Toward 2D and 3D imaging of magnetic nanoparticles using EPR measurementsA. Coene, G. Crevecoeur, J. Leliaert, and L. Dupré Citation: Medical Physics 42, 5007 (2015); doi: 10.1118/1.4927374 View online: http://dx.doi.org/10.1118/1.4927374 View Table of Contents: http://scitation.aip.org/content/aapm/journal/medphys/42/9?ver=pdfcov 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); 10.1063/1.4906948 Size-dependent ferrohydrodynamic relaxometry of magnetic particle imaging tracers in different environments Med. Phys. 40, 071904 (2013); 10.1118/1.4810962 Magnetic resonance imaging of microvessels using iron-oxide nanoparticles J. Appl. Phys. 113, 124701 (2013); 10.1063/1.4797484 A handheld fluorescence molecular tomography system for intraoperative optical imaging of tumor margins Med. Phys. 38, 5873 (2011); 10.1118/1.3641877 X -band EPR imaging as a tool for gradient dose reconstruction in irradiated bones Med. Phys. 36, 4223 (2009); 10.1118/1.3194775

Toward 2D and 3D imaging of magnetic nanoparticlesusing EPR measurements

A. Coenea) and G. CrevecoeurDepartment of Electrical Energy, Systems and Automation, Ghent University, Zwijnaarde 9052, Belgium

J. LeliaertDepartment of Electrical Energy, Systems and Automation, Ghent University, Zwijnaarde 9052, Belgiumand Department of Solid State Sciences, Ghent University, Ghent 9000, Belgium

L. DupréDepartment 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 distri-bution. A promising, noninvasive, and sensitive technique to visualize MNP distributions in vivo iselectron 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 simulationsto allow accurate imaging of MNP distributions.Methods: To find the MNP distribution belonging to EPR measurements, an inverse problem needsto be solved. The solution of this inverse problem highly depends on the stability of the inverseproblem. The authors adapt 1D EPR imaging to realize the imaging of multidimensional MNPdistributions. Furthermore, the authors introduce partial volume excitation in which only parts of thevolume are imaged to increase stability of the inverse solution and to speed up the measurements. Theauthors simulate EPR measurements of different 2D and 3D MNP distributions and solve the inverseproblem. The stability is evaluated by calculating the condition measure and by comparing the actualMNP distribution to the reconstructed MNP distribution. Based on these simulations, the authorsdefine requirements for the EPR system to cope with the added dimensions. Moreover, the authorsinvestigate how EPR measurements should be conducted to improve the stability of the associatedinverse problem and to increase reconstruction quality.Results: The approach used in 1D EPR can only be employed for the reconstruction of small volumesin 2D and 3D EPRs due to numerical instability of the inverse solution. The authors performed EPRmeasurements of increasing cylindrical volumes and evaluated the condition measure. This showed thata reduction of the inherent symmetry in the EPR methodology is necessary. By reducing the symmetryof the EPR setup, quantitative images of larger volumes can be obtained. The authors found that, byselectively exciting parts of the volume, the authors could increase the reconstruction quality evenfurther while reducing the amount of measurements. Additionally, the inverse solution of this activationmethod degrades slower for increasing volumes. Finally, the methodology was applied to noisy EPRmeasurements: using the reduced EPR setup’s symmetry and the partial activation method, an increasein 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 theEPR measurements showed a tremendous increase in noise robustness, thereby making EPR avaluable method for quantitative imaging of multidimensional MNP distributions. C 2015 AmericanAssociation of Physicists in Medicine. [http://dx.doi.org/10.1118/1.4927374]

Key words: inverse problems, electron paramagnetic resonance (EPR), magnetic nanoparticles, imagereconstruction

1. INTRODUCTION

Magnetic nanoparticles (MNPs) have interesting propertieswhich can be exploited in biomedical applications.1–3 Theirhigh saturation magnetization allows noninvasive detectionusing sensitive coils such as SQUIDs (Refs. 4–6) or Flux-gates.7–9 Due to their small diameter, they can reach almostevery area in the body making them ideal carriers for localizeddrug delivery.10–12 Furthermore, by applying a time-varying

magnetic field, they can generate heat and this way destroymalignant tissue, i.e., hyperthermia.13–15 To allow a safe andadequate working of these applications, an accurate knowl-edge of the spatial MNP distribution is required.

In this paper, we employ the promising electron para-magnetic resonance (EPR) technique for visualizing MNPdistributions. In an EPR experiment, a sample is placed ina homogeneous magnetic field and the unpaired electronsare excited by an incident wave, which alters the direction

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of their magnetic moments. This is similar to MRI, wherethe magnetic moments of nuclei instead of the electronsare investigated.16 Because MRI does not measure the MNPdirectly, but only their impact on the relaxation of the protons,challenges remain to be solved in order to determine thespatial MNP distribution in a quantitative way.17 Furthermore,EPR has a larger specificity with respect to MRI, as EPRcan only measure unpaired electrons. On the other hand,MRI can achieve high resolution anatomical images, anda multimodal approach together with EPR could be veryuseful.18 In traditional EPR, the incident wave frequency iscommonly around 9 GHz, which does not allow penetrationof the wave in the human body and is thus not suitable for invivo visualization of the MNP. Furthermore, an indirect effectthe absorption of the electromagnetic wave by the sample ismeasured. The EPR setup studied in this paper measures theMNP magnetization directly through a pickup coil.19 Becauseof the high magnetic susceptibility of the MNP, a largemagnetization can be achieved in a low magnetic field, so thatthe incident magnetic wave for the setup in this paper canhave a low frequency between 60 and 500 MHz. This way thepenetration depth in tissues is increased and in vivo imaging ispossible. The drawback of this technique is the lack of spatialinformation of the particles since only the MNP amount ina single voxel can be obtained. In Refs. 20 and 21, the EPRsetup recovers the 1D distribution of the MNP by moving thesample through the magnetic field, denoted as 1D EPR.

A so-called inverse problem needs to be solved to obtainthis 1D distribution. First, a forward model is evaluated whichcalculates for a certain MNP distribution the correspondingEPR measurement. We need to find the MNP distributionwhich minimizes the differences between the calculated EPRmeasurement and the actual measurement. This problem istypically ill-posed as the number of unknowns (the MNPdistribution) is much larger than the EPR measurement sites.Such a minimization is generally solved using truncatedsingular value decomposition (TSVD).22

In this paper, we present a general technique for recon-structing a spatial multidimensional MNP distribution usingEPR measurements. We analyze the condition of the linearinverse problem for different EPR implementations and wenumerically investigate how the associated inverse problemneeds to be adapted to cope with the added dimensions.These added dimensions require a different way of obtainingmeasurement data with the goal to achieve fast measurementsand to attain an increased stability of the inverse problem.Therefore, we introduce a measurement method in which weactivate only certain parts of the volume. Finally, we studythe impact of noise to assess the robustness of the proposed3D EPR technique.

2. MATERIALS AND METHODS2.A. Single voxel EPR

Single voxel EPR is a sensitive technique which allowsto determine the total MNP amount in a sample, typicallyexpressed in iron amounts. In our experiments, we employ

F. 1. Schematic representation of the placement of MNP sample and coilsin the EPR setup (not to scale).

Resovist® (Schering AG, Berlin, Germany) particles, a clini-cally approved contrast agent for MRI.23 The setup consists ofthree perpendicular coils, schematically shown in Fig. 1. OneHelmholtz coil pair for generating the homogeneous magneticfield, an excitation coil for producing the radio frequencywave, and a sensing coil for measuring the magnetization.The radio frequency wave flips the magnetization of theparticles toward the sensing coil. The signal from the sensingcoil is proportional to the amount of particles in the sample.A more elaborated explanation about the EPR setup can befound in Refs. 19 and 20.

2.B. EPR measurements

The unknown MNP distribution is denoted as the 1×Ndimensional vector c. It contains all the iron concentrationson different grid points, so that N is the total number of gridpoints. In the case of 1D EPR, this is a 1D grid, and in 2D and3D EPRs, these are 2D and 3D grids, respectively. An EPRmeasurement is represented by

SEPR= [S1, . . ., Sm, . . ., SM]. (1)

These M measurements originate from the M positions ofthe sample during a translation and/or rotation of the samplethrough the magnetic field (see Fig. 1). In 1D EPR, the sampleis only translated along a line through the magnetic field. In2D EPR, similar translations are performed but in a planeinstead of along one line. In 3D EPR, additionally rotations ofthe 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 theexcitation and sensing coil for every position m, a differentmeasurement value is obtained for an equal iron concentration.These varying measurements are incorporated in the systemresponse function R which contains the different measurementvalues as a function of the position m in the magnetic field fora unit iron concentration. A unit concentration is chosen as thesignal linearly scales with the iron concentration. m representsthe spatial variables, for example, R(z) in 1D EPR, R(x,z) in2D EPR, and R(r,θ,z) or R(x,y,z) in 3D EPR depending on

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F. 2. By placing only a part of the sample in the homogeneous magneticfield (and thereby limit the zone of particle response), more independent in-formation is available to solve the inverse problem which results in improvedreconstructions of the MNP distribution.

the type of coordinate system employed. To find the responseof a certain MNP amount on grid position m, the responsevalue on m, R(m), is multiplied with the MNP amount on gridposition m. S can then be modeled by

S(c)=F ·c, (2)

F is a M × N system matrix consisting of the associatedresponse values for every grid concentration for the M posi-tions of the sample. Row m of F, i.e., position m of the sam-ple, is constructed from all the corresponding response valuesR(mn) for every unknown local cn, n= 1,. . .,N on position mn.In 3D EPR measurements, M typically equals Z ∗Θ with Zthe number of translations of the sample and Θ the maximumnumber of rotational movements of the sample.

It is possible to place only a part of the volume in theregion Ω where R(m) is defined, see Fig. 2. This is referredto as partial volume excitation. In this case, R(mn) is zerofor cn with mn <Ω. In practice, the sample is moved to mea-surement positions such thatΩ comprises a different region ofthe sample. In the previous approach, the complete sample wasplaced in Ω. Partial volume excitation can reduce the numberof measurements by selectively activating parts of the volume,thus reducing the requirement of a complete translation of thevolume through the magnetic field. Magnetic gradient fieldscan further improve this measurement method by activelytargeting certain areas of the sample.

2.C. EPR for visualizing MNP distributions

The MNP distribution can be recovered by solving aninverse problem,

c∗= argminc

∥S(c)−SEPR∥, (3)

SEPR represents the EPR measurements and S(c) the forwardmodel solution. This inverse problem is solved by TSVD. In

TSVD, the Moore–Penrose inverse, F†, is calculated as

c∗=F†κSEPR= (UΣVT)†κSEPR=

κl=1

uTl

SEPR

σlvl . (4)

In TSVD, the eigenvalues from F, denoted by σl, are obtainedand are organized according to decreasing size. Only the κlargest eigenvalues are kept, because smaller eigenvalues arecommonly associated to noise sources. In Ref. 20, a methodwas devised to acquire the optimal κ for EPR imaging. Ageneral confidence interval for the recovered MNP distributioncan be obtained from numerical simulations for various MNPdistributions, different discretizations, noise levels, etc., inwhich the reconstructed MNP distribution is compared to thesimulated MNP distribution. Note that for performing accuratereconstructions of the MNP distribution, the response functionneeds 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. RESULTS3.A. Reconstruction quality

The reconstruction quality of the presented technique needsto be assessed. In this paper, we use the correlation coefficient,

CC=

Nn=1

(c∗n− c∗)(cn− c)Nn=1

(c∗n− c∗)2 Nn=1

(cn− c)2, (5)

c∗ is the reconstructed distribution and c is the actual dis-tribution, c∗ and c are their averages. A CC of one correspondsto a ‘perfect’ reconstruction of the actual distribution. Fur-thermore, we investigate the condition of the linear inverseproblem with the condition measure ρ,24

ρ=σ1

1/LLl=1

σl

. (6)

Smaller ρ signifies improved numerical stability. In contrastto the conventional condition measure, Eq. (6) depends lesson the smallest eigenvalues, which allows us to investigateand compare different sizes of F.

3.B. Synthetic 3D system response

We generated synthetic 3D response functions based onthe measured 1D system response [Fig. 3(a)]. The 1D systemresponse shows the fact that particles further away from theexcitation and sensing coil are moderately excited by the radiofrequency 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 theparticles are completely excited and sensed by the coils.

For the 3D response, we assume a similar behavior withrespect to the two coils, a decrease of the response value inthe directions further away from the two coils. Furthermore,

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F. 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 mmand reduced θ symmetry.

we assume a certain circular symmetry with respect tothe coils. The symmetry does not result in equal responsevalues due to noise and small displacements in hardware;therefore, white Gaussian noise with a noise level of 1% withrespect to the local response values was added to the model.These assumptions were also experimentally validated byperforming coarse measurements of the response function ona few positions in the EPR setup. We defined the 3D responsefunction in cylindrical coordinates to show the rotationalsymmetry along the θ dimension. Due to the current diameterof the sensing coil (≈9 mm) of the setup, only a 4 mmradius of the response is possible [Fig. 3(b)]. We calculatedtwo additional response functions with radii 8 and 12 mmto assess the influence of volumes with a larger diameter.We additionally modeled these three response functions withlarger variations with respect to the θ-direction, Radap, toinvestigate the impact of reduced θ symmetry [Fig. 3(c)].This reduced symmetry of the response can be implementedin the system by adding magnetic field gradients or by slightlychanging the positions of the coils, as the response of theparticles depends on the local magnetic field surrounding theparticle. These synthetic 3D responses are made to investigatethe necessary requirements for the system response functionand to show the impact of variations in system response onthe inverse problem.

3.C. Requirements of the system matrix

In a first approach, we used the symmetric 3D responseto generate system matrices. The sample was placed in themagnetic field and translated in steps of 1 mm, no partialactivation is employed. For each step of the translation, a fullrotation (360 in steps of 10) is performed. The resolutionof the volume is fixed at 1 mm. We performed noise-freereconstructions of different samples having dimensions ofa × a × a mm3 with a = 1,. . .,10. We obtain high-quality

reconstructions until N ≈ 200. Figure 4 shows an in-planereconstruction example of a 5×5×5 mm3 with N = 125, M=170 and a 10×10×10 mm3 volume with N = 1000, M = 729.For the smaller volume, a perfect reconstruction is obtained,while the larger volume is unrecognizable. This way ofperforming 3D EPR only allows the imaging of smallMNP volumes as the results deteriorate quickly for largervolumes. Furthermore, the measurement procedure is time-consuming.

Because the response function depends on the relativedistances to the two coils, a circular symmetry is observedwhere values do not differ a lot. By incorporating these valuesin F, it is possible that linearly dependent information isadded and thus the condition of the linear inverse problemdeteriorates. To investigate this, we made system matricesof different sizes corresponding to the reconstructions ofdifferent volume sizes. We start with the reconstruction ofa cylindrical volume with length 5 mm and radius 1 mmhaving only an angle of 10. The corresponding condition ofthis inverse problem is ρ(F)≈ 7 [Eq. (6)]. Then the volumewas increased each time along 1D (r , θ or z), and for everyincrease of the volume, the condition of the inverse problemwas evaluated. Figure 5(a) shows ρ for all these volumes.In this case the response function with a radius of 12 mmwas employed (full lines). It can be seen that increasing theθ dimension results in a decrease of system stability. Thesereconstructions were also made with Radap which does nothave this circular symmetry. Then an improved stability forthe system matrix is obtained (dashed line). For an increase inr and z dimensions, the dashed line is equal to the respectivefull lines as only the θ direction has changed. Similar resultswere obtained for the other two response functions withsmaller 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 forsamples with larger radii and lengths. Figure 5(b) depicts the

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F. 4. 3D EPR imaging of a small 5×5×5 mm3 volume and a larger 10×10×10 mm3 volume using the symmetrical response function. Using this firstapproach, only small volumes can be reconstructed.

normalized eigenvalue distribution [Eq. (4)] for the case ofr = 1 mm, θ = 360, and z = 5 mm for R and Radap. The slowerdecrease of the Radap eigenvalues and their relative larger sizesmakes the solution more robust compared to R. The cutoffvalue κ was determined according to Ref. 20. By reducing theEPR symmetry, quantitative images of larger MNP samples(N ≈ 800) from Fig. 4 can be obtained.

The numerical stability of the system matrix was furtherimproved by placing only a part of the sample in the homo-geneous field, so only a well-defined segment of the sam-ple 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 samplewas gradually translated, in steps of 1 mm, into the magneticfield generated by the Helmholtz coil, thereby increasing thepart of the sample inΩ. Note that the reconstruction quality islower compared to that in Fig. 4, this is due to the shape of thesample which allows less measurements compared to the num-ber of unknowns. The amount of measurements was adaptedso both measurement methods obtained the same number ofmeasurements (M = 28, N = 77). The reconstruction withpartial volume excitations shows an increase in reconstructionquality (a CC of 63% compared to a CC of only 10%) and

F. 5. (a) Numerical stability of F for increasing volume dimensions. Because of symmetry reasons, an increase in θ dimension results in a decrease in systemstability. Using Radap, this system instability is reduced. Furthermore, an increase in r and z dimensions has no large impact on system stability. (b) Theeigenvalue distribution of Radap shows a relative slower decrease which improves the stability of the system matrix.

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F. 6. Reconstruction of a 2D MNP distribution with and without partial volume excitation. When partial volume excitation is employed, improvedreconstructions of the MNP distribution are obtained.

in numerical stability (ρ(F)≈ 28–ρ(F)≈ 15). This increase instability was also observed in the system matrices of Fig. 5(a)and now a perfect reconstruction of the larger volume of Fig. 4is attainable.

Figure 7 depicts the averaged reconstruction scores andtheir standard deviation of 50 randomly generated MNPdistributions with and without partial volume excitation fora volume with fixed θ and r dimensions, but increasing zdimension. The larger the MNP sample and thus the moreill-posed the problem, the better the partial volume excitationperforms with respect to the conventional measurements.The partial volume excitation was performed by graduallyinserting the sample in the magnetic field. In these simulations,the total number of measurements compared to the totalnumber of unknowns was kept constant for both methods (byincreasing/decreasing the number of measurement positions)as the ratio of M to N also has an impact on the inverseproblem.21 The improved reconstructions thus only originatefrom adding information to the inverse problem to besolved, by only exciting parts of the sample instead of thecomplete sample. Because of the increased stability, we cantherefore speed up the measurement method while attaining anincrease in reconstruction quality. This way we can increasemeasurement speed by ≈10%. This makes us believe that theEPR methodology can be further improved by using magneticfield gradients that can actively target certain areas of thesample.

3.D. Noise robustness of 3D EPR

Previous noise-free simulations regarded the stability of theEPR matrix. We reduced the symmetry in the original responsefunction to obtain improved solutions for the inverse problem.Additionally, we introduced the concept of partly exciting thesample which further improved stability of the methodologyand showed improved reconstructions. In this section, we

investigate the robustness toward noise for these adaptations.White Gaussian noise was increased in steps of 1%. Forevery noise level, 200 noise measurements are performedwhich are then averaged. The noise levels of 1%–10% arewith respect to the system response amplitude and correspond

F. 7. Larger 3D MNP volumes deteriorate the condition of the inverseproblem. For every volume 50, randomly generated 3D MNP distributionswere reconstructed with and without partial volume excitation. When em-ploying partial volume excitation, the condition of the inverse problem de-grades slower.

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F. 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 distributionfrom (b) using a noise level of 10% and the adapted system response.

to actual noise values of the setup.20 Figure 8(a) shows theaverage CC and standard deviation for increasing noise levelsfor the different approaches used in this paper. As statedpreviously, the original response function R is not able toreconstruct larger MNP samples. Now that noise is added,the solution even deteriorates faster than before. Radap delaysthe deterioration due to the reduced symmetry but showssignificant noise impact (CC goes from ≈0.8 to ≈0.13) anda large standard deviation. When using the partial volumeexcitation, in which we gradually insert the sample in themagnetic field, we see a tremendous increase in reconstructionquality for both response functions (R≈ 0.82 and Radap≈ 0.92)and the standard deviation is now significantly lower.Figure 8(c) shows a reconstruction example of the MNPdistribution depicted in Fig. 8(b) using the adapted responsefunction with partial volume excitation and a noise levelof 10%.

4. DISCUSSION

In this paper, we presented a general technique toreconstruct multidimensional MNP samples using EPR. Theextension of the 1D approach from Ref. 20 to 2D and 3D worksfine for smaller volumes (N < 200), but numerical stabilityissues arise when imaging larger volumes. We generateddifferent system matrices for cylindrical volumes which wereincreased along 1D. The deterioration of the solution wasespecially visible for increasing θ dimensions, which is dueto the symmetrical property of the 3D response functionR. r and z dimensions only had a small effect on thestability. The stability of the inverse problem showed that areduction in symmetry is necessary to assure a system matrixwhich is sufficiently stable to allow sensitive reconstructionsof the MNP distribution. To reduce the symmetry of theresponse 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 firstapproach, this was done by gradually inserting/removing thesample in the magnetic field. This adds linearly independentinformation to the inverse problem to be solved and translatesinto improved reconstructions. Due to the increased stabilityand better imaging results, it is also possible to reducethe number of measurement positions and this way speedup our technique. A speedup of ≈10% can be achievedwhile still obtaining an increased reconstruction quality.We expect to further improve this measuring method ifmagnetic gradient fields are employed which can targetspecific areas of the volume. Finally, the robustness of ourmethod toward noise was investigated, which showed that ouradaptations realize an increase in reconstruction quality, andthe EPR setup maintains its stability with respect to largermultidimensional samples. In the future, we plan to speed upthe measurements by adding magnetic gradient fields. Thisway sample movement can be simulated. These gradient fieldsalso reduce the symmetry in the system’s response and thuswill increase the performance of the EPR imaging modalityeven further.

5. CONCLUSIONS

This paper extends the reconstruction of 1D magneticnanoparticle distributions using EPR signals toward 2D and3D and analyzes in-depth the feasibility of multidimensionalreconstruction using numerical computer simulations. Due tothe larger number of unknowns, the stability of the inverseproblem has to be improved. This is done by reducingthe inherent symmetry of the setup and by introducing anew measurement method in which the MNP sample canbe activated partially. The adaptations show a large noiserobustness (increase in reconstruction quality by ≈80%),enable 3D reconstructions of larger samples, and allow toreduce measurement time by ≈10%.

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ACKNOWLEDGMENT

J.L. acknowledges the support from BOF Project No.01J16113 UGent.

a)Electronic mail: annelies.coene@ugent.be1C. T. Lim, J. Han, J. Guck, and H. Espinosa, “Micro and nanotechnologyfor biological and biomedical applications,” Med. Biol. Eng. Comput. 48,941–943 (2010).

2Q. Pankhurst, N. Thanh, S. Jones, and J. Dobson, “Progress in applications ofmagnetic nanoparticles in biomedicine,” J. Phys. D: Appl. Phys. 42, 224001(15pp.) (2009).

3Y. Gao, Y. Liu, and C. Xu, “Magnetic nanoparticles for biomedical appli-cations: From diagnosis to treatment to regeneration,” in Engineering inTranslational Medicine (Springer, London, 2014), pp. 567–583.

4E. Flynn and H. Bryant, “A biomagnetic system for in vivo cancer imaging,”Phys. Med. Biol. 50, 1273–1293 (2005).

5F. Wiekhorst, U. Steinhoff, D. Eberbeck, and L. Trahms, “Magnetorelaxom-etry assisting biomedical applications of magnetic nanoparticles,” Pharm.Res. 29, 1189–1202 (2012).

6D. Baumgarten, R. Eichardt, G. Crevecoeur, E. Supriyanto, and J.Haueisen, “Magnetic nanoparticle imaging by random and maximumlength sequences of inhomogeneous activation fields,” in 2013 35thAnnual International Conference of the IEEE Engineering in Medicineand Biology Society (EMBC) (IEEE, Osaka, 2013), pp. 3258–3260.

7F. Ludwig, E. Heim, S. Mäuselein, D. Eberbeck, and M. Schilling, “Magne-torelaxometry of magnetic nanoparticles with fluxgate magnetometers forthe analysis of biological targets,” J. Magn. Magn. Mater. 293, 690–695(2005).

8J. Dieckhoff, M. Schilling, and F. Ludwig, “Fluxgate based detection ofmagnetic nanoparticle dynamics in a rotating magnetic field,” Appl. Phys.Lett. 99, 112501 (2011).

9B. W. Ficko, P. M. Nadar, P. J. Hoopes, and S. G. Diamond, “Developmentof a magnetic nanoparticle susceptibility magnitude imaging array,” Phys.Med. Biol. 59, 1047–1071 (2014).

10E. M. Cherry, P. G. Maxim, and J. K. Eaton, “Particle size, magnetic field,and blood velocity effects on particle retention in magnetic drug targeting,”Med. Phys. 37, 175–182 (2010).

11F. Wiekhorst, M. Liebl, U. Steinhoff, L. Trahms, S. Lyer, S. Drr, and C. Alex-iou, “Magnetorelaxometry for in-vivo quantification of magnetic nanoparti-cle distributions after magnetic drug targeting in a rabbit carcinoma model,”in Magnetic Particle Imaging, Springer Proceedings in Physics Vol. 140,

edited by T. M. Buzug and J. Borgert (Springer, Berlin, Heidelberg, 2012),pp. 301–305.

12L. Arcese, M. Fruchard, and A. Ferreira, “Endovascular magnetically guidedrobots: Navigation modeling and optimization,” IEEE Trans. Biomed. Eng.59, 977–987 (2012).

13M. Johannsen, U. Gneveckow, B. Thiesen, K. Taymoorian, C. Cho, N.Waldöfner, R. Scholz, A. Jordan, S. Loening, and P. Wust, “Thermother-apy of prostate cancer using magnetic nanoparticles: Feasibility, imaging,and three-dimensional temperature distribution,” Eur. Urol. 52, 1653–1662(2007).

14D. Piao, R. A. Towner, N. Smith, and W. R. Chen, “Magnetothermoacousticsfrom magnetic nanoparticles by short bursting or frequency chirped alternat-ing magnetic field: A theoretical feasibility analysis,” Med. Phys. 40, 063301(12pp.) (2013).

15Y. L. Raikher and V. Stepanov, “Physical aspects of magnetic hyperthermia:Low-frequency ac field absorption in a magnetic colloid,” J. Magn. Magn.Mater. 368, 421–427 (2014).

16J. Weil and J. Bolton, Electron Paramagnetic Resonance: Elementary The-ory and Practical Applications (Wiley, Hoboken, NJ, 2007).

17T.-H. Shin, Y. Choi, S. Kim, and J. Cheon, “Recent advances in magneticnanoparticle-based multi-modal imaging,” Chem. Soc. Rev. 44, 4501–4516(2015).

18S. Y. Lee, S. I. Jeon, S. Jung, I. J. Chung, and C.-H. Ahn, “Targeted multi-modal imaging modalities,” Adv. Drug Delivery Rev. 76, 60–78 (2014),targeted imaging.

19X. Li, G. Torfs, J. Vandewege, J. Bauwelinck, and J. Verbiest, “Sensitive andquantitative pEPR detection system for SPIO nanoparticles,” Electron. Lett.49, 1600–1601 (2013).

20A. Coene, G. Crevecoeur, L. Dupré, and P. Vaes, “Quantitative estimation ofmagnetic nanoparticle distributions in one dimension using low-frequencycontinuous wave electron paramagnetic resonance,” J. Phys. D: Appl. Phys.46, 245002 (2013).

21A. Coene, G. Crevecoeur, and L. Dupre, “Robustness assessment of 1-delectron paramagnetic resonance for improved magnetic nanoparticle recon-structions,” IEEE Trans. Biomed. Eng. 62, 1635–1643 (2015).

22P. Hansen, “The truncated SVD as a method for regularization,” BIT Numer.Math. 27, 534–553 (1987).

23Y.-X. J. Wang, “Superparamagnetic iron oxide based MRI contrast agents:Current status of clinical application,” Quant. Imaging Med. Surg. 1, 35–40(2011).

24R. Eichardt, D. Baumgarten, B. Petkovic, F. Wiekhorst, L. Trahms, and J.Haueisen, “Adapting source grid parameters to improve the condition of themagnetostatic linear inverse problem of estimating nanoparticle distribu-tions,” Med. Biol. Eng. Comput. 50, 1081–1089 (2012).

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