Magnetic nanoparticle density mapping from the magnetically induced displacement data: a simulation study
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Magnetic nanoparticles are gaining great roles in biomedical applications as targeted drug delivery agents or targeted imaging contrast agents. In the magnetic nanoparticle applications, quantification of the nanoparticle density deposited in a specified region is of great importance for evaluating the delivery of the drugs or the contrast agents to the targeted tissues. We introduce a method for estimating the nanoparticle density from the displacement of tissues caused by the external magnetic field.
We can exert magnetic force to the magnetic nanoparticles residing in a living subject by applying magnetic gradient field to them. The nanoparticles under the external magnetic field then exert force to the nearby tissues causing displacement of the tissues. The displacement field induced by the nanoparticles under the external magnetic field is governed by the Navier's equation. We use an approximation method to get the inverse solution of the Navier's equation which represents the magnetic nanoparticle density map when the magnetic nanoparticles are mechanically coupled with the surrounding tissues. To produce the external magnetic field inside a living subject, we propose a coil configuration, the Helmholtz and Maxwell coil pair, that is capable of generating uniform magnetic gradient field. We have estimated the coil currents that can induce measurable displacement in soft tissues through finite element method (FEM) analysis.
From the displacement data obtained from FEM analysis of a soft-tissue-mimicking phantom, we have calculated nanoparticle density maps. We obtained the magnetic nanoparticle density maps by approximating the Navier's equation to the Laplacian of the displacement field. The calculated density maps match well to the original density maps, but with some halo artifacts around the high density area. To induce measurable displacement in the living tissues with the proposed coil configuration, we need to apply the coil currents as big as 104A.
We can obtain magnetic nanoparticle maps from the magnetically induced displacement data by approximating the Navier's equation under the assumption of uniform-gradient of the external magnetic field. However, developing a coil driving system with the capacity of up to 104A should be a great technical challenge.
KeywordsMagnetic Force Magnetic Particle Divergence Term Magnetic Field Gradient Uniform Magnetic Field
magnetic force [N]
magnetic dipole moment [A·m2]
magnetic field density [Tesla]
displacement vector [m]
magnetic field intensity [A/m]
effective magnetization [A/m]
material density [kg/m3]
shear modulus [kPa)
magnetic permeability [wb/(A·m)]
coil radius [m]
spacing between upper and lower coils [m]
number of coil turns
coil current [A]
gradient field [Tesla/m]
0: in free space
Molecular imaging is known to be very powerful in early detection of cancers since molecular images show information about molecular or cellular level activities in a living body with much higher contrast of cancer tissues than conventional diagnostic images [1, 2]. In molecular imaging, securing high contrast of the targeted molecules or cells against the background tissues is crucial. Nuclear imaging devices like positron emission tomography (PET) or single photon emission computed tomography (SPECT) utilize radio-pharmaceuticals that tend to combine with targeted molecules or cells. Nuclear imaging devices have very high sensitivity and contrast, but they suffer from toxicity of the radio-pharmaceuticals and long scan time . Optical imaging devices, particularly fluorescence and bioluminescence imaging devices, are believed to be most versatile for molecular and cellular imaging since fluorescent and bioluminescent probes have unparalleled sensitivity and specificity in detecting biochemical activities . Optical imaging devices, however, suffer from very limited imaging depth, which is particularly problematic in human imaging. Magnetic resonance imaging (MRI) uses magnetic nanoparticles for molecular or cellular imaging. In MRI, the main magnetic field, usually an order of Tesla, magnetizes the nanoparticles injected to the living subject and the nanoparticles make detectable interference pattern in resulting magnetic resonance images . However, molecular MRI suffers from long scan time due to its low sensitivity.
Recently, it has been reported that ultrasound imaging can be used for molecular or cellular imaging with aids of magnetic nanoparticles . If we apply strong magnetic field to a living body into which magnetic nanoparticles are administered, the magnetic field produce magnetic propulsion force that can induce spatial displacement of the magnetic nanoparticles and surrounding tissues. If we measure the displacement using ultrasound imaging techniques, we can obtain information about migration of the magnetic nanoparticles in the living body. Feasibility of ultrasound molecular imaging using magnetic nanoparticles is suggested in a recent in vitro study of a mouse liver .
Quantification and visualization of molecular probe density at a region of interest is of crucial importance to understand the molecular or cellular activities in the living body. The previous studies on molecular ultrasound imaging with magnetic nanoparticles have been limited to discriminating existence of the nanoparticles at a region of interest. In this paper, we propose a method for quantitative magnetic-nanoparticle-density mapping from the magnetically induced displacement data. We also propose a magnetic coil configuration to realize the proposed method.
Application of magnetic force to the magnetic particles to induce displacement
When both m and B point to the same direction, for example m = m z az, B = B z az, in a region of interest as is often the case for magnetic propulsion of magnetic particles , the magnetic force becomes . This means that we need magnetic field gradient to apply magnetic force on the magnetic moment. Usually magnetic nanoparticles show superparamagnetism . Superparamagnetic nanoparticles are magnetized by the external magnetic field, and they have little hysteresis. If we apply spatially uniform magnetic field H0 = H0a z [A/m] to the nanoparticle containing region, the magnetic moment of a nanoparticle will be m = χH o [A·m2] where χ is the susceptibility of the magnetic nanoparticle. At a pixel of volume V [m3] containing multiple particles, we can define the effective magnetization of the pixel as [A/m] where m k is the magnetic moment of the k-th particle. If we apply the magnetic gradient field along with the uniform field H o , the magnetic force exerted on the pixel of interest will be F z = VM z (∂B z /∂z). Therefore, to make the magnetic force map be proportional to the magnetic particle density map, it is necessary to apply uniform magnetic field gradient over the region of interest.
Magnetic nanoparticle density calculation from the displacement data
This simplified equation implies that we can find the magnetic force field from the displacement field if we know the mechanical stiffness parameters G and v in advance. Since we usually perform molecular imaging on an interested region inside a living body, we may presume that we know the mechanical stiffness parameters beforehand. Otherwise, an elasticity imaging, such as ultrasound elastography, of the interested region will be needed before applying the magnetic field.
In equation (4), we neglect the double derivative in the x-direction (elevational direction in ultrasound imaging) only taking account of the y- (lateral) and z- (axial) directions. In the results section we will observe the effects of the omission of divergence term on the magnetic force map.
Magnetic coil system design
where a is the coil radius [m], d the interspacing [m] between the upper and lower coils, n1 the number of coil turns, I1 the coil current, μo the magnetic permeability [wb/(A·m)] in free space. A Helmholtz coil makes most uniform magnetic field at the center of the coil when the coil radius a and the coil interspacing d satisfies d = a. Then, the magnetic field at the center of the Helmholtz coil is given as B z (0, 0, 0) = 0.716 μ0n1I1/a.
FEM analysis of magnetic nanoparticle density mapping
With FEM analysis, we have first verified validness of equation (4) where we conveniently neglected the divergence term. We have also established FEM models to estimate the magnetic forces strong enough to make the measurable displacement. In ultrasound elastography, it is well known that the optimum strain level to be measured by ultrasound imaging is about 0.1 ~ 1.0% considering the decorrelation and SNR level of the ultrasound RF signals [16, 17].
Navier's equation tells that we can derive magnetic force maps from the displacement maps. To get precise force maps, we need to know all the three components of the displacement vector field. But, we think measuring all the three components is not feasible whatever types of imaging modalities we use considering their physical limitations like spatial resolution and scan time. Ultrasound imaging is a well known and most precise tool for measuring the axial displacement at tissues as thick as 10 cm. Optical coherence tomography can measure the axial displacement with much higher precision than ultrasound imaging, by an order of tens of nm, but it can measure axial displacements at only shallow region no deeper than several mm . If we use the axial displacement component only, we can have the magnetic force map with some halo-like artifacts around the boundary of the magnetic-particle-containing region. Despite the halo-like artifacts in the force maps, it has been found that the average intensity in the force maps is linearly proportional to the magnetic particle density. This implies that the magnetic force maps may give valuable information necessary for quantification of magnetic particle deliveries in a living body.
We need to consider some technical challenges for practical use of the proposed method. We need a coil driving system with high output current capacity along with proper coil cooling mechanism. Coil vibration during the coil driving may cause erroneous displacement in the imaging region if the imaging region is not well mechanically isolated from the coil system. But considering the recent report on the optical measurement of displacement with precision of tens of nm, induced by magnetic nanoparticles under time-varying magnetic field , it seems that magnetic coil vibration can be decoupled to a sufficient level.
In conclusion, magnetic force maps or magnetic nanoparticle density maps can be derived from the displacement induced by the external magnetic field. Even though the magnetic force maps have halo-like artifacts around the magnetic-particle-containing regions, we think the proposed method may find applications in real-time molecular imaging studies with an ultrasound scanner.
This work was supported by the National Research Foundation (NRF) of Korea funded by the Korean government (MEST) (No:2009-0078310).
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