IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 62, NO. 6, JUNE 2015 1635

Robustness Assessment of 1-D ElectronParamagnetic Resonance for ImprovedMagnetic Nanoparticle Reconstructions

Annelies Coene∗, Guillaume Crevecoeur, and Luc Dupré

Abstract—Electron paramagnetic resonance (EPR) is a sensitivemeasurement technique which can be used to recover the 1-D spa-tial distribution of magnetic nanoparticles (MNP) noninvasively.This can be achieved by solving an inverse problem that requiresa numerical model for interpreting the EPR measurement data.This paper assesses the robustness of this technique by includingdifferent types of errors such as setup errors, measurement errors,and sample positioning errors in the numerical model. The impactof each error is estimated for different spatial MNP distributions.Additionally, our error models are validated by comparing the sim-ulated impact of errors to the impact on lab EPR measurements.Furthermore, we improve the solution of the inverse problem byintroducing a combination of truncated singular value decompo-sition and nonnegative least squares. This combination enables torecover both smooth and discontinuous MNP distributions. Fromthis analysis, conclusions are drawn to improve MNP reconstruc-tions with EPR and to state requirements for using EPR as a 2-Dand 3-D imaging technique for MNP.

Index Terms—Electron paramagnetic resonance (EPR), imagereconstruction, inverse problems, magnetic nanoparticles (MNP),robustness.

I. INTRODUCTION

MAGNETIC nanoparticles (MNP) are convenient tools forbiomedical applications [1]–[3]. Their diameter in thenanometer region allows them to reach virtually every part in thebody. Additionally, they have a high saturation magnetizationmaking them detectable from a distance. These properties areextensively used in hyperthermia [4], [5], drug targeting [6]–[10], and disease detection [11], [12]. One drawback is the lackof a quantitative and qualitative reconstruction technique forthe spatial MNP distribution. A consequence thereof is that thepreviously mentioned applications do not operate efficiently andsafety issues arise. In order to improve the performance of thesebiomedical modalities, an accurate reconstruction technique isrequired.

A well-known reconstruction technique is magnetic particleimaging (MPI) [13], [14]. This method has a high resolution

Manuscript received April 28, 2014; revised October 14, 2014; acceptedFebruary 1, 2015. Date of publication February 3, 2015; date of current versionMay 18, 2015. Asterisk indicates corresponding author.

*A. Coene is with the Department of Electrical Energy, Systems andAutomation, Ghent University, 9000 Ghent, Belgium (e-mail: Annelies.Coene@ugent.be).

G. Crevecoeur and L. Dupré are with the Department of Electrical Energy,Systems and Automation, Ghent University.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TBME.2015.2399654

(

1636 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 62, NO. 6, JUNE 2015

Fig. 1. Schematic overview of the studied methodology. The dash-dotted rect-angles represent necessary model inputs. Additionally, lab EPR measurements(full line rectangle) can be employed as a model input so to validate the errormodels.

Fig. 2. Example of the EPR setup with the three perpendicular coils showinga lab EPR measurement. The MNP sample is moved through B along a line (inthis case along XY), so to obtain spatial information about the MNP. Adaptedfigure from [24].

(grid, forward model properties, etc.), MEs, and sample posi-tioning errors (PE) is explored. The simulated impact of thedifferent errors is compared to the impact of EPR lab measure-ments with deliberately added errors so to validate the errormodels. Furthermore, we assess the importance of the inversesolving method by introducing a combination of truncated sin-gular value decomposition (TSVD) [26] and nonnegative leastsquares (NNLS) [27] to solve the associated inverse problem.We test these inverse solvers on MNP distributions exhibitingsmooth and discontinuous behavior. Fig. 1 shows a schematicoverview of the studied methodology. The aim of this paper isto inspect the limits of the EPR setup and to understand the in-fluence of different types of errors on the MNP reconstructions.This allows us to optimize the EPR setup toward more accuratereconstructions of MNP distributions.

II. METHODS

A. EPR Setup

We briefly describe the EPR setup and how the EPR sig-nal is generated. A more detailed description can be found in[22]–[24]. The EPR setup has three perpendicular coils; an ex-citation coil to generate a radio frequency wave, a Helmholtzcoil pair to have a homogeneous magnetic field B, and fi-nally, a sensing coil to measure the magnetization of theparticles (see Fig. 2). Due to B, the MNP have their mag-netization oriented parallel to this field. When exciting theMNP with a radio frequency wave, perpendicular to B, theorientation of their magnetization flips so that their magne-

tizing component can be measured by the sensing coil. Thecorresponding signal is proportional to the MNP concentration,but does not contain spatial information. To recover this infor-mation, we move the sample through B as the amount of signalfrom the particles is related to their respective distances to theexcitation and sensing coil.

B. EPR Measurement

We denote the unknown spatial MNP distribution as c. Sincewe consider the 1-D reconstruction of MNP, c is a 1-D vectorof N elements. In an EPR measurement, the unknown MNPdistribution is measured for M adjacent positions along a linethrough the homogeneous magnetic field B in steps of ΔM (seeFig. 2). This results in M measurements that can be representedby:

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

This EPR measurement S can be modeled as

S(c) = F · c. (2)

F is the forward model, which is used to simulate the EPRmeasurement values. F is a M × N matrix which associates ameasurement Sm to position m of the concentration distributionc.

F is constructed by measuring an accurately known amount ofMNP with N = 1 (i.e., not spatially extended) for M positionsas follows:

R =1

ccalSEPR

= [R1 , . . . , Rm , . . . , RM ], m = 1, . . . , M (3)

where ccal is the MNP amount of the calibration sample and istypically in the order of micromol. In our experiments, Reso-vist (Schering AG, Berlin, Germany) particles are employed.More information about these particles can be found in [28].SEPR(m) represents the EPR measurement value for positionm of ccal . At every position m of ccal , a response value Rmis measured. Rm is normalized to obtain the response valueof a unit concentration (1 μmol) on m. These M normalizedresponse values constitute the response function R. F is thenconstructed by using the appropriate response function valuesfor the unknown cn (n = 1, . . . , N) of the MNP concentrationdistribution and their positions mn . mn represents the positionof cn in EPR measurement m. These positions change for everynew measurement.

C. EPR for Retrieving 1-D MNP Distributions

The spatial MNP distribution c is retrieved by minimizingthe differences between the model solution (S(c)) and the actualEPR measurement (SEPR )

c∗ = arg minc

‖S(c) − SEPR‖. (4)

c∗ is the solution of the inverse problem and represents the re-constructed MNP distribution. This inverse problem is tradition-ally solved using TSVD [24], [26]. In TSVD, the eigenvalues

COENE et al.: ROBUSTNESS ASSESSMENT OF 1-D EPR FOR IMPROVED MNP RECONSTRUCTIONS 1637

σ are organized according to decreasing size. Only the eigen-values equal to or larger than σr are retained, i.e., the followingequation is solved:

c∗ = F†rSEPR = (UΣVT )†rSEPR =

r∑

l=1

uTl SEPRσl

vl . (5)

F†r represents the pseudoinverse of F which is found by solvingthe SVD, and Σ is a diagonal matrix containing the eigenvaluesof F. ul and vl are the eigenvectors of the matrices U and V. Adetailed description of the EPR 1-D reconstruction methodologyand how to select the right r can be found in [24]. In NNLS, (4)is solved iteratively subject to the constraint that all elements ofthe solution are positive

w = FT (SEPR − S(c)). (6)Every iteration transforms the dual vector w until only elementssmaller than or equal to zero remain, so that the solution c∗ hasonly elements greater than or equal to zero [27]. Both algorithmsare combined in the following way: first, the inverse problemis solved using TSVD, then it is solved by NNLS. The positiveMNP amounts from the NNLS reconstructions are substitutedby the values of the TSVD algorithm at the same position.

D. EPR Sensitivity Analysis

A change in a setup parameter of the EPR setup results in achange of ΔF in the model

S = (F + ΔF) · c. (7)In case of a fixed setup, the setup parameters can vary andlead to errors in the reconstruction, i.e., setup errors. Examplesof setup parameters are the resolution of c∗, the resolution ofSEPR , and the ratio of M to N (ill-posedness of the problem).Moreover, R depends on hardware properties of the EPR setup.We investigate the effect of these parameters to assess in a firststage the robustness of the EPR setup and in a second stageto optimize the EPR setup for improved MNP reconstructionpurposes.

Moreover, the impact of ME and PEs on the reconstructeddistributions c∗ are examined. A ME λ is modeled as follows:

SME(m,κ) = S + λ(m,κ). (8)

SME(m,κ) represents the simulated measurement with anadded ME vector λ(m,κ), which expresses the ME λ on mea-surement position m with a relative error size of κ% comparedto the mean amplitude of S. This allows to carefully characterizethe consequences of a ME. A measurement with a PE can berepresented by

SPE(m,α) = F(Fm = F(m+α)) · c. (9)Row m of F, Fm , consists of all the response values for thepositions of the MNP at measurement position m. If we er-roneously measure at position m + α instead, these responsevalues are the values from m + α where α is the shift from theactual position m (here in mm units).

The spatial variation of SEPR depends on the shape of c. Weinvestigate the necessary requirements to allow a quantitative

detection of the MNP distributions. To this end, various MNPdistributions are considered in the analysis (see Section III-A).

Remark that the aggregation behavior of Resovist particlescan be neglected for the typical MNP amounts employed in oursetup, this is why we did not consider this to be an error. In caseof extremely high MNP amounts, aggregation needs to be takeninto account.

In this paper, we investigate the impact of the different typesof errors independently, but in practice, these errors coexist.The error models allow an estimate of the maximum error (i.e.,the summation of all the errors). These errors can compensateeach other, but by adding them, a worst-case scenario can beassessed.

III. RESULTS AND DISCUSSION

A. Measures of Reconstruction Quality

A certain quality is assigned to every reconstruction c∗ andis measured using the correlation coefficient (CC)

CC =∑N

n=1(c∗n − c̄∗)(cn − c̄)√∑N

n=1(c∗n − c̄∗)2∑N

n=1(cn − c̄)2. (10)

c∗n and cn , respectively, represent elements from the recon-structed MNP distribution and the actual distribution, while c̄and c̄∗ are their averages. A CC of 1 implies a “perfect” recon-struction. Additionally, the average difference between c∗ andc (denoted by μ) and its standard deviation (denoted by σ) areemployed. In most cases, the differences in correlation scoreallow us to distinct the reconstructions and no other parametersare necessary. In certain situations, the reconstruction qualitydepends on the numerical stability of the system matrix F from(2). We explore the numerical stability of F by employing thecondition measure ρ

ρ =σ1

1/L∑L

l=1 σl. (11)

ρ is defined in [29] as the ratio between the largest eigenvalueof F and the mean singular value size of F. Smaller ρ signifies abetter numerical stability. The conventional condition measure(σ1 · σ−1L ) depends to a great extent on the smaller eigenvalueswhich make it difficult to compare different matrix sizes.

The distributions considered in this paper are depicted inFig. 3. The upper row represents MNP distributions in the labmeasured according to the method described in Section II-B.Each distribution has specific properties to allow targeted testingof the setup. The first distribution is an example of a calibrationsample, ccalib (see Section II-B). In this way, we can investigatethe impact of different parameters and errors on our calibration.Errors in the calibration result in an inaccurate forward model[see (2), (3)] and, thus, in inaccurate reconstructions. The bot-tom row represents numerically simulated MNP distributions.We investigate constant (i.e., homogeneous) distributions, ccon ,with slope equal to zero and gradient distributions, cgrad withconstant slope different from zero to assess the impact of slopesizes. Multiple cgrad MNP distributions are tested, but only twoof them are shown for clarity reasons. The red cgrad distribution

1638 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 62, NO. 6, JUNE 2015

Fig. 3. Upper row shows lab made MNP distributions. The bottom row depictsnumerically simulated distributions. The red lines represent simulated distribu-tions of MNP Distributions 1 and 2, respectively, but without discontinuities.

Fig. 4. Impact of the setup noise level on the MNP distributions from Fig. 3.

corresponds to the measured MNP Distribution 2 (upper row),but without discontinuities. This is also the case for the ccon dis-tribution and MNP Distribution 1. These distributions are usedto investigate the impact of discontinuities whether they realizeimproved or deteriorated reconstructions compared to continu-ous distributions, which are found in biomedical applications.We also consider 100 random MNP distributions, crand . Thesedistributions are generated with uniformly distributed pseudo-random numbers in the open interval (0,1). Additionally, wesimulate a realistic MNP distribution, crea , corresponding to asmall MNP injection.

B. Impact of Setup Parameters

In this section, we consider parameters that change F byΔF (as in (7)). We regard noise as a setup parameter, whichoriginates, for example, from the coils and increases as the coilsheat up. This setup noise results in a different response functionand, thus, in a different setup behavior

S = (F + ΔFnoise) · c. (12)

ΔFnoise represents the changes in the final setup matrix dueto setup noise. Fig. 4 depicts the noise robustness of the EPRsetup toward the distributions from Fig. 3. Typical noise lev-els for this setup are between 1% and 10 % of the measuredEPR signals. We use noise levels relative to the signal inten-sity of the distributions as MNP distributions with higher MNPamounts are less sensitive to noise if absolute values are used.The white Gaussian noise is increased in steps of 1%, and for

every noise level, 200 simulations (noise measurements) areperformed which are then averaged. The shadow band aroundthe average represents the standard deviation due to the noise.The influence of the noise is similar as in [24]: The averageCC decreases for increasing noise levels. The presence of noiserequires a cutoff of the eigenvalues [r, from (5)]. To determinethe cutoff value, we use the method from [24]. A large standarddeviation (0.4 CC) is observed for Distribution 1 due to theinherent symmetry of the distribution. The calibration sampleremains largely unaffected by the noise, which means that thecalibration results will not differ in time due to small changes insetup noise levels. The random and constant distributions (omit-ted for clarity) reach a CC close to zero starting from noise levelsof 1%, suggesting a lesser performance for these type of distri-butions with this setup. The CC of the random distributions canalso be explained due to the combined randomness of the noiseand the distribution. Distributions 2 and 3 obtain similar CCscores, but are lower compared to Distribution 1 because of thesmaller spacing between the components. The cgrad distribu-tions obtain a CC between Distributions 2, 3 and Distribution 1.The best results are found for the realistic distribution, crea ,because of the changing steepness in the distribution on a largerspatial scale (compared to the lab distributions). This figure isused as a reference figure for observing the additional impact ofmeasurement and PEs.

The impact of the response function R on the reconstructionis further investigated, generating measurement

S = (F(R1) + ΔF) · c = F(R2) · c. (13)

R1 and R2 represent two different response functions. We as-sess the impact of R on the reconstruction quality and wantto define necessary requirements for R to improve reconstruc-tions. Two types of response functions were investigated in [24],but it remained unclear what properties of the response func-tion caused the differences in reconstruction quality. First, weregard R2 = D · R1 , D ∈ R (i.e., equal shape, but changedslope). R1 corresponds to a measured response function. An in-creased slope should show an increased robustness toward setupnoise and, thus, increased reconstruction performance. Startingfrom D = 4, the impact of the noise is negligible for all distribu-tions shown in Fig. 3. In this case, the noise level is consideredrelatively to R1 . Fig. 5 shows an example with D = 4 for MNPDistribution 1. However, no reconstructions are obtained with aCC of 1. This suggests that other properties such as the spatialvariation of R also have an impact.

We, therefore, introduce three synthetic response functionsRa , Rb , and Rc with following properties:

dRadm

= constant, m = 1, . . . ,M

⎧⎪⎨

⎪⎩

dRbdm

= constant, m = 1, . . . ,M/2

dRbdm

= -constant, m = M/2 + 1, . . . ,M

COENE et al.: ROBUSTNESS ASSESSMENT OF 1-D EPR FOR IMPROVED MNP RECONSTRUCTIONS 1639

Fig. 5. (a) Response functions R1 and R2 with R2 = 4 · R1 . The bluedotted line depicts a measured response, (b) CC for every response as functionof noise level, (c) standard deviation for every function due to the noise levels.

Fig. 6. (a) Depicts the different response functions. (b) CC for every typeof response function for MNP Distribution 1 as a function of noise level.(c) Standard deviation on the CC in (b) due to the noise.

⎧⎪⎨

⎪⎩

dRcdm

= t · constant, m = 1, . . . ,M/2, t ∈ RdRcdm

= -constant, m = M/2 + 1, . . . , M.

Fig. 6 depicts these response functions and the associated CCwhen reconstructing MNP Distribution 1 for increasing noiselevels. Here, the setup noise is relative to the mean amplitude ofeach R to remove the impact of the slope (as shown in Fig. 5).Additionally, two measured response functions from two EPRsetups, denoted as Setup A and Setup B, are plotted (for moredetailed information see [24]). Lab measurements of MNP Dis-tribution 1 obtain a CC of 80% for Setup A and a CC of 93% forSetup B. We observe that a constant slope is noise robust (Ra ).Having similar up and downward slopes (Rb ) result in similarresponse values and, therefore, a decreased noise robustness.However, for lower noise values, a higher CC is observed. Thisis also the case for Setup A, which has similar slopes. SetupB has two different slopes (similar to Rc ) and is, therefore,more noise robust. When extending the existing 1-D EPR to-ward the 2-D and 3-D reconstruction of MNP distributions inpossible further EPR setups, we have to take into account theserequirements on the response function.

Fig. 7. Impact of F resolutions on setup performance.

TABLE IPERFORMANCE OF THE RELATIVE RESOLUTIONS OF c∗ AND S FROM FIG. 7

HIGHLY DEPEND ON THE NUMERICAL STABILITY (ρ [−]) OF F

Resolution S\Resolution c∗ 0.01 0.1 1

1 23.569 23.491 8.0590.1 232.186 79.239 8.1060.01 790.321 79.244 8.106

F also depends on the required resolution (i.e., the step sizebetween consecutive values) of the measurement S and the MNPdistribution c∗. The resolutions of these parameters determinethe M to N ratio of F and the resolution of R. Our aim is to havethe smallest possible resolution for c∗, by performing the leastpossible amounts of measurements so to realize fast measure-ments. Lab measurements were performed of MNP Distribution2 with S having a resolution of 1 mm. Starting from this mea-surement, reconstructions, c∗, with resolutions of 1, 0.1, and0.01 mm were realized [see Fig. 7(a)]. A similar reconstruc-tion quality is observed for these reconstructions, suggestingno numerical impact of increasing the resolution. When S islinearly interpolated to smaller resolutions (0.01 and 0.1 mm),a decrease in reconstruction quality due to numerical instabil-ity of the matrix can be observed, as only linearly dependentinformation is added to the matrix. This decrease also dependson the resolution of c∗; for smaller resolutions, there are moreunknowns and the problem is ill-posed. The performance of theselected resolutions highly depends on the numerical stability[see (11)] of the matrix F in (2), see Table I. In Fig. 7(b), valuesof the measurement S are omitted to acquire resolutions of 2and 3 mm. Because less information is available to solve theinverse problem, their reconstruction quality decreases. Startingfrom a resolution of 3 mm for c∗, the problem is overdeterminedto such extent that the approximated solution does insufficientlycharacterize the spatial variations of the MNP distribution.

C. Impact of MEs

MEs of the form (8) were investigated through numericalsimulations. In a first stage, the impact of a single ME, λ, isinvestigated. λ is added to a noise-free measurement, Scon ,with dSc o ndm = 0, m = 1, . . . ,M , for investigating the position

1640 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 62, NO. 6, JUNE 2015

Fig. 8. (a) Impact of the position of λ for constant S and for increasing κ; (b) impact of 1 λ on the reconstruction quality (average CC) for changing κ for theMNP distributions from Fig. 3 without setup noise; (c) impact of ME on the reconstruction quality (average CC) for κ equal to 15% with setup noise; (d) 100simulations are performed of Distribution 1 for increasing number (1–25) of random MEs (random sizes, random positions) with setup noise, the simulated resultsare compared to an averaged lab measurement of Distribution 1 that is also subjected to random ME’s.

dependence of the introduced ME λ. In this case, neighboringmeasurement positions of λ, m − 1, and m + 1, differ alwaysκ% of Scon(m) in value. There is no need to investigate dif-ferent amplitudes of Scon as the κ parameter scales the sizeof λ compared to the average amplitude of Scon . For each m(m = 1, . . . , M ) and for fixed κ, SME(m,κ) [see (8)] is gener-ated, resulting in M simulated measurements each containingone error on position m. From each measurement, a reconstruc-tion of c with N = 16 unknowns is performed and a correlationscore is associated. Fig. 8(a) depicts the difference between themaximum and minimum score of the M corresponding correla-tion scores. Nevertheless, a dependence is observed due to thenature of the inverse problem, which increases for rising κ incase of an underdetermined (M = 8) or overdetermined (M= 24) problem. A slower increase is observed for the underde-termined problem compared to the overdetermined. For a well-defined problem (M = 16), there is no dependence on m. It isrecommended, therefore, to achieve an underdetermined prob-lem (M < N ) or to be close to the well-posed case (M ≈ N ).In biomedical applications, the first situation will most likelyoccur. The overdetermined problem shows larger differencesfor the M correlation scores due to the numerical instability ofthe larger matrix F. ρ(FM =24) = 13.2 and ρ(FM =8) = 7.3(11), suggesting an improved numerical stability for the under-determined problem. The average CC of the M reconstructionsis similar for the under- and overdetermined problem; therefore,we show in the subsequent results the average CC to diminishthe influence of the position of λ on the CC.

Fig. 8(b) shows the average CC for increasing κ. In general,larger variations (larger slopes) in the MNP distribution result

in a smaller influence of the ME λ. This can also be seen onFig. 8(b) as the spatial variations of MNP Distribution 1 > crea >MNP Distribution 3 > ccon . The lab MNP distributions (MNPdistributions 1, 2, and 3) have high slopes (sudden increase ofMNP values higher than 1 μmol), which result in improvedreconstructions. For the same reason, the simulated continuousdistributions (see Fig. 3, red) have a lower average CC than thelab distributions. Distributions 2 and 3 obtain similar results dueto an equal spacing of 2 mm between the compartments. Thisspacing is smaller (2 mm compared to 4 mm) compared to MNPDistribution 1, which makes them inherently more sensitive toerrors. Note that the influence of κ is small for ccalib whichdetermines F leading to an increase of the system’s stability.

In the next phase, white Gaussian setup noise was added to thesimulations. The effect of λ combined with noise is investigatedfor κ ranging from 1% to 50%. Fig. 8(c) shows an example forκ = 15%. The noise averages the signal and decreases the κimpact in the measurement. The influence of κ decreases forhigher noise levels because the ME is less distinct comparedto the noise. Distributions less sensitive to setup noise obtainhigher CC’s than in the noise-free case, changing the relativepositions now to crea > MNP Distribution 1 > ccalib . Thismeans that noise has a larger impact than the spatial variationsin c from the previous paragraph.

For the analysis of multiple MEs, we generated uniformlydistributed pseudorandom numbers for the position of the error(limited to the range of m) and the size of the error (rang-ing from 1% to 15 % of the measurement’s mean amplitude).We performed 100 simulations for increasing number of errors(1 to 25).

COENE et al.: ROBUSTNESS ASSESSMENT OF 1-D EPR FOR IMPROVED MNP RECONSTRUCTIONS 1641

Fig. 9. Dashed lines represent the overdetermined (red), well posed (green),and underdetermined (blue) CC differences for an Sgrad measurement withΔ = 2 μmol and 1 PE ranging from 1 to 5 mm. The full lines are for Δ =0.2 μmol.

Finally, five lab measurements were performed of MNP Dis-tribution 1. These five measurements were averaged and thenconsidered to be free of MEs. Next, we added multiple ran-dom MEs as described before and compared the results to thesimulated MEs [see Fig. 8(d)]. The shadow band shows thestandard deviation for 100 simulations. In the simulations, theeffect of setup noise was considered to be ≈0.05 CC. The cor-respondence (difference in average CC of only 0.02) betweenthe correlation scores of the actual measurement and the sim-ulated measurement suggests that aforementioned simulationsassess the impact of MEs correctly. The small differences stillobserved are due to the fact that only five measurements wereaveraged and MEs still might be present. Furthermore, the setupnoise cannot be perfectly characterized. However, the averagedimpact on the lab measurements is still within the standard de-viation of our simulated impact.

D. Impact of PEs

In this section, the impact of PEs [see (9)] is investigated. Wefirst assumeS to have a constant slope �= 0,Sgrad . Consequently,the impact of one PE (equal to dSg r a ddm ) should be identical forevery position m. Next, M reconstructions are calculated fromSPE for m = 1, . . . ,M and M correlation scores are associated.In the ideal case, these M CC’s should be identical. Based onthe posedness of the problem, a dependence of the positionerror on m is observed (see Fig. 9). The well-posed case hasno dependence on the position m of the error except for α =1 mm in case of 2 μmol (large error) instead of 0.2 μmol dueto the equal size (1 mm) of reconstruction grid and PE. This isalso observed for the underdetermined case. When a differentgrid size is taken, this dependence is shifted to the grid size.Therefore, the grid size should be smaller or larger than theestimated PE. The overdetermined problem is less numericallystable, i.e., ρ(FM =24) = 13.2, and is much more independent(difference in correlations of 0.03 for Δ10 times larger) on thegradient of Sgrad than the underdetermined case (differences incorrelations of 0.4). On the other hand, increasing the PE (α)from 1 to 5 mm increases the CC difference by 0.8 and for theunderdetermined case only by ≈ 0.15. We suppose a degree ofnumerical instability is needed to cope with the increase of the

Fig. 10. (a) PE depends on m (α = 1 mm), (b) average CC for 1 PE with α= 1 and 3 mm without setup noise, and (c) the impact of the PE is correlatedwith the local gradient (α = 1 mm).

Fig. 11. (a) Impact of 1 PE on the reconstruction quality (average CC) for αequal to 3 mm with setup noise. (b) 100 simulations are performed for increasingnumber (1–25) of random PEs (random sizes, random positions) with setupnoise; these simulated results are compared to an averaged lab measurement ofDistribution 1 that is also subjected to random PE’s.

error (i.e., the local gradient of the measurement results betweenm and m + α).

The current EPR setup is asymmetric due to the relative po-sitions of the hardware. This is also reflected in the impact ofthe PE. The first positions (smaller m) only have small relativechanges to one another (small gradient), and therefore, an er-ror of 1 mm has almost no impact, while the last measurementpositions (larger m) have a higher slope and, thus, a higher asso-ciated impact of the PE [see Fig. 10(a)]. This results in a loweraverage CC for PE’s [see Fig. 10(b)]. A similar relative sensi-tivity toward position errors as MEs is observed for the MNPdistributions from Fig. 3. The largest impact (lowest CC) is forthe ccon , cgrad , and crand distributions. The lab MNP distribu-tions 1, 2, and 3 obtain similar scores. ccalib has the highest CCscores and, therefore, is only limitedly affected by the PE. creaseems to be more sensitive (compared to the other distributions)to PEs than to MEs. The CC correlates directly to the gradientof the measurements [see Fig. 10(c)]. The measurement of creafor example, contains steeper gradients than the MNP lab dis-tributions. The crand and cgrad distributions are sensitive to thesize of α due to their larger gradients. For α = 3 and 5 mm,

1642 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 62, NO. 6, JUNE 2015

Fig. 12. Reconstruction of MNP Distribution 1 with TSVD, NNLS, and with TSVD and NNLS combined.

ccalib obtains similar scores as the distributions due to a largergradient for higher m. These results show that PEs are moreimportant than MEs [see Fig. 8(b)]. This means that automatingthe setup is a higher priority than stabilizing the measurementsystem.

Second, Gaussian white setup noise was added, analogousas in Section III-C [see Fig. 11(a)]. Comparison with Fig. 8(c)reveals that adding noise results in a similar relative performanceof MNP reconstructions than when adding a ME, however theimpact of the PE is larger. This suggests that the noise sensitivityof the distributions plays a larger role than the inserted errors.The standard deviation due to the noise is considerably smallerfor the PEs.

Fig. 11(b) presents the impact of multiple random PEs for α= 1–3 mm and increasing number of errors (1–25). The simu-lations are performed analogous as in Section III-C. This figureclearly shows that PEs have a two to three times greater impact(average CC goes faster to zero and starts lower) than MEs.Finally, we added PEs to the five averaged lab measurements ofMNP Distribution 1 and compared them to simulated randomerrors. We obtained similar results (average CC difference ofonly 0.05) for the simulated impact and the actual impact, sug-gesting that our method assesses PEs well and allows to considermeasurement and PEs independently.

E. Solving the Inverse Problem Using a NonnegativityConstraint

TSVD is a traditional inverse solution method but may ex-hibit negative reconstructed MNP concentrations. We, therefore,suggest solving the inverse problem using a NNLS algorithm.Both solution methods are compared and finally combined inorder to improve the reconstruction results. In this section, thereconstructions are performed on actual measurement data.

Fig. 12 depicts a reconstruction example of MNP Distribu-tion 1. NNLS is able to pinpoint the filled compartments inthe tube; however, a large discrepancy exists between the ac-tual MNP amounts in the compartment compared to the re-constructed amount (a mean difference of 0.3 μmol). TSVDon the other hand, shows a MNP concentration similar to theactual concentrations for the compartment positions (a meandifference of 0.06 μmol), but the compartments are broadenedby an average of 2 mm. By combining TSVD and NNLS, wecan achieve a reconstruction with CC = 99%, μ = 0.08 μmol,and σ = 0.04 μmol. This is an increase of 10% and 6% in CCcompared to the separate use of the algorithms.

Distribution 1 is discontinuous which improves the perfor-mance of the NNLS algorithm. Therefore, we investigate theperformance of the NNLS algorithm also on the more continu-ous Distribution 2. The reconstruction with TSVD is broadenedbecause of PEs of the sample during the movement throughthe magnetic field. When employing NNLS, the CC increaseswith an average of 12%. Combining both methods decreases theCC again with 7% because of the poor performance of TSVD.Nevertheless, a better reconstruction score is obtained by em-ploying NNLS and TSVD + NNLS instead of only TSVD inreconstructions. In the future, the movement of the setup willbe automated and this will increase the reconstruction qualityof TSVD and TSVD + NNLS. This is in correspondence withour numerical simulations where PEs had the largest impact.

IV. CONCLUSION

In this paper, we numerically investigated the robustness ofEPR toward setup, measurement, and PEs. We obtained re-sults showing that the EPR measurements need to be preferablycarried out in a region where the response function exhibits a

COENE et al.: ROBUSTNESS ASSESSMENT OF 1-D EPR FOR IMPROVED MNP RECONSTRUCTIONS 1643

high continuous slope. In the future, the slope of the response canbe increased by adding a gradient magnetic field to the alreadypresent homogeneous field. Furthermore, the measurement andPE models were compared with actual EPR measurements inwhich we deliberately introduced errors to validate the simulatedimpacts. Correspondence for the PEs is observed (difference inaverage CC of only 0.05) and MEs (difference in average CC ofonly 0.02). PEs have a two to three times larger impact than MEsso the automation of the sample positioning setup should be pri-oritized above improving the EPR measurement sensitivity. Thegradient magnetic field can additionally improve measurementand setup noise errors as lower spatial variations of the MNPdistribution are required while a larger variation in the EPRmeasurement is obtained. Additionally, we proposed a combi-nation of TSVD and NNLS, which improved the solution of theinverse problem with ≈10%. The reconstruction quality is fur-ther affected by the ill-posedness of the inverse problem and thenumber of measurements should therefore be kept comparableto or lower than the number of unknowns in the spatial distribu-tion. The EPR setup used for the reconstructions in this paperis not well suited for measuring homogeneous (constant) MNPdistributions, but in biomedical applications, these are rarely en-countered. High-quality reconstructions were achieved in caseof MNP distributions with large spatial variations.

ACKNOWLEDGMENT

The authors would like to thank PEPRIC NV for giving themthe opportunity to perform measurements on their EPR setup.A. Coene gratefully acknowledges J. Leliaert for a critical read-ing of the manuscript.

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Authors’ photographs and biographies not available at the time of publication.

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