Optimization of Deformable Magnetic-Sensitive Hydrogel-Based

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Optimization of Deformable Magnetic-Sensitive Hydrogel-Based Targeting System in Suspension Fluid for Site-Specific Drug Delivery Qimin Liu, Hua Li, and K. Y. Lam Mol. Pharmaceutics, Just Accepted Manuscript • DOI: 10.1021/acs.molpharmaceut.8b00626 • Publication Date (Web): 22 Aug 2018 Downloaded from http://pubs.acs.org on August 29, 2018

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Molecular Pharmaceutics

Optimization of Deformable Magnetic-Sensitive Hydrogel-Based Targeting System in Suspension Fluid for Site-Specific Drug Delivery Qimin Liu, Hua Li* and K. Y. Lam School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Republic of Singapore ABSTRACT: For optimization of the targeting performance of the magnetic hydrogel subject to the magneto-chemo-hydro-mechanical coupled stimuli, a multiphysics model for suspension fluid flow in blood vessel is developed, in which a deformable magneticsensitive hydrogel-based drug targeting system moves with fluid. In this model, the fluidstructure interaction of the movable and deformable magnetic hydrogel with surrounding fluid flow is characterized through the fully coupled arbitrary Lagrangian-Eulerian (ALE) algorithm. Moreover, the four physicochemical responsive mechanisms are considered, including hydrogel magnetization, solvent diffusion, fluid flow, and nonlinear large deformation. After the present model is examined by the experimental data in open literature, the transient behaviors of the motion and deformation of the magnetic hydrogel are investigated in suspension flow. It is found that the higher flow velocity and/or the larger hydrogel size accelerate the movement of the hydrogel, while the smaller hydrogel size contributes to the larger swelling ratio. Furthermore, the performance of the magnetic targeting system is optimized for delivering the drug-loaded hydrogel to the desired site by tuning the maximum magnetic field strength, the maximum inlet flow velocity, and the magnet position. Therefore, it is confirmed that the present optimizable magnetic hydrogel-based drug targeting system via the multiphysics model may provide a promising efficient platform for site-specific drug delivery. Keywords: Magnetic-sensitive hydrogel; site-specific drug delivery; multiphysics model

*

Corresponding author, Tel.: + 65 6790 4953; Fax: +65 6792 4062 Email address: [email protected] (Hua Li)

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1. INTRODUCTION The development of effective treatment for diseases is a major goal of the human.1 In recent decades, a great range of drugs are developed to fight against infectious disease, inflammatory illness, and systematic disorder or damage in the human body, while the treatments for several diseases including cancers still remain challenging, due to the low tumor targeting efficiency and the severe adverse effects on healthy tissue.2 Therefore, much effort is made to develop drug delivery system for overcoming the drawbacks, including active and passive drug targeting technologies. Among these, the magnetic drug targeting may be the most popular one, since the magnetic field does not require special environmental properties such as conductivity or transparency, and it is biocompatible to the human body even at relatively high field strength.3 Furthermore, it enables to manipulate the magnetic objects toward the specific organ or tissues of the body,4-6 resulting in increased drug concentration at the target tissue with low systematic toxicity. In general, the magnetic drug carrier is developed by adding highly permeable magnetic nanoparticles into a non-magnetic matrix,1, 7, 8 where usually the hydrophilic polymer is adopted,1, 8-10 due to its ability to increase the water solubility of hydrophobic drugs, extend the circulation of drugs in the blood, and suppress or eliminate fast renal excretion. The resulting particle-based composite is often referred to as magneticsensitive hydrogel. Subject to an external magnetic field, the drug-loaded magnetic hydrogel can be guided to the region of interest and simultaneously undergoes large and reversible deformation, making it very suitable as drug carrier in biomedical diagnose and therapy.


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A literature review reveals that various experimental works were carried out to characterize the performance of the magnetic-sensitive hydrogel-based drug targeting system. Chen et al.11 synthesized the magnetic polyvinyl pyrrolidone hydrogel microsphere loaded with the chemotherapeutic drug and then the microsphere was guided to the tumor site by an external magnet. After two weeks, the drugs were mostly accumulated at the tumor site and the tumor size was reduced significantly. Effective magnetic drug targeting was also demonstrated by Sun et al.8 when they injected magnetic polyacrylamide hydrogels into rabbits, and found that they were mainly retained at the right back leg of the rabbit, where the permanent magnet was positioned. In order to improve the antitumor efficacy of intravesical Bacillus Calmette-Guérin, the drug-loaded magnetic chitosan hydrogel was prepared by Zhang et al.,12 and it was attracted and attached to the bladder wall by an external magnetic field. After two days, the tumor number decreased about five times compared with that of the traditional therapy. Recently the multi-stimuli-responsive hydrogels were developed for efficient synergistic therapy, such as the simultaneous drug targeting and release, where the magnetic nanoparticles were generally embedded for site-specific targeting, and the pH,7, 13 temperature-,14, 15 or light-sensitive 16 functional groups incorporated for drug release at a desired rate.

Despite many experimental works on magnetic hydrogel-based drug targeting, a literature search shows that no effort was made in theoretical modeling for magneticsensitive hydrogel in suspension fluid. From the open literature, it is known that most of the reported models focused on movable magnetic particle-based drug targeting system where the particles are rigid without solvent diffusion. The particle response is described 3 ACS Paragon Plus Environment


by the mechanical equilibrium between the magnetic and viscous forces,17-19 where the former was associated with the gradient of the magnetic field, and the latter related to the fluid flow. However, the model developed in this paper focuses on the movable and deformable magnetic hydrogel-based targeting system, in which the hydrogel imbibes large amount of fluid due to its hydrophilic nature, and its deformation is enhanced, due to the driving forces from magnetic and hydrodynamic fields. Meanwhile, the hydrogel can move towards the specific site with the moving fluid, due to the magnetic field gradient. As the interface between hydrogel and surrounding fluid is dynamically altered during the hydrogel motion and deformation, the flow velocity, pressure, and the magnetic field are redistributed, which in turn change the motion and deformation of the hydrogel. Therefore, it is a full-loop magneto-chemo-hydro-mechanical coupled system, and this work thus develops a multiphysics model to investigate the performance of the drug targeting platform by elucidating the relationship of the targeting efficiency (e.g. targeting time and accuracy) with various conditions, such as hydrogel size and location, inlet flow velocity, maximum magnetic intensity, and magnet position.

In this work, the responsive behavior of the magnetic-sensitive hydrogel-based drug targeting system is investigated via a multiphysics model, in which the fluid-structure interaction is characterized between the movable and deformable magnetic hydrogel and moving fluid by the fully coupled arbitrary Lagrangian-Eulerian (ALE) algorithm. In the present model, the four physiochemical processes are integrated: (i) the magnetic field distribution by the Maxwell’s equation, (ii) the fluid diffusion by the conservation of mass, (iii) the fluid flows by the Navier-Stokes equation, and (iv) the nonlinear large deformation of the hydrogel by the conservation of momentum. The model developed is 4 ACS Paragon Plus Environment

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examined by the experimental data in open literature. After validation, various transient simulations are carried out to characterize and optimize the performance of the magnetic hydrogel-based drug targeting system, such as the hydrogel motion trajectory, the hydrogel deformation, as well as the magnetic field distribution, under the magnetochemo-hydro-mechanical coupled fields.

2. METHODS In the absence of free current, the magnetostatic field is governed by the Maxwell’s equations as 20

∇X ⋅ B = 0

∇X × H = 0 ,

(1)

where ∇ X × (•) and ∇X ⋅ (•) are the material curl and divergence operators with respect to Lagrangian coordinate X respectively. Vectors B and H are the magnetic induction and intensity respectively with respect to X , and they are connected via 20 B = µ 0 J C −1 ( H + M )

(2)

where M is the magnetization vector, µ0 the magnetic permeability at vacuum, C = F T F the right Cauchy-Green tensor, F = ∇ X x the deformation gradient, J = det(F) the determinant with the standard convention J > 0 .

At the surface of discontinuity, namely the interface between the magnetic hydrogel and fluid medium, the magnetic field is varied on the basis of the jump conditions 20



N ⋅ [[ B ]] = 0 ,

N × [[ H ]] = 0

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(3)

where N is the Lagrangian unit outward normal, and the double square bracket represents a quantity jump across the surface from the inside to outside of the material.

For the nonlinear large deformation of the magnetic hydrogel, the conservation of the momentum 21 is employed as

ρ 0 V& h = ∇ X ⋅ P + f b

(4)

where ρ0 is the nominal mass density of the polymer network, V h the deformation velocity, P the first Piola-Kirchhoff stress, and fb the external force density. Over the hydrogel-fluid interface, the stress P is required to satisfy the mechanical boundary condition 20 P ⋅ N = Ta + Tm

(5)

where Ta and Tm are the mechanical and magnetic tractions respectively, and Tm = Pm ⋅ N , where the Lagrangian Maxwell stress Pm is defined as 22

1 Pm = F −T ( H ⊗ B ) − ( H ⋅ B)F −T 2

(6)

Let C s ( X, t ) denote the number of solvent molecules per unit reference volume absorbed by the magnetic hydrogel, J s the number of solvent molecules per unit reference area entering into the hydrogel across the interface, Rs the number of solvent


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generation per unit reference volume per unit time, the chemical field with respect to the fluid diffusion is written in the local form of 23 C& s = −∇ X ⋅ J s + Rs

(7)

Over the hydrogel-fluid interface, the chemical boundary condition is given by 24 J s ⋅ N = −i

(8)

where i is the surface flux rate.

To describe the flow characteristics, the incompressible Navier-Stokes equations are employed, namely 25

∇x ⋅ v s = 0

ρs

∂v s + ρs (v s ⋅ ∇x )v s = ∇x ⋅ σ s + ρsg ∂t

(9)

(10)

where ∇ x (•) denotes the spatial gradient operator in Eulerian coordinate x . ρ s , vs , and g represent the fluid mass density, velocity, and gravitational acceleration respectively.

The fluid stress tensor σ s is defined as σ s = − pI + ϕ [∇ x v s + (∇ x v s )T ] ,25 where p and ϕ are the fluid pressure and dynamic viscosity respectively.

Over the hydrogel-fluid interface, the kinetic and dynamic conditions are given below 25 v s = u& ,

n ⋅ σ = n ⋅ (σ s + σ m ) 7

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(11)


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where u is the hydrogel displacement, σ the Cauchy stress of the hydrogel associated with the first Piola-Kirchhoff stress P via σ = J −1PFT , and σ m the Maxwell stress tensor in Eulerian form and associated with Pm by σ m = J −1Pm F T . According to thermodynamics,26, 27 it is required for the energy imbalance that the temporal increase in the free energy of the magnetic hydrogel is less than or equal to the external power plus the free energy due to fluid transport and magnetization, namely

d 1 2 { ∫ ( F + ρ 0 Vh )dv} ≤ ∫ PN ⋅ Vh ds + ∫ f b ⋅ Vh dv − ∫ µ s J s ⋅ Nds + ∫ µ s Rs dv dt V0 2 S0 V0 S0 V0 − ∫ [−(M ⋅ B ⋅ F ) : F& + (F −T ⋅ M ⊗ B) : F& + M ⋅ B& ]dv −T

(12)

V0

where µs is the chemical potential and F the free energy density.

By Equations (4) and (7), and the divergence theorem, the free energy imbalance (12) is rewritten in the local form of F& ≤ P : F& + ( µ s C& s − J s ⋅ ∇ X µ s ) − [− (M ⋅ B ⋅ F −T ) : F& + (F −T ⋅ M ⊗ B) : F& + M ⋅ B& ] (13)

If the magnetic induction B , solvent concentration Cs , and deformation gradient F are the independent variables, namely F = F (B, Cs , F) , we have

(P + M ⋅ B ⋅ F −T − F −T ⋅ M ⊗ B −

∂F & ∂F ∂F & ):F −( + M) ⋅ B& + ( µ s − )Cs − J s ⋅ ∇ X µ s ≥ 0 (14) ∂F ∂B ∂Cs


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The coefficient in any bracket vanishes, as a result of the arbitrariness of the variables.28 Moreover, to ensure the inequality held, the solvent flux J s is associated with the chemical potential gradient ∇ X µ s via 24 J s = − L∇ X µ s

(15)

where the mobility L is a positive-definite tensor and associated with the diffusion coefficient Ds and temperature in the unit of energy k BT via L = C s Ds C −1 /( k BT ) . Therefore, we have the following constitutive equations

P=

∂F ∂F ∂F − M ⋅ B ⋅ F −T + F −T ⋅ M ⊗ B, M = − , µs = ∂F ∂B ∂Cs

(16)

If the magnetic intensity H is employed as the independent variable, the augmented energy of F with partial Legendre transformation is required by 27 Ω( F, H, Cs ) = F ( F, B, C s ) + (C : B ⊗ B) /( 2 µ 0 J ) − H ⋅ B

(17)

such that the constitutive equation (16) is rewritten as

P=

∂Ω ∂Ω ∂Ω , B=− , µs = ∂F ∂H ∂Cs

(18)

The free energy density of the present magnetic hydrogel Ω is required to predict the hydrogel response. In many hydrogels, the density of the crosslinks is so low that the effect of crosslinks on mixing between the polymeric chain and the solvent molecules may be neglected.29,

30

Furthermore, if the magnetic hydrogel exhibits the liquid-like 9 ACS Paragon Plus Environment


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magnetization behavior,31 the free energy of the hydrogel system is assumed to be separated into contributions from stretching the network Ω ela , mixing the polymer and solvent molecules Ω mix , and the hydrogel magnetization Ω mag . In the present work, if the molecular incompressibility condition 1 + vs C s = det(F )

24

is considered, the free energy

density Ω is given by 32-34

Ω=

Jµ 1 G{tr (F T F ) − 3 − 2 ln[det(F )]} − m [(F −T H ) ⋅ (F −T H )] 2 2 χ H vs C s vs Cs k BT + [v s C s ln( )+ ] + Π[1 + v s C s − det(F )] vs 1 + vs Cs 1 + vs C s

(19)

where G is the hydrogel shear modulus, χ H the Flory-Huggins parameter to describe the interaction between the solvent and polymer networks, and vs the volume per solvent species, µ m the magnetic permeability and relates to the relative permeability µm r via

µm = µ0 µmr , and Π the Lagrange multiplier and also understood as the osmotic pressure.24, 35

From the free energy function (19) and the constitutive equation (18), the stress P , the magnetic induction B , and the chemical potential µs are expressed respectively as

1 P = G (F − F −T ) − JΠF −T + Jµ m F −T (H ⊗ H)C −1 − Jµ m F −T [(C −1H ) ⋅ H ] 2

(20)

B = Jµ m C − 1 H

(21)

µ s = k BT [ln(1 −

χH 1 1 )+ + ] + Πvs 1 + vs C s 1 + vs C s (1 + vs Cs ) 2 10


(22)

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Based on Equation (1), the magnetic intensity vector H relates to the magnetic scalar potential ψ by H = −∇ Xψ . By Equations (1) and (21), the magnetic potential ψ in both the magnetic hydrogel and the exterior fluid satisfies ∇ X ⋅ ( Jµ m C −1∇ Xψ ) = 0

(23)

So far the multiphysics model has been completed theoretically for the magnetic hydrogel moving with fluid, under the magnetic-chemo-hydro-mechanical coupled fields. It consists of the magnetostatic equation (23), the mechanical balance equation (4), the mass conservation equation (7), as well as the incompressible Navier-Stokes equations (9) and (10). In addition, the free energy density is given by Equation (19), and the constitutive relations given by Equations (15), (20), (21), and (22). The present model is implemented by the commercial finite element package, COMSOL Multiphysics 5.3. It is known that the hydrogel deformation is conveniently formulated by a Lagrangian description and a material frame, whereas the fluid motion formulated by an Eulerian description and a spatial frame. To connect the interface between the material frame of the hydrogel and spatial frame of the fluid, the fully coupled ALE method is employed for the fluid-structure interaction, which calculates the new mesh coordinates according to the movement of the hydrogel boundary and mesh smoothing of the fluid.36 Furthermore, the function of automatic remeshing is activated to ensure the calculation accuracy, when the mesh quality is below the critical value (e.g. 0.25).



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3. RESULTS AND DISCUSSION 3.1.

Validation of the Multiphysics Model

In order to examine the model, a comparison of theoretical simulation with the experimental data incorporation

of

37

is made, where the spherical magnetic droplet was synthesized by the

magnetite

nanoparticles

into

a

carboxylate-functional

polydimethylsiloxane (PDMS), and its motion through a highly viscous medium was recorded when subjected to a magnetic field gradient by a permanent magnet, as shown in Figure 1(a). In the validation, the magnetic boundary condition for the nonuniform magnetic field is reconstructed from the experimental measurement.37 In the absence of the magnetic hydrogel, the magnetic field H (0, Z ) was measured as a function of the distance Z from the magnet, which is fitted by a fifth-degree polynomial, H = −2.62 × 1010 Z 5 + 2.56 × 10 9 Z 4 − 9.71× 10 7 Z 3 + 1.8 × 10 6 Z 2 − 1.7 × 10 4 Z + 69.26 , as shown in Figure 1(b). Based on the relation H = −∇ Xψ , the magnetic scalar potential ψ is given by a sixth-degree polynomial ψ (0, Z ) = p6 ( Z ) along the axial direction. Because the magnetic scalar potential ψ satisfies the Laplace’s equation ∇ X 2ψ = 0 in the absence of magnetic droplet, the analytical solution of the magnetic potential ψ is obtained as

ψ ( R, Z ) = p6 ( Z ) − R 2 p6( 2 ) ( Z ) / 4 + R 4 p 6( 4 ) ( Z ) / 64 − R 6 p6( 6 ) ( Z ) / 2304 ,38 which is then imposed on the fluid domain boundary ∂S0 . The input parameters required by the present model are tabulated in Table 1. Table 2 shows the comparisons of the numerical and experimental travel times for droplets with various diameters of 1, 1.8, and 2 mm and positioned at distances of 12, 12, and 11 mm respectively, where good agreements are achieved between the present simulation and experimental data.37 As such, it is concluded 12 ACS Paragon Plus Environment

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that the multiphysics model can capture well the response of the magnetic hydrogel. It is also observed that the relative error of theoretical result by Mefford et al. exceeds about 50%, probably the simulation regarded the droplet as a point dipole without consideration of the deformation coupled with its motion towards the magnet. However, the droplet size and the distance between the droplet and the magnet may have a significant effect on the transit time, which will be further discussed in the following sections by the validated model.

3.2.

Performance of the Magnetic Hydrogel Subject to Varying Hydrogel Size,

Location, and Inlet Flow Velocity

To provide an insight into the effects of hydrogel and suspending fluid on the targeting performance in the blood vessel, it is worthwhile to characterize the distributive magnetic field, the hydrogel deformation, and the transit time for the hydrogel to the targeting area. Herein, the motion and deformation of the magnetic hydrogel are investigated with the coupled effects of magneto-chemo-hydro-mechanical fields. To the best of our knowledge, it is the first attempt to study the fluid-structure interaction of the movable and deformable hydrogel with the moving fluid under a nonuniform magnetic field.

As illustrated in Figure 2(a), a two-dimensional model is employed for the numerical simulation, since the variation along the direction perpendicular to the X-Y plane is assumed to be rather small during the transport of magnetic hydrogel.39 The magnetic hydrogel is submerged in the biological fluid that is delivered to the blood vessel with a parabolic flow profile. The permanent magnet is located vertically at the coordinate ( L X , 0.5 LY ) , where the magnitude of field H maximizes at the surface of the magnet,



and decays exponentially away along the X-axis direction, namely H = H max e −0.2 ( X − LX ) . Initially， the hydrogel is at its dry state, it then swells gradually and at the same time moves to the desired site, due to the magnetic and hydrodynamic effects. The inputs of the parameters required for the numerical simulation include: L X = 20 mm , LY = 3 mm , Ds = 2 × 10 −7 m 2 /s , χ H = 0.4 , ρ s = 1000 kg/m 3 , ϕ = 0.001 Pa ⋅ s , µ mr = 1.06 ，

H max = 20 kA/m , k BT = 4.11 × 10 −21 J , and G = 105 Pa .

Figure 2(b) is plotted to illustrate the time evolution of the motion and deformation of the three hydrogels with varying radii, 0.1 mm, 0.15 mm, and 0.2 mm, respectively, where the maximum inlet flow velocity vmax = 0.1 mm/s . The magnetic hydrogel moves fast with increasing radius, which is qualitatively consistent with the relevant experimental findings

40

and theoretical results

19, 39

for the magnetic particle-based drug

delivery system. It is because that the magnetic force is highly associated with the hydrogel radius,19,

41

where a larger hydrogel size results in stronger magnetic force

acting on the hydrogel and thus the shorter time to reach the site of interest. It is also found from the figure that, no significant change is found in the movement of the three hydrogels when the time t is less than 10 s, while their movements vary drastically if t is larger than 20 s.

The motion behavior of the hydrogel is demonstrated more clearly in Figure 2(c), which shows the influence of the hydrogel radius R0 on variation of the displacement u of the hydrogel center point with time. It reveals that: (i) initially the displacement u increases linearly until it reaches 7 mm; (ii) it then increases nonlinearly and sharply.


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This is because the magnetic field decays exponentially away from the permanent magnet, where it exhibits the highest field strength at the surface of the magnet. At the initial time step, the magnetic hydrogel is far away from the magnet, such that the magnetic effect is rather weak and the hydrogel movement is mainly determined by the fluid drag. When the hydrogel moves further to approach the magnet, the impact of the magnetic field increases gradually and then the hydrogel is accelerated towards the magnet. It is further found from Figure 2(b) that, the hydrogel swells during its motion, due to the diffusion of the fluid. In order to elucidate the swelling phenomenon, the time evolutions of the average volume swelling ratios are demonstrated in Figure 2(d), where the three curves have the similar profile patterns. The hydrogels swell sharply at the early stage of swelling, and then its rate decreases smoothly, until an equilibrium state is reached. Theoretically, the swelling of the hydrogel is highly associated with the chemical driving force, i.e. the gradient of the chemical potential, which decreases if more and more fluid enters into the hydrogel.42 It is noted that the transient swelling profile is consistent with experiment result by Zhuang et al.,43 which further verifies the capability of the developed model in predicting the hydrogel response. It is also found from the figure that the response rate of swelling is inversely proportional to the hydrogel radius, where the hydrogel with the smaller radius swells faster than that with the larger one. For example, the average swelling ratio increases from about 5.1 to 5.9 or 5.9 to 6.8 when the radius R0 decreases from 0.2 to 0.15 or 0.15 to 0.1 mm respectively, when t =10 s. Because the smaller hydrogel radius corresponds to a larger surface-to-volume ratio. When hydrogels are immersed in fluid, the hydrogel with smaller radius (i.e. higher surface-to-volume ratio) exhibits a faster stimuli response, due to the higher surface-to-volume ratio (i.e.



smaller radius) leads to more contact with the surrounding solvent, as experimentally reported by Czugala et al.44 Furthermore, Figure 2(d) also illustrates the contours for distribution of swelling ratio of the magnetic hydrogel with R0 = 0.1 mm . The swelling ratio is distributed homogeneously at the dry state when t = 0 . Immediately upon beginning of the swelling (e.g. t = 2 s ), the hydrogel consists of regions with different degree of swelling, namely the outside surface swells much faster than that at the center of the hydrogel. As the swelling process approximates the equilibrium stage (e.g. t = 32 s ), the swelling of the hydrogel tends to be homogeneous again. The interesting

transient phenomenon observed during the hydrogel swelling probably results from the relatively long diffusion time from the outside boundary to the inner core.42

It is commonly known that the fluid flow velocity has a pronounced impact on the motion and deformation of the magnetic hydrogel, because there is a direct relationship between the shear stress and velocity field. To clarify this effect, the range of the maximum inlet flow velocity vmax is taken within 0.05~0.5 mm/s. The motion and deformation of the hydrogel are studied with the three maximum inlet velocities vmax , 0.05 mm/s, 0.1 mm/s, and 0.5 mm/s, as shown in Figure 3(a). With increasing maximum inlet velocity vmax , the time decreases significantly from 31.1s to 25.1 s and to 11.9 s for the hydrogel moving from the initial position to the outlet. Because the shear stress is highly associated with the flow velocity, the hydrogel undergoes larger driving force from the hydrodynamic effect with increasing velocity vmax . Therefore, the hydrodynamic effect is also required to be considered in the design and optimization of the magnetic drug targeting system. Figure 3(b) demonstrates influence of the maximum 16 ACS Paragon Plus Environment

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inlet velocity vmax on the trajectory of the hydrogel center point with time. It is seen that the movement speed increases with increasing velocity vmax . When the displacement u reaches about 7 mm, the three hydrogels are suddenly accelerated towards the magnet, where the large magnetic field gradient occurs. To describe the transient swelling of the hydrogels during the hydrogel motion, Figure 3(c) is plotted for the time evolution of the average swelling ratio subject to different velocities vmax . It is interesting that no noticeable change is found for the swelling process, probably due to the significant timescale difference between the short-time mechanical deformation by the hydrodynamic and magnetic fields and the long-time solvent migration.28, 45

Figure 4 is plotted for time evolution of the motion and deformation of the magnetic hydrogels located at three initial coordinates, (1 mm, -0.625 mm), (1 mm, 0), and (1mm, 0.625 mm), respectively, where vmax = 0.1 mm/s and R0 = 0.2 mm . It is seen that the time t of 25.1 s is required for the hydrogel located at the centerline to reach the end, while 25.7 s required for those at other positions. Because the profile of inlet flow velocity is parabolic, the shear stress is inverse to the distance from the hydrogel to the centerline of the vessel. It is interesting that all the three hydrogels may eventually be immobilized at the center of the outlet, due to the strong attracting force by the high magnetic field gradient close to the magnet surface.

As well known, the distribution of the magnetic field has a considerable influence on the behavior of motion and deformation of the magnetic hydrogel. Herein the spatial distribution of the non-dimensional magnetic intensity H / H max at different times, 20, 25, and 25.1 s, along the horizontal line Y = 0 is visualized in Figure 5(a) to (c) 17 ACS Paragon Plus Environment


respectively, where the dash and solid lines denote the original magnetic field strength and the magnetic intensity in the presence of magnetic hydrogel respectively. The nondimensional magnetic intensity H / H max increases with increasing non-dimensional coordinate X / L X , it then increases to a maximum at the hydrogel-fluid interface, where the intensity H / H max follows a step function with a sharp decrease at the interface within a small range of the normalized coordinate X / L X . Subsequently, it rises up to another maximum until it reaches the interface again, where it increases drastically. After that, the normalized intensity H / H max increases linearly with increasing X / LX . As also observed from the figures, the magnetic intensity H / H max is discontinuous across the hydrogel-solution interface. We may understand the phenomena above by the following two reasons: (i) the different magnetic permeabilities between the hydrogel and the surrounding fluid, and (ii) the magnetic boundary condition at the interface (3), which shows that the normal component of magnetic induction B and the tangential component of the magnetic intensity H are continuous across the material boundary, while the tangential component of magnetic induction B and the normal component of the magnetic intensity H are discontinuous across the material boundary.28 Furthermore, the figures also show that no observable change is found between the magnetic field applied and magnetic intensity in the presence of magnetic hydrogel, when the coordinate X / L X is far away from the hydrogel. However, they deviate from each other significantly when the coordinate X / L X approaches the hydrogel-fluid interface, due to the prominent edge effect in the vicinity of the magnetic hydrogel.46


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3.3.

Optimization of the Targeting Performance of the Magnetic Hydrogel Subject to

Varying Inlet Flow Velocity, Magnetic Intensity, and Magnet Position

For manipulation of the drug-loaded magnetic hydrogel in a remote manner, the external magnet may be required to vary its field strength and location in real time.14, 47 Herein, the performance of the magnetic hydrogel is further studied under varying inlet velocity, magnetic field strength, and the field source position, in which the field strength is associated with the maximum magnetic intensity H max , and the field source with the magnet position horizontally located at the coordinate（LD , 0）, as illustrated in Figure 6(a). In the present simulation, the height of the vessel LY = 8 mm , the total height with the tissue LH = 10 mm , the other inputs of the parameters follow those in section 3.2.

Figure 6(b) shows the trajectories of the center point of magnetic hydrogels originally located at the coordinate (1 mm, 8 mm) , when subjected to different maximum magnetic intensity H max , 2, 4, 6, 8, and 10 kA/m respectively, where vmax = 5 mm/s , R0 = 0.2 mm , and LD = 10 mm . It is seen that the hydrogel moves out of the vessel if the maximum magnetic intensity H max of 2 and 4 kA/m are imposed. Because the maximum fields H max of 2 and 4 kA/m are relatively weak, the motion of the hydrogel is mainly determined by the fluid stress. However, with the increase of the magnetic intensity H max , the magnetic force gradually becomes the main driving source and thus the hydrogels tend to retain at the bottom of the vessel. It is also found that the hydrogel can be attracted to approach the magnet if H max = 10 kA/m , meaning that the hydrogel-based drug delivery system can be optimized and manipulated to the desired location by control 19 ACS Paragon Plus Environment


of the external magnetic field strength. Moreover, as observed from the inset in Figure 6(b), initially the hydrogel moves up slightly, and then it rises to a maximum, after which it goes down progressively. Probably the profile of the inlet flow velocity is parabolic and the initial position of the hydrogel deviates far away from the magnet, such that the hydrogels tend to move up with the hydrodynamic field at the early stage. As they move further and approach the magnet, the magnetic force increases large enough to pull them down.

In order to investigate the hydrodynamic effect, due to the maximum inlet velocity

vmax , on the design and optimization of the magnetic hydrogel-based targeting system, Figures 6(b) and (c) are plotted to demonstrate the trajectories of the center points of magnetic hydrogels at different maximum inlet velocities. It is observed that the hydrogel moves in a relatively higher position along the Y-axis with the increase of the inlet velocity vmax . Because the motion of the hydrogel is associated with the fluid shear stress

σ s , based on the boundary condition at the hydrogel-fluid interface (11). By increasing the maximum inlet velocity vmax , the fluid shear stress σ s enlarges and σ s tends to push the hydrogel towards the outlet, which results in a relatively higher position for the hydrogel at a larger velocity vmax . In addition, it is also observed that the optimized magnetic intensity H max for the hydrogel to reach the desired site (i.e. magnet position) is changed when subjected to different velocities vmax , where a higher magnetic intensity H max , e.g. from H max = 10 to 17 kA/m, is required if the velocity vmax of the inlet fluid


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flow increases from vmax = 5 to 8 mm/s, because the greater magnetic force is required to resist the force due to the faster fluid flow.

Figures 7(a) and (b) are illustrated for the influence of the maximum magnetic field

H max on trajectories of the center point of the magnetic hydrogels subject to different magnet positions located at LD = 5 and 15 mm respectively, where vmax = 5 mm/s . The hydrogel tends to move toward the magnet, which demonstrates that the magnet position influences the hydrogel trajectories significantly. For example, the hydrogel travels about 4.2 mm along the horizontal direction when the magnet moves from the coordinate (5 mm, 0) to (1 5 mm, 0) and H max = 6 kA/m . This finding is in line with experimental

result qualitatively,47 in which the magnetic hydrogel traveled in buffer solution following the magnetic guidance. This unique magnetic-guided directed motion is of great potential in numerous applications, such as cartilage defect therapy,47 in vitro cell culture,48 and microrobotic platform for therapeutic intervention.16 Moreover, it is found from the displacement contours that, the hydrogel morphology is varied from the initial circular shape to an irregular one, probably due to the inhomogeneous magnetic force and the fluid stress. For optimization of the performance of the present magnetic hydrogel for site-specific drug delivery, the magnet position is utilized as the targeting site, i.e. optimal objective. It is found that the drug-loaded hydrogels can eventually reach the targeting site via the optimization of the maximum magnetic field H max . When the magnet is located at the coordinate (5 mm, 0), the optimized H max is attained about 24 kA/m, while it is about 7.5 kA/m for magnet at the coordinate (15 mm, 0). Therefore, it is concluded that the magnetic-sensitive hydrogel-based drug targeting system can provide 21 ACS Paragon Plus Environment


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a noncontact and noninvasive platform to realize the site-specific drug delivery with desired accuracy.

4. CONCLUSION In this work, a magnetic-sensitive hydrogel-based drug targeting system has been studied via a multiphysics model for optimization of its targeting performance, with consideration

of

magneto-chemo-hydro-mechanical

coupled

fields.

The

model

characterizes the fluid-structure interaction of the deformable magnetic hydrogel moving with surrounding fluid flow through the fully coupled ALE algorithm. It can capture well the response of the magnetic hydrogel, including magnetic field distribution, fluid diffusion, fluid flow, and nonlinear large deformation of the hydrogel. After validated with the experimental data in open literature, the present model is further employed to predict the motion and deformation of the hydrogel. The transient results show that the higher flow velocity and the larger hydrogel size accelerate the movement of the hydrogel, and the smaller hydrogel size contributes to the larger hydrogel swelling degree. Furthermore, the magnetic targeting system is optimized for site-specific drug delivery by tuning the maximum magnetic field strength, the maximum inlet flow velocity, and the magnet position. Hence, the multiphysics model is a useful tool: (1) to elucidate the correlation of the targeting efficiency (e.g. targeting time and accuracy) with the hydrogel size and position, and (2) to get a great understanding on the responsive behavior under varying inlet velocity, magnetic field strength, and magnet position. Therefore, the magnetic-sensitive hydrogel-based drug targeting system may provide an efficient platform for site-specific drug delivery.


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ACKNOWLEDGMENT The authors gratefully acknowledge the financial support from Nanyang Technological University through the project (No: M4081151.050) and NTU Research Scholarships.

CONFLICT OF INTEREST The authors declare no competing financial interest.



REFERENCES

(1). Ulbrich, K.; Holá, K.; Šubr, V.; Bakandritsos, A.; Tuček, J.; Zbořil, R., Targeted drug delivery with polymers and magnetic nanoparticles: Covalent and noncovalent approaches, release control, and clinical studies. Chem. Rev. 2016, 116, 5338-5431. (2). Thevenot, J.; Oliveira, H.; Sandre, O.; Lecommandoux, S., Magnetic responsive polymer composite materials. Chem. Soc. Rev. 2013, 42, 7099-7116. (3). Gao, W.; Wang, L.; Wang, X.; Liu, H., Magnetic driving flowerlike soft platform: Biomimetic fabrication and external regulation. ACS Appl. Mater. Interfaces 2016, 8, 14182-14189. (4). Janikowska, A.; Matuszak, J.; Lyer, S.; Schreiber, E.; Unterweger, H.; Zaloga, J.; Groll, J.; Alexiou, C.; Cicha, I., A novel human artery model to assess the magnetic accumulation of spions under flow conditions. Sci. Rep. 2017, 7, 42314. (5). Licciardi, M.; Scialabba, C.; Fiorica, C.; Cavallaro, G.; Cassata, G.; Giammona, G., Polymeric nanocarriers for magnetic targeted drug delivery: Preparation, characterization, and in vitro and in vivo evaluation. Mol. Pharmaceutics 2013, 10, 4397-4407. (6). Cinteza, L. O.; Ohulchanskyy, T. Y.; Sahoo, Y.; Bergey, E. J.; Pandey, R. K.; Prasad, P. N., Diacyllipid micelle-based nanocarrier for magnetically guided delivery of drugs in photodynamic therapy. Mol. Pharmaceutics 2006, 3, 415-423. (7). Hao, L.; Gwangjun, G.; Seong Yong, K.; Jong-Oh, P.; Sukho, P., Magnetic actuated pH-responsive hydrogel-based soft micro-robot for targeted drug delivery. Smart Mater. Struct. 2016, 25, 027001. (8). Sun, H.; Zhang, L.; Zhang, X.; Zhang, C.; Wei, Z.; Yao, S., 188re-labeled mpegmodified superparamagnetic nanogels: Preparation and targeting application in rabbits. Biomed. Microdevices 2008, 10, 281-287. (9). Hamidi, M.; Azadi, A.; Rafiei, P., Hydrogel nanoparticles in drug delivery. Adv. Drug Del. Rev. 2008, 60, 1638-1649. (10). Caccavo, D.; Cascone, S.; Lamberti, G.; Barba, A. A., Modeling the drug release from hydrogel-based matrices. Mol. Pharmaceutics 2015, 12, 474-483. (11). Chen, J.; Yang, L.; Liu, Y.; Ding, G.; Pei, Y.; Li, J.; Hua, G.; Huang, J., Preparation and characterization of magnetic targeted drug controlled-release hydrogel microspheres. Macromol. Symp. 2005, 225, 71-80.


Page 24 of 38

Page 25 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(12). Zhang, D.; Sun, P.; Li, P.; Xue, A.; Zhang, X.; Zhang, H.; Jin, X., A magnetic chitosan hydrogel for sustained and prolonged delivery of bacillus calmette–guérin in the treatment of bladder cancer. Biomaterials 2013, 34, 10258-10266. (13). Jie, W.; Xiao‐Jie, J.; Xiao‐Yi, Z.; Rui, X.; Wei, W.; Ying‐Mei, L.; Liang‐Yin, C., Multi‐stimuli‐responsive microcapsules for adjustable controlled‐release. Adv. Funct. Mater. 2014, 24, 3312-3323. (14). Liu, Y.-M.; Wu, W.; Ju, X.-J.; Wang, W.; Xie, R.; Mou, C.-L.; Zheng, W.-C.; Liu, Z.; Chu, L.-Y., Smart microcapsules for direction-specific burst release of hydrophobic drugs. RSC Adv. 2014, 4, 46568-46575. (15). Purushotham, S.; Chang, P. E. J.; Rumpel, H.; Kee, I. H. C.; Ng, R. T. H.; Chow, P. K. H.; Tan, C. K.; Ramanujan, R. V., Thermoresponsive core–shell magnetic nanoparticles for combined modalities of cancer therapy. Nanotechnology 2009, 20, 305101. (16). Fusco, S.; Sakar, M. S.; Kennedy, S.; Peters, C.; Bottani, R.; Starsich, F.; Mao, A.; Sotiriou, G. A.; Pane, S.; Pratsinis, S. E.; Mooney, D.; Nelson, B. J., An integrated microrobotic platform for on-demand, targeted therapeutic interventions. Adv. Mater. 2014, 26, 952-957. (17). Shaw, S.; Murthy, P. V. S. N., Magnetic drug targeting in the permeable blood vessel—the effect of blood rheology. J. Nanotechnol. Eng. Med. 2010, 1, 021001. (18). Mikkelsen, C.; Fougt Hansen, M.; Bruus, H., Theoretical comparison of magnetic and hydrodynamic interactions between magnetically tagged particles in microfluidic systems. J. Magn. Magn. Mater. 2005, 293, 578-583. (19). M., C. E.; G., M. P.; K., E. J., Particle size, magnetic field, and blood velocity effects on particle retention in magnetic drug targeting. Med. Phys. 2010, 37, 175-182. (20). Dorfmann, A.; Ogden, R. W., Some problems in nonlinear magnetoelasticity. Z. angew. Math. Phys. 2005, 56, 718-745. (21). Gurtin, M. E.; Fried, E.; Anand, L., The mechanics and thermodynamics of continua. Cambridge University Press: Cambridge, U.K., 2010. (22). Nedjar, B., A coupled bem-fem method for finite strain magneto-elastic boundaryvalue problems. Comput. Mech. 2016, 59, 795–807. (23). Coussy, O.; Dormieux, L.; Detournay, E., From mixture theory to biot's approach for porous media. Int. J. Solids Struct. 1998, 35, 4619-4635. (24). Hong, W.; Zhao, X.; Zhou, J.; Suo, Z., A theory of coupled diffusion and large deformation in polymeric gels. J. Mech. Phys. Solids 2008, 56, 1779-1793. 25 ACS Paragon Plus Environment


(25). Zhang, Y.; Liu, Z.; Swaddiwudhipong, S.; Miao, H.; Ding, Z.; Yang, Z., pHsensitive hydrogel for micro-fluidic valve. J. Funct. Biomater. 2012, 3, 464-479. (26). Gurtin, M. E.; Fried, E.; Anand, L., The mechanics and thermodynamics of continua. Cambridge University Press: 2010. (27). Nedjar, B., A theory of finite strain magneto-poromechanics. J. Mech. Phys. Solids 2015, 84, 293-312. (28). Liu, Q.; Li, H.; Lam, K. Y., Development of a multiphysics model to characterize the responsive behavior of magnetic-sensitive hydrogels with finite deformation. J. Phys. Chem. B 2017, 121, 5633-5646. (29). Cai, S.; Suo, Z., Equations of state for ideal elastomeric gels. Epl 2012, 97, 34009. (30). Hong, W.; Zhao, X.; Suo, Z., Large deformation and electrochemistry of polyelectrolyte gels. J. Mech. Phys. Solids 2010, 58, 558-577. (31). Han, Y. I.; Hong, W. E. I.; Faidley, L., Coupled magnetic field and viscoelasticity of ferrogel. Int. J. Appl. Mech. 2011, 03, 259-278. (32). Flory, P. J., Thermodynamics of high polymer solutions. J. Chem. Phys 1942, 10, 51-61. (33). Flory, P. J., Principles of polymer chemistry. Cornell University: New York, 1953. (34). Pelteret, J.-P.; Davydov, D.; McBride, A.; Vu, D. K.; Steinmann, P., Computational electro- and magneto-elasticity for quasi-incompressible media immersed in free space. Int. J. Numer. Meth. Eng 2016, 108, 1307–1342. (35). Bouklas, N.; Huang, R., Swelling kinetics of polymer gels: Comparison of linear and nonlinear theories. Soft Matter 2012, 8, 8194-8203. (36). Xu, X.; Li, Z.; Nehorai, A., Finite element simulations of hydrodynamic trapping in microfluidic particle-trap array systems. Biomicrofluidics 2013, 7, 054108. (37). Mefford, O. T.; Woodward, R. C.; Goff, J. D.; Vadala, T. P.; St. Pierre, T. G.; Dailey, J. P.; Riffle, J. S., Field-induced motion of ferrofluids through immiscible viscous media: Testbed for restorative treatment of retinal detachment. J. Magn. Magn. Mater. 2007, 311, 347-353. (38). Afkhami, S.; Renardy, Y.; Renardy, M.; Riffle, J. S.; St Pierre, T., Field-induced motion of ferrofluid droplets through immiscible viscous media. J. Fluid Mech. 2008, 610, 363-380.


Page 26 of 38

Page 27 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(39). Lunnoo, T.; Puangmali, T., Capture efficiency of biocompatible magnetic nanoparticles in arterial flow: A computer simulation for magnetic drug targeting. Nanoscale. Res. Lett. 2015, 10, 426. (40). Zhu, T.; Lichlyter, D. J.; Haidekker, M. A.; Mao, L., Analytical model of microfluidic transport of non-magnetic particles in ferrofluids under the influence of a permanent magnet. Microfluid. Nanofluid. 2011, 10, 1233-1245. (41). Asfer, M.; Prasad Prajapati, A.; Kumar, A.; Kumar Panigrahi, P., Visualization and motion of curcumin loaded iron oxide nanoparticles during magnetic drug targeting. J. Nanotechnol. Eng. Med. 2015, 6, 011004. (42). Liu, Y.; Zhang, H.; Zhang, J.; Zheng, Y., Transient swelling of polymeric hydrogels: A new finite element solution framework. Int. J. Solids Struct. 2016, 80, 246-260. (43). Yinfeng, Z.; Liuwei, C.; Zhongqi, Z.; Hua, Y., Preparation and separation function of n‐isopropylacrylamide copolymer hydrogels. Polym. Adv. Technol. 2000, 11, 192197. (44). Czugala, M.; O’Connell, C.; Blin, C.; Fischer, P.; Fraser, K. J.; Benito-Lopez, F.; Diamond, D., Swelling and shrinking behaviour of photoresponsive phosphonium-based ionogel microstructures. Sensors Actuators B: Chem. 2014, 194, 105-113. (45). Chester, S. A.; Anand, L., A coupled theory of fluid permeation and large deformations for elastomeric materials. J. Mech. Phys. Solids 2010, 58, 1879-1906. (46). Liu, Q.; Li, H.; Lam, K. Y., Transition of magnetic field due to geometry of magneto-active elastomer microactuator with nonlinear deformation. J. Microelectromech. Syst. 2018, 27, 127-136. (47). Zhang, N.; Lock, J.; Sallee, A.; Liu, H., Magnetic nanocomposite hydrogel for potential cartilage tissue engineering: Synthesis, characterization, and cytocompatibility with bone marrow derived mesenchymal stem cells. ACS Appl. Mater. Interfaces 2015, 7, 20987-20998. (48). Rödling, L.; Volz, E. M.; Raic, A.; Brändle, K.; Franzreb, M.; Lee‐Thedieck, C., Magnetic macroporous hydrogels as a novel approach for perfused stem cell culture in 3d scaffolds via contactless motion control. Adv. Healthc. Mater. 2018, 7, 1701403. (49). Li-Heng, C.; E., K. T.; E., G. R.; F., P. A.; Michael, R.; A., W. D., Soft poly(dimethylsiloxane) elastomers from architecture-driven entanglement free design. Adv. Mater. 2015, 27, 5132-5140.



List of tables Table 1. Input parameters for validation of the present model. ....................................... 30 Table 2. Comparison of the travel times between the present numerical simulation and experimental, and theoretical results by Mefford et al.,37 for droplets positioned at different initial distances with various diameters. ........................................ 30


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List of figures Figure 1. (a) Schematic of a spherical magnetic droplet immersed in viscous fluid under a nonuniform magnetic field. (b) Fitting curve of the experimental measurement on magnetic field distribution along axial direction by a fifthdegree polynomial. ........................................................................................... 31 Figure 2. Schematic of the magnetic-sensitive hydrogel placed in a moving fluid subject to a nonuniform magnetic field by a permanent magnet (a). Influence of the hydrogel radius R0 on motion and deformation of the magnetic hydrogel (b), time evolution of the hydrogel displacement u (c), and time evolution of the swelling ratio (d). .............................................................................................. 32 Figure 3. Influence of the maximum inlet velocity vmax on (a) motion and deformation of the magnetic hydrogel, (b) time evolution of the hydrogel displacement u, and (c) time evolution of the swelling ratio. ........................................................... 33 Figure 4. Influence of the initial hydrogel position on motion and deformation of the magnetic hydrogel with time. ........................................................................... 34 Figure 5. Distribution of the non-dimensional magnetic field H / H max along X-axis direction for the magnetic hydrogel at different times, t = 20 s (a), 25 s (b), and 25.1 s (c), where the dash and solid lines denote the distributions of the magnetic field applied and magnetic field in the presence of magnetic hydrogel respectively. ...................................................................................................... 35 Figure 6. Schematic of the magnetic-sensitive hydrogel placed in a moving fluid subject to a nonuniform magnetic field by a horizontally movable magnet (a). Influence of the maximum magnetic field H max on trajectory of the magnetic hydrogel when LD = 10 mm , vmax = 5 mm/s (b), and vmax = 8 mm/s (c). ...... 36 Figure 7. Influence of the maximum magnetic field H max on trajectory of the magnetic hydrogel when LD = 5 mm (a), and 15 mm (b). .............................................. 37



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Table 1. Input parameters for validation of the present model. Parameters

Symbol

Value

density of droplet

ρ0

1320 kg/m3 37

dynamic viscosity

ϕ

50 Pa s 37

fluid density

ρs

998 kg/m3 37

shear modulus

G

10 4 Pa

length of the whole domain

LR

0.004 m

width of the whole domain

LH

0.016 m

vacuum magnetic permeability

µ0

4π × 10 −7 N/A 2

relative magnetic permeability

µ mr

2.4

49

Table 2. Comparison of the travel times between the present numerical simulation and experimental, and theoretical results by Mefford et al.,37 for droplets positioned at different initial distances with various diameters. Diameter of droplet (mm) 1.0 1.8 2.0

Initial distance from droplet to magnet (mm) 12 12 11

Simulation by Mefford et al. 37 (min) 21.2 6.6 3.2


Simulation by present model (min) 13.2 5.3 2.8

Experimental result (min) 15 4.0 2.5

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(b) (a) Figure 1. (a) Schematic of a spherical magnetic droplet immersed in viscous fluid under a nonuniform magnetic field. (b) Fitting curve of the experimental measurement on magnetic field distribution along axial direction by a fifth-degree polynomial.



Page 32 of 38

(a)

(b)

(c)

(d)

Figure 2. Schematic of the magnetic-sensitive hydrogel placed in a moving fluid subject to a nonuniform magnetic field by a permanent magnet (a). Influence of the hydrogel radius R0 on motion and deformation of the magnetic hydrogel (b), time evolution of the hydrogel displacement u (c), and time evolution of the swelling ratio (d).


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(a)

(b)

(c)

Figure 3. Influence of the maximum inlet velocity vmax on (a) motion and deformation of the magnetic hydrogel, (b) time evolution of the hydrogel displacement u, and (c) time evolution of the swelling ratio.



Figure 4. Influence of the initial hydrogel position on motion and deformation of the magnetic hydrogel with time.


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(a)

(b)

(c)

Figure 5. Distribution of the non-dimensional magnetic field H / H max along X-axis direction for the magnetic hydrogel at different times, t = 20 s (a), 25 s (b), and 25.1 s (c), where the dash and solid lines denote the distributions of the magnetic field applied and magnetic field in the presence of magnetic hydrogel respectively.



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(a)

(b)

(c)

Figure 6. Schematic of the magnetic-sensitive hydrogel placed in a moving fluid subject to a nonuniform magnetic field by a horizontally movable magnet (a). Influence of the maximum magnetic field H max on trajectory of the magnetic hydrogel when LD = 10 mm , vmax = 5 mm/s (b), and vmax = 8 mm/s (c).


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(b)

(a)

Figure 7. Influence of the maximum magnetic field H max on trajectory of the magnetic hydrogel when LD = 5 mm (a), and 15 mm (b).



55x30mm (300 x 300 DPI)


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Optimization of Deformable Magnetic-Sensitive Hydrogel-Based

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