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. 2025 Nov 19;97(47):25999–26008. doi: 10.1021/acs.analchem.5c03537

Comparative Analysis of Sample Loop and Counting Bead–Based Methods for Size-Dependent Bias in Flow Cytometry

Hye Ji Shin †,‡, Subeen Kim §, Minjeong Kwak ∥, Inchul Yang †, Sang-Ryoul Park †, Jihwan Song §,*, Ji Youn Lee †,‡,*
PMCID: PMC12676513  PMID: 41259737

Abstract

Reliable and accurate particle number concentration measurements are essential across various fields, including clinical diagnostics, environmental monitoring, and industrial applications. Flow cytometry is widely used for these measurements, where the use of counting beads is a common approach. However, this method can introduce size-dependent bias when the target particles differ in size from the counting beads. To evaluate size-dependent bias, this study systematically compares the conventional counting bead–based method with a sample loop–based method that relies on total counting with a defined sample volume. Experimental results show that while both methods yield similar concentrations for beads of comparable size, discrepancies arise when there are significant size differences between the counting beads and target particles. To investigate the cause of this bias, simulations based on force balance analysis were conducted, revealing that larger beads experience stronger forces that facilitate their movement toward the detection area, while smaller beads are more influenced by Brownian motion, which impedes their overall motion. These findings provide a mechanistic explanation for the observed size-dependent bias, confirming that differences in hydrodynamic behavior contribute to variations in bead distribution and motion. By using the sample loop method, which minimizes size-dependent bias, and employing an empirical equation derived from the results, this study offers a reliable approach for predicting and mitigating bias in particle concentration measurements. This work therefore contributes to the development of more precise and traceable methods for particle number concentration measurement, with implications for a wide range of biological and industrial applications.

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Reliable and traceable concentration measurement of particles in suspension is crucial across diverse fields for both regulatory and industrial needs, ranging from water quality control to diagnostics and industrial processes. For example, particle counting plays a routine role in clinical laboratories for blood cell counting, , quality control for cell therapy products, in environmental monitoring for particulate analysis, , and in industries for insoluble particle inspections. Number concentration is conceptually straightforward, calculated as the total particle count divided by the sample volume. However, in most instruments used for number concentration measurements, typically in flow systems, it is challenging to accurately determine the sample volume passing through the detection area. This uncertainty in volume measurement often results in inaccurate concentration determinations. To address this, current approaches include estimating volume based on steady-state sample flow or performing total counting on a fixed volume or weight of sample. These methods require a thorough understanding and mitigation of uncertainty sources to ensure reliable and traceable results.

Several research groups in metrology institutes have made significant strides in this area. One group, for example, has developed methods for liquid-borne particle counting using optical particle counters, incorporating strategies such as repetitive measurements and syringe-pump-based mass reduction to estimate steady-state sample volumes. , They later extended these methods to submicron particles and identified associated uncertainty sources. Validation was conducted using orthogonal techniques such as scanning electron microscope (SEM) imaging to compare static counting results with the optical methods. Another research group has employed modified particle counters, such as light obscuration and flow imaging systems, to achieve accurate number concentration measurements, addressing challenges like coincidence errors and optical effects. Further, a team has developed impedance-based reference methods for blood cell counting and also explored a calibrated syringe and gas-buffered injection system for precise volume determination via flow cytometry. , Our group has designed methods for exhaustive counting using defined capillaries , and sample loop–based injection with a commercial flow cytometer, which we validated for biomolecules such as DNA and RNA as well as microbeads. Here, a sample loop refers to a component in flow systems used to introduce a defined volume of sample into the analytical path, ensuring accurate and reproducible measurements. The volumes of the capillaries and the sample loop are determined gravimetrically for traceable number concentration measurements.

Flow cytometry offers distinct advantages for particle counting and number concentration measurements, as it can handle a wide range of analytes including cells, ,,− bacteria, − extracellular vesicles, , and nanoparticles. , While flow cytometry is often used for qualitative comparisons based on signal intensity, achieving highly reliable number concentration measurements could make it an even more informative tool for diverse applications. One widely adopted method for number concentration measurements in biological applications involves the use of counting beads. , This approach assumes that counting beads and target particles exhibit similar hydrodynamic behaviors, enabling ratiometric concentration determination. Specifically, the concentration of the target particles is calculated by using the ratio of target particles to counting beads, based on the known count or concentration of the counting beads. While the counting bead method is simple and effective, requiring only mixing and measuring to calculate the target particle concentration, its limitations include cost and the assumption that the beads and target particles behave similarly during sampling and detection. Moreover, differences in hydrodynamic properties (such as size or shape) can lead to biases in measurements, which are often overlooked. In addition, uncertainties in sample volume and particle behavior during detection can compromise the reliability and traceability of the measurements. In light of these issues, comparing the counting bead method with the sample loop method could provide valuable insights into reducing the biases and improving measurement accuracy.

In this work, our goal is to identify and quantify potential size-dependent biases in number concentration measurements and investigate their potential sources. To do so, we systematically compare the traditional counting bead method with the sample loop method, which we previously developed using a sample loop for total particle counting. Through the results, we aim to contribute to the achievement of more reliable and traceable number concentration measurements.

Experimental Section

Bead Solution Preparation

Materials

The beads used for measurement were either yellow-green fluorescent polystyrene beads (0.5, 0.75, 1, 3, and 6 μm, Polysciences; 2 and 10 μm, Thermo Fisher Scientific) or nonfluorescent polystyrene beads (5 μm, streptavidin-coated, Spherotech; 8 μm, Sigma-Aldrich). The diluent used was a solution of deionized water with 0.01% or 0.1% Tween 20, which was filtered through a 20 nm syringe filter.

Bead Mixture Solution

Bead mixture solutions were freshly prepared before each experiment. The bead mixture was adjusted to a final concentration of 1–2 × 106 particles/mL, with similar concentrations for each bead type. To prevent aggregation, the bead solutions were sonicated for approximately 30 s prior to use.

Dilution Series Test Samples

To verify linearity, 1, 5, and 8 μm bead solutions of approximately 1 × 106 particles/mL were prepared. Based on these solutions, dilution series with three to five dilution fractions (DFs) between 0.1 to 1 were then prepared. The weight of each solution was measured using an analytical balance (XS204, Mettler Toledo) to determine the actual DF.

Bead Mixture with Counting Beads

A bead solution of DFs of 0.6 or 1 was prepared by mixing the bead stock solution with a diluent. Then 0.4 to 0.5 mL of the resulting bead solution was added directly to Trucount tubes (BD Biosciences). Three or four tubes were combined and aliquoted for use in either sample loop–based or conventional counting bead–based measurements. The weight of the added solution was measured with an analytical balance for an accurate concentration measurement.

Flow Cytometer Measurement

The employed setup using a sample loop is described in a previously published paper. In brief, a six-port injection valve equipped with 20 μL sample loops was connected to the sample line of a FACSVerse (BD Biosciences) flow cytometer. In injection mode, a fixed volume of the sample inside the sample loop was delivered to the detection area by a vacuum-driven carrier medium (0.2 μm filtered deionized water in this study), under the manufacturer’s high flow rate setting of 120 μL/min (nominal value). An excess volume of carrier medium was flowed through to perform exhaustive counting. Data analysis was conducted using FlowJo (version 10.9, FlowJo LLC). For bead counting, the nonfluorescent 5 and 8 μm bead populations were gated using forward scatter (FSC) versus side scatter (SSC), while the fluorescent 0.5 μm, 0.75 μm, 1 μm, 3 μm, and counting bead populations were analyzed by gating the fluorescence channel (e.g., FITC or APC) versus the SSC channel.

Bead Size Measurement

The particle size distribution of the polystyrene beads was measured using an SEM (GeminiSEM 500, Carl Zeiss Microscopy GmbH). For bead solutions containing residual proteins, dispersants, or other organic components, the samples were purified using a dialysis cassette with a 10,000 molecular weight cutoff (Slide-A-Lyzer Dialysis Cassette, Thermo Fisher Scientific) for 24 h prior to analysis. Bead solutions were applied to 1 cm × 1 cm silicon wafer pieces, which were cleaned with ethanol and deionized water, followed by UV/ozone treatment to remove residual contaminants. A 5 μL aliquot of the bead suspension was evenly distributed on the wafer surface using a spin coater (800 rpm, 20 min) and air-dried.

Image processing was performed using ImageJ (NIH), employing Otsu thresholding for particle segmentation. For accuracy, only well-separated, distinct single particles (minimum of 90) were analyzed. Particle size and distribution were determined based on the Feret diameter.

Simulations for Flow Characteristics and Bead Tracing

A multiphysics simulation coupling computational fluid dynamics (CFD) and particle tracing simulation for the flow characteristics of bead-based counting and bead tracing was performed using commercial software (COMSOL Multiphysics 5.4; Comsol, Inc.) (Supporting Information Figure S1). For particle tracing simulations of bead motion, forces including drag (F D), buoyancy (F B), gravity (F G), and Brownian motion (F Brownian) were considered. Force balance analysis was performed on a region near the bottom of the tube (4.5 mm from the bottom), where the difference in the bead motion was prominent (Figure S2). The detailed equations and parameters are described in the Supporting Information.

Results and Discussion

Study Design and Bead Mixture Preparation

In our previous work, we developed and validated a volumetric counting method using a sample loop. The primary advantage of this method is exhaustive counting, which hypothetically enables unbiased sampling and uniform measurement regardless of sample type, whether single substances or mixtures with different hydrodynamic properties. For method validation, we compared our approach with the conventional counting bead–based measurement. Although the results were generally comparable, a consistent offset of 8–9% was observed between the two methods, with the sample loop method yielding lower values. The counting bead method calculates the concentration of the target substance in a ratiometric manner, assuming that both the counting beads and the target particles are measured equivalently after being mixed together. However, if the size of the counting beads differs from that of the target particles, differences in hydrodynamic properties can introduce bias. While such differences can also depend on factors like flow rate, particle density, and sample injection configurations, we presume that the primary cause of the observed bias is the size difference between the counting beads and target particles.

To investigate this, in the present work we aimed to systematically compare the volumetric sample loop method and the ratiometric counting bead method for particles of various sizes, as illustrated in Figure . We prepared a polystyrene bead mixture comprising different sizes to enable efficient analysis through the simultaneous measurement of all sizes, and compared the concentrations measured using the two methods. For counting beads, we used Trucount tubes, which contain a freeze-dried pellet with a known number of fluorescent beads. These beads are reported to be approximately 3–5 μm in size, and we selected target beads for the mixture with both smaller and larger diameters relative to the counting beads.

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(A) Schematic illustration of two approaches to number concentration measurement: the sample loop method and the counting bead method. In the sample loop method, the bead solution is delivered to a sample loop by loading with syringe, and the concentration (C B,SL) is determined by dividing the total bead count (N B) by the sample loop volume (V). In the counting bead method, the bead solution is filled in a sample tube together with counting beads, and the concentration (C B,CB) is calculated using the ratio of target bead count (N B) to counting bead count (N CB), multiplied by the known concentration of counting beads (C CB). (B) Bead sizes used in the study. Yellow-green fluorescent polystyrene (PS) beads with nominal diameters of 0.5, 0.75, 1, and 3 μm were used, as well as nonlabeled PS beads of 5 and 8 μm. The counting beads are also shown for reference.

Before the main experiments with the bead mixture, we performed preliminary tests to ensure the suitability of our approach. We examined whether the bead populations appeared in the same positions on the cytogram for both methods, regardless of whether the sample consisted of single beads or the bead mixture, and confirmed that all beads were counted exhaustively. Additionally, we verified that the concentrations measured during acquisition were consistent between single beads and bead mixture, with no significant differences. For five bead sizes ranging from 1 to 10 μm, we performed acquisitions both with single beads and the bead mixture. Regardless of the measurement method or sample type, the same cytograms were obtained, and the same gating strategy was applicable (Figure S3). For the 5 min acquisition, more than 95.6% of bead events for all bead sizes ranging from 0.5 to 8 μm were acquired within the first minute, while less than 0.3% were detected during the final minute of acquisition (Figure S4). To rule out potential event loss due to smaller beads being obscured by larger beads in the detection region, we compared measurements from 1, 2, and 5 μm single beads and their mixture (Figure S5). We observed no significant differences, likely due to the moderate total particle concentration used in our experiments (about 1–1.5 × 106 particles/mL or 1000–1500 events/s). These results demonstrated that the use of the bead mixture did not lead to event loss or bias, confirming its suitability for the present study.

Then we selected a set of target beads for the main bead mixture experiments. The criteria for bead selection included the presence of a well-defined main population with minimal doublets, clear distinction from noise, and unambiguous separation of each bead population, even when multiple bead sizes were mixed. Based on these criteria, we selected six bead sizes ranging from 0.5 to 8 μm, ensuring sufficient size differences for clear gating during simultaneous measurements. For smaller beads, yellow-green fluorescent beads were selected to achieve effective separation from noise. Figure summarizes our approach and the bead mixture composition, and Figures A and S6 display the gating strategy used for the bead mixture. Since we acquire events from multiple beads simultaneously, coincidence is indispensable. However, the average coincidence level was around 0.3%, which is a relatively small value and does not significantly affect the differences between the methods we investigated. To minimize the impact of coincidence events, we performed gating to include only the main bead populations.

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(A) Representative gating strategy for bead populations in the sample loop (SL) method. Beads with diameters of 0.5, 0.75, 1, and 3 μm are gated using the FITC-H vs SSC-H cytogram, while 5 and 8 μm beads are gated using the FSC-A vs SSC-A cytogram. (B) Measured bead concentrations (C B,SL and C B,CB) for each bead size using the SL and CB methods. The SL method estimates concentration based on sample volume, while the CB method uses the ratiometric approach with counting beads. Bars represent the mean concentration with error bars indicating standard deviation. (C) Comparison of the CB/SL ratio (C B,CB /C B,SL) across different bead sizes, showing size-dependent bias between the two methods. Each data point represents an independent measurement, with error bars indicating variability.

Comparison of Number Concentration Measurements by the Sample Loop and Counting Bead Methods

We compared the concentration measurements obtained using the sample loop method and the counting bead method for beads of each size (Figure B). Interestingly, the measured concentrations varied depending on the target bead size. Relative to the counting beads, opposite trends were observed based on bead size. When the target bead size was smaller than the counting beads, the concentrations measured by the sample loop method were higher than those by the counting bead method. Conversely, when the target bead size was larger than the counting beads, the opposite trend was observed. Furthermore, the degree of this size-dependent bias increased proportionally with the size difference.

To quantify the bias across different experiments, we introduced a new parameter: the relative ratio of counting bead–based measurement compared to sample loop–based measurement, denoted as the CB/SL ratio. This was needed to facilitate comparison of different sets of experiments because some variation in the concentration of individual beads was inevitable (Figure S7), even though we prepared bead mixtures freshly before each experiment to maintain similar concentrations across beads of different sizes. As shown in Figure C, the size-dependent bias was reproducibly observed, in terms of CB/SL ratio, across independent experiments. All the beads used in this study were polystyrene beads, presumably sharing identical physical properties (e.g., density) except for size and fluorescence. Therefore, bead size was the sole factor influencing their hydrodynamic properties.

Investigating Potential Sources for the Bias

While it could be expected that differing hydrodynamic behaviors between beads of varying sizes may potentially lead to the discrepancies between the two methods, the observed trend was not immediately explainable. Additionally, it remained unclear which method is more accurate, necessitating further investigation into the bias. We hypothesized that both methods could contribute to bias and outlined potential sources as follows.

For the sample loop method, one possible source of bias is the presence of dead volume in the sample loop setup, caused by additional fittings. In this method, concentration is calculated as the number of events divided by the sample volume, which is assumed to be the volume of the sample loop. If dead volume is present, additional volume may be loaded, leading to an overestimation of concentration. Furthermore, if a vortex occurs within the dead volume, smaller beads might remain in the system longer than larger beads, potentially causing sample loss and lower concentration measurements for smaller beads. While this effect is challenging to observe directly, it is questionable whether a significant dead volume exists in practice. Another possible source of bias is incomplete exhaustive counting. Failure to achieve exhaustive counting due to particle loss could result in an overall underestimation of concentration. However, the time histogram data described in the previous section suggest this is unlikely in our setup, since only a very small portion of events were observed in the final minute of acquisition. Additionally, introducing a sample loop increases the path length from the sample reservoir to the detection point, which could lead to particle loss by adsorption. For instance, when using a 20 μL loop, the sample travel distance increases by approximately 50% compared to a setup without a sample loop.

To investigate potential biases caused by event loss due to adsorption in the sample loop setup, we examined the effects of detergent concentration and linearity. Variations in surfactant concentration could impact particle recovery, so we tested different concentrations of Tween 20 in the diluent. While the typical range is 0.01–0.1%, we expanded the range to 0.001–0.5% for this test. When relative counts were plotted against the 0.1% concentration, the results showed a slight increase in bead count as the Tween 20 concentration increased up to 0.1%, followed by a decrease at 0.5%. The variation was within 5% (Figure S8). For consistency, we used a 0.1% concentration of Tween 20 concentration in all main experiments. Additionally, we evaluated the linearity of the bead concentration across different dilutions. If adsorption were a significant issue, linearity would not be maintained. In the results, 1, 5, and 8 μm beads showed a high linearity, confirming that no noticeable adsorption effects were present (Figure S9).

For the counting bead method, one plausible source of bias is sedimentation. In this setup, the bead mixture is contained within a sample tube, and a sample line extends near the bottom of the tube to draw up the mixture. This raises the question of whether sedimentation could lead to size-dependent differences in how the beads are taken up. Specifically, heavier particles may sediment more quickly, potentially causing beads of different sizes to be drawn into the sample line at varying rates, thereby introducing bias. The impact of sedimentation is expected to vary with acquisition time, and therefore extended acquisition durations could help reveal its effects.

We therefore tested whether sedimentation might affect the composition of beads entering the sample line. Typically, we prepared 0.3–0.4 mL of sample and conducted relatively short acquisitions lasting 30 to 60 s, but for this experiment, we extended the acquisition time to 600 s by preparing a sufficient volume of sample. In the results, the bead ratios remained nearly constant throughout the measurement, despite the extended time (Figure S10). This consistency is likely because the beads used in our study, all smaller than 10 μm, are less prone to sedimentation over short timeframes. Given that our acquisitions were typically completed within a few minutes, we concluded that sedimentation is unlikely to introduce significant bias in our measurements.

Since sedimentation was ruled out as a source of bias, the size-dependent bias is likely introduced during the process of beads traveling from the sample line injection point to the detection unit. In other words, particles of different sizes may exhibit distinct hydrodynamic behaviors as they move through the system, as we hypothesized in the study design. This aspect is further examined through simulations in the next section.

Additionally, potential sources of bias in bead counting also include the possible loss of counts due to coincidence counting and instrument dead time. ,, These factors are relevant to both methods, but as noted earlier about the gating strategy, the coincidence level was low, and therefore these factors were not specifically addressed in this work. However, for accurate number concentration measurements, especially when high event rates are maintained during the measurements, these factors must be carefully considered.

Computational Analysis of Bead Motion in the Counting Bead Method

Figure A shows the magnitude of sample velocity and fluid flow in the sample tube and flow cell. The average magnitude of velocity in the sample tube (≈1.7 × 10–5 m/s) was significantly lower than in the capillary (≈0.02 m/s). Streamlines in the figure display the formation of fluid flow aspirating fluid from the sample tube into the capillary as well as hydrodynamic focusing by the sheath flow. Using the velocity field results from the CFD simulation, bead motion was evaluated using particle tracing simulation. Target beads of various sizes (diameter d = 0.5, 0.75, 1, 3, 5, and 8 μm, the same as those used in the bead mixture experiment) and counting beads (d = 4 μm) were included. Beads were assumed to be randomly distributed within the sample tube at the start of the simulation. Figure B shows the accumulated bead count of each bead type at the detection point. As shown in the graph, the slope of the bead count profile increased with d. Consequently, more beads reached the detection point as d increased, suggesting that the beads have different hydrodynamic behaviors depending on d.

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Computational analysis of the counting bead–based measurement system through multiphysics modeling. (A) Streamlines and magnitude of flow velocity in the flow cell (left) and sample tube (right). The red and white lines represent the streamlines in the flow cell and capillary, respectively. (B) Bead count profiles from the number of beads counted at the detection point (i.e., capillary outlet) over time according to d (0.5, 0.75, 1, 3, 4, 5, and 8 μm). (C) Ratio of the number of counted beads according to d (N B) with respect to those of d = 4 μm (N CB) at t = 60 s. (D) Spatial distribution of d = 0.5 μm (top) and d = 8 μm (bottom) beads in the sample tube when aspirated from the sample tube into the capillary. Dark blue and light blue indicate beads in the top and bottom of the sample tube, respectively. (E) Bead velocity (v bead) depending on d with ρbead = 1.06 g/cm3 along the streamline shown in the inset. (F) The z-component of the forces (i.e., F D,z, F B,z – F G,z, and F Brownian,RMS) acting on the bead according to d.

The bead count ratio was then evaluated (Figure C), which is the number of target beads (N B) relative to the number of counting beads (N CB), consistent with the CB/SL ratio obtained in the experiments. The bead count ratio increased with the size of the target beads, ranging from 0.95 for the smallest beads to 1.07 for the largest beads. This trend matches well with the CB/SL ratio from the experimental results (Figure C). Additionally, the spatial distributions of 0.5 and 8 μm beads (Figure D) indicate that beads located near the bottom of the sample tube (i.e., light blue beads in the figure) moved more rapidly toward the capillary inlet as d increased under the fluid velocity field of flow cytometry. A consistent trend of bead motion with increasing d was also observed from 0.75 to 5 μm beads (Figure S11). However, in the capillary, the dominant influence of fluid velocity overrode the size effect (Figure S12), resulting in uniform bead motion regardless of bead size.

Force Balance Analysis of the Beads in the Counting Bead Method

To quantitatively assess the differences in motion based on bead size (d), the bead velocity (v bead) in the sample tube was calculated. Figure E shows the magnitude of v bead depending on d over time as the bead travels along the streamline from the CFD simulation (inset of Figure E). The results indicate that v bead increases with d from 51 μm/s (d = 0.5 μm) to 65 μm/s (d = 8 μm) at t = 40 s. This behavior is attributed to larger beads moving more rapidly toward the bottom of the tube, where fluid velocity is higher. To assess the effect of bead sedimentation on the size-dependent variation in v bead, the settling velocity of the beads (v settle) was calculated in the absence of fluid flow in flow cytometry (Figure S13). At d = 8 μm, v settle was approximately 2 μm/s, which is significantly lower than the aspirated v bead in flow cytometry (≈65 μm/s). This indicates that v settle is negligible compared to the inflow velocity of the beads. To elucidate the cause of the size-dependent difference in v bead, a force balance analysis was conducted. Four primary forces acting on the beads were considered for the analysis: drag (F D), buoyancy (F B), gravity (F G), and Brownian motion (F Brownian). Particularly, we focused on the z-component of the forces (i.e., F D,z, F B,z, F G,z, and F Brownian,z) as a function of d because beads travel downward to be taken up into the capillary (Figure F). As d increases, F D,z increased significantly from 4 × 10–17 N (d = 0.5 μm) to 1.6 × 10–13 N (d = 8 μm). F D,z is proportional to the relative velocity between the fluid and beads. As shown in Figure S14, the relative velocity increased as d increased. Furthermore, the relative velocity was comparable to v settle across all bead sizes. This indicates that the settling due to the difference between buoyancy and gravity (i.e., F B,z – F G,z) causes beads to move further downward relative to the fluid, leading to the difference in hydrodynamic behavior with d. This difference between buoyancy and gravity, F B,z – F G,z acting in the – z direction, increased from −6.6 × 10–17 N (d = 0.5 μm) to −1.6 × 10–13 N (d = 8 μm) with increasing d. Among the forces analyzed, F B,z – F G,z had the largest magnitude, with its negative values indicating that this term is the dominant factor in the bead motion approaching the bottom of the tube. To compare the nondirectional Brownian motion (Figure S15) with the deterministic forces (i.e., F D,z and F B,z – F G,z), the root-mean-square (RMS) value of Brownian motion (F Brownian,RMS) was used as a representative magnitude. As d increased, F Brownian,RMS increased from 2.0 × 10–17 N (d = 0.5 μm) to 8.2 × 10–17 N (d = 8 μm). The influence of F Brownian,RMS, which impedes bead motion from the tube to the capillary, is prominent for smaller d due to the low magnitude of F B,z – F G,z in the −z direction. Consequently, F Brownian,RMS counteracts F B,z – F G,z, the major driver of the size-dependent bias, thereby diminishing the effect of the settling velocity with small d.

Lastly, the z-component of the relative contribution of each force (F relative,z) acting on the beads was evaluated (Figure S16), as defined in eq 10 of the Supporting Information. As d increased, F relative,z in the – z direction increased significantly from −1.05 × 10–17 N (d = 0.5 μm) to −5 × 10–16 N (d = 8 μm). Notably, the increase in F relative,z induced by F B,z – F G,z for larger d facilitated the downward bead motion of larger beads toward the capillary, resulting in the size-dependent bias.

Force balance analysis confirmed that the difference in hydrodynamic behavior with d was induced by F B,z – F G,z. Although bulk-scale sedimentation did not occur, F B,z – F G,z generated subtle differences in the z-position of beads at the microscale (Figure S17). These subtle differences were amplified when coupled with the steeply increasing fluid velocity field, ultimately leading to the size-dependent bias.

Confirmation of Size-Dependent Bias in the Counting Bead Method

The simulation results suggested that the counting bead method introduces bias when particles with different hydrodynamic behaviors are mixed, likely due to sampling issues. To further investigate this, we conducted an experiment combining the target bead mixture with counting beads for measurement via the sample loop method. In this case, the bead mixture containing counting beads is not continuously drawn from a sample tube into the sample line but instead exhaustively counted in the sample loop setup, as illustrated in Figure A, thereby eliminating the potential for sampling-related bias. If the bias observed in the counting bead method arises from sampling, it is likely to disappear under this setup.

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(A) Schematic illustrations of three different bead concentration measurement methods: sample loop–based, sample loop with counting beads, and counting bead–based. In the sample loop with counting beads method, concentrations were calculated in two different ways. (B) Relative bead concentrations (normalized to the sample loop method) for various bead sizes (d = 0.5, 0.75, 1, 3, 5, and 8 μm) using three measurement methods. Conc.ratio1, Conc.ratio2, and Conc.ratio3 represent the ratios of bead concentration measurements for each method compared to the sample loop–based method.

Particle concentrations in the sample loop setup were calculated using two approaches: one based on the sample loop volume (C B,SL1 ) and the other using the relative concentration of counting beads (C B,SL2 ). Results were compared with the concentration from the standard counting bead method (C B,CB ). These three concentrations were expressed relative to that from a standard sample loop–based measurement without counting beads (C B,SL ), denoted Conc.ratio1, Conc.ratio2, and Conc.ratio3, respectively (Figure B). For all beads, Conc.ratio1 and Conc.ratio2 showed similar results, while Conc.ratio3 showed distinctive bead size-dependent differences. This result indicates that if target beads are mixed with counting beads in a sample loop, the size-dependent bias does not occur; i.e., the bias only arises when target beads are mixed with counting beads in a sample tube. Accordingly, it is confirmed that the bias in the counting bead method arises from the sampling process as particles are drawn from the sample tube into the sample line. These findings strongly support the simulation results.

Investigation of Size-Dependent Bias Using SEM and Empirical Analysis

Given that the observed bias appeared to be size-dependent, we hypothesized that it might be possible to establish an empirical correlation to predict the extent of the bias based on bead size. However, this required precise bead size measurements, as the sizes used in the current experiments were nominal values provided by the manufacturers. To address this, we performed SEM measurements to more precisely determine the bead sizes. The measured diameters were found to be 3.9–9.1% larger than their nominal values. As an example, an SEM image of a 0.5 μm bead and the corresponding Feret diameter histogram are shown in Figure A,B, respectively, while the complete size measurements are provided in Table S1.

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Analysis of size-dependent bias using SEM measurements and correlation plots. (A) Representative SEM image of 0.5 μm polystyrene beads. (B) Feret diameter distribution of the 0.5 μm beads based on SEM measurements. (C) CB/SL ratio as a function of bead size, using both nominal and measured diameters. (D) Correlation between the size difference of the target beads (d B) and counting beads (d CB) and the concentration difference between the two methods (C B,CB – C B,SL).

Using these measured values, we plotted the CB/SL ratio as a function of both nominal and measured bead size (Figure C). The results revealed a relatively linear trend, allowing us to derive an empirical equation to predict the bias. At least within the parameters of our experimental setup, the equation appears to provide a reliable estimation of size-dependent bias. While using the measured diameters resulted in a slight overall shift in the graph and a marginal improvement in correlation as compared to the nominal values, no substantial deviations were observed. To further explore the correlation, we analyzed the relationship between the size difference between the target beads (d B) and counting beads (d CB) and the concentration difference between the two methods, C B,CB – C B,SL (Figure D). This direct comparison demonstrated that when the nominal target bead size matched the nominal counting bead size, the bias between the two methods was minimal. Yet, the use of the measured diameters further reduced the observed bias, reinforcing the importance of precise size determination. These findings align well with our simulation results and suggest that, with accurate knowledge of both the target bead size and counting bead size, the bias in the counting bead method can be reasonably predicted.

Conclusions

In this study, we compared the conventional counting bead–based method with the sample loop–based method for particle number concentration measurement in flow cytometry, focusing on size-dependent bias. While the counting bead method is widely used and relatively simple, our results demonstrate that it can introduce size-dependent bias, particularly when the target particles differ significantly in size from the counting beads. Through systematic experiments using bead mixtures of varying sizes, we observed that the sample loop method, which relies on total counting with a defined sample volume, minimizes such bias.

The force balance analysis conducted via simulations supported these findings, showing that the total force acting on the beads varies significantly with bead size. Larger beads experience stronger forces in the vertical direction, driven by the difference between buoyancy and gravity, which enables them to move more rapidly toward the capillary. In contrast, smaller beads, experiencing lower total force, are more susceptible to the randomizing effects of Brownian motion that impede overall bead motion. As a result, F B,z – F G,z induces subtle differences in the z-position of beads at the microscale, which are amplified with the steeply increasing fluid velocity field, leading to size-dependent bias. These results provide a mechanistic explanation for the size-dependent bias observed in the system, confirming that differences in the total force contribute to variations in bead motion and distribution.

Our findings underscore the importance of considering size-dependent bias for accurate particle concentration measurements when employing the counting bead method, especially when the target particles significantly vary in size from the counting beads. With volumetric methods, such as the sample loop method having minimal or no significant size-related bias, the derived empirical equation can be used to predict the bias. This study contributes to the advancement of more precise and traceable methods for particle concentration measurement, with implications for a wide range of biological and industrial applications.

Supplementary Material

ac5c03537_si_001.pdf (1.7MB, pdf)

Acknowledgments

This work was supported by the Korea Research Institute of Standards and Science (KRISS-GP2024-0006), the National Research Council of Science & Technology (NST) (GTL24022-000), and the National Research Foundation of Korea (NRF) (RS-2025-00558156 and RS-2025-24533867), all funded by the Korean government (MSIT).

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.analchem.5c03537.

  • Additional experimental and simulation results, along with detailed simulation methodologies, including the overall workflow of the multiphysics modeling, schematic illustrations of the forces acting on the beads, and computational details for CFD and particle tracing simulations (PDF)

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Department of Mechanical Engineering, Sogang University 35 Baekbeom-ro, Mapo-gu, Seoul 04107, Republic of Korea

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H.J.S. and S.K. contributed equally. Idea and study design: J.Y.L. and J.S. FCM data generation and analysis: H.J.S. and J.Y.L. Simulation: S.K. and J.S. SEM bead size measurement: M.J.K. Data discussion and manuscript writing: H.J.S., S.K., I.C.Y., S.R.P., J.S., and J.Y.L.

The authors declare no competing financial interest.

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Associated Data

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Supplementary Materials

ac5c03537_si_001.pdf (1.7MB, pdf)

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