured within a study were always determined consistently using one of the OPD approaches and a systematic error would merely manifest itself as a constant offset to the measured RI values.

#### 4.2. Outlook

The presented combination of the 2D fitting algorithm and the systematically corrected Rytov approximation for a sphere is efficient, but still offers room for improvement. First, graphical processing units (GPUs) could significantly speed-up the Fourier transform and the image interpolation steps of our current implementation. Second, the choice for the sampling of the radius with 42 points to compute the Rytov approximation, as discussed in Appendix A.1, is conservative and could be reduced to about 20 points. In this case, however, the correction formulas (Eqs. (3) and (4)) need to be updated. With these modifications, the fitting algorithm could in principle achieve real-time performance for live QPI analysis.

The fit residuals shown in Fig. 4 exhibit small systematic phase errors, visible as blue and red circles around the object perimeter. These can be attributed to the fact that the point spread function (PSF) of the imaging setups used is not taken into account in the simulation. At the cost of more computation time, the simulation images could in principle be convolved with the PSF which would yield lower residuals. However, we do not expect higher fitting accuracies from this approach, because the residuals shown are already low and because the number of pixels in the described region is small compared the total number of pixels that resemble the imaged object.

To resolve intracellular compartments such as the nucleus, nucleoli, and lipid droplets or to analyze cells that have an approximately ellipsoidal shape, our algorithm could be extended to support a superposition of spheres [49] and ellipsoids. This would allow to resolve subcellular compartments from a single phase image or enable a frame-by-frame analysis of cell volume and RI of elongated cells in an optical stretcher experiment. It should be noted that simplifying the cell as a set of spheres and ellipses is an inherently limiting factor for data interpretation. Nevertheless, such an extended approach would allow to paint an enhanced picture of the imaged cell from a single phase image at the sole cost of additional fitting parameters.

An accurate measurement of the RI of spherical objects is an important prerequisite for emerging topics in adjacent fields of research. For instance, our approach could allow valuable insights into the aging process of FUS [43] by tracking and characterizing the fit residuals during the liquid to solid phase transition of individual protein droplets. This would allow to determine whether the aging process starts at the center of the drop, at its surface, or homogeneously throughout. Furthermore, our algorithms could be used to assess the quantitative imaging quality of ODT and QPI techniques, both of which frequently employ microgel or polymer beads as reference samples (e.g. [27, 33, 37, 38, 42]). Finally, complementing the optical analysis of microgel beads as presented here with a mechanical description using, for instance, atomic force microscopy, will allow to establish a well-characterized microgel bead toolbox [47] which would be an invaluable reference for the optomechanical analysis of cells using techniques such as Brillouin microscopy or the optical stretcher. Thus, the optical characterization of spheres is fundamentally important to address topical questions in cell biology and biophysics.

## 5. Conclusion

The methods presented here resemble an important foundation for the accurate characterization of spherical micro-objects, including the optomechanical phenotyping of live cells, the study of protein droplets, or the classification of artificial beads. The presented approach is complementary to tomographic techniques, limited to spherical objects but able to deliver average values for RI and radius using only a single quantitative phase image. The derivation of an efficient model for light scattering by a sphere and the implementation of an automated 2D phase-fitting algorithm resemble a major advancement in accuracy, stability, and throughput. Hence, the presented approach is an attractive tool for many emerging techniques that rely on an exact optical characterization of spherical objects to address biophysical questions in basic and applied research.

## A. Appendix

The algorithms presented here are implemented in the Python package qpsphere version 0.1.3, available at https://pypi.python.org/pypi/qpsphere.

## A.1. Rytov near-field of a sphere

We obtained the field scattered by a sphere in the Rytov approximation by computing its analytical 2D Fourier transform followed by an inverse Fourier transform using the Fourier diffraction theorem. The use of an analytical solution in Fourier space has the advantage that it reduces artifacts that arise from the sharp boundaries of the spherical volume in real space and that the 3D problem is broken down to a 2D problem. According to the Fourier diffraction theorem, the Fourier transform of the 2D background-corrected scattered field *Û*_{B}(**k**) is mapped onto a semi-spherical surface in the Fourier transform of the 3D object potential *F̂*_{B} (**k**) according to [26,50,51]

**k**, the distance between the center of the object potential and the detector plane

_{D}*l*

_{D}, the wave number

*k*

_{m}= 2

*πn*

_{med}/

*λ*(refractive index of medium

*n*

_{med}and vacuum wavelength

*λ*), the factor $M=\sqrt{1-{k}_{\text{x}}^{2}-{k}_{\text{y}}^{2}}/{k}_{\text{m}}$, and the unit vector representing the direction of illumination, i.e. the rotational position of the sample relative to the detector normal,

**s**. The rotationally symmetric Fourier transform of a homogeneous sphere with radius

_{0}*R*is given by where

*j*

_{1}is the spherical Bessel function of the first kind of order one. Evaluating Eq. (9) at the frequencies that correspond to the spherical surface described by Eq. (8) and performing an inverse Fourier transform of the resulting

*Û*

_{B,sph}yields the background-corrected scattered field component in the Born approximation

*u*

_{B,sph}. The Rytov approximation of the scattered field component

*u*

_{R,sph}is then obtained by computing the exponential of

*u*

_{B,sph}[50,52]

*ℱ*

^{−1}. Assuming a incident plane wave, the full complex field then computes to In practice, we computed the Rytov approximation in two steps. First, the scattered field is computed with a resolution that samples the sphere radius with approximately 42 points. Second, the amplitude and phase data are linearly interpolated to match the resolution of the images recorded in the experiment. This two-step approach ensures that the field computation is resolution-independent, allowing to derive a systematic error correction for the Rytov approximation to yield results comparable to Mie theory as presented and discussed in section 2.3.

## A.2. 2D fitting algorithm for spheres in QPI

The 2D fitting of a phase image with a sphere-scattering model is burdened by many calls of the modeling function. In practice, this bottleneck becomes problematic when the scattering model is computationally expensive which is the case for e.g. Mie simulations. Here, we used a custom fitting algorithm that relies on interval-based phase image interpolation to reduce the number of calls to the modeling function. The algorithm reliably fits Mie, Rytov, and OPD projection models to a 2D phase image of a sphere with the five parameters background phase image offset *ϕ*_{bg}, x-coordinate of the center of the sphere *x*, y-coordinate of the center of the sphere *y*, refractive index (RI) of the sphere *n*_{sph}, and radius of the sphere *r*_{sph}. Given initial estimates for center (*x*_{0}, *y*_{0}), RI *n*_{0}, and radius *r*_{0} of the sphere, which are obtained with the OPD edge-detection approach, the algorithm iteratively fits a sphere model in six steps.

**Vary radius.**First, compute three phase images using the sphere model at the radii,*r*−_{i}*b*,_{r}*r*, and_{i}*r*+_{i}*b*, where initially_{r}*i*= 0 and*b*= 0.05_{r}*r*_{0}. Second, generate 47 phase images in the interval [*r*−_{i}*b*,_{r}*r*+_{i}*b*] by interpolation. Third, compare the interpolated phase images with the measured phase image using the root mean square (RMS) error. The radius with the lowest RMS error_{r}*r*_{i+1}is used for the following steps.**Vary RI.**As in step one, compute three phase images with the sphere model at the RI values,*n*−_{i}*b*,_{n}*n*, and_{i}*n*+_{i}*b*, where initially_{n}*b*= 0.1(_{n}*n*_{0}−*n*_{med}) (RI of the medium*n*_{med}). Then, interpolate 47 phase images in the interval and select the one with the lowest RMS error*n*_{i+1}.**Vary center.**Initially, the interval parameter for the center coordinate*b*_{c}is set to one wavelength (*λ*) or 5% of the radius, depending on which is larger Then, 13×13 phase images are computed for the intervals [*x*−_{i}*b*_{c},*x*+_{i}*b*_{c}] and [*y*−_{i}*b*_{c},*y*+_{i}*b*_{c}]. The phase image with the lowest RMS defines the center position (*x*_{i+1},*y*_{i+1}) for subsequent iterations.**Phase background estimation.**Experimental phase images can contain a constant phase offset. We estimate the phase offset*ϕ*by averaging the background phase values of the experimental phase image. To determine the pixel locations of the background phase image, the intersection of two pixel sets (a) and (b) is used:_{i}**Scale down interval parameters.**The interval parameters*b*,_{r}*b*, and_{n}*b*_{c}are individually divided by two, depending on the following conditions:*b*: The radius_{r}*r*_{i+1}is in the interval [*r*− 0.1_{i}*b*,_{r}*r*+ 0.1_{i}*b*]._{r}*b*: The RI_{n}*n*_{i+1}is in the interval [*n*− 0.1_{i}*b*,_{n}*n*+ 0.1_{i}*b*]._{n}*b*: The change in the location of the center position $\sqrt{{\left({x}_{i}-{x}_{i+1}\right)}^{2}+{\left({y}_{i}-{y}_{i+1}\right)}^{2}}$is smaller than_{c}*b*._{c}

Scaling down the interval parameters leads to a parameter refinement in each iteration of the algorithm.

**Stopping criteria.**If the stopping conditions are not all met, then the algorithm proceeds with step one. The algorithm stops iterating when all of the following conditions are met:

## Funding

European Union’s Seventh Framework Programme, Starting Grant “Light Touch” (282060); Alexander-von-Humboldt Stiftung, Humboldt-Professorship.

## Acknowledgments

The authors want to thank Tony Hyman, Simon Alberti, and their respective research groups for discussions as well as the Protein Expression and Purification Facility of the Max Planck Institute of Molecular Cell Biology and Genetics (MPI-CBG) for the provision of the FUS protein sample. We thank the BIOTEC/CRTD Microstructure Facility (partly funded by the State of Saxony and the European Fund for Regional Development - EFRE) for the production of the microgel beads. The HL60/S4 cells were a generous gift from Donald and Ada Olins (University of New England). This project has received funding from the European Union’s Seventh Framework Programme for the Starting Grant “Light Touch” (grant agreement no. 282060) and from the Alexander-von-Humboldt Stiftung (Humboldt-Professorship to J.G.).

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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