# Influence of erythrocyte density on aggregability as a marker of cell age: Dissociation dynamics in extensional flow

Midhun Puthumana Melepattu, Guillaume Maîtrejean, Christian Wagner, Thomas Podgorski

► To cite this version:

Midhun Puthumana Melepattu, Guillaume Maîtrejean, Christian Wagner, Thomas Podgorski. Influence of erythrocyte density on aggregability as a marker of cell age: Dissociation dynamics in extensional flow. Journal of Biomechanics, 2025, 183, pp.112603. 10.1016/j.jbiomech.2025.112603. hal-04996395

HAL Id: hal-04996395

https://hal.science/hal-04996395v1

Submitted on 19 Mar 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

# Influence of erythrocyte density on aggregability as a marker of cell age: Dissociation dynamics in extensional flow

Midhun Puthumana Melepattu<sup>a,†</sup>,*, Guillaume Maîtrejean<sup>a,†</sup>, Christian Wagner<sup>b,†</sup>, Thomas Podgorski<sup>a,†</sup>,**

## ARTICLE INFO

Keywords:

Aggregation of RBCs
Erythrocyte aging
Microvascular flow
Extensional flow
Cell dynamics

## ABSTRACT

Blood rheology and microcirculation are strongly influenced by red blood cell (RBC) aggregation. The aggregability of RBCs can vary significantly due to factors such as their mechanical and membrane surface properties, which are affected by cell aging in vivo. In this study, we investigate RBC aggregability as a function of their density, a marker of cell age and mechanical properties, by separating RBCs from healthy donors into different density fractions using Percoll density gradient centrifugation. We examine the dissociation rates of aggregates in a controlled medium supplemented with Dextran, employing an extensional flow technique based on hyperbolic microfluidic constrictions and image analysis, assisted by a convolutional neural network (CNN). In contrast to other techniques, our microfluidic experimental approach highlights the behavior of RBC aggregates in dynamic flow conditions relevant to microcirculation. Our results demonstrate that aggregate dissociation is strongly correlated with cell density and that aggregates formed from the denser fractions of RBCs are significantly more robust than those from the average cell population. This study provides insight into the effect of RBC aging in vivo on their mechanical properties and aggregability, underscoring the importance of further exploration of RBC aggregation in the context of cellular senescence and its potential implications for hemodynamics. Additionally, it suggests that this technique can complement existing methods for improved evaluation of RBC aggregability in health and disease.

## 1. Introduction

Red blood cell (RBC) aggregation is a critical factor influencing blood rheology and circulation. Due to the presence of plasma proteins, RBCs tend to reversibly aggregate, forming mostly linear structures known as *rouleaux* favored by their flat biconcave shape. These linear structures can assemble into larger, gel-like networks at low shear rates and are mostly responsible for the strong shear-thinning of blood observed at low shear rates (Chien, 1970), in conjunction with RBC deformability (Chien et al., 1967; Lanotte et al., 2016).

Although shear rates in blood circulation generally exceed 100 s⁻¹ (Robertson et al., 2008), where aggregation does not impact macroscopic rheology measurements (Chien, 1970), various studies have shown that aggregation interactions play a crucial role in shaping the structure of RBC suspensions in narrow channels (Bishop et al., 2001; Zhang et al., 2009), stabilizing clusters within capillaries (Brust et al., 2014), and affecting blood perfusion in vascular networks (Reinhart

et al., 2017). Indeed, in narrow vessels or channels, small RBC aggregates made up of a few cells can resist to strain rates up to a few 100 s⁻¹ (Brust et al., 2014; Clavería et al., 2016; Puthumana Melepattu et al., 2024).

RBC aggregation is induced by plasma proteins, notably fibrinogen (Baskurt et al., 2011), via two well-supported mechanisms. The first is a bridging mechanism, where macromolecules like fibrinogen adsorb onto RBC membranes and form intercellular bridges (Chien and Sung, 1987; Chien and Jan, 1973a). The second is an entropic mechanism involving depletion forces (Asakura and Oosawa, 1958) due to the finite size of macromolecules, as quantitatively predicted by Neu and Meiselman (2002). While plasma composition variations (e.g., increased fibrinogen production in inflammatory conditions (Castell et al., 1990), typically reflected in ESR tests (Bedell and Bush, 1985)) affect aggregation, cellular factors – often termed *aggregability* (Baskurt and Meiselman, 2013) – also play a crucial role. For example, modeling

<footnote><sup>a</sup> Université Grenoble Apes, CNRS, Grenoble INP, LRP, Grenoble, 38000, France</footnote>

<footnote><sup>b</sup> Universität des Saarlandes, Department of Experimental Physics, Saarbrücken, 66123, Germany</footnote>

<footnote>* Correspondence to: Domaine Universitaire - BP 53, 38041 Grenoble Cedex 9, France.</footnote>

<footnote>** Corresponding author.</footnote>

<footnote>E-mail addresses: Midhun.Puthumana-Melepattu@univ-grenoble-alpes.fr (M. Puthumana Melepattu), Guillaume.Maîtrejean@univ-grenoble-alpes.fr (G. Maîtrejean), christian.wagner@uni-saarland.de (C. Wagner), Thomas.Podgorski@univ-grenoble-alpes.fr (T. Podgorski).</footnote>
and experimental studies show that parameters such as cell volume, membrane bending, and shear moduli significantly influence the equilibrium shape of erythrocyte aggregates (Flormann et al., 2017; Hoore et al., 2018).

RBCs are highly deformable under normal conditions, but in pathological states – such as sickle cell anemia and bacterial infections – their deformability is markedly reduced and aggregability increased (Tripette et al., 2009; Baskurt et al., 1997; Chen et al., 1996; Chong-Martinez et al., 2003). However, reduced deformability is not the sole factor; changes in membrane surface properties (e.g. charge or protein distribution) also modify aggregability via altered electrostatic and binding interactions. For instance, elevated plasma sodium impairs the erythrocyte glycocalyx and alters the zeta potential, resulting in higher aggregability (McNally et al., 2023; Oberleithner and Wilhelmi, 2013), while experiments with artificially rigidified cells indicate that even a small fraction of rigid cells mixed with healthy ones can reduce the aggregation index and mean aggregate size (Kuck et al., 2022).

Even in healthy conditions, red blood cells (RBCs) display significant heterogeneity that evolves over time. RBCs have a lifespan of about 120 days during which they become denser, stiffer, and undergo other age-related changes before being phagocytosed by splenic macrophages (Waugh et al., 1992; Bosch et al., 1994; Franco, 2012; Franco et al., 2013). As they circulate, RBCs encounter varying mechanical stresses (Caro et al., 2011; Nichols and O'Rourke, 2005) that can elongate cells, alter membrane protein distribution, and induce membrane fatigue—ultimately affecting functionality and lifespan (Lipowsky, 2005; Pivkin and Karniadakis, 2008). Aging RBCs also experience physical and chemical alterations, such as the loss of water, (2,3)-bisphosphoglyceric acid, ATP, proteins, hemoglobin, sialoglycoproteins, and sialic acids, leading to structural modifications of the cell membrane (Huang et al., 2011; Lutz and Stringaro-Wipf, 1983; Lutz et al., 1987). Thus, cellular, mechanical, and chemical variations are intrinsic to RBC heterogeneity even within a healthy sample.

Many studies have used Percoll or Optiprep (iodixanol) density gradients to separate RBCs or other cells without evidence of cell alteration. Recent works concluded that exposure to Percoll does not lead to alterations in RBC deformability (Barshtein et al., 2024), suggesting that it is a generally innocuous with few specific exceptions (Wakefield et al., 1982). These separation techniques show that RBC density distributions are relatively wide, ranging from 1.08 to 1.12 g/mL. Recent works have shown that RBC density significantly correlates with their dynamics, demonstrating a significant impact on RBC shape in flow (Nouaman et al., 2023), as well as migration and axial dispersion in capillaries (Losserand et al., 2019, 2023).

While there is no consensus on a strict equivalence between RBC density and age, and density varies widely among cells of the same age, it is generally accepted that, on average, density increases with cell age (Morrison et al., 1983). For instance, Franco et al. (2013) have shown that the density distribution of biotin-labeled RBCs, determined at successive time points by density fractionation, was significantly shifted to higher densities over the course of 120 days. A robust feature is that cell density is related to hemoglobin concentration, which tends to increase as cells age due to dehydration (Mohandas et al., 1986; Vaysse et al., 1988). In addition, density increase was found to be correlated to loss of membrane surface area. Both phenomena reduce static (deformability index) and dynamic deformability (increase of relaxation time), as is also observed in some RBC disorders (Mohandas et al., 1989).

A few studies have shown that RBC aging leads to increased aggregation (Rampling et al., 2004). In one of the earliest studies, Nordt (1983) demonstrated that in autologous plasma, denser cells aggregated more than twice as much as younger, less dense cells, with intermediate-density cells exhibiting moderate aggregation. These findings were later extended to the aggregation of density-separated cells in 70 kDa dextran solutions (Nash et al., 1987). However, these early

results are relatively qualitative and do not provide a detailed assessment of aggregation characteristics beyond variations in a global aggregation index (Chien and Jan, 1973b). Various techniques have been proposed to quantify aggregation at the suspension scale. These include the classic erythrocyte sedimentation rate (ESR) test (Bedell and Bush, 1985), techniques based on light scattering such as ekta-cytometry (Da Costa et al., 2016), LORCA (Hardeman et al., 1994), and ultrasound scattering (Franceschini et al., 2010), all of which are applied to dense suspensions. A limitation of these techniques is that they provide average measurements over the whole sample but fail to offer insight into the distribution of properties within a sample. Furthermore, these methods typically operate in simple shear flow, which does not fully represent conditions in the microcirculation. At the cellular scale, force measurements or controlled force application methods, such as micropipette aspiration techniques (Tozeren et al., 1989; Buxbaum et al., 1982), AFM (Steffen et al., 2013), and more recently optical tweezers (OT) (Lee et al., 2017; Ermolinskiy et al., 2020), allow direct investigation of cell-cell interactions. Recently, Ermolinskiy et al. (2020) showed that the aggregation force measured by OT increases with the density of erythrocytes, both in autologous plasma and in 50 mg/ml Dextran 70 kDa solutions, confirming an increase of RBC aggregability with cell age. However, this type of time-consuming aggregation force measurement only allows to measure a limited number of aggregates and does not account for hydrodynamic stresses that RBCs experience in vivo, limiting the applicability of their results. Finally, flow cytometry has been proposed as a way to measure RBC aggregation (Zhao et al., 2017) but also operates over a limited range of flow stresses that may not adequately represent in vivo flow conditions. Therefore, there remains a notable absence of quantitative techniques capable of analyzing the aging-induced variations in these interactions across a large number of aggregates in realistic flow conditions.

Recently, a new microfluidic technique combined with machine learning analysis was proposed to investigate the dissociation of small RBC aggregates in confined elongational flows relevant to the hydrodynamic stresses experienced in microcirculatory networks (Puthumana Melepattu et al., 2024). This high-throughput technique provides detailed information about dissociation statistics as a function of strain rate for aggregates of 2 to 4 cells over large samples, complementing other techniques.

In this work, we use this technique to quantify the variations in dissociation probabilities of small aggregates of healthy erythrocytes in extensional flow as a function of cell density after separating them using Percoll density gradients. We provide a detailed and quantitative description of RBC aggregate strength as a function of RBC density, confirming that aggregates formed from the denser fractions of red blood cells are significantly more robust than the average cell population and require significantly higher flow stresses for dissociation, and providing the dispersion of these values in a given sample. These results go beyond a simple correlation between RBC age and aggregation and add substantial value to the understanding of RBC aggregation dynamics in microcirculatory conditions. In addition, they demonstrate the relevance of the technique for improved evaluation of RBC aggregability in health and disease and provide valuable experimental data for the validation of numerical models.

## 2. Methods

### 2.1. Blood sample collection and preparation

Blood samples were collected from three healthy male volunteers with informed consent (convention: Declaration of Helsinki, approved by the ethics committee of “Ärztekammer des Saarlandes”, permission number - 51/18) through venipuncture, using tubes containing EDTA (1.6 mg/mL, SARSTEDT, Nümbrecht, Germany) as anticoagulant. Plasma and buffy coat were then separated from red blood cells by
<figure><img src="image_3.png" /><figcaption>**Fig. 1.** RBC density fractionation using percoll solution gradient. (a) 2 mL of 60% Hematocrit along with PBS is carefully placed above the 5 × 2 mL Percoll density gradient solution. The densities of each layer is shown in the figure. (b) Separated RBC layers, after centrifugation for 30 min at 4000 × g.</figcaption></figure>

centrifugation (3000 g, 9 min) prior to washing 3 times (3000 g, 3 min) using PBS (Phosphate Buffered Saline, Gibco, pH 7.4, Fisher Scientific, Germany).

The separation of RBCs according to cell age was conducted by utilizing the density variations that arise during the aging process (Waugh et al., 1992). To obtain density fractions of RBCs from a whole blood sample, we employed the Percoll density gradient centrifugation method outlined by Ermolinskiy et al. (2020). The different density media are mixtures of Percoll (Cytiva 17-0891-01, Sigma-Aldrich, Taufkirchen, Germany), distilled water, and a 1.5M NaCl solution in varying proportions adjusted to maintain isotonicity with a constant measured osmolarity of 300 ± 5 mOsm. Five solutions, with densities of 1.085 g/mL, 1.092 g/mL, 1.101 g/mL, 1.107 g/mL, and 1.122 g/mL, were prepared (see Supplemental Material for detailed compositions) and carefully layered one over the other in a 15 mL centrifuge tube (with the highest density solution at the bottom and the lowest at the top, each layer having a volume of 2 mL). Finally, a 2 mL suspension of washed red blood cells, diluted to a hematocrit of 60% in PBS, was carefully layered onto the Percoll gradient solution. The sample was then centrifuged at 4000 g for 30 min at a controlled temperature of 4 °C. Centrifugation results in cells being fractionated and forming bands labeled $L_0 - L_4$ in the transition zones between Percoll layers as shown in Fig. 1, with $L_2$ corresponding to the peak of RBC density distribution in blood samples (Mohandas et al., 1989; Ermolinskiy et al., 2020; Nouaman et al., 2023). These different layers were then pipetted, washed, centrifuged and resuspended in a solution of 3 g/dL Dextran 70kDa to promote RBC aggregation with an interaction energy that is comparable to fibrinogen concentrations in the 4–5 g/L range (Neu and Meiselman, 2002; Steffen et al., 2013; Brust et al., 2014) before using them separately in experiments. Due to the very low cell concentrations in layers $L_0$ and $L_1$, only layers $L_2$, $L_3$ and $L_4$ were used in experiments. Establishing a direct relationship

between RBC density and age is challenging due to the wide dispersion of densities among cells of the same age, as shown in previous studies. However, age ranges for each density layer can be estimated using data from Franco et al. (2013) and cell lifespan models like those in Shrestha et al. (2016), applying Bayes' theorem to generate histograms of age versus density. Due to the discrete nature of the datasets, these estimates are imprecise and should be interpreted with caution. Given these limitations, we estimate that the studied layers roughly correspond to the interquartile age ranges in Table 1.

After suspending in Percoll for no more than 2 h, RBC layers were randomly studied at 4 °C in the microfluidic setup for 2–4 h, ensuring a total delay from collection to analysis of no more than 6 h.

## 2.2. Microfluidics

The core component of the microfluidics setup, whose principle was detailed in a previous study and shown to reliably characterize RBC disaggregation (Puthumana Melepattu et al., 2024), is a hyperbolic constriction geometry (Fig. 2 a). This design has been effectively implemented in prior studies (Bento et al., 2018; Faustino et al., 2019) to induce an extensional flow utilizing a channel width that decreases inversely with the coordinate $x$ in the flow direction ($w(x) \sim 1/x$) and a constant thickness $h$. Mass conservation implies that the cross-sectional average axial velocity $\langle u_x(x) \rangle$ increases linearly along $x$ i.e. the average extension rate $d\langle u_x(x) \rangle/dx$ is constant. Consequently, particles along the central line experience an extensional strain rate $\dot{\epsilon}_f = dv_x/dx$, where $v_x = u_x(x, 0, 0)$. A series of such constrictions with constant thickness $h = 20 \ \mu\text{m}$, in the $z$ direction, inlet width($w_0 = 200 \ \mu\text{m}$) and outlet width ($w_L = 20 \ \mu\text{m}$) was implemented, with decreasing lengths, $L$ (1000, 500, 250, 125 $\ \mu\text{m}$), inducing increasing extension rates $\dot{\epsilon}_f$ for a given flow rate. The measurements of aggregate dissociation statistics presented here were conducted in the hyperbolic channel with $L = 250 \ \mu\text{m}$ as in Puthumana Melepattu et al. (2024).

The microfluidic chip features a 3-inlet flow-focusing section where the concentrated RBC suspension is focused in the $y$-direction between two perpendicular inlets of the suspending medium. A $\sim 5$ cm channel follows, allowing RBC aggregates to relax and center in the $z$-direction due to migration forces (Losserand et al., 2019; Grandchamp et al., 2013; Podgorski et al., 2011). To facilitate image acquisition, each constriction is followed by a straight channel (cross section $w(L) \times h$, length 150 $\ \mu\text{m}$) As demonstrated in Puthumana Melepattu et al. (2024), RBC aggregates in this confined setup are shear-insensitive, and changes in aggregate size statistics result solely from extensional stresses in the constriction. Flow in the setup is driven by applying pressure at the three inlets using an Elveflow OB1 controller.

## 2.3. Microscopy and image processing

RBC cluster images were acquired using an inverted microscope (Eclipse TE2000-S, Nikon) equipped with a x20 objective lens and a high-speed camera (Fastec HiSpec 2G, FASTEC Imaging), providing a 2.05 pixels/$\mu\text{m}$ resolution and a frame rates ranging 250–3000 fps. A typical aggregate dissociation sequence is shown in Fig. 2(b). The actual extension rate experienced by aggregates $\dot{\epsilon} = dv_p/dx$, where $v_p$ is the particle velocity, was evaluated for each input pressure by tracking several particle (cells/aggregates) positions $x(t)$ along the constriction and fitting them to the exponential function $x(t) = a + b \exp(\dot{\epsilon}t)$.

The cluster size distributions before and after the constriction were evaluated from image sets containing between 1000 and 10,000 objects (cells and clusters) for each experiment sequence, using a combination of classical image processing and a custom convolutional neural network (CNN) algorithm described in Puthumana Melepattu et al. (2024) and based on the TensorFlow framework (Abadi et al., 2016). After training the CNN with a carefully assembled set of images, achieving an impressive accuracy of approximately 95%, the algorithm classifies flowing objects into single cells, doublets, triplets, quadruplets, and
<table><caption>**Table 1**
Density layers studied in the experiment and corresponding density ranges and estimated age ranges.</caption><thead><tr><th>Layer</th><th>Density Range (g/mL)</th><th>Age Range (days)<br>(Franco et al., 2013;<br>Shrestha et al.,<br>2016)</th></tr></thead><tbody><tr><td>L<sub>2</sub> (light)</td><td>1.092–1.101</td><td>40 ± 30</td></tr><tr><td>L<sub>3</sub> (Intermediate)</td><td>1.101–1.107</td><td>53 ± 28</td></tr><tr><td>L<sub>4</sub> (dense)</td><td>1.107–1.122</td><td>58 ± 28</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Hyperbolic flow geometry producing extensional flow: (a) Design of the central part of the microfluidic channel and notations. Blue areas are image processing ROIs; (b) Example temporal sequence of a 2-cell aggregate breaking into 1 + 1 cells under extensional flow. Time-step: 1.1 ms; Extension rate $\dot{\epsilon} = 118$ s$^{-1}$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Dissociation probabilities of aggregates of 2, 3 and 4 cells (doublets (a), triplets (b) and quadruplets (c) for RBCs from layers $L_2$ to $L_4$ obtained by density fractionation and from whole blood (WB) samples (data from Puthumana Melepattu et al. (2024)). Continuous lines represent fitting curves using Eq. (1). All series represent data from 3 different donors.</figcaption></figure>

larger objects, with their respective relative populations denoted as $S_i$, $D_i$, $T_i$, $Q_i$, and $N_i$ (where $i = 0$ before and $i = 1$ after the constriction as defined in Fig. 2a). These populations are normalized by the total number of detected RBCs (aggregated or not), ensuring that $S_i + 2D_i + 3T_i + 4Q_i = 1$. Aggregates of five cells or more are very rare and are not included in this normalization.

Assuming that aggregates of 5 cells or more are negligible ($N_i \ll 1$), mass conservation between the initial size distribution before the constriction ($S_0$, $D_0$, $T_0$, $Q_0$) and after the constriction ($S_1$, $D_1$, $T_1$, $Q_1$) allows to compute dissociation probabilities $P_d$ (for doublets), $P_t$ (triplet breaking into a doublet and a single cell), $P_{q1}$ and $P_{q2}$ (quadruplet breaking into respectively two doublets or a single cell and a triplet), with $P_q = P_{q1} + P_{q2}$, with the additional assumption that $P_{q1} = 2P_{q2}$ (assuming that all cell-cell bonds in a quadruplet are equivalent) as done previously (Puthumana Melepattu et al., 2024).

### 3. Results and discussion

The results are summarized in Fig. 3 in which the dissociation probabilities $P_d$, $P_t$, $P_q$ of aggregates of cells from density layers $L_2$ to $L_4$ and from whole blood (WB) are plotted as a function of extension rate $\dot{\epsilon}$, with a 3g/dL Dextran buffer as suspending medium.

As previously reported (Puthumana Melepattu et al., 2024), all dis-aggregation probabilities follow a characteristic sigmoid curve with in-creasing extension rate. An assessment of inter-donor variability shows minimal differences between healthy samples, with all data aligning on the same master curve, regardless of donor or study order. This also confirms that fluctuations in the delay between sample collection and analysis had no significant effect within our experimental time-frame (see Supplemental Material). A strong jump observed around extension rates of 80–200 s$^{-1}$ defines a critical extension rate $\dot{\epsilon}_c$, corresponding to the maximum rate of change in dissociation probabilities. To determine
<figure><img src="image_6.png" /><figcaption>Fig. 4. Dissociation probabilities of aggregates of 2, 3 and 4 cells (■: doublets, ♦: triplets, ●: quadruplets) of RBCs from layers (a) L₂ (b) L₃ (c) L₄ obtained by density fractionation vs. extension rate, with fitting curves using Eq. (1). (d): Data from all layers vs. ˙εN/2 as suggested by Eq. (3) and compared to data for whole blood (from Puthumana Melepattu et al., 2024).</figcaption></figure>

this, experimental data in Fig. 3 is fitted with a sigmoid curve defined as:

$$P(\dot{\epsilon}) = A + B \tanh(C(\dot{\epsilon} - \dot{\epsilon}_c)) + D\dot{\epsilon} \quad (1)$$

where A, B, C, D, ˙εc are fitting parameters. The determination coefficient R² for all data series in Fig. 3 is higher than 0.99, allowing a reliable determination of ˙εc.

In Puthumana Melepattu et al. (2024), an analysis of the forces acting on cells within a cluster, performed in a reference frame centered on the center of mass of the clusters, established that the dissociation dynamics are primarily governed by the balance between the work of viscous drag forces on the cells and the interaction energy between them. This led to the proposal of a scaling law for the critical extension rate required to dissociate a two-cell aggregate (doublet):

$$\dot{\epsilon}_c \approx \frac{\epsilon_{ad}}{12\eta\delta l} \quad (2)$$

which can be generalized for a (linear) aggregate of N cells:

$$\dot{\epsilon}_c(N) \approx \frac{\epsilon_{ad}}{12(N/2)\eta\delta l} \quad (3)$$

where εad is the interaction energy per unit area, η is the suspending medium's viscosity and δl is the typical displacement required to separate cells. Note that for a doublet (N = 2) with εad = 5 µJ/m² (Steffen et al., 2013), η = 2 mPa s (viscosity of 3g/dL dextran solution), and δl ≈ 1 µm (displacement to separate cells), Eq. (3) gives a critical extension rate of about 200 s⁻¹, which is in the order of magnitude observed in experimental measurements (Fig. 3).

As suggested by the scaling ˙εc(N) ~ 1/N, larger aggregates tend to dissociate at lower extension rates compared to smaller aggregates, as

the former experience higher drag in the extensional field, as seen in Fig. 4(a-c), where layer-wise data is represented.

This is also reflected in Fig. 4(d), where dissociation probabilities are plotted as a function of ηεN/2 (Eq. (3)). Data for triplets (Pᵢ) and quadruplets (Pq) align well with doublets (Pᵣ) from L₂, L₃, and WB, though the collapse is weaker for L₄ (older, denser cells). For L₄, ˙ε depends weakly on N, as seen in Fig. 4(c), where lateral shift is smaller and dissociation probabilities for triplets and quadruplets are nearly identical. We will discuss this further below.

Comparing data from different density layers in Figs. 3 and 4 shows that RBCs from denser layers form more robust aggregates requiring higher extensional stresses to dissociate. As ˙ε increases, cells from L₂ dissociate first, followed by L₃ and L₄. The experimental critical extension rate ˙εc, defined as the inflection point in the dissociation curves (Figs. 3–4), is shown in Fig. 5. On average, aggregates from L₄ require extensional stresses approximately twice as high as those for L₂ to reach a similar dissociation probability (e.g., for quadruplets), with a significant increase (up to 65%) in ˙εc. Data for WB lies between L₂ and L₃ in Figs. 3 and 4(d), consistent with L₂ being at the peak of the RBC density distribution, as shown in Fig. 1 and previous works (Mohandas et al., 1989; Ermolinskiy et al., 2020; Nouaman et al., 2023).

The strong shift in extension rate for L₄ compared to the more modest shift between L₂ and L₃ resembles the variation in deformability reported by Nouaman et al. (2023), who found that deformability indexes (DI) and DI variations with strain rate are drastically reduced for L₄ using the same density fractionation, or in earlier ektacytometry measurements showing more pronounced changes for the densest RBC fractions (Mohandas and Groner, 1989). In comparison, optical tweezer measurements (Ermolinskiy et al., 2020) revealed a gradual
<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Critical extension rates for aggregates of different layers (■: doublets, ●: triplets, ●: quadruplets) estimated by fitting Eq. (1) to the data represented in Fig. 4. Error bars represent 95% confidence intervals defined as $1.96\sigma$ where $\sigma$ is the standard error of the fitting parameter $\dot{\epsilon}_c$.</figcaption></figure>

20% increase in aggregation force between $L_2$ and $L_4$ in autologous plasma and 10% in a 5g/dL Dextran 70 kDa solution. Despite the different suspending media, the measured effects are comparable to the evolution of $\dot{\epsilon}_c$ for doublets in Fig. 5, where the critical extension rate for doublets is about 10% higher in $L_4$ compared to $L_2$. Meanwhile, the value of $P_d$ is significantly lower in $L_4$ at any $\dot{\epsilon}$ (Fig. 3), reflecting stronger aggregability. Aggregate sizes of $L_4$ cells are smaller than those from other layers in cytometry measurements of Percoll-separated samples (Huang et al., 2011). Following the analysis leading to Eq. (3), aggregates of older cells require a higher extension rate to experience the same total drag force as slightly larger young cell aggregates.

The evolution with cell density is however much more contrasted with aggregates of 3 and 4 cells that are much more robust in $L_4$. This is also consistent with the fact that in this density layer, aggregates of different sizes (doublets, triplets, quadruplets) dissociate at about the same extension rate and do not obey the scaling with $N$ proposed in Eq. (3). This can be related to morphological changes that can be observed and have been documented for denser and older cells:

* cell aging and density increase is associated to dehydration, membrane loss and other alterations that can lead to smaller, less deformable and possibly more spherical cells. A higher occurrence of echinocytes in older cells has also been reported (Huang et al., 2011). These alterations at the cell level have consequences on aggregate morphology.

* for the same reason as mentioned above, the higher compactivity or smaller volume of aggregates of denser cells means that the scaling with $N$ in Eq. (3) is less applicable, as it is based on the assumption that aggregates are linear rouleaux.

* differences in membrane properties such as surface charge are also likely to contribute (Hadengue et al., 1998) in the variations observed in between younger (lighter) and older (denser) cell aggregates.

## 4. Conclusions

We investigated the dissociation of RBC aggregates under extension, focusing on cell aging with cell density as an age indicator. A significant variation in aggregate dissociation was observed, in terms of critical hydrodynamic stress and dissociation probability. This aligns with previous studies on aggregation strength differences across cell ages and confirms that RBC aggregability increases with cell density and age.

In contrast to other aggregability measurement methods, our microfluidic technique provides hydrodynamic conditions relevant to flow in the microcirculation, particularly in capillary bifurcations. Our findings contribute to understanding aggregation dynamics in microcirculation and the effects of cell aging. While aging in vivo decreases cell deformability, leading to their removal in the spleen (Moreau et al., 2023), the increased RBC aggregability and aggregate robustness we quantified likely have significant microcirculatory consequences (e.g., increased resistance and clogging), potentially more severe than the simple decrease in deformability of single cells. This underscores the physiological importance of removing aging cells from circulation.

Altered properties through which aggregability and robustness of aggregates may increase for denser and older cells were mentioned, including RBC shape, volume, surface changes, decreased deformability, and altered membrane properties. While these effects are not dissociable in our study, we suggest that the same technique can be used to study each parameter separately, for instance, by modifying RBC deformability using diamide or glutaraldehyde, or altering RBC volume with hyper or hypo-osmotic media.

## CRediT authorship contribution statement

**Midhun Puthumana Melepattu:** Writing – original draft, Visualization, Software, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. **Guillaume Maître-jean:** Software, Methodology. **Christian Wagner:** Writing – review & editing, Supervision, Resources, Methodology, Funding acquisition, Conceptualization. **Thomas Podgorski:** Writing – original draft, Supervision, Resources, Methodology, Funding acquisition, Formal analysis, Conceptualization.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

The authors thank M. Van Melle-Gateau (LIPhy, CNRS-UGA) for technical assistance (microfabrication), T. John, M. Nouaman, A. Darras and S. Reckenwald for experimental advice; M. P M and T.P. thank G. Ghigliotti, E. Franceschini and G. Coupier for discussions and suggestions. This work was supported by CNES (Centre National d'Etudes Spatiales, France, DAR ID 8106 "Rhéologie sanguine"), and ANR project ANR-15-IDEX-02 (IDEX Université Grenoble Alpes, "Aide à la mobilité internationale des doctorants").

## Appendix A. Supplementary data

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.jbiomech.2025.112603.

## References

Abadi, M., Barham, P., Chen, J., Chen, Z., Davis, A., Dean, J., Devin, M., Ghe-mawat, S., Irving, G., Isard, M., Kudlur, M., Levenberg, J., Monga, R., Moore, S., Murray, D.G., Steiner, B., Tucker, P., Vasudevan, V., Warden, P., Wicke, M., Yu, Y., Zheng, X., 2016. TensorFlow: A system for large-scale machine learning. In: Proceedings of OSDI'16: 12Th Usenix Symposium on Operating Systems Design and Implementation. USENIX Assoc; ACM SIGOPS; Google; Microsoft; Huawei; NSF; Facebook; NetApp; VMWare; Amazon Web Serv; Hewlett Packard Enterprise; IBM Res; Symantec; Oracle; Microsoft Res, pp. 265–283, https://www.usenix.org/system/files/conference/osdi16/osdi16-abadi.pdf, 12th USENIX Symposium on Operating Systems Design and Implementation (OSDI), Savannah, GA, NOV 02-04, 2016.

Asakura, S., Oosawa, F., 1958. Interaction between particles suspended in solutions of macromolecules. J. Polym. Sci. 33 (126), 183–192. http://dx.doi.org/10.1002/pol.1958.1203312618.
Barshtein, G., Arbell, D., Gural, A., Livshits, L., 2024. Distribution of red blood cells deformability: Study on density-separated cell subpopulations. Colloids Interfaces 8 (4), http://dx.doi.org/10.3390/colloids8040407.

Baskurt, O.K., Meiselman, H.J., 2013. Erythrocyte aggregation: basic aspects and clinical importance. Clin. Hemorheol. Microcirc. 53 (1-2), 23-37. http://dx.doi.org/10.3233/CH-2012-1573.

Baskurt, O., Neu, B., Meiselman, H.J., 2011. Red Blood Cell Aggregation. CRC Press, Boca Raton, http://dx.doi.org/10.1201/b11221.

Baskurt, O.K., Temiz, A., Meiselman, H.J., 1997. Red blood cell aggregation in experimental sepsis. J. Lab. Clin. Med. 130 (2), 183-190. http://dx.doi.org/10.1016/S0022-2143(97)90094-9.

Bedell, S.E., Bush, B.T., 1985. Erythrocyte sedimentation-rate - from folklore to facts. Am. J. Med. 78 (6), 1001-1009. http://dx.doi.org/10.1016/0002-9343(85)90224-4.

Bento, D., Rodrigues, R.O., Faustino, V., Pinho, D., Fernandes, C.S., Pereira, A.I., Garcia, V., Miranda, J.M., Lima, R., 2018. Deformation of red blood cells, air bubbles, and droplets in microfluidic devices: Flow visualizations and measurements. Micromachines 9 (4), 151. http://dx.doi.org/10.3390/mi9040151.

Bishop, J.J., Nance, P.R., Popel, A.S., Intaglietta, M., Johnson, P.C., 2001. Effect of erythrocyte aggregation on velocity profiles in venules. Am. J. Physiol.-Hear. Circ. Physiol. 280 (1), H222-H236. http://dx.doi.org/10.1152/ajpheart.2001.280.1.H222.

Bosch, F.H., Werre, J.M., Schipper, L., Roederinkholder-Stoelwinder, B., Huls, T., Willekens, F., Wichers, G., Halie, M.R., 1994. Determinants of red blood cell deformability in relation to cell age. Eur. J. Haematol. 52 (1), 35-41. http://dx.doi.org/10.1111/j.1600-0609.1994.tb01282.x.

Brust, M., Aouane, O., Thiébaud, M., Flormann, D., Verdier, C., Kaestner, L., Laschke, M., Selmi, H., Benyoussef, A., Podgorski, T., Coupier, G., Misbah, C., Wagner, C., 2014. The plasma protein fibrinogen stabilizes clusters of red blood cells in microcapillary flows. Sci. Rep. 4, 4348. http://dx.doi.org/10.1038/srep04348.

Buxbaum, K., Evans, E., Brooks, D.E., 1982. Quantitation of surface affinities of red blood cells in dextran solutions and plasma. Biochemistry 21 (13), 3235-3239. http://dx.doi.org/10.1021/bi00256a032.

Caro, C.G., Pedley, T.J., Schroter, R.C., Seed, W.A., Parker, K.H., 2011. The Mechanics of the Circulation, second ed. Cambridge University Press, http://dx.doi.org/10.1017/CBO9781139013406.

Castell, J.V., Gómez-lechón, M.J., David, M., Fabra, R., Trullenque, R., Heinrich, P.C., 1990. Acute-phase response of human hepatocytes: regulation of acute-phase protein synthesis by interleukin-6. Hepatology 12 (5), 1179-1186. http://dx.doi.org/10.1002/hep.1840120517.

Chen, S., Eldor, A., Barshtein, G., Zhang, S., Goldfarb, A., Rachmilewitz, E., Yedgar, S., 1996. Enhanced aggregability of red blood cells of beta-thalassemia major patients. Am. J. Physiol.-Hear. Circ. Physiol. 270 (6), H1951-H1956. http://dx.doi.org/10.1152/ajpheart.1996.270.6.H1951, PMID: 8764243.

Chien, S., 1970. Shear dependence of effective cell volume as a determinant of blood viscosity. Science 168, 977-979. http://dx.doi.org/10.1126/science.168.3934.977.

Chien, S., Jan, K.M., 1973a. Red cell aggregation by macromolecules: roles of surface adsorption and electrostatic repulsion. J. Supramol. Struct. 1 4, 385-409. http://dx.doi.org/10.1002/jss.400010418.

Chien, S., Jan, K.m., 1973b. Ultrastructural basis of the mechanism of rouleaux formation. Microvasc. Res. 5 (2), 155-166. http://dx.doi.org/10.1016/0026-2862(73)90068-X.

Chien, S., Sung, L.A., 1987. Physicochemical basis and clinical implications of red cell aggregation. Clin. Hemorheol. Microcirc. 7 (1), 71-91. http://dx.doi.org/10.3233/CH-1987-7108.

Chien, S., Usami, S., Dellenback, R.J., Gregersen, M.I., 1967. Blood viscosity: influence of erythrocyte deformation. Science 157 (3790), 827-829. http://dx.doi.org/10.1126/science.157.3790.827.

Chong-Martinez, B., Buchanan, T.A., Wenby, R.B., Meiselman, H.J., 2003. Decreased red blood cell aggregation subsequent to improved glycaemic control in Type 2 diabetes mellitus. Diabetic Med. 20 (4), 301-306. http://dx.doi.org/10.1046/j.1464-5491.2003.00926.x.

Clavería, V., Aouane, O., Thiébaud, M., Abkarian, M., Coupier, G., Misbah, C., John, T., Wagner, C., 2016. Clusters of red blood cells in microcapillary flow: hydrodynamic versus macromolecule induced interaction. Soft Matter 12, 8235-8245. http://dx.doi.org/10.1039/C6SM01165A.

Da Costa, L., Suner, L., Galimand, J., Bonnel, A., Pascreau, T., Couque, N., Fen-neteau, O., Mohandas, N., 2016. Diagnostic tool for red blood cell membrane disorders: Assessment of a new generation ektacytometer. Blood Cells Mol. Dis. 56 (1), 9-22. http://dx.doi.org/10.1016/j.bcmd.2015.09.001.

Ermolinskiy, P., Lugovtsov, A., Yaya, F., Lee, K., Kaestner, L., Wagner, C., Priezzhev, A., 2020. Effect of red blood cell aging in vivo on their aggregation properties in vitro: Measurements with laser tweezers. Appl. Sci. 10 (21), 7581. http://dx.doi.org/10.3390/app10217581.

Faustino, V., Rodrigues, R.O., Pinho, D., Costa, E., Santos-Silva, A., Miranda, V., Ama-ral, J.S., Lima, R., 2019. A microfluidic deformability assessment of pathological red blood cells flowing in a hyperbolic converging microchannel. Micromachines 10 (10), 645. http://dx.doi.org/10.3390/mi10100645.

Flormann, D., Aouane, O., Kaestner, L., Ruloff, C., Misbah, C., Podgorski, T., Wagner, C., 2017. The buckling instability of aggregating red blood cells. Sci. Rep. 7 (1), http://dx.doi.org/10.1038/s41598-017-07634-6.

Franceschini, E., Yu, F.T.H., Destrempes, F., Cloutier, G., 2010. Ultrasound characterization of red blood cell aggregation with intervening attenuating tissue-mimicking phantoms. J. Acoust. Soc. Am. 127 (2), 1104-1115. http://dx.doi.org/10.1121/1.3277200.

Franco, R.S., 2012. Measurement of red cell lifespan and aging. Transfus. Med. Hemother. 39 (5), 302-307. http://dx.doi.org/10.1159/000342232.

Franco, R.S., Puchulu-Campanella, M.R., Barber, L.A., Palascak, M.B., Joiner, C.H., Low, P.S., Cohen, R.M., 2013. Changes in the properties of normal human red blood cells during in vivo aging. Am. J. Hematol. 88 (1), 44-51. http://dx.doi.org/10.1002/ajh.23344.

Grandchamp, X., Coupier, G., Srivastav, A., Minetti, C., Podgorski, T., 2013. Lift and down-gradient shear-induced diffusion in red blood cell suspensions. Phys. Rev. Lett. 110 (10), http://dx.doi.org/10.1103/PhysRevLett.110.108101.

Hadengue, A.L., Del-Pino, M., Simon, A., Levenson, J., 1998. Erythrocyte disaggregation shear stress, sialic acid, and cell aging in humans. Hypertension 32 (2), 324-330. http://dx.doi.org/10.1161/01.HYP.32.2.324.

Hardeman, M.R., Goedhart, P.T., Dobbe, J.G.G., Lettinga, K.P., 1994. Laser-assisted optical rotational cell analyser (LORCA); I. A new instrument for measurement of various structural hemorheological parameters. Clin. Hemorheol. Microcirc. 14 (4), 605-618. http://dx.doi.org/10.3233/CH-1994-14416.

Hoore, M., Yaya, F., Podgorski, T., Wagner, C., Gompper, G., Fedosov, D.A., 2018. Effect of spectrin network elasticity on the shapes of erythrocyte doublets. Soft Matter 14, 6278-6289. http://dx.doi.org/10.1039/C8SM00634B.

Huang, Y.X., Wu, Z.J., Mehrishi, J., Huang, B.T., Chen, X.Y., Zheng, X.J., Liu, W.J., Luo, M., 2011. Human red blood cell aging: correlative changes in surface charge and cell properties. J. Cell. Mol. Med. 15 (12), 2634-2642. http://dx.doi.org/10.1111/j.1582-4934.2011.01310.x.

Kuck, L., McNamee, A.P., Simmonds, M.J., 2022. Impact of small fractions of abnormal erythrocytes on blood rheology. Microvasc. Res. 139, 104261. http://dx.doi.org/10.1016/j.mvr.2021.104261.

Lanotte, L., Mauer, J., Mendez, S., Fedosov, D.A., Fromental, J.-M., Claveria, V., Nicoud, F., Gompper, G., Abkarian, M., 2016. Red cells dynamic morphologies govern blood shear thinning under microcirculatory flow conditions. Proc. Natl. Acad. Sci. 113, 13289-13294. http://dx.doi.org/10.1073/pnas.1608074113.

Lee, K., Wagner, C., Priezzhev, A.V., 2017. Assessment of the "cross-bridge"-induced interaction of red blood cells by optical trapping combined with microfluidics. J. Biomed. Opt. 22 (9), 091516. http://dx.doi.org/10.1117/1.JBO.22.9.091516.

Lipowsky, H.H., 2005. Microvascular rheology and hemodynamics. In: Handbook of Physiology: Microcirculation, Vol. 2. American Physiological Society, pp. 87-116. http://dx.doi.org/10.1080/10739680590894966.

Losserand, S., Coupier, G., Podgorski, T., 2019. Migration velocity of red blood cells in microchannels. Microvasc. Res. 124, 30-36. http://dx.doi.org/10.1016/j.mvr.2019.02.003.

Losserand, S., Coupier, G., Podgorski, T., 2023. Axial dispersion of red blood cells in microchannels. Phys. Rev. Fluids 8 (4), 043102. http://dx.doi.org/10.1103/PhysRevFluids.8.043102.

Lutz, H.U., Bussolino, F., Flepp, R., Fasler, S., Stammler, P., Kazatchkine, M.D., Arese, P., 1987. Naturally occurring anti-band-3 antibodies and complement together mediate phagocytosis of oxidatively stressed human erythrocytes. Proc. Natl. Acad. Sci. 84 (21), 7368-7372. http://dx.doi.org/10.1073/pnas.84.21.7368.

Lutz, H.U., Stringaro-Wipf, G., 1983. Senescent red cell-bound IgG is attached to band 3 protein. Biomed. Biochim. Acta 42 (11-12), S117-21, http://europepmc.org/abstract/MED/6675681.

McNally, R.J., Morselli, F., Farukh, B., Chowienzyk, P.J., Faconi, L., 2023. A pilot study to evaluate the erythrocyte glycolycaly sensitivity to sodium as a marker for cellular salt sensitivity in hypertension. J. Hum. Hypertens. 37 (4), 286-291. http://dx.doi.org/10.1038/s41371-022-00683-z.

Mohandas, N., Groner, W., 1989. Cell membrane and volume changes during red cell development and aging. Ann. New York Acad. Sci. 554, 217-224. http://dx.doi.org/10.1111/j.1749-6632.1989.tb22423.x.

Mohandas, N., Johnson, A., Wyatt, J., Croisille, L., Reeves, J., Tycko, D., Groner, W., 1989. Automated quantitation of cell-density distribution and hyperdense cell fraction in rbc disorders. Blood 74 (1), 442-447. http://dx.doi.org/10.1182/blood.V74.1.442.442.

Mohandas, N., Kim, Y.R., Tycko, D.H., Orlik, J., Wyatt, J., Groner, W., 1986. Accurate and independent measurement of volume and hemoglobin concentration of individual red cells by laser light scattering. Blood 68 (2), 506-513. http://dx.doi.org/10.1182/blood.V68.2.506.506.

Moreau, A., Yaya, F., Lu, H., Surendranath, A., Charrier, A., Dehapiot, B., Helfer, E., Viallat, A., Peng, Z., 2023. Physical mechanisms of red blood cell splenic filtration. Proc. Natl. Acad. Sci. USA 120 (44), http://dx.doi.org/10.1073/pnas.2300095120.

Morrison, M., Jackson, C.W., Mueller, T.J., Huang, T., Dockter, M.E., Walker, W.S., Singer, J.A., Edwards, H.H., 1983. Does cell density correlate with red cell age? Biomed. Biochim. Acta 42 (11-12), S107-11, http://europepmc.org/abstract/MED/6675678.

Nash, G., Wenby, R., Sowemimo-Coker, S., Meiselman, H., Stoltz, J., 1987. Influence of cellular properties on red cell aggregation. Clin. Hemorheol. Microcirc. 7 (1), 93-108. http://dx.doi.org/10.3233/CH-1987-7109.

Neu, B., Meiselman, H., 2002. Depletion-mediated red blood cell aggregation in polymer solutions. Biophys. J. 83 (5), 2482-2490. http://dx.doi.org/10.1016/S0006-3495(02)75259-4.
Nichols, W.W., O'Rourke, M.F., 2005. McDonald's Blood Flow in Arteries: Theoretical, Experimental and Clinical Principles, fifth ed. CRC Press, http://dx.doi.org/10.1201/b13568.

Nordt, F.J., 1983. Hemorheology in cerebrovascular diseases: Approaches to drug development. Ann. New York Acad. Sci. 416 (1 Surface Pheno), 651–661. http://dx.doi.org/10.1111/j.1749-6632.1983.tb35218.x.

Nouaman, M., Darras, A., John, T., Simionato, G., Rab, M.A.E., van Wijk, R., Laschke, M.W., Kaestner, L., Wagner, C., Recktenwald, S.M., 2023. Effect of cell age and membrane rigidity on red blood cell shape in capillary flow. Cells 12 (11), 1529. http://dx.doi.org/10.3390/cells12111529.

Oberleithner, H., Wilhelmi, M., 2013. Determination of erythrocyte sodium sensitivity in man. Pflügers Arch. -Eur. J. Physiol. 465 (10), 1459–1466. http://dx.doi.org/10.1007/s00424-013-1289-x.

Pivkin, I.V., Karniadakis, G.E., 2008. Accurate coarse-grained modeling of red blood cells. Phys. Rev. Lett. 101 (11), 118105. http://dx.doi.org/10.1103/PhysRevLett.101.118105.

Podgorski, T., Callens, N., Minetti, C., Coupier, G., Dubois, F., Misbah, C., 2011. Dynamics of vesicle suspensions in shear flow between walls. Microgravity Sci. Technol. 23 (2), 263–270. http://dx.doi.org/10.1007/s12217-010-9212-y.

Puthumana Melepattu, M., Maîtrejean, G., Podgorski, T., 2024. Dissociation of red blood cell aggregates in extensional flow. Phys. Rev. Fluids 9 (7), L071101. http://dx.doi.org/10.1103/PhysRevFluids.9.L071101.

Rampling, M., Meiselman, H., Neu, B., Baskurt, O., 2004. Influence of cell-specific factors on red blood cell aggregation. Biorheology 41 (2), 91–112. http://dx.doi.org/10.1177/0006355X2004041002002.

Reinhart, W.H., Piety, N.Z., Shevkoplyas, S.S., 2017. Influence of red blood cell aggregation on perfusion of an artificial microvascular network. Microcirculation 24 (5), e12317. http://dx.doi.org/10.1111/micc.12317.

Robertson, A., Sequeira, A., Kameneva, M., 2008. Hemorheology. In: Galdi, G.P., Robertson, A., R., R., S., T. (Eds.), Hemodynamical Flows: Modeling, Analysis and Simulation. Birkhäuser Basel, Basel, pp. 63–120. http://dx.doi.org/10.1007/978-3-7643-7806-6_2.

Shrestha, R.P., Horowitz, J., Hollot, C.V., Germain, M.J., Widness, J.A., Mock, D.M., Veng-Pedersen, P., Chait, Y., 2016. Models for the red blood cell lifespan. J. Pharmacokinet. Pharmacodyn. 43, 259–274. http://dx.doi.org/10.1007/s10928-016-9470-4.

Steffen, P., Verdier, C., Wagner, C., 2013. Quantification of depletion-induced adhesion of red blood cells. Phys. Rev. Lett. 110 (1), 018102. http://dx.doi.org/10.1103/PhysRevLett.110.018102.

Tozeren, A., Sung, K.L., Chien, S., 1989. Theoretical and experimental studies on cross-bridge migration during cell disaggregation. Biophys. J. 55 (3), 479–487. http://dx.doi.org/10.1016/S0006-3495(89)82841-3.

Tripette, J., Alexy, T., Hardy-Dessources, M.-D., Mougenel, D., Beltan, E., Chalabi, T., Chout, R., Etienne-Julan, M., Hue, O., Meiselman, H.J., Connes, P., 2009. Red blood cell aggregation, aggregate strength and oxygen transport potential of blood are abnormal in both homozygous sickle cell anemia and sickle-hemoglobin C disease. Haematologica 94 (8), 1060. http://dx.doi.org/10.3324/haematol.2008.005371.

Vaysse, J., Vassy, R., Eclache, V., Gattegno, L., Bladier, D., Pilardeau, P., 1988. Some characteristics of human red blood-cells separated according to their size - A comparison with density-fractionated red blood-cells. Am. J. Hematol. 28 (4), 232–238. http://dx.doi.org/10.1002/ajh.2830280405.

Wakefield, J.S., Gale, J.S.J., Berridge, M.V., Jordan, T.W., Ford, H.C., 1982. Is percoll innocuous to cells? Biochem. J. 202 (3), 795–797. http://dx.doi.org/10.1042/bj2020795.

Waugh, R.E., Narla, M., Jackson, C.W., Mueller, T.J., Suzuki, T., Dale, G.L., 1992. Rheologic properties of senescent erythrocytes: loss of surface area and volume with red blood cell age. Blood 79 (5), 1351–1358. http://dx.doi.org/10.1182/blood.V79.5.1351.1351.

Zhang, J., Johnson, P.C., Popel, A.S., 2009. Effects of erythrocyte deformability and aggregation on the cell free layer and apparent viscosity of microscopic blood flows. Microvasc. Res. 77 (3), 265–272. http://dx.doi.org/10.1016/j.mvr.2009.01.010.

Zhao, L., Kaewprayoon, W., Zhou, H., Georgieva, R., Baeumler, H., 2017. RBC aggregation in dextran solutions can be measured by flow cytometry. Clin. Hemorheol. Microcirc. 65 (1), 93–101. http://dx.doi.org/10.3233/CH-15121.