Predicting the vascular adhesion of deformable drug carriers in narrow capillaries traversed by blood cells

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Predicting the vascular adhesion of deformable drug carriers in

narrow capillaries traversed by blood cells

A. Coclitea, G. Pascaziob, M. D. de Tulliob, P. Decuzzia,∗

aLaboratory of Nanotechnology for Precision Medicine, nPMed, Fondazione Istituto Italiano di Tecnologia,

Via Morego 30-16163, Genova, Italy

bDipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via Re David 200 – 70125

Bari, Italy

Abstract

In vascular targeted therapies, blood-borne carriers should realize sustained drug release

from the luminal side towards the diseased tissue. In this context, such carriers are required

to ﬁrmly adhere to the vessel walls for a suﬃcient period of time while resisting force per-

turbations induced by the blood ﬂow and circulating cells. Here, a hybrid computational

model, combining a Lattice Boltzmann (LBM) and Immersed Boundary Methods (IBM),

is proposed for predicting the strength of adhesion of particles in narrow capillaries (7.5

µm) traversed by blood cells. While ﬂowing down the capillary, globular and biconcave

deformable cells ( 7µm) encounter 2µmdiscoidal particles, adhering to the vessel walls.

Particles present aspect ratios ranging from 0.25 to 1.0and a mechanical stiﬀness varying

from rigid (Ca = 0) to soft (Ca = 10−3). Cell-particle interactions are quantitatively pre-

dicted over time via three independent parameters: the cell membrane stretching δp; the

cell-to-particle distance r, and the number of engaged ligand-receptor bonds NL. Under

physiological ﬂow conditions (Re = 10−2), rigid particles are detached and displaced away

from the wall by blood cells. This is associated with a signiﬁcant cell membrane stretching

(up to 10%) and rapid breaking of molecular bonds (tumax/H<1). Diﬀerently, soft parti-

cles deform their shape as cells pass by, thus reducing force perturbations and extending

the life of molecular bonds. Yet, only the thinnest deformable particles (2 ×0.5µm) ﬁrmly

adhere to the walls under all tested conﬁgurations. These results suggest that low aspect

ratio deformable particles can establish long-lived adhesive interactions with capillary walls,

arXiv:2210.06043v1 [physics.flu-dyn] 12 Oct 2022

enabling de facto vascular targeted therapies.

Keywords: Drug delivery, Lattice Boltzmann, Immersed boundary, Computational

modeling, Computational nanomedicine

1. Introduction

Targeting the diseased vasculature is an attractive strategy for the diagnosis and treat-

ment of a variety of pathologies. In cancer, endothelial cells express unique receptor molecules,

such as integrins αvβ3and αvβ5, tumor endothelial markers, vascular epidermal growth factor

receptors, and so on (Neri and Bicknell (2005) and Atukorale et al. (2017)). In cardiovas-

cular and chronic inﬂammatory diseases, the endothelium presents inﬂammatory molecules,

such as P- and E-selectins, ICAM and VCAM-1, higher densities as compared to the normal

vasculature (Libby, 2002; Ta et al., 2018; Soriano et al., 2000). Even, within the white

adipose tissue, speciﬁc receptors are exposed on the surface of endothelial cells (Kolonin et

al., 2004; Daquinag et al., 2011; Anselmo and Mitragotri, 2017). Although some of these

receptors could be directly used as targets for small therapeutic molecules, most of them

would serve as docking sites for blood-borne, vascular targeted particles (Kolhar. et al.,

2013; Decuzzi and Ferrari, 2008). Engineered micro- and nano-carriers for the precise deliv-

ery of multiple agents are entering clinical trials for the diagnosis and treatment of a variety

of diseases, including cancer, cardiovascular and neurological (Peer et al., 2007; Mulder et

al., 2014). These carriers are realized using diﬀerent techniques where attributes such as the

size, the composition, the surface properties and, more recently, the shape and mechanical

stiﬀness can be ﬁnely and independently tuned (Euliss et al., 2006; Godin et al., 2012; Muro

et al., 2008; Palange et al., 2017; Palomba et al., 2018; Anselmo et al., 2015). Vascular

targeted carriers can deposit at the luminal side large amounts of imaging and therapeutic

molecules whose controlled release towards the diseased tissue can be triggered via a number

of mechanisms, including chemical and mechanical stimuli (Mura et al., 2013).

∗Corresponding author

Email addresses: alessandro.coclite@iit.it (A. Coclite), giuseppe.pascazio@poliba.it (G.

Pascazio), marcodonato.detullio@poliba.it (M. D. de Tullio), paolo.decuzzi@iit.it (P. Decuzzi)

Preprint submitted to Journal of Fluids and Structures October 13, 2022

Several reports have investigated the vascular transport of micro and nano-carriers in

the attempt to identify the optimal conﬁguration that could favor their deposition on en-

dothelial walls. For instance, the works of Liu and collaborators showed that red blood cells

would favor the lateral drifting and vascular binding of suﬃciently large nanoparticles (Tan

et al., 2011). Similarly, the authors demonstrated that the contribution of red blood cells

is mostly ampliﬁed for micron-sized particles over conventional 100 nm nanoparticles (Lee

et al., 2013). A more comprehensive analysis was provided by Vahidkhah and Bagchi, who

considered the vascular transport and adhesion of spherical, prolate and oblate spheroids

with diﬀer aspect ratios (Vahidkhah and Bagchi, 2015). Their numerical results conﬁrmed

that oblate spheroids with moderate aspect ratios would more eﬃciently marginate and ad-

here to the wall, as compared to spherical particles and prolate spheroids. Beyond size and

shape, the eﬀect of particle deformability on margination dynamics was documented in the

work of Muller et al. (2016). These authors conﬁrmed that micrometer carriers marginate

better than their sub-micrometer counterparts and that deformable carriers are less prone

to marginate as compared to rigid particles and demonstrated, in complex whole blood ﬂow,

that 2D and 3D simulations of red blood cells tend to return qualitatively similar results.

Fish M. B. and colleagues explored the ability of micro- and nanosized particles to identify

and bind to diseased endothelium, conﬁrming that microcarriers outperforms nanoconstructs

in margination and adhesive properties (Fish et al., 2017). In particular, rigid particles show

improved adhesive abilities for large shear rate, while soft particles for low shear rates. 3D

blood ﬂow modeling and the accurate description of the viscoelastic properties of red blood

cells (RBCs) has allowed the scientiﬁc community at large to learn more about molecular,

nano and microscale transport within capillary networks, under physiological and patholog-

ical conditions (Fedosov et al., 2010; Ahmed et al., 2018; Li et al., 2017). Nonetheless, 2D

models can still be very eﬀective in dissecting some basic mechanisms regulating the interac-

tion between deformable RBCs and nano/micro-particles. For instance, a recently published

paper used a 2D blood ﬂow model to predict the dispersion of nanoparticles, as a function

of the RBC motion and deformation, in good agreement with experimental data (Tan et al.,

2016). Despite these seminal works, at authors’ knowledge, no study has ever systematically

analyzed the interaction between particles deposited onto a wall and circulating blood cells.

Carrier margination and adhesion are necessary but not suﬃcient conditions for fully

realizing drug delivery through vascular targeting. Indeed, carriers should adhere to the

vessel walls for suﬃciently long times, without being dislodged away by hemodynamic forces

and circulating cells, in order to support the continuous and controlled release of therapeutic

agents towards the diseased tissue. In this work, the authors propose a hybrid computational

scheme, combining an ImmersedBoundary (IBM) and a Lattice Boltzmann-BGK (LBM)

method, for studying cell-particle interactions in narrow capillaries. While the IBM serves

to characterize the capillary transport of deformable objects (cells and particles), the LBM

provides an Eulerian description of the ﬂuid evolution (Coclite et al., 2016). While ﬂowing

within a narrow capillary, 7µmdeformable cells encounter particles adhering to the wall,

which act as partial occlusions. Cell and particle dynamics are predicted under diﬀerent

conditions, including four cell shapes, resembling leukocytes and erythrocytes; four aspect

ratios for the adhering particles; two values of particle aﬃnity with the vascular walls and two

diﬀerent values of the mechanical stiﬀness of the particles. Three parameters are introduced

for quantitatively described the physical problem, namely the stretching ratio of the cell;

relative distance between particles and cells; and adhesive strength of the particles to the

wall.

2. Computational method

2.1. The lattice Boltzmann method

The evolution of the ﬂuid is deﬁned in terms of a set of Ndiscrete distribution functions

{fi},(i= 0, . . . , N −1), which obey the dimensionless Boltzmann equation

fi(x+ei∆t, t + ∆t)−fi(x, t) = −∆t

τ[fi(x, t)−feq

i(x, t)] ,

in which xand tare the spatial and time coordinates, respectively; [ei],(i= 0, . . . , N −1)

is the set of discrete velocities; ∆t is the time step; and τis the relaxation time given by the

unique non-null eigenvalue of the collision term in the BGK approximation (Bhatnagar et

al., 1954). The kinematic viscosity of the ﬂow is strictly related to the single relaxation time

τas v=c2

sτ−1

2∆tbeing cs=1

√3

∆x

∆tthe reticular speed of sound. The moments of the

distribution functions deﬁne the ﬂuid density ρ=Pifi, velocity u=Pifiei/ρ, and the

pressure p=c2

sρ=c2

sPifi. The local equilibrium density functions [feq

i] (i= 0, . . . , N −1)

are expressed by the Maxwell-Boltzmann distribution

feq

i(x, t) = ωiρ1 + 1

(ei·u) + 1

2c4

(ei·u)2−1

2c2

u2.

On the two-dimensional square lattice with N=9speeds (D2Q9) (Qian et al., 1992), the

set of discrete velocities is given by

ei=









(0,0),if i= 0

cos (i−1)π

2,sin (i−1)π

2,if i= 1 −4

√2cos (2i−9)π

4,sin (2i−9)π

4,if i= 5 −8,

with the weight, ωi= 1/9for i=1−4, ωi= 1/36 for i=5−8, and ω0= 4/9. Here,

we adopt a discretization in the velocity space of the equilibrium distribution based on the

Hermite polynomial expansion of this distribution (Shan et al., 2006).

2.2. Immersed-boundary treatment

Deformable body models are commonly based on continuum approaches using strain en-

ergy functions to compute the membrane response (Pozrikidis, 2001; Skalak et al., 1973;

Krüger, 2012). However, a particle-based model governed by molecular dynamics has

emerged due to its mathematical simplicity while providing consistent predictions (Dao et

al., 2006; Fedosov et al., 2011; Nakamura et al., 2013; Ye et al., 2014). Following this, here a

particle-based model is adopted via the Immersed-Boundary (IB) technique. The immersed

body consists in a network of nv vertices linked with nl linear elements, whose centroids are

usually referred as Lagrangian markers. An eﬀective forcing term Fi(i= 0,...,8), account-

ing for the immersed boundary, is included as an additional contribution on the right-hand

side of Eq. (1). Fiis expanded in term of the reticular Mach number, ei

cs, resulting in

Fi=1−1

2τωiei−u

+ei·u

ei·fib,

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PredictingthevascularadhesionofdeformabledrugcarriersinnarrowcapillariestraversedbybloodcellsA.Coclitea,G.Pascaziob,M.D.deTulliob,P.Decuzzia,aLaboratoryofNanotechnologyforPrecisionMedicine,nPMed,FondazioneIstitutoItalianodiTecnologia,ViaMorego30-16163,Genova,ItalybDipartimentodiMeccanica,Matematica...

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Predicting the vascular adhesion of deformable drug carriers in narrow capillaries traversed by blood cells

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