- Email: [email protected]
Observation and modelling of capillary flow occlusion resulting from the capture of superparamagnetic nanoparticles in a magnetic field
Chemical Engineering Science 63 (2008) 3960 -- 3965
Contents lists available at ScienceDirect
Chemical Engineering Science journal homepage: w w w . e l s e v i e r . c o m / l o c a t e / c e s
Observation and modelling of capillary flow occlusion resulting from the capture of superparamagnetic nanoparticles in a magnetic field B. Hallmark, N.J. Darton, X. Han, S. Palit, M.R. Mackley, N.K.H. Slater ∗ Department of Chemical Engineering, New Museums Site, Pembroke St, Cambridge CB2 3RA, UK
A R T I C L E
I N F O
Article history: Received 22 November 2007 Received in revised form 24 April 2008 Accepted 29 April 2008 Available online 6 May 2008 Keywords: Biochemical engineering Biomedical engineering Fluid mechanics Numerical analysis
A B S T R A C T
The magnetic field mediated capture of 10 nm diameter superparamagnetic nanoparticles, in the form of agglomerates of mean diameter 330 and 580 nm, from microcapillary flows has been observed and modelled. The steady state thickness of the captured layer in microcapillaries of diameter 400--800 m could be predicted for both the 330 and the 580 nm diameter agglomerates at flow rates of between 0.1 and 0. 4 ml min−1 . The model provides insight into blockage formation at a constant flow rate as a precursor to the prediction of thrombotic embolism in magnetic directed therapies. Capillary constriction was particularly acute for the 580 nm agglomerates in large microcapillaries (800 m) with flow rates of 0. 1 ml min−1 . From this model, agglomerates of diameter 330 nm or less offer the potential for minimal microcapillary occlusion in a range of flow rates. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction The magnetic field mediated targeting of superparamagnetic nanoparticle linked therapeutic agents to specific sites of disease in the body is an exciting prospect. Intentional embolization with magnetic particles has been used therapeutically with some success to selectively stop blood supply to tumours, causing them to necrotize (Sako et al., 1986). However, in drug delivery settings excessive nanoparticle accumulation might block small blood capillaries, causing a thrombotic embolism, which is undesirable. Magnetic nanoparticles are already in use clinically as contrast agents in magnetic resonance imaging (Pankhurst et al., 2003). However, to date there appear to have only been three early stage clinical trials of magnetic targeting of paramagnetic nanoparticles in humans. In two of these trials, the chemotherapeutic drug epirubicin linked to 100 nm paramagnetic nanoparticles was administered intravenously. These nanoparticles were successfully targeted with minimal side effects to tumours with an externally applied 0.2--0.8 T magnet placed by the tumour (Lubbe et al., 1999, 1996). Wilson et al. (2004) delivered a sample of intra-arterially administered doxorubicin linked 0.5--5 m superparamagnetic nanoparticles to tumours in patients with an external 1.5 T magnet placed adjacent to the tumour. Although patients reported short lived abdominal pain during injection of the drug bearing nanoparticles, embolization was not detected in post procedure angiography (Wilson et al., 2004). Whilst
∗
Corresponding author. Tel.: +44 7786232492. E-mail address: [email protected] (N.K.H. Slater).
0009-2509/$ - see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2008.04.057
tissue penetration of pharmaceutical linked nanoparticles is important in this paper we develop and test a model to assess the effect that the size of the administered paramagnetic nanoparticles has on the constriction of capillaries as a result of magnetic targeting. 2. Materials FeCl3 · 6H2 O, FeCl2 · 4H2 O, NH3 · H2 O and HCl were obtained from Fisher Scientific (UK) and PMAA from Sigma Aldrich. The capillary arrays were manufactured in-house from a commercially available plastomer (Dow Affinity䉸 ) using a novel extrusion-based process that has been previously described (Hallmark et al., 2005a, b). 3. Methods A 10 nm superparamagnetic magnetite (Fe3 O4 ) nanoparticles were synthesized by co-precipitation of Fe2+ and Fe3+ aqueous salt solutions upon addition of a base in an oxygen-free, non-oxidizing, environment as described by Xia et al. (2005) using overhead mixing rather than ultrasonication to mix the reaction solution (Darton et al., 2008). To reduce agglomeration of the superparamagnetic nanoparticles, 3% by weight PMAA was added and the pH adjusted to 7.4 (Mendenhall et al., 1996). This resulted in agglomerates, assumed to be spherical, of the 10 nm superparamagnetic nanoparticles of 575 ± 8. 0 nm in diameter as measured by a Brookhaven Zeta-sizer. To obtain smaller agglomerates of diameter 328 ± 3. 5 nm a sample of nanoparticles in 3% by weight PMAA was sonicated with a 330 W ultrasonic cell crusher (Heat Systems XL-2020) on full power for 10 min. These particle sizes were considered suitable for this work
B. Hallmark et al. / Chemical Engineering Science 63 (2008) 3960 -- 3965
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Fig. 2. Photomicrograph of a layer of captured nanoparticles in a microcapillary. The flow direction is from right to left, and the magnet is positioned on the upper wall of the microcapillary.
Fig. 1. Schematic diagram of the apparatus to measure magnetic in-flow capture of superparamagnetic nanoparticles.
because they were large enough to enable in-flow capture with a 0.5 T NdFeB magnet but still relatively small to potentially improve tissue penetration at target site. To reduce possible surface coat interactions or unevenness no encapsulation of particles was used. The magnetic properties of the nanoparticles were measured using SQUID magnetometery at 293 K from −1 to 1 T, which showed that the particles were superparamagnetic with a magnetic susceptibility of 1. 57 × 10−4 . Nanoparticle capture was studied in a novel plastic capillary array, termed a microcapillary film or MCF (Hallmark et al., 2005a,b) containing capillaries of 410 m diameter. The MCFs were fabricated in-house from a commercially available polymer resin, Dow Affinity Plastomer, using a novel extrusion process (Hallmark et al., 2005b). MCFs provide low refraction for optical microscopy and previous research has characterized fluid flows in the material (Hornung et al., 2006). Only one of the 19 available capillaries was used in each film. A typical experiment would proceed as follows. A steady flow of an aqueous solution of PMAA (3% w/w) was first established in the test capillary and its flow rate was accurately controlled by an HPLC pump (Kontron 422). An electro-mechanical injection valve unit (VICI Valco) was used to introduce into the flow 2 ml slugs of 40 mg ml−1 superparamagnetic nanoparticles suspended in PMAA solution (3% w/w). Superparamagnetic particles were magnetically targeted with a 0.5 T NdFeB permanent magnet (e-magnets UK, Sheffield). The solution leaving the film was not recycled. The setup of the apparatus is illustrated schematically in Fig. 1. Experiments were carried out on two sizes of nanoparticle aggregate (330 and 580 nm). Flow-rates less than 1 ml min−1 proved optimal for capture and hence flows between 0.1 and 0. 5 ml min−1 were examined in steps of 0. 05 ml min−1 . These volumetric flow rates correspond to superficial linear velocities of 1.3 and 6. 3 cm s−1 , respectively, with steps of 0. 6 cm s−1 . The Reynolds numbers corresponding to these flow velocities are 1.7 and 8.0, respectively, using a measured fluid viscosity of PMAA solution (3% w/w) of 2. 5×10−3 Pa s and a density of 1024 kg m−3 . Additionally, the physical properties of the PMAA solution (3% w/w) are close to those of blood, with values of blood viscosity reported circa 3. 3 × 10−3 Pa s (Lowe et al., 2000)
and density between 1043 and 1051 kg m−3 (Hinghofer-Szalkay and Greenleaf, 1987). Nanoparticle capture was observed until one of two criteria were met; either a steady thickness of captured nanoparticles had been held for over an hour after the injected pulse, or the initial layer was eroded by the flow. When a stable layer of captured nanoparticles formed its thickness depended on the flow rate and particle size (Darton et al., 2008). Erosion was the dominant behaviour at higher flow-rates. A photomicrograph illustrating a layer of captured nanoparticles in a microcapillary is shown in Fig. 2. 4. Model The stability of a steady-state layer of magnetically captured nanoparticles within a flowing fluid depends on the relative magnitude of the forces acting on each nanoparticle within the layer. Two dominant forces act on a stationary nanoparticle at the wall of a capillary: the hydrodynamic force resulting from the flow field around the nanoparticle and the magnetic force due to the presence of a magnetic field gradient. Electrostatic forces and Van de Waal’s forces have been neglected as in other work in this area (Ebner et al., 1997). A key assumption in the model is that the captured nanoparticle layer is stable when the magnetic forces are greater than the hydrodynamic forces. When the two forces become equal the nanoparticle layer is assumed to become unstable and subject to erosion. 4.1. Hydrodynamic analysis Experimental observations reveal that nanoparticles are captured by the magnetic field and form a layer on the capillary wall. With a constant volumetric flow-rate this layer constricts the capillary, resulting in higher fluid velocities and shear stresses on the surface of the nanoparticle layer. The aim of this analysis is to estimate the magnitude of the shear force acting on the nanoparticles resulting from flow field in a constricted capillary. Fig. 3 illustrates schematically the constriction of the flow within a capillary due to a captured layer of particles at the capillary wall. The cross sectional area of the captured layer is assumed to take the form of a segment of the capillary’s circular cross section; its area can hence be shown to be Rc − h Adeposit = R2c Cos−1 − (Rc − h) h(2Rc − h), (1) Rc where Rc is the capillary radius and h is the maximum thickness of the captured layer. If this area is subtracted from the total cross
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B. Hallmark et al. / Chemical Engineering Science 63 (2008) 3960 -- 3965
Fig. 3. Schematic diagram showing the flow constriction of a capillary due to nanoparticle capture, highlighting the forces acting on a layer of magnetic nanoparticles.
sectional area of the capillary, then an equivalent hydraulic radius for the area available for the fluid flow can be found, 2 Rc − h Rc − R2c Cos−1 + (Rc − h) h(2Rc − h) Rc rh = . (2) For low Reynolds number flows, the flow in the capillary is laminar and unidirectional, having a parabolic velocity profile,
2Q r2 u(r) = 1− , (3) rh2 rh2 where Q is the volumetric flow-rate of fluid through the capillary and r is a radial co-ordinate. If the fluid in the capillary is assumed to be Newtonian, then the maximum shear stress due to the flow is at the wall of the capillary, and is equal to du max = dr r=r h √ 4Q = − (4) 3/2 . Rc −h −1 2 2 Rc −Rc Cos +(Rc −h) h(2Rc −h) Rc The hydrodynamic force, Fh , acting on a single nanoparticle of radius rp in the flow direction resulting from the shear stress can be thought of as the maximum shear stress acting on the surface area of a circle of the same diameter of the nanoparticle, i.e. 4Q 3 rp2 Fh = − (5) 3/2 . Rc − h R2c − R2c Cos−1 + (Rc − h) h(2Rc − h) Rc 4.2. Magnetic analysis The magnetic field is assumed to be that of a dipole acting from a point in space, with a magnetic field, H, described in spherical polar co-ordinates as (see, for example, Duffin, 1998), ⎡ − ⎢ 2r3
Cos
⎤
⎥ ⎥ − H(r, , ) = ⎢ ⎣ Sin ⎦ . 3 4r 0
(6)
Here, is the strength of dipole moment, which for a bar magnet is defined as Bi Vm = , 0
(7)
where Bi is the intrinsic induction of the magnet, Vm is the magnet volume and 0 is the permittivity of a vacuum.
Fig. 4. Schematic diagram of the location of the capillary and layer of captured nanoparticles with respect to the magnetic dipole.
To evaluate the force on a magnetic nanoparticle due to an applied magnetic field, an approach has been used that is similar to that used by others (Ebner et al., 1997). For simplicity, the effect of an existing layer of nanoparticles on enhancing the magnetic field gradient has been neglected. Firstly, it is assumed that the nanoparticle is a pointlike dipole and that the force, Fm , on this dipole, of strength m, with a magnetic induction, B, is Fm = (m · ∇)B.
(8)
For a single nanoparticle, the strength of the dipole moment, m, is assumed to be m=
Vp B . 0
(9)
Here, Vp is the particle volume, and the difference between the susceptibilities of the medium in which the nanoparticles are immersed and the material from which the nanoparticles are made from. Combining Eqs. (8) and (9) leads to an expression for the force on a single nanoparticle, Fm =
Vp (B · ∇)B. 0
(10)
Using a standard identity, ∇(B · B) = 2B × (∇ × B) + 2(B. ∇)B
(11)
and assuming that there are no time varying currents or fields, such that ∇ × B = 0, then Eq. (10) becomes Fm = Vp ∇( 12 B · H),
(12)
B. Hallmark et al. / Chemical Engineering Science 63 (2008) 3960 -- 3965
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Fig. 5. Schematic diagram showing the main forces acting on a captured magnetic nanoparticles along with the three conditions that: (i) result in a particle remaining captured; (ii) result in the release of a captured particle into the flow; and (iii) result in a particle being on the point of release into the flow.
since B = 0 H. Using the expression for the magnetic field given in Eq. (6), the expression for the force on one nanoparticle subject to the magnetic field from a dipole becomes ⎤ ⎡ −3 1 Cos2 + Sin2 ⎥ 4 4 2 ⎢ ⎥ ⎢ (13) Fm = Vp 0 −3 ⎥. ⎢ 7 ⎦ Cos Sin r ⎣ 16 0 The nanoparticle capture and erosion was viewed at an angle of = /2, such that the magnetic force experienced by the particle is only in the radial direction, as shown in Fig. 4. This results in a magnetic force on the nanoparticles that corresponds to Fm =
−3Vp 0 2 rˆ . 16(R + h)7
(14)
0
In this equation, R0 is the distance of the capillary wall from the magnetic dipole and h is the thickness of the nanoparticle layer. 4.3. Model validation and predictions Nanoparticles are eroded from the layer if the shear forces due to the flow field dominate the magnetic force. The transition from retention to erosion is assumed to happen when the hydraulic shear force and the magnetic force are equal. This is shown schematically in Fig. 5. The thickness of the nanoparticle layer at transition, h, was calculated from Eqs. (5) and (14). The volume of the magnet was measured to be 37. 5 cm3 and the surface flux 0.5 T using a Hall-effect magnetometer. This allowed calculation of the strength of the magnetic dipole, . The diameter of the capillary in which the fluid was flowing was measured as 410 m using optical microscopy. The thickness of the 580 nm particle layer was computed numerically using a simple linear search and tuned using the parameter R0 , the distance from the capillary wall to the magnet dipole, such that an acceptable fit
Fig. 6. Plot showing the experimentally measured thickness of the 580 and 330 nm nanoparticles layers (open symbols) and the predictions of layer thickness made by the model.
was obtained to the experimental data. The thickness of the 330 nm particle layer was then calculated using the same value for R0 . The best fit to experimental data was obtained for a value of R0 equal to 5.68 mm. As the distance between the magnet and the capillary wall in the experimental apparatus was roughly 1.5 mm, then this estimate of R0 seems reasonable given the assumption that the form of the magnetic field is that due to a dipole. Model predictions for the variation of h with flow-rate are compared with the experimental data in Fig. 6. The trend of decreasing nanoparticle layer thickness with increasing flow-rate for the 580 nm nanoparticles shown in the plot in Fig. 6 is predicted by the model to a precision within experimental error. The model does not perform as well for the 330 nm nanoparticles at high flow rates as the experimental data shows that the trend of decreasing layer thickness with increasing flow-rate is weaker than with the 580 nm nanoparticles. The model has been used to investigate the steady-state layer thicknesses formed by the magnetic capture of 580 and 330 nm
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Fig. 7. Contour plots showing the prediction percentage constriction of a capillary as a function of volumetric flow-rate and capillary diameter for: (A) 330 nm nanoparticles and (B) 580 nm nanoparticles.
nanoparticles in capillaries of diameters between 400 and 800 m and flow rates between 0.1 and 0. 4 ml min−1 . The results from these simulations are expressed as a percentage constriction of the capillary (i.e. expressing the thickness of the layer of the nanoparticles with respect to the diameter of the capillary). This represents a first step towards the evaluation of risk of embolism due to nanoparticle capture in a system where the volumetric flow-rate of fluid flowing along a capillary is constant. The contour plot in Fig. 7(a) shows the predicted percentage constriction as a function of volumetric flow-rate and capillary diameter for 330 nm nanoparticles. As expected, the area of the plot that corresponds to a small amount of constriction is high volumetric flowrates through narrow capillaries, which results in high shear forces acting on the nanoparticles. The opposite is also true in that the highest predicted constrictions correspond to low volumetric flow-rates through large capillaries. Comparing Fig. 7(a) to (b), which predicts the percentage constriction due to 580 nm nanoparticles, the same trend can be seen. The smallest amount of constriction results from high volumetric flow-rates through smaller diameter capillaries, and the highest amount of constriction from low volumetric flow-rates through larger diameter capillaries. In absolute terms, however, the 580 nm nanoparticles result in a greater amount of predicted constriction as compared to the 330 nm nanoparticles. This result is not unexpected since the larger diameter nanoparticles contain a greater volume of magnetic material, hence resulting in larger magnetic forces, thus enabling a thicker nanoparticle layer to be captured. 5. Conclusion Observations of the magnetic field capture of superparamagnetic nanoparticle agglomerates in microcapillary flow were used to assess a model, which estimates the steady state thickness of a captured nanoparticle layer. This was achieved by balancing the shear stress on the surface of the captured nanoparticle layer with the magnetic force on the superparamagnetic nanoparticles. The model takes into account the strength of the magnetic field used to capture the particles as well as the hydrodynamics of the flow through the microcapillary, rheological properties of the fluid and the size and magnetic properties of the nanoparticle agglomerates. The rheological properties of the PMAA solution (3% w/w) used were comparable to those published for blood (Lowe et al., 2000). The model considers a constant flow rate system, which is an inaccurate approximation to flows in a living organism where capillaries are part of a large network, allowing the flow to redistribute itself should one
capillary, or a number of capillaries, become blocked. Modelling this more complex phenomenon, however, firstly requires the dynamics of the steady state layer in a single capillary to be understood. The results of the model indicate the potentially crucial role that superparamagnetic nanoparticle agglomerate diameter has on the risk of thrombotic embolism in magnetic directed therapy. A significant difference can be seen in the predicted percentage microcapillary occlusion when particles just 250 nm different in mean diameter are used in magnetic targeting. Of the superparamagnetic nanoparticle agglomerates investigated here, those of 330 nm in diameter are predicted to result in the least microcapillary occlusion with the best magnetic capture characteristics (Darton et al., 2008) that could potentially deliver a high dose of therapeutic compound to the site of disease.
Notation B Bi Fm Fh h H m Q rˆ r rh R0 Rc U Vm Vp
magnetic flux, T intrinsic inductance, T magnetic force vector, N hydrodynamic force, N thickness of nanoparticle layer, m magnetic field vector, A m−1 dipole moment due to nanoparticles, A m−1 volumetric flow-rate, m3 s−1 radial direction unit vector radial co-ordinate, m equivalent hydraulic radius, m distance of capillary wall from magnet, m capillary radius, m velocity, m s−1 volume of magnet, m3 nanoparticle volume, m3
Greek letters 0 max
susceptibility difference, dimensionless fluid viscosity, Pa s angular co-ordinate, rad strength of dipole moment, T m A−1 vacuum permittivity, T m A−1 maximum wall shear stress, Pa angular co-ordinate, rad
B. Hallmark et al. / Chemical Engineering Science 63 (2008) 3960 -- 3965
References Darton, N.J., Hallmark, B., Palit, S., Han, X., Slater, N.K.H., Mackley, M.R., 2008. Nanomedicine, 4(1), 19–29. Duffin, W.J., 1998. Electricity and Manetism. McGraw Hill, London. Ebner, A.D., Ritter, J.A., Ploehn, H.J., 1997. Feasibility and limitations of nanolevel high gradient magnetic separation. Separation and Purification Technology 11, 199--210. Hallmark, B., Gadala-Maria, F., Mackley, M.R., 2005a. The melt processing of polymer microcapillary film (MCF). Journal of Non-Newtonian Fluid Mechanics 128 (2--3), 83--98. Hallmark, B., Mackley, M.R., Gadala-Maria, F., 2005b. Hollow microcapillary arrays in thin plastic films. Advanced Engineering Materials 7, 545--547. Hinghofer-Szalkay, H.G., Greenleaf, J.E., 1987. Continuous monitoring of blood volume changes in humans. Journal of Applied Physiology 63, 1003--1007. Hornung, C.H., Hallmark, B., Hesketh, R.P., Mackley, M.R., 2006. The fluid flow and heat transfer performance of thermoplastic microcapillary films. Journal of Micromechanics and Microengineering 16, 434--447. Lowe, G., Rumley, A., Norrie, J., Ford, I., Shepherd, J., Cobbe, S., Macfarlane, P., Packard, C., 2000. Blood rheology, cardiovascular risk factors, and cardiovascular disease: the west of Scotland coronary prevention study. Thrombosis and Haemostasis 84, 553--558.
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Lubbe, A.S., Bergemann, C., Riess, H., Schriever, F., Reichardt, P., Possinger, K., Matthias, M., Dorken, B., Herrmann, F., Gurtler, R., et al., 1996. Clinical experiences with magnetic drag targeting: a phase I study with 4 -epidoxorubicin in 14 patients with advanced solid tumors. Cancer Research 56 (20), 4686--4693. Lubbe, A.S., Bergemann, C., Brock, J., McClure, D.G., 1999. Physiological aspects in magnetic drug-targeting. Journal of Magnetism and Magnetic Materials 194 (1--3), 149--155. Mendenhall, G.D., Geng, Y., Hwang, J., 1996. Optimization of long-term stability of magnetic fluids from magnetite and synthetic polyelectrolytes. Journal of Colloid Interface Science 184 (2), 519--526. Pankhurst, Q.A., Connolly, J., Jones, S.K., Dobson, J., 2003. Applications of magnetic nanoparticles in biomedicine. Journal of Physics D---Applied Physics 36 (13), R167--R181. Sako, M., Hirota, S., Ohtsuki, S., 1986. Clinical evaluation of ferromagnetic microembolization for the treatment of hepatocellular carcinoma. Annals of Radiology (Paris) 29 (2), 200--204. Wilson, M.W., Kerlan, R.K., Fidelman, N.A., Venook, A.P., LaBerge, J.M., Koda, J., Gordon, R.L., 2004. Hepatocellular carcinoma: regional therapy with a magnetic targeted carrier bound to doxorubicin in a dual MR imaging/conventional angiography suite---initial experience with four patients. Radiology 230 (1), 287--293. Xia, Z.F., Wang, G.B., Tao, K.X., Li, J.X., 2005. Preparation of magnetite--dextran microspheres by ultrasonication. Journal of Magnetism and Magnetic Materials 293 (1), 182--186.
Contents lists available at ScienceDirect
Chemical Engineering Science journal homepage: w w w . e l s e v i e r . c o m / l o c a t e / c e s
Observation and modelling of capillary flow occlusion resulting from the capture of superparamagnetic nanoparticles in a magnetic field B. Hallmark, N.J. Darton, X. Han, S. Palit, M.R. Mackley, N.K.H. Slater ∗ Department of Chemical Engineering, New Museums Site, Pembroke St, Cambridge CB2 3RA, UK
A R T I C L E
I N F O
Article history: Received 22 November 2007 Received in revised form 24 April 2008 Accepted 29 April 2008 Available online 6 May 2008 Keywords: Biochemical engineering Biomedical engineering Fluid mechanics Numerical analysis
A B S T R A C T
The magnetic field mediated capture of 10 nm diameter superparamagnetic nanoparticles, in the form of agglomerates of mean diameter 330 and 580 nm, from microcapillary flows has been observed and modelled. The steady state thickness of the captured layer in microcapillaries of diameter 400--800 m could be predicted for both the 330 and the 580 nm diameter agglomerates at flow rates of between 0.1 and 0. 4 ml min−1 . The model provides insight into blockage formation at a constant flow rate as a precursor to the prediction of thrombotic embolism in magnetic directed therapies. Capillary constriction was particularly acute for the 580 nm agglomerates in large microcapillaries (800 m) with flow rates of 0. 1 ml min−1 . From this model, agglomerates of diameter 330 nm or less offer the potential for minimal microcapillary occlusion in a range of flow rates. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction The magnetic field mediated targeting of superparamagnetic nanoparticle linked therapeutic agents to specific sites of disease in the body is an exciting prospect. Intentional embolization with magnetic particles has been used therapeutically with some success to selectively stop blood supply to tumours, causing them to necrotize (Sako et al., 1986). However, in drug delivery settings excessive nanoparticle accumulation might block small blood capillaries, causing a thrombotic embolism, which is undesirable. Magnetic nanoparticles are already in use clinically as contrast agents in magnetic resonance imaging (Pankhurst et al., 2003). However, to date there appear to have only been three early stage clinical trials of magnetic targeting of paramagnetic nanoparticles in humans. In two of these trials, the chemotherapeutic drug epirubicin linked to 100 nm paramagnetic nanoparticles was administered intravenously. These nanoparticles were successfully targeted with minimal side effects to tumours with an externally applied 0.2--0.8 T magnet placed by the tumour (Lubbe et al., 1999, 1996). Wilson et al. (2004) delivered a sample of intra-arterially administered doxorubicin linked 0.5--5 m superparamagnetic nanoparticles to tumours in patients with an external 1.5 T magnet placed adjacent to the tumour. Although patients reported short lived abdominal pain during injection of the drug bearing nanoparticles, embolization was not detected in post procedure angiography (Wilson et al., 2004). Whilst
∗
Corresponding author. Tel.: +44 7786232492. E-mail address: [email protected] (N.K.H. Slater).
0009-2509/$ - see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2008.04.057
tissue penetration of pharmaceutical linked nanoparticles is important in this paper we develop and test a model to assess the effect that the size of the administered paramagnetic nanoparticles has on the constriction of capillaries as a result of magnetic targeting. 2. Materials FeCl3 · 6H2 O, FeCl2 · 4H2 O, NH3 · H2 O and HCl were obtained from Fisher Scientific (UK) and PMAA from Sigma Aldrich. The capillary arrays were manufactured in-house from a commercially available plastomer (Dow Affinity䉸 ) using a novel extrusion-based process that has been previously described (Hallmark et al., 2005a, b). 3. Methods A 10 nm superparamagnetic magnetite (Fe3 O4 ) nanoparticles were synthesized by co-precipitation of Fe2+ and Fe3+ aqueous salt solutions upon addition of a base in an oxygen-free, non-oxidizing, environment as described by Xia et al. (2005) using overhead mixing rather than ultrasonication to mix the reaction solution (Darton et al., 2008). To reduce agglomeration of the superparamagnetic nanoparticles, 3% by weight PMAA was added and the pH adjusted to 7.4 (Mendenhall et al., 1996). This resulted in agglomerates, assumed to be spherical, of the 10 nm superparamagnetic nanoparticles of 575 ± 8. 0 nm in diameter as measured by a Brookhaven Zeta-sizer. To obtain smaller agglomerates of diameter 328 ± 3. 5 nm a sample of nanoparticles in 3% by weight PMAA was sonicated with a 330 W ultrasonic cell crusher (Heat Systems XL-2020) on full power for 10 min. These particle sizes were considered suitable for this work
B. Hallmark et al. / Chemical Engineering Science 63 (2008) 3960 -- 3965
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Fig. 2. Photomicrograph of a layer of captured nanoparticles in a microcapillary. The flow direction is from right to left, and the magnet is positioned on the upper wall of the microcapillary.
Fig. 1. Schematic diagram of the apparatus to measure magnetic in-flow capture of superparamagnetic nanoparticles.
because they were large enough to enable in-flow capture with a 0.5 T NdFeB magnet but still relatively small to potentially improve tissue penetration at target site. To reduce possible surface coat interactions or unevenness no encapsulation of particles was used. The magnetic properties of the nanoparticles were measured using SQUID magnetometery at 293 K from −1 to 1 T, which showed that the particles were superparamagnetic with a magnetic susceptibility of 1. 57 × 10−4 . Nanoparticle capture was studied in a novel plastic capillary array, termed a microcapillary film or MCF (Hallmark et al., 2005a,b) containing capillaries of 410 m diameter. The MCFs were fabricated in-house from a commercially available polymer resin, Dow Affinity Plastomer, using a novel extrusion process (Hallmark et al., 2005b). MCFs provide low refraction for optical microscopy and previous research has characterized fluid flows in the material (Hornung et al., 2006). Only one of the 19 available capillaries was used in each film. A typical experiment would proceed as follows. A steady flow of an aqueous solution of PMAA (3% w/w) was first established in the test capillary and its flow rate was accurately controlled by an HPLC pump (Kontron 422). An electro-mechanical injection valve unit (VICI Valco) was used to introduce into the flow 2 ml slugs of 40 mg ml−1 superparamagnetic nanoparticles suspended in PMAA solution (3% w/w). Superparamagnetic particles were magnetically targeted with a 0.5 T NdFeB permanent magnet (e-magnets UK, Sheffield). The solution leaving the film was not recycled. The setup of the apparatus is illustrated schematically in Fig. 1. Experiments were carried out on two sizes of nanoparticle aggregate (330 and 580 nm). Flow-rates less than 1 ml min−1 proved optimal for capture and hence flows between 0.1 and 0. 5 ml min−1 were examined in steps of 0. 05 ml min−1 . These volumetric flow rates correspond to superficial linear velocities of 1.3 and 6. 3 cm s−1 , respectively, with steps of 0. 6 cm s−1 . The Reynolds numbers corresponding to these flow velocities are 1.7 and 8.0, respectively, using a measured fluid viscosity of PMAA solution (3% w/w) of 2. 5×10−3 Pa s and a density of 1024 kg m−3 . Additionally, the physical properties of the PMAA solution (3% w/w) are close to those of blood, with values of blood viscosity reported circa 3. 3 × 10−3 Pa s (Lowe et al., 2000)
and density between 1043 and 1051 kg m−3 (Hinghofer-Szalkay and Greenleaf, 1987). Nanoparticle capture was observed until one of two criteria were met; either a steady thickness of captured nanoparticles had been held for over an hour after the injected pulse, or the initial layer was eroded by the flow. When a stable layer of captured nanoparticles formed its thickness depended on the flow rate and particle size (Darton et al., 2008). Erosion was the dominant behaviour at higher flow-rates. A photomicrograph illustrating a layer of captured nanoparticles in a microcapillary is shown in Fig. 2. 4. Model The stability of a steady-state layer of magnetically captured nanoparticles within a flowing fluid depends on the relative magnitude of the forces acting on each nanoparticle within the layer. Two dominant forces act on a stationary nanoparticle at the wall of a capillary: the hydrodynamic force resulting from the flow field around the nanoparticle and the magnetic force due to the presence of a magnetic field gradient. Electrostatic forces and Van de Waal’s forces have been neglected as in other work in this area (Ebner et al., 1997). A key assumption in the model is that the captured nanoparticle layer is stable when the magnetic forces are greater than the hydrodynamic forces. When the two forces become equal the nanoparticle layer is assumed to become unstable and subject to erosion. 4.1. Hydrodynamic analysis Experimental observations reveal that nanoparticles are captured by the magnetic field and form a layer on the capillary wall. With a constant volumetric flow-rate this layer constricts the capillary, resulting in higher fluid velocities and shear stresses on the surface of the nanoparticle layer. The aim of this analysis is to estimate the magnitude of the shear force acting on the nanoparticles resulting from flow field in a constricted capillary. Fig. 3 illustrates schematically the constriction of the flow within a capillary due to a captured layer of particles at the capillary wall. The cross sectional area of the captured layer is assumed to take the form of a segment of the capillary’s circular cross section; its area can hence be shown to be Rc − h Adeposit = R2c Cos−1 − (Rc − h) h(2Rc − h), (1) Rc where Rc is the capillary radius and h is the maximum thickness of the captured layer. If this area is subtracted from the total cross
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B. Hallmark et al. / Chemical Engineering Science 63 (2008) 3960 -- 3965
Fig. 3. Schematic diagram showing the flow constriction of a capillary due to nanoparticle capture, highlighting the forces acting on a layer of magnetic nanoparticles.
sectional area of the capillary, then an equivalent hydraulic radius for the area available for the fluid flow can be found, 2 Rc − h Rc − R2c Cos−1 + (Rc − h) h(2Rc − h) Rc rh = . (2) For low Reynolds number flows, the flow in the capillary is laminar and unidirectional, having a parabolic velocity profile,
2Q r2 u(r) = 1− , (3) rh2 rh2 where Q is the volumetric flow-rate of fluid through the capillary and r is a radial co-ordinate. If the fluid in the capillary is assumed to be Newtonian, then the maximum shear stress due to the flow is at the wall of the capillary, and is equal to du max = dr r=r h √ 4Q = − (4) 3/2 . Rc −h −1 2 2 Rc −Rc Cos +(Rc −h) h(2Rc −h) Rc The hydrodynamic force, Fh , acting on a single nanoparticle of radius rp in the flow direction resulting from the shear stress can be thought of as the maximum shear stress acting on the surface area of a circle of the same diameter of the nanoparticle, i.e. 4Q 3 rp2 Fh = − (5) 3/2 . Rc − h R2c − R2c Cos−1 + (Rc − h) h(2Rc − h) Rc 4.2. Magnetic analysis The magnetic field is assumed to be that of a dipole acting from a point in space, with a magnetic field, H, described in spherical polar co-ordinates as (see, for example, Duffin, 1998), ⎡ − ⎢ 2r3
Cos
⎤
⎥ ⎥ − H(r, , ) = ⎢ ⎣ Sin ⎦ . 3 4r 0
(6)
Here, is the strength of dipole moment, which for a bar magnet is defined as Bi Vm = , 0
(7)
where Bi is the intrinsic induction of the magnet, Vm is the magnet volume and 0 is the permittivity of a vacuum.
Fig. 4. Schematic diagram of the location of the capillary and layer of captured nanoparticles with respect to the magnetic dipole.
To evaluate the force on a magnetic nanoparticle due to an applied magnetic field, an approach has been used that is similar to that used by others (Ebner et al., 1997). For simplicity, the effect of an existing layer of nanoparticles on enhancing the magnetic field gradient has been neglected. Firstly, it is assumed that the nanoparticle is a pointlike dipole and that the force, Fm , on this dipole, of strength m, with a magnetic induction, B, is Fm = (m · ∇)B.
(8)
For a single nanoparticle, the strength of the dipole moment, m, is assumed to be m=
Vp B . 0
(9)
Here, Vp is the particle volume, and the difference between the susceptibilities of the medium in which the nanoparticles are immersed and the material from which the nanoparticles are made from. Combining Eqs. (8) and (9) leads to an expression for the force on a single nanoparticle, Fm =
Vp (B · ∇)B. 0
(10)
Using a standard identity, ∇(B · B) = 2B × (∇ × B) + 2(B. ∇)B
(11)
and assuming that there are no time varying currents or fields, such that ∇ × B = 0, then Eq. (10) becomes Fm = Vp ∇( 12 B · H),
(12)
B. Hallmark et al. / Chemical Engineering Science 63 (2008) 3960 -- 3965
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Fig. 5. Schematic diagram showing the main forces acting on a captured magnetic nanoparticles along with the three conditions that: (i) result in a particle remaining captured; (ii) result in the release of a captured particle into the flow; and (iii) result in a particle being on the point of release into the flow.
since B = 0 H. Using the expression for the magnetic field given in Eq. (6), the expression for the force on one nanoparticle subject to the magnetic field from a dipole becomes ⎤ ⎡ −3 1 Cos2 + Sin2 ⎥ 4 4 2 ⎢ ⎥ ⎢ (13) Fm = Vp 0 −3 ⎥. ⎢ 7 ⎦ Cos Sin r ⎣ 16 0 The nanoparticle capture and erosion was viewed at an angle of = /2, such that the magnetic force experienced by the particle is only in the radial direction, as shown in Fig. 4. This results in a magnetic force on the nanoparticles that corresponds to Fm =
−3Vp 0 2 rˆ . 16(R + h)7
(14)
0
In this equation, R0 is the distance of the capillary wall from the magnetic dipole and h is the thickness of the nanoparticle layer. 4.3. Model validation and predictions Nanoparticles are eroded from the layer if the shear forces due to the flow field dominate the magnetic force. The transition from retention to erosion is assumed to happen when the hydraulic shear force and the magnetic force are equal. This is shown schematically in Fig. 5. The thickness of the nanoparticle layer at transition, h, was calculated from Eqs. (5) and (14). The volume of the magnet was measured to be 37. 5 cm3 and the surface flux 0.5 T using a Hall-effect magnetometer. This allowed calculation of the strength of the magnetic dipole, . The diameter of the capillary in which the fluid was flowing was measured as 410 m using optical microscopy. The thickness of the 580 nm particle layer was computed numerically using a simple linear search and tuned using the parameter R0 , the distance from the capillary wall to the magnet dipole, such that an acceptable fit
Fig. 6. Plot showing the experimentally measured thickness of the 580 and 330 nm nanoparticles layers (open symbols) and the predictions of layer thickness made by the model.
was obtained to the experimental data. The thickness of the 330 nm particle layer was then calculated using the same value for R0 . The best fit to experimental data was obtained for a value of R0 equal to 5.68 mm. As the distance between the magnet and the capillary wall in the experimental apparatus was roughly 1.5 mm, then this estimate of R0 seems reasonable given the assumption that the form of the magnetic field is that due to a dipole. Model predictions for the variation of h with flow-rate are compared with the experimental data in Fig. 6. The trend of decreasing nanoparticle layer thickness with increasing flow-rate for the 580 nm nanoparticles shown in the plot in Fig. 6 is predicted by the model to a precision within experimental error. The model does not perform as well for the 330 nm nanoparticles at high flow rates as the experimental data shows that the trend of decreasing layer thickness with increasing flow-rate is weaker than with the 580 nm nanoparticles. The model has been used to investigate the steady-state layer thicknesses formed by the magnetic capture of 580 and 330 nm
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Fig. 7. Contour plots showing the prediction percentage constriction of a capillary as a function of volumetric flow-rate and capillary diameter for: (A) 330 nm nanoparticles and (B) 580 nm nanoparticles.
nanoparticles in capillaries of diameters between 400 and 800 m and flow rates between 0.1 and 0. 4 ml min−1 . The results from these simulations are expressed as a percentage constriction of the capillary (i.e. expressing the thickness of the layer of the nanoparticles with respect to the diameter of the capillary). This represents a first step towards the evaluation of risk of embolism due to nanoparticle capture in a system where the volumetric flow-rate of fluid flowing along a capillary is constant. The contour plot in Fig. 7(a) shows the predicted percentage constriction as a function of volumetric flow-rate and capillary diameter for 330 nm nanoparticles. As expected, the area of the plot that corresponds to a small amount of constriction is high volumetric flowrates through narrow capillaries, which results in high shear forces acting on the nanoparticles. The opposite is also true in that the highest predicted constrictions correspond to low volumetric flow-rates through large capillaries. Comparing Fig. 7(a) to (b), which predicts the percentage constriction due to 580 nm nanoparticles, the same trend can be seen. The smallest amount of constriction results from high volumetric flow-rates through smaller diameter capillaries, and the highest amount of constriction from low volumetric flow-rates through larger diameter capillaries. In absolute terms, however, the 580 nm nanoparticles result in a greater amount of predicted constriction as compared to the 330 nm nanoparticles. This result is not unexpected since the larger diameter nanoparticles contain a greater volume of magnetic material, hence resulting in larger magnetic forces, thus enabling a thicker nanoparticle layer to be captured. 5. Conclusion Observations of the magnetic field capture of superparamagnetic nanoparticle agglomerates in microcapillary flow were used to assess a model, which estimates the steady state thickness of a captured nanoparticle layer. This was achieved by balancing the shear stress on the surface of the captured nanoparticle layer with the magnetic force on the superparamagnetic nanoparticles. The model takes into account the strength of the magnetic field used to capture the particles as well as the hydrodynamics of the flow through the microcapillary, rheological properties of the fluid and the size and magnetic properties of the nanoparticle agglomerates. The rheological properties of the PMAA solution (3% w/w) used were comparable to those published for blood (Lowe et al., 2000). The model considers a constant flow rate system, which is an inaccurate approximation to flows in a living organism where capillaries are part of a large network, allowing the flow to redistribute itself should one
capillary, or a number of capillaries, become blocked. Modelling this more complex phenomenon, however, firstly requires the dynamics of the steady state layer in a single capillary to be understood. The results of the model indicate the potentially crucial role that superparamagnetic nanoparticle agglomerate diameter has on the risk of thrombotic embolism in magnetic directed therapy. A significant difference can be seen in the predicted percentage microcapillary occlusion when particles just 250 nm different in mean diameter are used in magnetic targeting. Of the superparamagnetic nanoparticle agglomerates investigated here, those of 330 nm in diameter are predicted to result in the least microcapillary occlusion with the best magnetic capture characteristics (Darton et al., 2008) that could potentially deliver a high dose of therapeutic compound to the site of disease.
Notation B Bi Fm Fh h H m Q rˆ r rh R0 Rc U Vm Vp
magnetic flux, T intrinsic inductance, T magnetic force vector, N hydrodynamic force, N thickness of nanoparticle layer, m magnetic field vector, A m−1 dipole moment due to nanoparticles, A m−1 volumetric flow-rate, m3 s−1 radial direction unit vector radial co-ordinate, m equivalent hydraulic radius, m distance of capillary wall from magnet, m capillary radius, m velocity, m s−1 volume of magnet, m3 nanoparticle volume, m3
Greek letters 0 max
susceptibility difference, dimensionless fluid viscosity, Pa s angular co-ordinate, rad strength of dipole moment, T m A−1 vacuum permittivity, T m A−1 maximum wall shear stress, Pa angular co-ordinate, rad
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