Toward 2D and 3D imaging of magnetic nanoparticles using EPR measurements

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1 Toward 2D and 3D imaging of magnetic nanoparticles using EPR measurements A. Coene, G. Crevecoeur, J. Leliaert, and L. Dupré Citation: Medical Physics 42, 5007 (2015); doi: / View online: View Table of Contents: Published by the American Association of Physicists in Medicine Articles you may be interested in Magnetic nanoparticle imaging using multiple electron paramagnetic resonance activation sequences J. Appl. Phys. 117, 17D105 (2015); / Size-dependent ferrohydrodynamic relaxometry of magnetic particle imaging tracers in different environments Med. Phys. 40, (2013); / Magnetic resonance imaging of microvessels using iron-oxide nanoparticles J. Appl. Phys. 113, (2013); / A handheld fluorescence molecular tomography system for intraoperative optical imaging of tumor margins Med. Phys. 38, 5873 (2011); / X -band EPR imaging as a tool for gradient dose reconstruction in irradiated bones Med. Phys. 36, 4223 (2009); /

2 Toward 2D and 3D imaging of magnetic nanoparticles using EPR measurements A. Coene a) and G. Crevecoeur Department of Electrical Energy, Systems and Automation, Ghent University, Zwijnaarde 9052, Belgium J. Leliaert Department of Electrical Energy, Systems and Automation, Ghent University, Zwijnaarde 9052, Belgium and Department of Solid State Sciences, Ghent University, Ghent 9000, Belgium L. Dupré Department of Electrical Energy, Systems and Automation, Ghent University, Zwijnaarde 9052, Belgium (Received 24 February 2015; revised 6 June 2015; accepted for publication 14 July 2015; published 5 August 2015) Purpose: Magnetic nanoparticles (MNPs) are an important asset in many biomedical applications. An effective working of these applications requires an accurate knowledge of the spatial MNP distribution. A promising, noninvasive, and sensitive technique to visualize MNP distributions in vivo is electron paramagnetic resonance (EPR). Currently only 1D MNP distributions can be reconstructed. In this paper, the authors propose extending 1D EPR toward 2D and 3D using computer simulations to allow accurate imaging of MNP distributions. Methods: To find the MNP distribution belonging to EPR measurements, an inverse problem needs to be solved. The solution of this inverse problem highly depends on the stability of the inverse problem. The authors adapt 1D EPR imaging to realize the imaging of multidimensional MNP distributions. Furthermore, the authors introduce partial volume excitation in which only parts of the volume are imaged to increase stability of the inverse solution and to speed up the measurements. The authors simulate EPR measurements of different 2D and 3D MNP distributions and solve the inverse problem. The stability is evaluated by calculating the condition measure and by comparing the actual MNP distribution to the reconstructed MNP distribution. Based on these simulations, the authors define requirements for the EPR system to cope with the added dimensions. Moreover, the authors investigate how EPR measurements should be conducted to improve the stability of the associated inverse problem and to increase reconstruction quality. Results: The approach used in 1D EPR can only be employed for the reconstruction of small volumes in 2D and 3D EPRs due to numerical instability of the inverse solution. The authors performed EPR measurements of increasing cylindrical volumes and evaluated the condition measure. This showed that a reduction of the inherent symmetry in the EPR methodology is necessary. By reducing the symmetry of the EPR setup, quantitative images of larger volumes can be obtained. The authors found that, by selectively exciting parts of the volume, the authors could increase the reconstruction quality even further while reducing the amount of measurements. Additionally, the inverse solution of this activation method degrades slower for increasing volumes. Finally, the methodology was applied to noisy EPR measurements: using the reduced EPR setup s symmetry and the partial activation method, an increase in reconstruction quality of 80% can be seen with a speedup of the measurements with 10%. Conclusions: Applying the aforementioned requirements to the EPR setup and stabilizing the EPR measurements showed a tremendous increase in noise robustness, thereby making EPR a valuable method for quantitative imaging of multidimensional MNP distributions. C 2015 American Association of Physicists in Medicine. [ Key words: inverse problems, electron paramagnetic resonance (EPR), magnetic nanoparticles, image reconstruction 1. INTRODUCTION Magnetic nanoparticles (MNPs) have interesting properties which can be exploited in biomedical applications. 1 3 Their high saturation magnetization allows noninvasive detection using sensitive coils such as SQUIDs (Refs. 4 6) or Fluxgates. 7 9 Due to their small diameter, they can reach almost every area in the body making them ideal carriers for localized drug delivery Furthermore, by applying a time-varying magnetic field, they can generate heat and this way destroy malignant tissue, i.e., hyperthermia To allow a safe and adequate working of these applications, an accurate knowledge of the spatial MNP distribution is required. In this paper, we employ the promising electron paramagnetic resonance (EPR) technique for visualizing MNP distributions. In an EPR experiment, a sample is placed in a homogeneous magnetic field and the unpaired electrons are excited by an incident wave, which alters the direction 5007 Med. Phys. 42 (9), September /2015/42(9)/5007/8/$ Am. Assoc. Phys. Med. 5007

3 5008 Coene et al.: Multidimensional imaging of magnetic nanoparticles using EPR 5008 of their magnetic moments. This is similar to MRI, where the magnetic moments of nuclei instead of the electrons are investigated. 16 Because MRI does not measure the MNP directly, but only their impact on the relaxation of the protons, challenges remain to be solved in order to determine the spatial MNP distribution in a quantitative way. 17 Furthermore, EPR has a larger specificity with respect to MRI, as EPR can only measure unpaired electrons. On the other hand, MRI can achieve high resolution anatomical images, and a multimodal approach together with EPR could be very useful. 18 In traditional EPR, the incident wave frequency is commonly around 9 GHz, which does not allow penetration of the wave in the human body and is thus not suitable for in vivo visualization of the MNP. Furthermore, an indirect effect the absorption of the electromagnetic wave by the sample is measured. The EPR setup studied in this paper measures the MNP magnetization directly through a pickup coil. 19 Because of the high magnetic susceptibility of the MNP, a large magnetization can be achieved in a low magnetic field, so that the incident magnetic wave for the setup in this paper can have a low frequency between 60 and 500 MHz. This way the penetration depth in tissues is increased and in vivo imaging is possible. The drawback of this technique is the lack of spatial information of the particles since only the MNP amount in a single voxel can be obtained. In Refs. 20 and 21, the EPR setup recovers the 1D distribution of the MNP by moving the sample through the magnetic field, denoted as 1D EPR. A so-called inverse problem needs to be solved to obtain this 1D distribution. First, a forward model is evaluated which calculates for a certain MNP distribution the corresponding EPR measurement. We need to find the MNP distribution which minimizes the differences between the calculated EPR measurement and the actual measurement. This problem is typically ill-posed as the number of unknowns (the MNP distribution) is much larger than the EPR measurement sites. Such a minimization is generally solved using truncated singular value decomposition (TSVD). 22 In this paper, we present a general technique for reconstructing a spatial multidimensional MNP distribution using EPR measurements. We analyze the condition of the linear inverse problem for different EPR implementations and we numerically investigate how the associated inverse problem needs to be adapted to cope with the added dimensions. These added dimensions require a different way of obtaining measurement data with the goal to achieve fast measurements and to attain an increased stability of the inverse problem. Therefore, we introduce a measurement method in which we activate only certain parts of the volume. Finally, we study the impact of noise to assess the robustness of the proposed 3D EPR technique. 2. MATERIALS AND METHODS 2.A. Single voxel EPR Single voxel EPR is a sensitive technique which allows to determine the total MNP amount in a sample, typically expressed in iron amounts. In our experiments, we employ FIG. 1. Schematic representation of the placement of MNP sample and coils in the EPR setup (not to scale). Resovist (Schering AG, Berlin, Germany) particles, a clinically approved contrast agent for MRI. 23 The setup consists of three perpendicular coils, schematically shown in Fig. 1. One Helmholtz coil pair for generating the homogeneous magnetic field, an excitation coil for producing the radio frequency wave, and a sensing coil for measuring the magnetization. The radio frequency wave flips the magnetization of the particles toward the sensing coil. The signal from the sensing coil is proportional to the amount of particles in the sample. A more elaborated explanation about the EPR setup can be found in Refs. 19 and B. EPR measurements The unknown MNP distribution is denoted as the 1 N dimensional vector c. It contains all the iron concentrations on different grid points, so that N is the total number of grid points. In the case of 1D EPR, this is a 1D grid, and in 2D and 3D EPRs, these are 2D and 3D grids, respectively. An EPR measurement is represented by S EPR = [S 1,..., S m,..., S M ]. (1) These M measurements originate from the M positions of the sample during a translation and/or rotation of the sample through the magnetic field (see Fig. 1). In 1D EPR, the sample is only translated along a line through the magnetic field. In 2D EPR, similar translations are performed but in a plane instead of along one line. In 3D EPR, additionally rotations of the sample are allowed. For every step of the 1D translation, a full rotation of the sample is performed. Because of the changing distances with respect to the excitation and sensing coil for every position m, a different measurement value is obtained for an equal iron concentration. These varying measurements are incorporated in the system response function R which contains the different measurement values as a function of the position m in the magnetic field for a unit iron concentration. A unit concentration is chosen as the signal linearly scales with the iron concentration. m represents the spatial variables, for example, R(z) in 1D EPR, R(x,z) in 2D EPR, and R(r,θ,z) or R(x,y,z) in 3D EPR depending on

4 5009 Coene et al.: Multidimensional imaging of magnetic nanoparticles using EPR 5009 FIG. 2. By placing only a part of the sample in the homogeneous magnetic field (and thereby limit the zone of particle response), more independent information is available to solve the inverse problem which results in improved reconstructions of the MNP distribution. the type of coordinate system employed. To find the response of a certain MNP amount on grid position m, the response value on m, R(m), is multiplied with the MNP amount on grid position m. S can then be modeled by S(c) = F c, (2) F is a M N system matrix consisting of the associated response values for every grid concentration for the M positions of the sample. Row m of F, i.e., position m of the sample, is constructed from all the corresponding response values R(m n ) for every unknown local c n, n = 1,...,N on position m n. In 3D EPR measurements, M typically equals Z Θ with Z the number of translations of the sample and Θ the maximum number of rotational movements of the sample. It is possible to place only a part of the volume in the region Ω where R(m) is defined, see Fig. 2. This is referred to as partial volume excitation. In this case, R(m n ) is zero for c n with m n Ω. In practice, the sample is moved to measurement positions such that Ω comprises a different region of the sample. In the previous approach, the complete sample was placed in Ω. Partial volume excitation can reduce the number of measurements by selectively activating parts of the volume, thus reducing the requirement of a complete translation of the volume through the magnetic field. Magnetic gradient fields can further improve this measurement method by actively targeting certain areas of the sample. 2.C. EPR for visualizing MNP distributions The MNP distribution can be recovered by solving an inverse problem, c = argmin c S(c) S EPR, (3) S EPR represents the EPR measurements and S(c) the forward model solution. This inverse problem is solved by TSVD. In TSVD, the Moore Penrose inverse, F, is calculated as κ c = F κs EPR = (UΣV T ) κs u T l EPR = S EPR v l. (4) σ l In TSVD, the eigenvalues from F, denoted by σ l, are obtained and are organized according to decreasing size. Only the κ largest eigenvalues are kept, because smaller eigenvalues are commonly associated to noise sources. In Ref. 20, a method was devised to acquire the optimal κ for EPR imaging. A general confidence interval for the recovered MNP distribution can be obtained from numerical simulations for various MNP distributions, different discretizations, noise levels, etc., in which the reconstructed MNP distribution is compared to the simulated MNP distribution. Note that for performing accurate reconstructions of the MNP distribution, the response function needs to be a priori measured accurately. As given in Eq. (4), the reconstructions depend on the response values. In Ref. 21, the impact of this type of error was investigated. 3. RESULTS 3.A. Reconstruction quality The reconstruction quality of the presented technique needs to be assessed. In this paper, we use the correlation coefficient, N (cn c )(c n c) n=1 CC =, (5) N (cn c ) 2 N (c n c) 2 n=1 n=1 c is the reconstructed distribution and c is the actual distribution, c and c are their averages. A CC of one corresponds to a perfect reconstruction of the actual distribution. Furthermore, we investigate the condition of the linear inverse problem with the condition measure ρ, 24 σ 1 ρ =. (6) 1/L L σ l l=1 Smaller ρ signifies improved numerical stability. In contrast to the conventional condition measure, Eq. (6) depends less on the smallest eigenvalues, which allows us to investigate and compare different sizes of F. l=1 3.B. Synthetic 3D system response We generated synthetic 3D response functions based on the measured 1D system response [Fig. 3(a)]. The 1D system response shows the fact that particles further away from the excitation and sensing coil are moderately excited by the radio frequency wave and only partly registered by the sensing coil, resulting in a lower response value. The maximum value, which is equal to one, corresponds to the position where the particles are completely excited and sensed by the coils. For the 3D response, we assume a similar behavior with respect to the two coils, a decrease of the response value in the directions further away from the two coils. Furthermore,

5 5010 Coene et al.: Multidimensional imaging of magnetic nanoparticles using EPR 5010 FIG. 3. (a) Measured 1D response function; (b) synthetic 3D response function with radius of 4 mm; (c) synthetic 3D response function with radius of 4 mm and reduced θ symmetry. we assume a certain circular symmetry with respect to the coils. The symmetry does not result in equal response values due to noise and small displacements in hardware; therefore, white Gaussian noise with a noise level of 1% with respect to the local response values was added to the model. These assumptions were also experimentally validated by performing coarse measurements of the response function on a few positions in the EPR setup. We defined the 3D response function in cylindrical coordinates to show the rotational symmetry along the θ dimension. Due to the current diameter of the sensing coil ( 9 mm) of the setup, only a 4 mm radius of the response is possible [Fig. 3(b)]. We calculated two additional response functions with radii 8 and 12 mm to assess the influence of volumes with a larger diameter. We additionally modeled these three response functions with larger variations with respect to the θ-direction, R adap, to investigate the impact of reduced θ symmetry [Fig. 3(c)]. This reduced symmetry of the response can be implemented in the system by adding magnetic field gradients or by slightly changing the positions of the coils, as the response of the particles depends on the local magnetic field surrounding the particle. These synthetic 3D responses are made to investigate the necessary requirements for the system response function and to show the impact of variations in system response on the inverse problem. 3.C. Requirements of the system matrix In a first approach, we used the symmetric 3D response to generate system matrices. The sample was placed in the magnetic field and translated in steps of 1 mm, no partial activation is employed. For each step of the translation, a full rotation (360 in steps of 10 ) is performed. The resolution of the volume is fixed at 1 mm. We performed noise-free reconstructions of different samples having dimensions of a a a mm 3 with a = 1,...,10. We obtain high-quality reconstructions until N 200. Figure 4 shows an in-plane reconstruction example of a mm 3 with N = 125, M =170 and a mm 3 volume with N = 1000, M = 729. For the smaller volume, a perfect reconstruction is obtained, while the larger volume is unrecognizable. This way of performing 3D EPR only allows the imaging of small MNP volumes as the results deteriorate quickly for larger volumes. Furthermore, the measurement procedure is timeconsuming. Because the response function depends on the relative distances to the two coils, a circular symmetry is observed where values do not differ a lot. By incorporating these values in F, it is possible that linearly dependent information is added and thus the condition of the linear inverse problem deteriorates. To investigate this, we made system matrices of different sizes corresponding to the reconstructions of different volume sizes. We start with the reconstruction of a cylindrical volume with length 5 mm and radius 1 mm having only an angle of 10. The corresponding condition of this inverse problem is ρ(f) 7 [Eq. (6)]. Then the volume was increased each time along 1D (r, θ or z), and for every increase of the volume, the condition of the inverse problem was evaluated. Figure 5(a) shows ρ for all these volumes. In this case the response function with a radius of 12 mm was employed (full lines). It can be seen that increasing the θ dimension results in a decrease of system stability. These reconstructions were also made with R adap which does not have this circular symmetry. Then an improved stability for the system matrix is obtained (dashed line). For an increase in r and z dimensions, the dashed line is equal to the respective full lines as only the θ direction has changed. Similar results were obtained for the other two response functions with smaller radii (r is limited to their respective radii). Figure 5(a) also clearly shows that for increasing radius (r) and length (z), the stability is not reducing, making the technique suitable for samples with larger radii and lengths. Figure 5(b) depicts the

6 5011 Coene et al.: Multidimensional imaging of magnetic nanoparticles using EPR 5011 FIG. 4. 3D EPR imaging of a small mm 3 volume and a larger mm 3 volume using the symmetrical response function. Using this first approach, only small volumes can be reconstructed. normalized eigenvalue distribution [Eq. (4)] for the case of r = 1 mm, θ = 360, and z = 5 mm for R and R adap. The slower decrease of the R adap eigenvalues and their relative larger sizes makes the solution more robust compared to R. The cutoff value κ was determined according to Ref. 20. By reducing the EPR symmetry, quantitative images of larger MNP samples (N 800) from Fig. 4 can be obtained. The numerical stability of the system matrix was further improved by placing only a part of the sample in the homogeneous field, so only a well-defined segment of the sample generated the particles response (Fig. 2). In Fig. 6(a), 2D reconstruction example is shown with partial volume excitation [Fig. 6(b)] and without partial volume excitation [Fig. 6(c)]. In the case of partial volume excitation, the sample was gradually translated, in steps of 1 mm, into the magnetic field generated by the Helmholtz coil, thereby increasing the part of the sample in Ω. Note that the reconstruction quality is lower compared to that in Fig. 4, this is due to the shape of the sample which allows less measurements compared to the number of unknowns. The amount of measurements was adapted so both measurement methods obtained the same number of measurements (M = 28, N = 77). The reconstruction with partial volume excitations shows an increase in reconstruction quality (a CC of 63% compared to a CC of only 10%) and FIG. 5. (a) Numerical stability of F for increasing volume dimensions. Because of symmetry reasons, an increase in θ dimension results in a decrease in system stability. Using R adap, this system instability is reduced. Furthermore, an increase in r and z dimensions has no large impact on system stability. (b) The eigenvalue distribution of R adap shows a relative slower decrease which improves the stability of the system matrix.

7 5012 Coene et al.: Multidimensional imaging of magnetic nanoparticles using EPR 5012 FIG. 6. Reconstruction of a 2D MNP distribution with and without partial volume excitation. When partial volume excitation is employed, improved reconstructions of the MNP distribution are obtained. in numerical stability (ρ(f) 28 ρ(f) 15). This increase in stability was also observed in the system matrices of Fig. 5(a) and now a perfect reconstruction of the larger volume of Fig. 4 is attainable. Figure 7 depicts the averaged reconstruction scores and their standard deviation of 50 randomly generated MNP distributions with and without partial volume excitation for a volume with fixed θ and r dimensions, but increasing z dimension. The larger the MNP sample and thus the more ill-posed the problem, the better the partial volume excitation performs with respect to the conventional measurements. The partial volume excitation was performed by gradually inserting the sample in the magnetic field. In these simulations, the total number of measurements compared to the total number of unknowns was kept constant for both methods (by increasing/decreasing the number of measurement positions) as the ratio of M to N also has an impact on the inverse problem. 21 The improved reconstructions thus only originate from adding information to the inverse problem to be solved, by only exciting parts of the sample instead of the complete sample. Because of the increased stability, we can therefore speed up the measurement method while attaining an increase in reconstruction quality. This way we can increase measurement speed by 10%. This makes us believe that the EPR methodology can be further improved by using magnetic field gradients that can actively target certain areas of the sample. investigate the robustness toward noise for these adaptations. White Gaussian noise was increased in steps of 1%. For every noise level, 200 noise measurements are performed which are then averaged. The noise levels of 1% 10% are with respect to the system response amplitude and correspond 3.D. Noise robustness of 3D EPR Previous noise-free simulations regarded the stability of the EPR matrix. We reduced the symmetry in the original response function to obtain improved solutions for the inverse problem. Additionally, we introduced the concept of partly exciting the sample which further improved stability of the methodology and showed improved reconstructions. In this section, we FIG. 7. Larger 3D MNP volumes deteriorate the condition of the inverse problem. For every volume 50, randomly generated 3D MNP distributions were reconstructed with and without partial volume excitation. When employing partial volume excitation, the condition of the inverse problem degrades slower.

8 5013 Coene et al.: Multidimensional imaging of magnetic nanoparticles using EPR 5013 FIG. 8. (a) Noise robustness of the different approaches used in this paper; (b) example of a 3D MNP distribution; (c) reconstruction of the 3D MNP distribution from (b) using a noise level of 10% and the adapted system response. to actual noise values of the setup. 20 Figure 8(a) shows the average CC and standard deviation for increasing noise levels for the different approaches used in this paper. As stated previously, the original response function R is not able to reconstruct larger MNP samples. Now that noise is added, the solution even deteriorates faster than before. R adap delays the deterioration due to the reduced symmetry but shows significant noise impact (CC goes from 0.8 to 0.13) and a large standard deviation. When using the partial volume excitation, in which we gradually insert the sample in the magnetic field, we see a tremendous increase in reconstruction quality for both response functions (R 0.82 and R adap 0.92) and the standard deviation is now significantly lower. Figure 8(c) shows a reconstruction example of the MNP distribution depicted in Fig. 8(b) using the adapted response function with partial volume excitation and a noise level of 10%. 4. DISCUSSION In this paper, we presented a general technique to reconstruct multidimensional MNP samples using EPR. The extension of the 1D approach from Ref. 20 to 2D and 3D works fine for smaller volumes (N < 200), but numerical stability issues arise when imaging larger volumes. We generated different system matrices for cylindrical volumes which were increased along 1D. The deterioration of the solution was especially visible for increasing θ dimensions, which is due to the symmetrical property of the 3D response function R. r and z dimensions only had a small effect on the stability. The stability of the inverse problem showed that a reduction in symmetry is necessary to assure a system matrix which is sufficiently stable to allow sensitive reconstructions of the MNP distribution. To reduce the symmetry of the response function, magnetic field gradients can be employed. The stability of the system matrix was further increased by only exciting a part of the MNP sample. In this first approach, this was done by gradually inserting/removing the sample in the magnetic field. This adds linearly independent information to the inverse problem to be solved and translates into improved reconstructions. Due to the increased stability and better imaging results, it is also possible to reduce the number of measurement positions and this way speed up our technique. A speedup of 10% can be achieved while still obtaining an increased reconstruction quality. We expect to further improve this measuring method if magnetic gradient fields are employed which can target specific areas of the volume. Finally, the robustness of our method toward noise was investigated, which showed that our adaptations realize an increase in reconstruction quality, and the EPR setup maintains its stability with respect to larger multidimensional samples. In the future, we plan to speed up the measurements by adding magnetic gradient fields. This way sample movement can be simulated. These gradient fields also reduce the symmetry in the system s response and thus will increase the performance of the EPR imaging modality even further. 5. CONCLUSIONS This paper extends the reconstruction of 1D magnetic nanoparticle distributions using EPR signals toward 2D and 3D and analyzes in-depth the feasibility of multidimensional reconstruction using numerical computer simulations. Due to the larger number of unknowns, the stability of the inverse problem has to be improved. This is done by reducing the inherent symmetry of the setup and by introducing a new measurement method in which the MNP sample can be activated partially. The adaptations show a large noise robustness (increase in reconstruction quality by 80%), enable 3D reconstructions of larger samples, and allow to reduce measurement time by 10%.

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