Huilin
Ye
^{a},
Zhiqiang
Shen
^{a},
Mei
Wei
^{b} and
Ying
Li
*^{ac}
^{a}Department of Mechanical Engineering, University of Connecticut, 191 Auditorium Road, Unit 3139, Storrs, Connecticut 06269, USA. E-mail: yingli@engr.uconn.edu; Fax: +1 860 4865088; Tel: +1 860 4867110
^{b}Department of Mechanical Engineering, Ohio University, Athens, OH 45701, USA
^{c}Polymer Program, Institute of Materials Science, University of Connecticut, 97 North Eagleville Road, Unit 3136, Storrs, Connecticut 06269, USA

Received
10th September 2020
, Accepted 4th November 2020

First published on 6th November 2020

We investigate the circulation of nano- and micro-particles, including spherical particles and filamentous nanoworms, with red blood cells (RBCs) suspension in a constricted channel that mimics a stenosed microvessel. Through three-dimensional simulations using the immersed boundary-based Lattice Boltzmann method, the influence of channel geometries, such as the length and ratio of the constriction, on the accumulation of particles is systematically studied. Firstly, we find that the accumulation of spherical particles with 1 μm diameter in the constriction increases with the increases of both the length and ratio of the constriction. This is attributed to the interaction between spheres and RBCs. The RBCs “carry” the spheres and they accumulate inside the constriction together, due to the altered local hydrodynamics induced by the existence of the constriction. Secondly, nanoworms demonstrate higher accumulation than that of spheres inside the constriction, which is associated with the escape of nanoworms from RBC clusters and their accumulation near the wall of main channel. The accumulated near-wall nanoworms will eventually enter the constriction, thus enhancing their concentration inside the constriction. However, an exceptional case occurs in the case of constrictions with large ratio and long length. In such circumstances, the RBCs aggregate together tightly and concentrate at the center of the channel, which makes the nanoworms hardly able to escape from RBC clusters, leading to a similar accumulation of nanoworms and spheres inside the constriction. This study may provide theoretical guidance for the design of nano- and micro-particles for biomedical engineering applications, such as drug delivery systems for patients with stenosed microvessels.

To improve the delivery systems for the treatment of microvascular stenosis, extensive efforts have been made from biological and chemical aspects.^{13,14} Recently, researchers have endeavored to explore the physical approach in terms of fluid mechanics, since the local hydrodynamics are severely altered and exhibit different physical characteristics as a result of microvascular stenosis compared to those in normal vasculature.^{15–17} Based on these distinguishable characteristics, corresponding drug containers can be designed to target diseased sites with higher efficacy. For instance, Korin et al. have designed shear-activated nanotherapeutics for drug targeting to obstructed blood vessels, by using the high shear stress caused by vascular narrowing as a targeting mechanism.^{9} These shear-activated nanotherapeutics can maintain their compacted structure in normal blood flow and break up into nanoscale components under abnormally high fluid shear stress, thereby minimizing side effects while maximizing delivery efficacy. Nevertheless, due to the intricate structure of blood vasculature throughout the body and the complex dynamics of blood flow involving vast numbers of blood cells, the hydrodynamic motion of particles such as drug containers in stenosed microvasculature has not yet been fully explored.

Since blood flow plays an essential role during the circulation of particles in a blood vessel, extensive efforts have been devoted to the hemodynamics of blood.^{18–23} Compared to that in a large artery, the blood in the microvasculature behaves as a non-Newtonian fluid and manifests cellular characteristics such as Fahraeus–Lindqvist and Fahraeus effects.^{19,20} When circulating in the vessel, the red blood cells (RBCs) that occupy the major part of the blood vessel tend to migrate towards the center of the vessel, due to deformation-induced lift force.^{21} This migration of RBCs results in a typical distribution in microvasculature: a RBC-free layer (CFL) near the wall and a RBC-rich core at the center. The formation of CFL acts as a lubricant layer near the wall and is responsible for the Fahraeus–Lindqvist and Fahraeus effects, which refer to the reduction of apparent blood viscosity and the RBC's volume fraction with decreasing vessel diameter, respectively. Recently, it is found that both Fahraeus–Lindqvist and Fahraeus effects can be significantly affected by microvascular stenosis.^{24–27} For example, Vahidkhah et al. found that the Fahraeus–Lindqvist effect was largely enhanced as the apparent viscosity of blood increased by several fold, by virtue of the asymmetric distribution of RBCs in stenosed microvasculature.^{24} Moreover, the hemodynamics in stenosed microvasculature are studied through investigating the RBC's volume fraction,^{25} membrane stiffness, shape,^{26} and aggregation.^{27}

Based on the findings of hemodynamics in stenosed microvasculature, the hydrodynamic motion of the immersed particles is further studied with the applications to design effective drug containers for the treatment of stenosis.^{9,10,28–30} Bacher et al. found, through three-dimensional simulations, that rigid microparticles circulating in the constricted channel will concentrate in front of the constriction, which may have critical physiological consequences such as the formation of microthrombi. Therefore, the clustering of microparticles can be applied to drug containers that deposited at the site of stenosis.^{9–11} Furthermore, since the fluid shear stress can be increased locally by one to two orders of magnitude in the constricted region, the drug containers can be designed to break up into nanoscale components that help mitigate the stenosis when exposed to high-level fluid stress. This system, from circulation along with blood flow, concentrating in front of the constriction to finally releasing drug components is called a shear-activated system.^{10} With the lenticular vesicles as drug containers, Holme et al. found that the vesicles were stable under static conditions and released their contents under elevated shear stress.^{10} Utilizing this unique property, they designed a shear-stress sensitive drug delivery system for the treatment of stenosis.

Apart from the local hydrodynamics, the RBC hitchhiking phenomenon opens a new avenue of exploration for drug delivery systems (DDS).^{3–7,31–36} The RBC exhibits great potential to introduce unprecedented changes in pharmacokinetics, pharmacodynamics, and immunogenicity. The drugs can be loaded into RBC either via encapsulation or surface coupling, depending on the target applications. For instance, Brenner et al.^{32} reported that when the nanocarriers loaded with drugs were adsorbed into RBC and then entered the blood stream through intravascular injection, the liposome uptake in the first downstream organ, the lungs, was found to increase by 40-fold compared with that of freely administrated nanocarriers. Additionally, RBC hitchhiking has been applied to anti-thrombotic drug delivery. It is found that after targeting thrombomodulin to the circulating RBC, the pharmacokinetics and antithrombotic effects are improved without increasing bleeding in mouse models.^{3} Also, directly targeting a mutant plasminogen activator to circulating RBCs can remarkably prolong the intravascular circulation and fibrinolytic activity.^{5} These studies reveal that RBC hitchhiking is a clinically translatable technology to augment DDS in lung disease, thrombosis, and several other diseases.

Nevertheless, the geometry of stenosis should also influence the motion of particles in the stenosed microvasculature. Although many studies have made efforts to understand the effects of stenosis on RBC motion,^{37,38} platelet motion and thrombus formation,^{39,40} the investigation of the effect of stenosis geometry is still limited. For example, recently, Carboni et al. conducted a series of experiments to understand the margination of micro-particles with a diameter of 2.11 μm in a constricted microfluidic channel with different occlusions, constriction lengths, and eccentricities.^{30} They found that the margination of micro-particle-migration towards the channel wall increased with increasing occlusion and length of constriction. However, due to the limitations in tracking RBCs and particles with low spatial and temporal resolutions, the underlying mechanisms remain to be explored in detail. Inspired by this experimental work, we systematically study the geometrical effects, including constriction length and ratio, on the circulation of nano- and micro-particles in the blood flow. Furthermore, we consider two typical particle shapes: spheres and filamentous nanoworms. The RBC hitchhiking phenomenon is also observed. We find the distribution of spheres in the flow direction follows that of RBCs: dipping before and after the constriction while remaining high inside the constriction. The accumulation inside the constriction is found to increase with the increment of both constriction length and ratio. By comparing the distributions in the flow and radial directions, we find nanoworms exhibit higher accumulation than do spheres inside the constriction. This is related to the escape of nanoworms from RBC clusters and subsequent accumulation near the wall. However, this rationale cannot be applied to the case of severe a constriction with long length, in which the RBCs concentrate at the center of channel and aggregate tightly so that the nanoworms can no longer penetrate them. Therefore, the nanoworms show the same distribution as spheres in this scenario.

This paper is organized as follows. The first part describes the computational system that we adopt to model the microvascular stenosis and corresponding computational methods including fluid, RBC and particle models. In the results and discussion part, we first demonstrate the distribution of spheres in different cases, in which the geometry of the constriction varies. Then we compare the distributions between nanoworms and spheres in detail. This is followed by a detailed discussion to explain the difference of distributions between these two kinds of particles. Lastly, we conclude this study with some final remarks.

Here, we study the effect of constriction geometry on the circulation of particles in blood vessels. The blood vessel is modeled as a cylindrical channel with a constricted part, as presented in Fig. 1(a). Periodic boundary condition is applied along the flow direction (y-direction). The length and diameter of the two main channels are fixed at L = 36 μm and D = 18 μm respectively. The constriction is characterized by the constriction length (CL) L_{c} and constriction ratio (CR) , where d is the diameter of the constriction. Three CLs L_{c} = 2, 18 and 34 μm and three CRs β = 17%, 27.8% and 50% are examined as shown in Fig. 1(b). The two main channels and the constriction are linked through a transition zone (TZ). To have a smooth transition, the diameter of this cone-like zone follows a third-order polynomial with coefficients as functions of L_{c} and β. Specifically, the first order derivative of the polynomial at the two ends (linking points between the main channel and constriction) are 0. To simplify this comparison, we fix the hematocrit (volume fraction of RBCs) Ht = 20% (in the human body's microvasculature network, the hematocrit is within the range 20–40%^{41}) such that the number of RBCs varies from 147 to 195 according to specific geometries of constriction. In addition, 96 particles (spheres or nanoworms) are randomly placed among RBCs in the blood flow, which is adequate for statistical analysis. The diameter of the spherical particles ranges from 1 μm to 3 μm and the length of the nanoworms is fixed at l = 8 μm. The nanoworm is assumed inextensible and here we only consider a semi-flexible nanoworm with a bending stiffness as shown in Table 1. We assume the flow is driven by a constant pressure gradient between inlet and outlet of the channel. This assumption is based on the fact that for a specific part of the blood vessel, the pressures at the inlet and outlet should not be changed, since the blood is pumped by the heart beating at a specific rate, which is not relevant to the shape of the constriction formed within this part of blood vessel. To model the pressure-driven blood flow, we apply a spatially constant body force that mimics an external pressure gradient in the flow direction. The magnitude of the body force is controlled to keep the wall shear rate in the main channel in the order of 1000 s^{−1}.

Parameters | Simulation | Physical | Ref. |
---|---|---|---|

RBC diameter (D_{0}) |
32.0 | 8 × 10^{−6} m |
55 |

RBC shear modulus (μ_{r}) |
0.01 | 6.3 × 10^{−6} N m^{−1} |
56 |

Energy scale (k_{B}T) |
1.1 × 10^{−4} |
4.14 × 10^{−21} N m |
— |

Viscosity of fluid (η) | 0.167 | 0.0012 Pa s | — |

Area constant (k_{a}) |
0.0075 | 4.72 × 10^{−6} N m^{−1} |
50 |

Local area constant (k_{d}) |
0.367 | 2.31 × 10^{−4} N m^{−1} |
50 |

Volume constant (k_{v}) |
0.096 | 249 N m^{−2} |
50 |

RBC bending constant (k_{b}) |
0.013 | 5 × 10^{−19} N m |
50 |

Nanoworm stretching constant (K^{p}_{s}) |
1.0 | 6.3 × 10^{−4} N m^{−1} |
51 |

Nanoworm bending constant (κ) | 5.5 × 10^{−3} |
2.07 × 10^{−19} N m |
51 |

Sphere diameter (sphere D_{0}) |
4.0–12 | 1–3 × 10^{−6} m |
— |

Sphere shear modulus (μ_{r}) |
1.0 | 6.3 × 10^{−4} N m^{−1} |
51 and 54 |

Sphere area constant (k_{a}) |
0.075 | 4.72 × 10^{−5} N m^{−1} |
51 and 54 |

Sphere local area constant (k_{d}) |
3.67 | 2.31 × 10^{−3} N m^{−1} |
51 and 54 |

Sphere volume constant (k_{v}) |
0.96 | 2490 N m^{−2} |
51 and 54 |

Sphere bending constant (k_{b}) |
0.13 | 5 × 10^{−18} N m |
51 and 54 |

Morse energy well width (β) | 0.96 | 3.84 μm^{−1} |
53 and 57 |

Equilibrium distance (r_{0}) |
2.0 | 0.5 μm | 53 and 57 |

Morse cutoff distance (r_{c}) |
6.0 | 1.5 μm | 53 and 57 |

LJ depth of well (ε) | 1.1 × 10^{−4} |
4.14 × 10^{−21} N m |
51 and 54 |

LJ zero potential distance (σ) | 2.0 | 0.5 μm | 51 and 54 |

LJ cutoff distance (r_{LJ}) |
2.24 | 0.56 μm | 51 and 54 |

∇·u = 0, | (1) |

(2) |

(3) |

The advance of f_{i}(x,t) in the above equation is split into two processes: streaming and collision. In the streaming process, the L.H.S. of eqn (3) is discretized as f_{i}(x + e_{i},t + 1) − f_{i}(x,t), in which the f_{i}(x,t) updates in both time and spatial spaces. In the collision process that corresponds to the term in the R.H.S of eqn (3), the particle residing at (x,t) relaxes towards its equilibrium state through collision behavior. Many collision models can be chosen, and here we adopt the most popular and simplest Bhatnagar–Gross–Krook (BGK) scheme,^{44} in which only one parameter-relaxation time τ is controlled. The last term F_{i} represents the external forcing term. In the simulation, D3Q19 model is used, which means that each point in the three-dimensional Eulerian system has 19 lattice velocities with different directions.^{44} The possible discrete velocities are

(4) |

The equilibrium distribution function f^{eq}_{i}(x,t) in LBM can be approximated by Maxwell distribution and written as

(5) |

(6) |

The external forcing term can be discretized by the form^{45}

(7) |

After each time step, f_{i} in the whole domain are collected and we can calculate the fluid density and momentum according to the relations:

(8) |

(9) |

(10) |

(11) |

(12) |

(13) |

(14) |

(15) |

The sphere also adopts the same settings with RBC. The rigidity of the sphere is fulfilled by enlarging the spring constants in the potential functions. The coefficients involved in the potential functions for RBC are increased by one order. In particular, the shear modulus is two orders higher for the sphere. Additionally, 2% inflation of the volume for the sphere is made to generate pre-tension in the membrane network, which makes the sphere barely to deform. The nanoworm is described by a polymer chain with beads connected by harmonic springs. A linear spring and an angle spring are adopted to characterize the stretching and bending of the nanoworm, respectively.

U^{n}_{stretching} = k^{p}_{s}(l − l_{0})^{2}, U^{n}_{bending} = κ(θ − θ_{0})^{2}, | (16) |

In addition to the above potentials, inter-molecular interactions among RBCs, and between RBCs and particles, should be employed. The Morse potential has been extensively used to model the interaction among RBCs.^{52,53} It has the form U_{morse} = D_{0}[e^{−2β(r−r0)} − 2e^{−β(r−r0)}], r < r_{c}. D_{0} represents the energy well depth and β controls the width of the potential well, r is the distance between two points and r_{0} is the equilibrium distance, r_{c} is the cutoff distance. For the interaction between RBC and particles or among particles, a short-range and pure repulsive Lennard-Jones (LJ) potential is introduced to prevent the overlapping, ε and σ are depth of the well and zero potential distance, respectively. All the coefficients used in the current work are listed in Table 1. Among those, the coefficients associated with the model of RBC, e.g., shear modulus and Morse potential, are provided in ref. 50 and further validated in our previous work.^{47} The parameters introduced to maintain the rigidity of sphere are also verified in ref. 54, in which the deformation of the sphere is negligible. The LJ potential coefficients are chosen to prevent the overlapping of particles without intrusion of the interaction between particles, as only the repulsive part is applied.

(17) |

(18) |

(19) |

This stencil involves 64 fluid nodes, which has demonstrated stability and fewer artifacts,^{59,65} in comparison to the 2-points stencil with 8 fluid nodes. The force f^{fsi}(x,t) is then utilized as a body force in the fluid solver. It is necessary to use the same interpolation function to obtain the velocities of the structure on the moving boundary through

(20) |

It should be noted that the channel wall is also considered as a stationary immersed boundary in the simulations. This fluid–structure interaction (FSI) computational framework has been validated by our previous studies.^{47,54,64} For example, Ye et al.^{47} validated the coarse-grained model of RBC together with the dynamics of a single RBC in a simple shear flow. Also, the rheology of RBCs inside a straight tube was investigated by Ye et al.^{64} and good consistency with previous experiments and simulations was found, such as Fahraeus–Lindqvist and Fahraeus effects. These benchmarks confirm that our FSI computational framework is accurate enough to study the circulation of particles in RBC suspension.

Fig. 2 Distribution of spheres in constricted channel with different geometries: (a) length effect (CR is fixed at β = 27.8%); and (b) ratio effect (CL is fixed at L_{c} = 18 μm). |

Besides, the effect of CR is further investigated by fixing the CL at L_{c} = 18 μm and varying the CR from 17% to 50%. Fig. 2(b) presents the results of the distribution of spheres under the constricted channel with different CRs. The shadow region represents the constriction including two transition zones with β = 27.8%. Note that the transition zones for different constrictions will be different, since the same polynomial function is applied for the transition, while the radii of the constrictions are different. Therefore, it is obvious that the valleys at the outlet are slightly different in Fig. 2(b). All the cases show the same pattern of sphere distribution along the channel: one peak inside the constriction near the entrance, and two valleys at the entrance and outlet. However, both the peak value and the overall distribution of Φ_{NP} are found to increase with the increment of β. In particular, Φ_{NP} increases significantly for the case β = 50%. Also, as the total number of particles is fixed, the high distribution in the constriction results in the significant reduction of distribution in main channels for the larger β.

To illustrate how the distribution of spheres depends on the geometry of constriction, two aspects are examined: RBC's distribution and flow field. Note that the spheres are surrounded by many RBCs that also circulate along with the fluid flow. The interactions between RBCs and spheres should influence the motion of spheres. In addition, the profile of the Poiseuille flow in the normal vessel is severely altered due to the existence of constriction. The perturbation of the local flow properties, such as velocity and pressure, may affect the distribution of spheres.

Firstly, the cases of channels with different CLs (β = 27.8% and L_{c} = 2, 18, and 34 μm) are studied. The distribution of RBCs in the flow is obtained by the same statistical method as that used for spheres, and the results are presented in Fig. 3(a) together with those of spheres. It is found that, for the same case with specific CL, the RBCs and spheres have the same distribution tendency. Due to the smaller size of spheres (D_{p} = 1 μm) compared to that of RBCs (D_{RBC} = 8 μm), the interaction between spheres and RBCs has a remarkable influence on the motion of spheres, while it has a negligible effect on the motion of RBCs. Therefore, the same tendency presents that the distribution of spheres along the channel is highly dependent on that of RBCs. Instead of examining spheres, we focus on the effects of constriction geometries on the distribution of RBCs.

The distribution of RBCs is confined by the altered flow due to the existence of constriction. The velocity and pressure along the flow direction are also shown in Fig. 3(b) for the case β = 27.8%, L_{c} = 18 μm. The relative pressure is defined as , where p_{y} represents the pressure at the point with coordinate y in the flow direction, N_{p} is the number of points at the cross-section with coordinate y, and is the averaged pressure of the flow in the corresponding straight channel. The velocity along the y direction is defined as the averaged velocity at the position with coordinate y, and ū is the averaged velocity of the flow in the corresponding straight channel. We find that the pressure and velocity profiles have the same pattern as those without RBCs and particles inside the channel presented by ref. 29. Due to the mass conservation of the flow, the compression in radial direction will be compensated by the axial acceleration of flow. It results in the faster movement of RBCs that locate at the center of the channel, and thus leads to the larger distance between RBCs inside the constriction and those in the main channel. The depletion of RBCs is characterized by the dip of RBC distribution at the entrance of constriction. Inside the constriction, the pressure quickly reaches a peak due to the accumulation of RBCs at the location near the entrance. Whereas, the effect of the acceleration of the velocity, inside the constriction, is compensated by the reduced radius of the channel, and the corresponding velocity distribution approximately reaches a plateau.

When approaching the outlet of the constriction, the distribution of velocity decreases to a valley and then restores the same distribution in the main channel. It also results in a reduction of the RBC distribution near the outlet and recovery of the uniform distribution in the main channel. The corresponding distribution of the pressure follows that of RBCs closely. The slight asymmetry of the distribution for RBCs at the entrance and outlet of the constriction may be relevant to the deformation of RBCs, which is also observed in ref. 24, 28 and 29. We further plot the pressure and velocity distributions for the channels with different CLs in Fig. 3(c) and (d), respectively. From the pressure distribution, we can see that if the constriction length L_{c} increases from 2 μm to 18 μm, the pressure at the entrance of the constriction increases. This is caused by the larger concentration of RBCs at the entrance of the constriction. However, further increase of L_{c} cannot change the pressure due to the saturation of RBCs accumulated at the entrance of the constriction. Besides, an opposite effect is found on the velocity profile: the larger concentration of RBCs means more RBCs block the flow and results in a lower velocity.

Secondly, we examine the CR effect on the relationship of distributions between RBCs and spheres. From Fig. 4(a), we find the same effect as that of CL. The distribution of spheres follows the same trend as that of RBCs. In addition, the pressure and velocity distributions are presented in Fig. 4(b) and (c), respectively. With the increment of CR, the peak of the pressure occurs at the entrance of the constriction is higher, while the valley of the pressure happens at the outlet of the constriction is lower. For the velocity, the higher CR leads to the lower plateau of velocity inside the constriction. To have a better understanding of pressure distribution along the flow direction, we take the averaged value of pressure in axial direction and show them in Fig. 4(d) for channels with different CRs. It is found that the pressure at the entrance of constriction with higher CR is significantly higher than those of the constrictions with lower CRs.

To summarize the above results, we find that the flow in the channel with different geometries of constrictions will determine the distribution of RBCs. And the interaction between RBCs and spheres will alter the local flow properties around the spheres. All of them lead to the same distributions between spheres and RBCs.

Firstly, we study the distribution of nanoworms in the channels with different CLs, and compare them with those of spheres. The results are given in the Fig. 5. It is found that, for the short constriction (L_{c} = 2 μm, Fig. 5 (a)), the accumulation of nanoworms is slightly higher than that of spheres inside the constriction. This results in a lower concentration of nanoworms in the main channels due to the conservation of the total number of NPs. The higher accumulation of nanoworms inside the constriction becomes more significant for the case with moderate constriction (L_{c} = 18 μm, Fig. 5(b)). It should be noted that the distribution of nanoworms is not smooth compared to that of spheres. Two possible reasons are considered, one having to do with the statistics of the number of nanoworms inside a small bin, which is based on the center of mass of the nanoworms. The other one is the change of the volume of a small bin in the transition zone. The small bin near the main channel is larger than that near the constriction. Moreover, we find that the further increase of the CL (L_{c} = 34 μm, Fig. 5(c)) magnifies the difference between distributions of spheres and nanoworms. However, this difference is only significant near the entrance of the constriction. The distribution of nanoworms rapidly increases to a peak when entering the constriction, and then sharply decreases towards a plateau, the same as that of spheres. This will be discussed later in the following section.

Fig. 5 Comparison of distributions of spheres and nanoworms under moderate CR β = 27.8% with different CLs (a) L_{c} = 2 μm, (b) L_{c} = 18 μm and (c) L_{c} = 34 μm. |

Secondly, the distributions of nanoworms in the channels with different CRs are examined and shown in Fig. 6. The CL is fixed at L_{c} = 18 μm, and the CRs β vary from 17% to 50%. It is also found that the accumulations of nanoworms inside the constriction are higher than those of spheres in small (β = 17%) and moderate (β = 27.8%) CRs, and the difference is significant for the moderate one, similar to the CL effect. However, it is interesting that in the channel with high CR (β = 50%), the spheres and nanoworms manifest the same distributions in both the main channel and the constriction. Furthermore, we find that this exceptional phenomenon only exists for the channels with high CR (β = 50%) and large CL (L_{c} ≥ 18 μm).

Fig. 6 Comparison of sphere and nanoworm distributions under moderate CL L_{c} = 18 μm with different CRs (a) β = 17%, (b) β = 27.8% and (c) β = 50%. |

To unravel the higher accumulation of nanoworms inside the constriction and the exceptional phenomenon discussed above, the RBC effect is first investigated. Based on the results in Section 3.1, the spheres circulate along with RBCs in the channel like that RBCs “carry” spheres and move along with the flow. Therefore, we conduct additional simulations by removing RBCs in the channel and maintain the other conditions the same, such as flow strength and total particle numbers. The distributions of spheres and nanoworms without RBCs are presented to compare with those with RBCs inside the channel, as given in Fig. 7. Two cases are demonstrated: (i) β = 27.8%, L_{c} = 34 μm and (ii) β = 50%, L_{c} = 34 μm, which correspond to the typical cases of higher accumulation of nanoworms and an exceptional phenomenon, respectively. It is found that, in both cases, spheres and nanoworms have the same distributions without RBCs inside the channel. This indicates that the existence of RBCs results in the higher accumulation of nanoworms inside the constriction. In the following, we will further explore how RBCs affect the different behaviors of spheres and nanoworms during their circulation in the constricted channel.

Fig. 7 Comparison of sphere and nanoworm distributions with and without RBCs inside the channel for (a) β = 27.8%, L_{c} = 34 μm and (b) β = 50%, L_{c} = 34 μm. |

The above discussions are limited to the distribution of particles in the flow direction, while the radial distribution should be also considered. We study the radial distribution of particles in both the main channel and the constriction. Fig. 8(a) and (b) present the radial distribution in the main channel. r = 0 μm and r = 18 μm represent the center line and the wall of the channel, respectively. The shadow region is adopted to denote the extra part of the main channel compared to the constriction, and the left boundary of the shadow points to the position of constriction. We find that the spheres localize at the central part of the channel, while the nanoworms accumulate near the wall. The snapshots in Fig. 9 show the comparison of radial distribution of spheres and nanoworms under moderate CR β = 27.8% and moderate CL L_{c} = 18 μm. As shown by the red arrows, spheres tend to localize at the center of the channel, while nanoworms migrate towards the channel wall, leaving a nanoworm-depletion region at the center of the channel. Furthermore, for the case of moderate CR β = 27.8% (cf.Fig. 8(a)), the concentration of spheres in the main channel decreases with the increase of CL. Besides, the accumulation of nanoworms near the wall demonstrates the same tendency that it decreases with the increase of CL. In addition to the CL effect, we fix the CL at L_{c} = 18 μm and vary the CR. As shown in Fig. 8(b), we find that the concentration of spheres in the main channel decreases with the increase of CR. The accumulation of nanoworms near the wall with a larger CR is higher. However, the exceptional case occurs at β = 50%, in which nanoworms localize at the central part of the channel like spheres, instead of accumulating near the wall.

Fig. 9 Snapshots for radial distributions of (a) spheres and (b) nanoworms under moderate CR β = 27.8% and moderate CL L_{c} = 18 μm. |

Furthermore, the radial distributions inside the constriction are examined. In the Fig. 8(c), we find that, for the case with moderate CR (β = 27.8%), the spheres localize at the central part of the constriction, while the concentration increases with the increment of CL which is different from that in the main channel. This also applies to the accumulation of nanoworms near the wall that the distribution is higher in the longer constriction. Here, we should emphasize that we ignore the case of L_{c} = 2 μm, since it is difficult for a very short constriction to calculate the averaged radial distribution. For the constriction with fixed length L_{c} = 18 μm, we find the concentration of spheres has no obvious difference between the cases with different CRs as shown in Fig. 8(d). While the distribution of nanoworms inside the constriction is found to slightly increase with the increment of CR. It should be noted that the case with β = 50% is not shown here. In this extreme case (β = 50%), RBCs aggregate together and lead to almost identical distributions between spheres and nanoworms inside the constriction. This aspect will be further discussed in the next section.

Although nanoworms can accumulate near the wall, they are driven by the flow and will finally enter the constriction. They will be affected by the RBCs that locate near the constriction wall, since the nanoworms accumulating at the near-wall region of the main channel will enter the constriction together with RBCs. Therefore, we also show the radial distribution of RBCs in Fig. 10(c) and (d). It is found that, for moderate CR (β = 27.8%), the longer constriction leads to more RBCs concentrating at the near-wall region, which results in the greater number of nanoworms accumulated at the near-wall region of the main channel being carried by RBCs into the constriction. This also applies to the cases with moderate CL (L_{c} = 18 μm). Larger CR will lead to greater accumulation of RBCs at the near-wall region of the main channel, thus carrying more nanoworms into the constriction.

The exceptional phenomenon of spheres and nanoworms having the same distribution when CR is high (β = 27.8%) and CL is large (L_{c} ≥ 18 μm) is observed. To understand this phenomenon, we examine the radial distribution of particles and RBCs under these circumstances. Fig. 11(a) shows the snapshots of two typical cases: moderate CR (β = 27.8%) and high ratio (β = 50%) for the longest CL (L_{c} = 34 μm). With moderate CR, RBCs are almost uniformly distributed inside the channel. However, for the case with high CR, RBCs concentrate at the central part of the channel and some RBCs are blocked at the entrance of the constriction. Associated with these two cases, we present the radial distribution of spheres, nanoworms and RBCs in Fig. 11(b) and (c), respectively. Under moderate CR, we observe the same results as in the previous section. The spheres localize at the central part, while nanoworms accumulate near the wall of the main channel. We also find that RBCs concentrate at the location near the wall of the constriction, thus carrying more nanoworms into the constriction. Under these circumstances, nanoworms demonstrate higher accumulation in the front of the constriction than spheres do. Fig. 11(c) shows that both spheres and nanoworms are subjected to the same distribution of RBCs that concentrate at the central part of the channel. In this case, RBCs aggregate together and the gap between RBCs is very limited. Therefore, it is not only spheres that cannot escape from the RBC cluster, but also the nanoworms cannot penetrate into the RBC-depletion layers near the channel wall. Only a small amount of spheres or nanoworms are found to accumulate at the entrance of the constriction due to the obstruction by RBCs. Although the accumulation is small, it still promotes the development of the constriction if these spheres or nanoworms are platelets, which is the main reason for the plaque accumulation.^{16}

Fig. 11 (a) Snapshots for distribution of RBCs inside the channel. The radial distribution of spheres, nanoworms and RBCs for the cases: (b) β = 27.8%, L_{c} = 34 μm and (c) β = 50%, L_{c} = 34 μm. |

To support our analysis that the difference between the distributions of spheres and nanoworms in the constricted channel is caused by the shape-induced penetration in RBCs suspension, we conduct more simulations for spheres with different diameters. The distributions in the flow direction are shown in Fig. 12(a). We observe that the accumulation of spheres will increase with the increment of the diameter D_{p}. Fig. 12(b) presents the distributions in the radial direction. It is found that the larger the sphere's diameter is, the more spheres localize at the central part of the channel. Besides, we also show the radial distribution of nanoworms in Fig. 12(b). We find that, as the diameter of the spheres decreases, the radial distribution of spheres approaches that of nanoworms. This indicates that the smaller spheres will behave like nanoworms and escape from the RBC cluster and accumulate near the main channel wall. The accumulated spheres will eventually enter the constriction, resulting in higher accumulation inside the constriction. Whereas, the larger spheres are stuck in the gaps among RBCs and they normally circulate with the RBCs in the core region of the channel.

Secondly, we examine the distribution of nanoworms and compare it with that of spheres. It is found that nanoworms demonstrate stronger accumulation inside the constriction. This difference is explained by analyzing the radial distributions of spheres, RBCs and nanoworms. We find that RBCs uniformly distribute in the main channels with a depletion layer near the wall, and spheres move along with the RBCs. However, nanoworms can escape from RBC clusters and tend to migrate towards the channel wall. The accumulated nanoworms near the wall will eventually enter the constriction, leading to a higher accumulation of nanoworms inside the constriction. Nevertheless, an interesting and exceptional phenomenon happens for the case with a constriction of high ratio and long length. Under these circumstances, passing through the constriction makes the RBCs aggregate together and focus at the core region of the channel, as the main channel is not long enough for RBCs to disperse in the radial direction. This means that the nanoworms are hardly able to escape from the RBC clusters, like spheres can, resulting in almost the same accumulation inside the constriction as that of spheres. This mechanism is further supported by performing additional simulations of spheres with different diameters. The larger sphere is hardly to escape from the RBC cluster and demonstrates lower accumulation inside the constriction.

In the present study, we only consider the nanoworm with semi-flexibility, since the goal is to compare a spherical particle with a filamentous one. However, the flexibility of the nanoworm is also another important factor to change the dynamics of nanoworms in the blood flow.^{67–69} In our previous study,^{51} we find that the soft nanoworm collapses in the blood flow and acts as a spherical particle in terms of the margination. While the rigid nanoworm demonstrates strong accumulation on the vessel wall, due to the exclusion of RBCs. Therefore, we consider only the semi-flexible nanoworm can demonstrate the above RBC hitchhiking behavior.

For simplification, the properties of RBCs such as stiffness are fixed in all the simulations. However, in reality, the nanoparticles and the thrombo-inflammatory conditions can alter their mechanical properties, leading to different behaviors of the RBCs in the DDS.^{70–74} For example, the polystyrene NP can increase the RBC stiffness during the interaction,^{70} and a high NP loading can damage RBCs by mechanical, osmotic and oxidative stress.^{72} Also, during the thrombo-inflammatory process, due to the compression of activated platelets pulling on fibrin, the shape of RBCs can largely change within contracting blood clots. These biological aspects should be considered in the application of RBC hitchhiking in DDS.

This study unravels the underlying mechanism of nano- and micro-particles circulating in the constricted channel through comparing two typical shapes: spherical versus filamentous. The simulation results may guide the rationalized design of these particles with applications in biomedical engineering, such as targeting drugs at the site of stenosis in microvessels.

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