The Effects of Particle Agglomeration in Magnetic Particle Imaging

Transcription

1 The Effects of Particle Agglomeration in Magnetic Particle Imaging Mathew Solomon Advisor: Dr. Michael Martens Department of Physics Case Western Reserve University Cleveland Ohio, July 31 st, 2013

2 Abstract Magnetic particle imaging is a new medical imaging technique that offers many advantages when used in combination with MRI over just using MRI, including increased measurement sensitivity and real-time images of blood flow. Further applications of the technique include cancer detection and cancer thermotherapy. Iron oxide particles of diameter nm with a polymer coating are put in a body and used as tracers. By inducing a magnetic response from the nanoparticles with an oscillating magnetic field, as well as isolating the response to specific regions using a larger static magnetic field, we can generate a spatial density map of the particles in real time. However, gaps in our current theoretical model for magnetic nanoparticles regarding particle agglomeration and the effect of Neel s relaxation on smaller particles limit the potential of the technology. This research attempts to understand how particles agglomerate, and propose a model that allows us to predict the behavior of agglomerated samples. This is done through a series of statistical simulations to determine the modification of characteristics because of agglomeration. It was found that agglomeration gives the illusion of an inflated relaxation constant, which could be interpreted as a smaller particle density or too high of a solution viscosity. However in the adiabatic limit, the opposite is true, and the measurements give an illusion of a deflated relaxation constant. 2

3 Contents LIST OF FIGURES... 4 ACKNOWLEDGEMENTS INTRODUCTION Magnetic Particle Imaging (MPI) Motivation for MPI MODELING MAGNETIC NANOPARTICLES Single Particle in an Applied Field Distribution of Particles in an Applied Field Solving the FPE with Matrix Continued Fractions Ratio of the 5 th /3 rd Harmonic PARTICLE AGGLOMERATION Motivation for studying particle agglomeration Simulating Agglomeration Effects of Agglomeration on Magnetization Effects of Agglomeration on 5 th /3 rd Harmonics DISCUSSION FUTURE DIRECTION REFERENCES

4 List of Figures FIGURE 1: MAGNETIZATION RESPONSE OF THE MAGNETIC NANOPARTICLES IN AN OSCILLATING APPLIED FIELD. THE NON-LINEAR RESPONSE GENERATES A RESULTING OSCILLATING MAGNETIZATION SIGNAL WITH HIGHER HARMONICS... 9 FIGURE 2: ORIENTATION ANGLE OF THE MAGNETIC MOMENT OF A PARTICLE WITH RESPECT TO THE DIRECTION OF THE MAGNETIC FIELD FIGURE 3: RATIO OF 5TH TO 3RD HARMONIC OF MAGNETIZATION RESPONSE SIGNAL AS A FUNCTION OF VERTICAL AXIS IS SCALED BY 5/3, SUCH THAT A RATIO OF 1 CORRESPONDS TO A SQUARE WAVE THE FIGURE 4: GAMMA DISTRIBUTIONS FOR DIFFERENT VALUES OF N FIGURE 5: MAGNETIZATION RESPONSE IN THE ADIABATIC APPROXIMATION WITH AGGLOMERATIONS N=1, 2, 3, 4, 5, 10, AND FIGURE 6: MAGNETIZATION VS. SCALED TIME (ΩT) FOR VARIOUS ωτ_b VALUES WITH AGGLOMERATIONS N=1, 2, 3, 4, 5, 10, AND FIGURE 7: RATIO OF 5TH TO 3RD HARMONIC AS A FUNCTION OF FOR VARIOUS N VALUES OF AGGLOMERATION FIGURE 8: ZOOMED IN 5TH/3RD PLOT TO SEE ADIABATIC LIMIT. NOTE THAT THE AGGLOMERATION RATIOS ARE HIGHER THAN THE UN-AGGLOMERATED RATIO AT THE ADIABATIC LIMIT, BUT CROSS AT ~

5 Acknowledgements Thank you to Dr. Martens for his guidance, support, and help throughout the entire project. Thank you to Dr. Brown for his enthusiasm and excitement for MPI, which sparked my interest to learn more and become involved in this project Thank you to Dr. Petschek and Dr. Singer for their patience as I struggled through my over-scheduled senior year 5

6 1 Introduction 1.1 Magnetic Particle Imaging (MPI) Magnetic particle imaging is a new imaging modality developed in 2005 by B. Gleich and J. Weizenecker at Philips research in Hamburg. In MPI, magnetic nanoparticles are used as tracers in a body. By measuring the non-linear magnetic response of these nanoparticles in an external magnetic field, a spatial particle density distribution can be generated in realtime. This can be used to image the flowing of any solution with these nanoparticles in any substance that does not interfere with the magnetization of the particles. The first in-vivo experiment with MPI was done by J. Weizenecker and B. Gleich in 2009, where they made a 3D real-time scan of a beating mouse heart [4]. Further in-vivo experiments have been done since then, but MPI is still primarily a research topic, not fully ready for medical application. A magnetic response from the nanoparticles is induced through the use of two external magnetic fields: a saturation selection field, and an oscillating field. The oscillating field induces a non-linear magnetic response from the nanoparticles, the flux of which can be easily measured through a pick-up coil on the instrument. The magnitude of the response signal is proportional to the density of particle in the region being measured. A larger static field is layered on top of the oscillating field, saturating the magnetization of almost all particles, except at the field free point, where the static field is zero. Thus, the measured magnetic response is caused only by the nanoparticles in the field free point. By moving the field free point through three dimensions, a three dimensional particle density 6

7 mapping of a region can be generated, with the resolution being the size of the field free point. The nanoparticles used are typically super paramagnetic iron oxide cores, between nm in diameter [3], with a polymer coating. The type of coating varies for different experiments, but is generally around 10 nm thick; thick enough such that any magnetic interaction between nanoparticles will be insignificant compared to the average thermal energy in solution. 1.2 Motivation for MPI While current MPI instruments are not precise enough yet to make it worth switching to MPI over other imaging modalities, such as MRI, MPI has the potential to be an extremely significant imaging technique in the future. Unlike MRI, which is heavily limited by the noise caused by thermal energy, MPI has a very high signal to noise ratio with an easily measurable signal with a particle concentration of 40µmols (measurable signals have been tested down to 8µmols) [5]. This signal strength has been measured at sub-millimeter resolutions [4]. Combined with the very fast scan time on the order of milliseconds, and the relatively simple and cheap instruments required to perform MPI scans, MPI has a lot of potential for affordable high temporal resolution imaging. Extensions of MPI techniques involving manipulation of super-paramagnetic ironoxide nanoparticles also have exciting medical applications. The polymer coatings of the nanoparticles can be made to target specific cells in the body, such as cancer cells, so the particles will agglomeration around the target cell, which will be very easily detectable with MPI. In addition, a main area of interest in early cancer treatment is in Hyperthermia 7

8 therapy, where nanoparticles are driven to vibrate by an external oscillating magnetic field to locally heat up whatever region is being targeted. If the nanoparticles could be made to agglomerate onto cancer cells, Hyperthermia may become an important tool for treating early cases of cancer. 2 Modeling Magnetic Nanoparticles 2.1 Single Particle in an Applied Field The magnetic response of super paramagnetic iron oxide nanoparticles in an external magnetic field can be modeled using the Langevin theory of paramagnetism [5], which states that the particle s magnetization as a function of applied field will go along the Langevin Equation. ( ) ( ( ) ) (1) Here, is the scaled oscillating magnetic field ( ) B (ω ) (ω ) (2) where is the magnetic moment of a single nanoparticle, is Boltzmann s constant, is the temperature, B is the magnitude of the external oscillating field, and ω is the frequency of that field. We define the z-axis as aligning with the direction of the oscillating field. The resulting magnetization, because of the non-linear magnetic response of the particles, is an oscillating signal with higher harmonics, seen in Figure 1, and described by the function 8

9 ( ) ( ( B (ω )) B (ω ) ) (3) Figure 1: Magnetization response of the magnetic nanoparticles in an oscillating applied field. The non-linear response generates a resulting oscillating magnetization signal with higher harmonics As we can see in the non-linear response, given by the Langevin equation, the larger the magnitude of the applied field is, the more the magnetization of the nanoparticles will saturate. In other words, the larger the magnitude of the applied field is, the more square the resulting magnetization signal will appear, because the particles will saturate quicker on every up-stroke of the field. An extremely important distinction to understand is that the Eq. (3) only describes the magnetization of the nanoparticles in equilibrium with the external oscillating field. In other words, this model describes the low frequency limit, or adiabatic approximation, where the particles are able to physically rotate fast enough remain at equilibrium with the applied field at any given point in time. We will look at how to model particles in a more general sense in section

10 2.2 Distribution of Particles in an Applied Field Of course, MPI measurements deal with a large distribution of magnetic nanoparticles. Rather than applying the Langevin model for numerous molecules (although I will demonstrate later in this section that the Langevin Model still works and is consistent with the model I am about to introduce), we can model this system much more elegantly with a statistical distribution model. In the statistical model, we look at the distribution of orientation angles, θ, of the particles off of the z-axis (aligned with the static magnetic field), as shown in Figure 2. Assuming that the primary particle relaxation mechanism is Brownian, with other relaxations negligible (I will discuss the prospect of introducing other relaxations in Section 5), we can define the relaxation constant as τ (4) where V is the volume of a nanoparticle and is the viscosity of the fluid the particles are in. z θ m B Figure 2: Orientation angle of the magnetic moment of a particle with respect to the direction of the magnetic field. 10

11 If we define ( ) as the probability density of the orientation angles, where ( ), we can use the corresponding Fokker-Planck equation (FPE) to describe the time evolution of this distribution [2] τ ( ) [( ) ( ( ) ( ) ( ))] (5) where ( ) is the scaled oscillating field, as described earlier. This FPE allows us to define the net magnetization from the expectation value of the orientation angles. ( ) ( ) ( ) (6) To show that this model is consistent with the single particle Langevin model, we can solve the FPE in the adiabatic approximation. Recalling that the adiabatic approximation is where the particles are at equilibrium with the applied fields at all times, we can solve for this state by setting the time derivative of the probability density,, to zero, and setting the applied field, ( ), to a constant value. These conditions yield the expected result; the expectation value of the orientation angle corresponds with the Langevin model [2]. ( ) ( ) (7) And by using (6), we can see that this statistical Fokker-Planck adiabatic approximation agrees with the single particle Langevin adiabatic approximation. 11

12 ( ) ( ( ) ) (8) An important relationship to keep in mind, which will come into play in Section 2.4, is the product of the field oscillation frequency and the Brownian relaxation constant, ωτ. We can see that the left side of the FPE is analogous to this product, and that the adiabatic approximation, which sets the left side of the FPE to zero, is analogous to this product equaling zero. Thus, we can interpret the adiabatic approximation as being when the product of the oscillation frequency and relaxation constant approaches zero. Adiabatic Approx: τ ( ) ωτ (9) This description of the adiabatic approximation has some important implications. If ωτ is approaching zero, that can mean two things: that the oscillation is getting slower, or that the relaxation constant is getting smaller. The oscillation frequency getting slower is what we would expect for an adiabatic approximation, because remember that our initial definition of the approximation is that the applied field was oscillating slowly enough for the particles to always be in equilibrium. But if the approximation also holds true for when the relaxation constant is getting smaller, that means it can be used to describe scenarios where the particles are sufficiently small, the solution is sufficiently thin, or the environment is sufficiently hot (because τ ). These are important conditions to remember during the qualitative analysis in Chapter 4. 12

13 2.3 Solving the FPE with Matrix Continued Fractions So far, I have only discussed ways to solve for the magnetic behavior of nanoparticles in the adiabatic approximation. This approximation is a useful analysis tool, but with typical particle sizes being between nm, and typical oscillation speeds of 25kHz [4], the adiabatic approximation is too far from any real experiments to predict real results. Using a numerical method called the Matrix Continued Fraction technique, as described by Coffey et. all, the Fokker-Planck equation for Brownian relaxation can be solved numerically [1]. This method uses multiple recursion relationships with the orientation angle probability density to solve for individual Fourier coefficients of the magnetization signals. This method will prove to yield important insights in Section Ratio of the 5 th /3 rd Harmonic Rather that only looking at the magnetic response signal as a function of time, a very useful analysis tool is to look at the ratio of the 5 th to 3 rd harmonic of the magnetic response signals. Because of the particles non-linear magnetic response, there are plenty of higher harmonics in their response signal. Also, looking at the 5 th and 3 rd harmonics eliminates any potential background noise in the fundamental frequency, which could likely come from the applied oscillating field. Specifically, we look at the 5 th to 3 rd harmonic as a function of the product of the oscillating frequency and the relaxation time, ωτ, as see in Figure 3. This plot was generated by a script written by Martens et all [2]. What we see makes sense, because if we 13

14 Scaled Ratio of 5th to 3rd Harmonic increase ωτ (for example, speeding up the oscillating field), then the magnetic response signal would not have as much of a chance to saturate. The saturation is what creates most of the higher harmonics, so the 5 th to 3 rd harmonic ratio would decrease ωτ B Figure 3: Ratio of 5th to 3rd harmonic of magnetization response signal as a function of vertical axis is scaled by 5/3, such that a ratio of 1 corresponds to a square wave. The 14

15 3 Particle Agglomeration 3.1 Motivation for studying particle agglomeration So far, I have discussed ways to represent a system of completely ideal, uniform magnetic nanoparticles in a closed system with no inter-particle interaction and no relaxation mechanism besides Brownian. Obviously this is not how the particles truly act. And one of the most prominent examples of non-ideal particle behavior in current experiments is particle agglomeration. Solutions of magnetic nanoparticles have been seen to passively clump up over time, without any real explanation as to why. In addition, deviations from expected particle behavior have almost blindly been accredited to particle agglomeration as the most likely source of error. However, no formal study of how particle agglomerations behave, or how they might alter the expected measurements, has been done. Furthermore, there might be practical applications beyond error analysis for a theoretical model of particle agglomeration behavior. One experimental method for nanoparticle study is to freeze the particles in a film in order to isolate certain behaviors and characteristics. This freezing is essentially forming particle agglomerations. Another use of agglomerations lies in one of the many potential medical applications of magnetic nanoparticles. Dr. Weaver at University of Dartmouth is attempting to use agglomerated strings of nanoparticles in order to target cancer cells in the body. Once again, a formal theoretical model for particle agglomerations might prove useful. In the rest of this paper, I will explore the theoretical effect of magnetic nanoparticle agglomerations on the previously mentioned analysis methods. 15

16 3.2 Simulating Agglomeration First, I will propose a model to simulate the effect of agglomeration on our established terms and variables. I will assume spherical agglomerations of N number of particles. I will also assume that there is no inter-particle interaction biasing the magnetic moment configuration in any particular way in an agglomeration. In other words, the particle orientations will be random with respect to each other inside an agglomeration. And now, if we treat the agglomerations as a static unit, we can assign it properties based on the properties of the N particles that make it up. τ τ (10) The most important aspect of the agglomeration is that the net magnetic moment is dependent on how the particles randomly stuck together. The maximum value the net magnetic moment would be, if every particle was constructively aligned, is the moments of the individual particles times the number of particles in the agglomeration:. The minimum value of the net magnetic moment the magnetic would be zero, if every particle was deconstructively aligned. Likely, the net moment will lie somewhere inbetween, and its value will be determined by the probability distribution, which I define as 16

17 where (11) = 1 represents the particles moments adding completely, and represents the particles moments completely canceling each other out. Thus, it follows that ( ) B (ω ) ( ) (12) To determine what the distributions where, I simulated the process of N number of particles sticking together. I did this by randomly generating N number of 3-dimensional Figure 4: Gamma distributions for different values of N 17

18 unit vectors, and adding them together. I then divided the magnitude of the resulting vector by N determine one value of. For each value of N, I iterated this simulation ten thousand times, generating a statistical probability distribution for as a function of N, as seen in Figure 4. The results that we see make sense. The higher the N value, that is, the more particles that clump up, the more likely it is for the magnetic moments to cancel each other. In other words, the bigger agglomerations there are, the more net magnetization that is lost in the overall system. 3.3 Effects of Agglomeration on Magnetization Using the modified terms for agglomeration, we can use the Langevin and the Fokker- Planck models to determine the actual effect of the agglomeration on the magnetization signal. In order to modify the magnetization signal, we must integrate the magnetization over all the values for for a given N. And because the distributions are discrete distributions, we perform this integral as a Riemann sum. ( ) ( ) ( ) ( ) ( ) (13) ( ) In the adiabatic approximation, we can modify the Langevin model to determine 18

19 ( ) [ ( ( )) ( ) ] (14) Using this, we can easily plot the effect of particle agglomeration on the magnetic response signal in the adiabatic approximation, seen in Figure 5. Figure 5: Magnetization response in the adiabatic approximation with agglomerations N=1, 2, 3, 4, 5, 10, and 20 We can see that, in the adiabatic approximation, as the particles form larger agglomerations, the overall magnitude of the signal decreases, and the shape of the signal becomes squarer. By modifying the matrix continued fractions technique to include particle agglomeration parameters, we simulate what actual measurements in the non-adiabatic model would look like with agglomeration for different values of ωτ, as seen in Figure 6. In the non-adiabatic approximation, we see a distinctly different result. Particularly noticeable with the ωτ plot, as the agglomerations get bigger, the signal becomes rounder, rather than squarer. 19

20 Figure 6: Magnetization vs. scaled time (ωt) for various ωτ_b values with agglomerations N=1, 2, 3, 4, 5, 10, and 20 20

21 3.4 Effects of Agglomeration on 5 th /3 rd Harmonics By looking at the effect of agglomeration on the ratio of the 5 th to 3 rd harmonic, we will illuminate how agglomeration modifies the behavior of the signal, as well as the key differences between the adiabatic and non-adiabatic model. I incorporated the modified agglomeration terms into the 5 th to 3 rd script in order to generate plots of the 5 th to 3 rd ratio for different values of agglomeration. These plots can be seen in Figures 7 and 8. Figure 7: Ratio of 5th to 3rd harmonic as a function of ωτ B for various N values of agglomeration Figure 8: Zoomed in 5th/3rd plot to see adiabatic limit. Note that the agglomeration ratios are higher than the un-agglomerated ratio at the adiabatic limit, but cross at ωτ B ~.3. 21

22 4 Discussion We see that particle agglomeration has a very noticeable effect on the measurements made for MPI, and that further understanding of the cause and behavior of particle agglomerations is necessary to make accurately measurements with magnetic nanoparticles. In the basic magnetization measurements, we see that the signal looks significantly weaker with more agglomeration, which makes it look like the particle density is lower than it is. When looking at the ratio of the 5 th to 3 rd harmonic of particles with ωτ values between 4 and 20, as is most frequently the case in current experiments, the ratio is lower with agglomerations than without. More specifically, if a measurement is taken with no knowledge of there being agglomerations, the result will appear to be further to the right on the above ratio plot, thus giving the effect of there being a higher viscosity or frequency than there actually is. This is also reflected in the magnetization signal plots, where the agglomeration signals are rounder, both of which reflects a higher viscosity. Interestingly, near the adiabatic limit, the opposite is true; the agglomeration ratios are higher than the un-agglomerated ratio. This is reflected in the adiabatic magnetization response plot, as the agglomerated responses are squarer. Low frequency or low viscosity experiments will see agglomerations giving the effect of lower viscosities. In more general terms, the relaxation constant seems inflated in the non-adiabatic model, and it seems deflated in the adiabatic approximation. This could result in misinformation in an experiment if it was not understood that agglomeration was the cause. 22

23 5 Future Direction There are many directions that I would like to take this research in the future. First, I would like to develop a way to quantitatively describe the particle agglomeration, so it can be quantitatively predicted and corrected for. I did some initial work attempting to fit the ratio of 5 th to 3 rd harmonics to an exponential, and was getting reasonable results. I believe with some extra work, a fairly accurate quantitative model for agglomeration effects could be developed. Not only would allow for accurate error correcting, but it would allow the amount of agglomeration to be calculated from experimental data. Another unexplored area is how to accommodate for when a sample is only partially agglomerated. This would involve introducing another probability function, depicting how much of the sample would agglomerate. However, the parameters that would affect this are unknown, as we do not currently have a good grasp on what causes particle agglomeration in the first place. Additional corrections could be made for if the particles did bias each other in one way or another as they agglomerated, such that their magnetic moments were not randomly oriented against each other. This factor would likely depend on the thickness of the coating on the particles, Finally, if we are to consider using smaller particles in the future, such as <20nm, we need to understand how Neel s relaxation affects the behavior of the particles in combination with Brownian relaxation. We need a Fokker-Planck equation that fully incorporates both relaxation methods. With that, agglomeration effects could be easily calculated for both relaxation methods, allowing for a much more comprehensive theoretical model for all potential MPI applications 23

24 References [1] Coffey, W. T., Kalmykov, Y. P., & Waldron, J. T. (2004). The Langevin Equation, 2nd ed. Singapore: World Scientific. [2] Martens, M. A., Deissler, R. J., Wu, Y., Yau, Z., Brown, R., & Griswold, M. (2012). Modeling the Brownian relaxation of nanoparticle ferrofluids: Comparison with experiment. Cleveland, OH: Department of Physics, Case Western Reserve University. [3] Weaver, J. B., & Kuehlert, E. (2012). Measurement of magnetic nanoparticle relaxation time. Med. Phys. 39. [4] Weizenecker, J., Gleich, B., Rahmer, J., Dahnke, H. (2009). Three-dimensional real-time in vivo magnetic particle imaging. Phys. Med. Biol 54. [5] Weizenecker, J., Borgert, J., and Gleich, B. (2007). A Simulation Study on the Resolution and Sensitivity of Magnetic Particle Imaging. Phys. Med. Biol