Study of nanolayer on red blood cells as drug carrier in an artery with stenosis

: This article discusses a novel idea from cell therapy in which nanoparticles (NPs) are adsorbed on red blood cells (RBCs). RBCs serve as a drug carrier for NPs or nanodrugs adsorbed on the cell membrane of RBC. For the purpose of examination, Fe O 3 4 NPs are adsorbed on RBCs, collectively called NP-RBC complex. RBCs being a natural vascular carrier, have high transfusion rates and biocompatibility. This mathematical study provides a basis to attempt nanodrug delivery via RBCs, as carriers for nanodrugs, to the stenosed sites in an artery. The mathematical model is developed for an artery with stenosis and a catheter that regards the temperature and velocity of the NP-RBC complex. Catheter coated with the NP-RBC complex is inserted into the lumen of the stenosed artery. The mathematical problem is solved numerically using Bernstein polynomials. The physical features were discussed through graphs plotted using MATLAB. The in ﬂ uence of parameters such as volume fraction, radius of the NP-RBC complex in blood, and the thickness of the nanolayer on RBCs was studied. A noticeable outcome states that the nanolayer of optimum thickness about 50 – 40 nm is suitable for this purpose. Thus, this is an attempt to study the delivery of NPs adsorbed on the surface of RBCs to develop newfangled strategies in nanomedicine bearing high precision and e ﬃ ciency.


Introduction
Cardiovascular diseases are the major reason for reducing life expectancy globally.Cardiovascular diseases have resulted in 17.9 million deaths in 2019, which represents 32% of deaths occurring worldwide.They also accounted for 38% premature deaths or deaths below the age of 70 years in 2019.It is estimated that more than 23 million deaths will occur due to cardiovascular diseases annually, by 2030 [14].
Atherosclerosis is a highly occurring cardiovascular ailment.It is an inflammation in the artery that causes a progressive buildup of plaques in the lining of arterial wall.The arterial lumen narrows due to the development of plaques that obstruct the flow of blood; these depositions are called stenosis [12].The development of stenosis at the wall of artery is accompanied by significant changes in blood flow.If not treated properly, it may develop into myocardial infraction because of plaque rupture with thrombus formation.
Catheters are a standard medical tool for the diagnosis and treatment of stenosis.They effectively clear the blockages in the artery and can also be used to monitor blood pressure and blood velocity.Introduction of a catheter, thus, alters the mechanics in the stenosed artery, and hence, it is an interesting area of mathematical investigation.
Despite the developments in medical and therapeutic management of cardiovascular diseases, the outcomes do not go very well with the treatments.This occurs owing to the lack of targeted theranostic.
Biomedicine aims to continuously evolve and bring about improvements in the therapeutic index of the drug.The ability of any drug to survive in physiological environment determines the success of drug.Cell therapy using red blood cells (RBCs) is a novel method to answer the discrepancies of conventional methods [17].Cell therapy offers advantages over conventional therapies because of better cellular networks.RBCs are the blood cells occurring in highest abundance, approximately × ∕ 5 10 L 12 , in the blood.Thus, they make about 45% of the total blood volume and hence are widely distributed throughout the circulation.The use of RBCs as drug carriers has been a keen area of research since the 1980s.The RBCs are biconcave disc-shaped without cell organelles that allow for high surface area in comparison with volume, which accredits to a higher loading of drugs.RBCs possess longevity in circulation, for instance, 45 days in mice and 120 days in humans.Thus, RBCs have possible use in vascular delivery of surface-bound drugs as carriers.Also, unlike other carriers, loading drug on RBCs does not interfere with the normal functioning of RBCs.But instead, the drugs can be released more conveniently [5].Such extraordinary mechanical and structural characteristics of RBCs make them a remarkable agent for the delivery of drug.
The adsorption or assimilation of drugs on RBCs' membrane can prevent the phagocytosis of drugs and can also reduce their possible biotoxicity [16].RBC membrane has a unique combination of mechanical flexibility to squeeze through microvessels and physical sturdiness to tolerate high stress conditions in circulation.Thus, RBCs are less fragile and more resistant cells.The RBC membrane renders a platform that encompasses the advantages of native RBCs and drugs.Coupling RBCs and drugs enables their transport in the vasculature without their untimely clearance.A number of studies over the past three decades have shown that encapsulating drugs on RBCs remarkably influences to improve the drug dynamics by prolonging their circulation in body [23].Due to these likeable properties, the adsorption of drugs on RBCs is developing and progressing, in methodologies as well as experimentally, to aid in their clinical applications.
Cell therapy in combination with nanoparticles (NPs) is a new idea of drug delivery.RBCs are the most suitable carrier of NPs due to their enhanced compatibility in the system owing to their safe transfusion.The use of NPs in medicine holds promising advantages for the delivery of drugs at the specified site [10].Loading of NPs on the RBCs has emerged as a promising strategy, because it was reported that when NPs alone were injected in the blood, they accumulated in liver or were cleared by macrophages [4].Nanoformulations using RBCs increase the tolerability of drug in the immune system offering drug stability.
Pan et al. [17] studied the effect of NPs on the biocompatibility of RBCs as drug carrier.Their observation on mouse's RBCs stated that RBCs can house millions of the NPs per microlitres of blood.Pan et al. [16] studied the pharmacokinetics of the NP-RBC complex and proved its usefulness in drug delivery.Xia et al. [23] used RBC membrane-camouflaged NPs in the treatment of tumours.
In order to achieve the adsorption of nanolayer on RBCs, RBCs are first separated and collected from a blood sample.They are then washed with ice-cold water and dipped in a hypotonic solution containing the required NPs.NPs then adsorb on the surface of RBCs.This solution is centrifuged, and the supernatant is discarded.The process is repeated for a number of times depending on the nature of nanodrug.The drug so formed after nanolayer is adsorbed on RBCs is called the NP-RBC complex.NP-RBC complex can be administered into circulation in many ways.Here, it is considered that the complex is coated on the catheter and then inserted into the artery.These complexes imitate the movement of RBCs' in the blood and thus can reach the desired site.
NPs are loaded on the RBCs and then inoculated back into circulation via coating on the catheter [21].These therapeutics using RBCs as nanocarriers promote guided, sustained, and selective drug release to the diseased site whilst maintaining the homogeneous distribution of drug.To achieve the purpose, various kinds of inorganic NPs such as Fe O 3 4 , Si, and Au are used.Fe O 3 4 NPs have proven to possess excellent targeting ability after their coating on the surface of RBC.Rao et al. [20] had designed Fe O 3 4 NPs coated on RBCs that exhibited superior biocompatibility and degradability.
Once the NPs reach the blood-stream, it is important to assess their behaviour.Nanofluids are nanomaterials dispersed in a base fluid such as water and organic solvents [19].Nanofluids display anomalous thermal properties and thus have gained significant interest over the last 10 years [9].The two factors responsible for this are surface area and quantum effects.As the size of particle reaches the nanoscale, the nanofluids display different thermal properties.It has been shown that small amounts of nanosized materials in a fluid enhance its thermal conductivity, thermal expansion, and viscosity to a large extent.The NP-RBC complex in blood is treated as nanofluid and the traits of which have been discussed applying the thermal properties of nanofluid.
Furthermore, the mathematical solution to the problem has been found using Bernstein polynomials.Bernstein polynomials solve differential equations approximately [2,6].These positive polynomials are defined on the interval such that their sum is always unity.This method reduces the mathematical model into a set of algebraic equations that reduces the complexity.
Based on the aforementioned literature survey, a mathematical model to analyse cell therapy via the deposition of NPs on RBCs has not been conducted so far.In this mathematical study, the NPs coated on RBCs, collectively identified as the NP-RBC complex, are coated on the catheter, which is inserted in the lumen of the artery.The continuity equation, Navier-Stokes equation for fluid flow, and temperature diffusion equation are used to design the mathematical model.Thermophysical properties such as viscosity, thermal conductivity, and thermal expansion have been considered.Bernstein polynomials are used to solve the problem.The temperature and velocity of NP-RBC in the blood have been found and investigated by plotting graphs.The influence of volume fraction of the NP-RBC complex and thickness of the NP layer on RBC has been monitored graphically using MATLAB.The aforementioned model has probable applications for the treatment of cardiovascular diseases using targeted drug delivery.

Formulation of the model
The geometrical representation of the mathematical model of a catheterized artery with stenosis is shown in Figure 1.The flow is described [3] ).The geometry of stenosis is , otherwise.
The equations governing the motion are as follows: Study of nanolayer on red blood cells as drug carrier  3 Continuity equation: Navier-Stokes equation: (4) where ′ F is the body forces in various indices, μ nf is the viscosity, and ρ nf is the nanofluid density.Temperature diffusion equation: where D nf is the thermal diffusivity, k nf is the nanofluid thermal conductivity, and H is the constant heat absorption or generation parameter.Equations ( 1)-( 6) assume the following considerations: (1) Flow is two-dimensional and axisymmetric.
(3) The insertion of catheter considers constant heat generation/absorption.Now, the equation of continuity The Navier-Stokes equation and its boundary conditions where γ nf is the thermal expansion of the NP-RBC complex.
No slip condition is assumed at the catheter: At the boundary of artery, Darcy's law is applied as the NP-RBC complex shall be absorbed at the stenosed site: in which is the velocity at permeable boundary , p nf (12) where ′ u B is the velocity of slip, ′ σ is the parameter of slip, and Da is the Darcy number.The temperature diffusion equation and conditions at boundary are as follows: The temperature at the surface of artery is T 0 and on the surface of the catheter T c : As described earlier, the nanofluids display enhanced thermal properties such as thermal conductivity and viscosity.Over the time, many studies have described various ideas to comprehend the properties of nanofluids.The prime model in this regard for nanofluid thermal conductivity was given by Maxwell [11].Yu and Choi [24] reformed this model by allowing for the traits of nanolayer on the membrane of dispersed particles, described mathematically as: where is the thickness of nanolayer, r p is the radius of NP-RBC complex, k f is the thermal con- ductivity of the base fluid, k p is the thermal conductivity of the NP-RBC complex, and ϕ is the volume fraction of the NP-RBC complex in blood.
Viscosity of nanofluid is another important property that directly affects the motion of the NP-RBC complex.The classical model for viscosity of nanofluid was given by Einstein [7].Hosseini et al. [8] gave a more advanced model for predicting the viscosity in the presence of nanolayer.This model holds consistency to experimental verifications on water-based nanofluids: where ′ T is the temperature, and a c , 1 , c 2 , and c 3 are the empirical constants.The principle of conservation of mass gives the density of nanofluid as follows: where ρ f is the density of blood and ρ p is the density of the NP-RBC complex.Thus, the thermal expansion of nanofluid ργ nf ( ) is expressed as: where γ f is the thermal expansion of blood and γ p is the thermal expansion of the NP-RBC complex .
The scheme for non-dimensionalization is as follows: where u o is the reference velocity, Re is the Reynolds number, Da is the Darcy number, Gr is the Grashof number, and h is the heat source parameter.
The final equations after non-dimensionalizing the geometry of stenosis, Navier-Stokes equation, temperature diffusion, and respective boundary conditions are given as follows: Study of nanolayer on red blood cells as drug carrier  5 where

Solution
The solution to equations ( 22) and ( 23) using the boundary conditions (24) to (29) is found numerically using Bernstein polynomials.Bernstein polynomials are defined as basis functions in the Hilbert space L 2 [0,1] expressed as: where Thus, we write a polynomial u r m ( ) of degree m as Bernstein polynomial B r k m . ( ) as: where c k 1. are the constants.Now, finding the s-order differential with respect to r of the polynomial u r m ( ) Now, Thus, Now, after allocating r j , = − j m 0, 1,…, 2, and z l , = l q 0, 1,… for framing our equations and boundary condi- tions in terms of Bernstein polynomials, we obtain Now, the solution to these algebraic equations is found using Gauss elimination method, in MATLAB.

Graphical results and discussions
RBCs are profusely occurring blood cells circulating in the body.Thus, they can be used on drug delivery system with certain modifications.Camouflaging the membrane of RBC to adsorb NPs as layer renders a platform that clubs the advantage of NPs with RBC, thus achieving enhanced biocompatibility and bioavailability.The changes in the pharmacokinetics when this NP-RBC complex enters the blood circulation via intravenous injection is the major concern of this study.The mathematical model is framed for an artery with stenosis.A catheter coated with Fe O 3 4 NP-RBC complex is inserted into the lumen of the artery.The model uses the continuity equation, the Navier-Stokes equation for fluid flow, and the temperature diffusion equations to describe the characteristics of the NP-RBC complex in blood.The temperature and velocity of the resultant nanofluid is found numerically using Bernstein polynomial approximations.The graphical analysis is performed using MATLAB.The effect of temperature of nanofluid θ is examined on different values of volume fraction ϕ of the NP-RBC complex, radius r p of the NP-RBC complex, thickness of nanolayer d nl , and heat source parameter h, against radial direction r (Figures 2-5), while the effect of velocity u of nanofluid is observed on different values of volume fraction ϕ of the NP-RBC complex, radius r p of the NP-RBC complex, thickness of nanolayer d nl , stenosis depth δ, and Darcy number Da against radial direction r (Figures 6-10).The results are discussed in detail henceforth that have shown a good agreement with the existing literature.Table 1 lists the values of thermophysical quantities used at 300 K.
The trend of temperature θ against radial direction r with respect to the volume fraction ϕ of the NP-RBC complex in blood (Figure 2) shows that the temperature rises for an increase in volume fraction.Volume fraction is the representative of the number of solute particles in a solvent.Thus, an increase in the number of solute particles enhances the net Brownian motion that brings about an increase in temperature.Kumar et al. [9] reported a similar conclusion in their study of nanofluids.Also, this aspect of volume fraction is important to decide the amount of dosage that should be administered.
The graph of temperature θ against radial direction r with respect to the radius r p of the NP-RBC complex (Figure 3) displays a rise in temperature for an enhancement in radius of the NP-RBC complex.This happens  because of the reduction in interparticle distance, which causes the temperature to rise because of increased Brownian motion.Wang et al. [22] in their study of shear kinetics of such complexes concluded that an average size of about 5,045 nm of the NP-RBC complex is apt for the drug delivery.
Similarly, the graph of temperature θ against radial direction r with respect to the thickness d nl of nanolayer (Figure 4) displays a rise in temperature for an increased thickness.The reason is the same as above.Ajdnik et al. [1] in their experimental study of use of nanolayers for medicine gave homogenous observations.Gradually, under the effect of varying shear stress, the nanodrug shall be released upon entering the capillaries [22].
The temperature θ profile against radial direction r with respect to heat source parameter h (Figure 5) depicts that as the temperature increases for an increase in heat source parameter.Heat source parameter typifies the amount of heat generated or absorbed due to the insertion of catheter.Thus, there is a greater heat generation for an increased heat source parameter.This factor shall determine the optimum temperature of nanodrug to function.Study of nanolayer on red blood cells as drug carrier  9 The trend of velocity u against radial direction r with respect to the volume fraction ϕ of the NP-RBC complex in blood (Figure 6) shows that velocity reduces for the rise in volume fraction.Since volume fraction is representative of the number of NP-RBC particles in blood, the increase in its value causes the viscous drag to increase because of an increase in Brownian motion, which decreases the net movement.Thus, the velocity decreases.Kumar et al. [9] also came up with the same conclusions.Hence, volume fraction physically regulates the drug dosage variability, working on the type of ailment.The profile of velocity u versus radial direction r with respect to radius r p of the NP-RBC complex (Figure 7) shows that the velocity declines for an increase in radius.With the increase in radius, the interparticle distance decreases that results in increasing the random Brownian motion.Thus, the flow slows down.Wang et al. [22] also probed into similar kinetic details, experimentally.
The profile of velocity u versus radial direction r with respect to thickness d nl of nanolayer (Figure 8) shows a decline in velocity for greater thickness.Similar reason as earlier can be attributed to the behaviour of velocity.Identical results were reported by Ajdnik et al. [1] to develop nanomedicines.Also, Brenner et al. [5] proclaimed the usefulness of nanolayer and NP-RBC complex with alike analysis.
The trend in velocity u against radial direction r with respect to depth δ (Figure 9) shows an increment in velocity with greater stenosis depth.Greater stenosis depth means lesser cross-sectional area of the artery available for blood flow.Thus, the velocity rises in a smaller cross-sectional area owing to Bernoulli's principle.Xia et al. [23] also established that the shape of tumour or stenosis is important to design the characteristics of the NP-RBC complex.The trend in velocity u against radial direction r with respect to the Darcy number (Figure 10) shows that velocity rises for increased Darcy number Da.The Darcy number quantifies the effect of medium's permeability against the area of flow.The rise in the Darcy number enhances the convections on the medium due to higher permeability.Thus, the velocity rises.and Shaw [13] also showed that the Darcy number boosted the drug targeting efficiency at the diseased site.The Darcy number controls the amount of drug transferred and absorbed by the diseased cells, thus eventually help in developing targeted theranostic.

Conclusion
Drug delivery via NPs has aroused as the latest and highly effective therapeutic technique in the treatment of cardiovascular diseases.The nanodrugs are targeted at the diseased site and accumulated there.This increases their retention at the site, reduces dosage, and also side effects.However, the rapid clearance of nanodrugs because of phagocytosis reduces the effectiveness of the drug.
Blood cells have provided a new avenue for drug delivery.RBCs occur in the highest number in blood; thus, they provide a suitable platform for drug loading.Loading NPs on blood cells such as RBCs increases the circulation time of nanodrugs.These benefits have leveraged the use of NP-RBC complex in delivery of drug for the treatment of cardiovascular diseases such as atherosclerosis.
In the current mathematical study, Fe O 3 4 NP-RBC complex is coated on the catheter that is inserted into the lumen of the stenosed artery.The results have been found for temperature and velocity of NP-RBC.The findings can be summarized as: (1) The temperature of the NP-RBC complex in blood increases for an increase in volume fraction ϕ of the NP- RBC complex, radius r p of the NP-RBC complex, thickness of nanolayer d nl , and heat source parameter h.(2) The velocity of the NP-RBC complex in blood decreases for an increase in volume fraction ϕ of the NP-RBC complex, radius r p of the NP-RBC complex, and thickness of nanolayer d nl .
(3) The velocity of the NP-RBC complex in blood increases for an increase in values of stenosis depth δ and the Darcy number Da.
The NP-RBC-derived complex has been found to be beneficial in many respects such as immune-compatibility and controlled drug release.Certain aspects such as removal of these complexes and cost-effective production need to be still explored and taken care of.Addressing these challenges holds great potential for nanodrug delivery using RBCs.

Figure 2 :
Figure 2: Behaviour of temperature θ of NP-RBC complex in blood against radial direction r with respect to volume fraction Θ.

Figure 3 :
Figure 3: Behaviour of temperature θ of the NP-RBC complex in blood against radial direction r with respect to radius of NP-RBC complex rp.

Figure 5 :
Figure 5: Behaviour of temperature θ of the NP-RBC complex in blood against radial direction r with respect to heat source parameter h.

Figure 4 :
Figure 4: Behaviour of temperature θ of the NP-RBC complex in blood against radial direction r with respect to thickness of nano- layer dnl.

Figure 6 :
Figure 6: Behaviour of velocity u of the NP-RBC complex in blood against radial direction r with respect to volume fraction Θ.

Figure 7 :
Figure7: Behaviour of velocity u of the NP-RBC complex in blood against radial direction r with respect to radius of NP-RBC complex rp.

Figure 8 :
Figure 8: Behaviour of velocity u of the NP-RBC complex in blood against radial direction r with respect to thickness of nanolayer dnl.

Figure 9 :
Figure 9: Behaviour of velocity u of the NP-RBC complex in blood against radial direction r with respect to stenosis depth δ.

Figure 10 :
Figure 10: Behaviour of velocity u of the NP-RBC complex in blood against radial direction with respect to the Darcy number Da.

Table 1 :
The values of thermophysical quantities of blood and nanoparticle used for calculations