Magnetorelaxometry Assisting Biomedical Applications of ... · Magnetorelaxometry Assisting Biomedical Applications of Magnetic Nanoparticles Frank Wiekhorst & Uwe Steinhoff & Dietmar

PERSPECTIVE

Magnetorelaxometry Assisting Biomedical Applicationsof Magnetic Nanoparticles

Frank Wiekhorst & Uwe Steinhoff & Dietmar Eberbeck & Lutz Trahms

Received: 2 August 2011 /Accepted: 16 November 2011 /Published online: 8 December 2011# The Author(s) 2011. This article is published with open access at Springerlink.com

ABSTRACT Due to their biocompatibility and small size, ironoxide magnetic nanoparticles (MNP) can be guided to virtuallyevery biological environment. MNP are susceptible to externalmagnetic fields and can thus be used for transport of drugs andgenes, for heat generation in magnetic hyperthermia or for contrastenhancement in magnetic resonance imaging of biological tissue. Atthe same time, their magnetic properties allow one to developsensitive and specific measurement methods to non-invasivelydetect MNP, to quantify MNP distribution in tissue and to determinetheir binding state. In this article, we review the application ofmagnetorelaxometry (MRX) for MNP detection. The underlyingphysical properties of MNP responsible for the generation of theMRX signal with its characteristic parameters of relaxation amplitudeand relaxation time are described. Existing single and multi-channelMRX devices are reviewed. Finally, we thoroughly describe someapplications of MRX to cellular MNP quantification, MNP organdistribution and MNP-based binding assays. Providing specific MNPsignals, a detection limit down to a few nanogram MNP, in-vivocapability in conscious animals and measurement times of a fewseconds, MRX is a valuable tool to improve the application ofMNP for diagnostic and therapeutic purposes.

INTRODUCTION

Physics of Magnetic Nanoparticles

Iron oxide nanoparticles represent a major class of particlesused for novel magnetic applications in biomedicine, such

as magnetic drug targeting, magnetic thermoablation,biosensors or molecular imaging. An overview over theseapplications is given in a number of review articles (1–8).Magnetic nanoparticles that are used for medical applica-tions consist of a magnetite or maghemite core surroundedby a stabilizing and biocompatible coating. Due to thesmall core diameter of typically 4-30 nm, 103–104 atomicmagnetic moments within one particle are coupled togetherresulting in a single domain of uniform magnetization witha total magnetic moment m given by

m ¼ MpVp ð1Þ

where Mp denotes the magnetization of the core materialand Vp the core volume of the particle. Note that Mp isusually smaller than the saturation magnetization MS of thebulk material.

Without an external field, the magnetic moments of anensemble of magnetic nanoparticles in thermal equilibriumwill be randomly oriented and no net magnetic momentwill be seen in a distance from the assembly. In the presenceof a magnetic field H, the magnetic moments of thenanoparticles will tend to align along the field direction,and a net magnetic moment will be generated. Thisphenomenon is commonly referred to by the notionsuperparamagnetism.

For isotropic and non-interacting particles in thermalequilibrium at a temperature T the fraction of the magneticmoments that are aligned give rise to the net magnetizationM(H,T ) that is described by the Langevin function relatingthe magnetic energy μ0H to the thermal energy kBT

MðH ;T Þ ¼ Mp cothmm0H

kBT

� �� kBT

mm0H

� �ð2Þ

The equilibrium magnetization arising from a change ofthe external magnetic field H is not reached instantaneously

F. Wiekhorst :U. Steinhoff (*) :D. Eberbeck : L. TrahmsPhysikalisch-Technische BundesanstaltBerlin, Germanye-mail: [email protected]

Pharm Res (2012) 29:1189–1202DOI 10.1007/s11095-011-0630-3

KEY WORDS magnetic binding assay . magnetic drugtargeting . magnetic nanoparticles . magnetofection .nanoparticle biodistribution

but after a characteristic time τD by rotational diffusion ofthe magnetic moments within the particles, where (9).

tD ¼ 3hmVp

kBTð3Þ

The parameter hm ¼ MS=ð6agÞ denotes some kind of“magnetic” viscosity which hampers the rotational move-ment of the magnetic moments. The damping α and thegyromagnetic ratio γ are internal parameters characteristicof the particular nanoparticles’ crystal structure.

The actual magnetization behaviour of most magneticnanoparticles is not isotropic, because of the presence of aneffective magnetic anisotropy K, generally arising from acombination of magnetocrystalline and shape anisotropy.Due to this anisotropy, the magnetic moment has preferredorientations within the particle, along so-called easy axesrepresenting local energy minima. For the simplest case ofuniaxial anisotropy the energy barrier separating two minimais proportional to the volume of the particle, EA = K Vp andthe probability at a temperature T to surmount this energybarrier is given by the Néel relaxation time τΝ (10)

tN ¼ t0 expKVp

kBT

� �ð4Þ

where t0 ¼ 3hm=K is a damping or extinction time withvalues between 10−8 s and 10−12 s (11) and kB is theBoltzmann constant.

An additional relaxation mechanism appears if themagnetic nanoparticles are suspended in a carrier fluid ofviscosity η, where the particles may rotate together withtheir magnetic moments. This is described by the Brownianrelaxation time τB

tB ¼ 3hVh

kBTð5Þ

with Vh denoting the hydrodynamic volume. Note that therotating structure is not necessarily a single particle; itmight be composed of several nanoparticles or of amagnetic nanoparticle bound to other colloidal objects. Inany case, Vh in Eq. 5 refers to the overall hydrodynamicvolume of the rotating colloidal structure.

If both relaxation mechanisms are present at the sametime, the overall relaxation behaviour is described by aneffective relaxation time τeff (12)

teff ¼ tNtBtN þ tB

ð6Þ

with the shortest relaxation time prevailing. Figure 1 showsa logarithmic 2D plot of the effective relaxation time fordifferent Néel and Brownian relaxation times according toEq. 6 together with the corresponding core and hydrody-namic diameter assuming spherical magnetite MNP with

an effective anisotropy K~104 J/m3. Additionally, theincrease of relaxation time by increasing the shell thick-nesses from 5 nm (white line) to 10 nm (red line) isvisualized. This effect is much less pronounced when thetotal diameter is increased from 150 nm (orange line) to160 nm (black line).

Real nanoparticle systems usually exhibit a distributionof particle sizes and thereby also a distribution of relaxationtimes. Often, the log-normal distribution PP(Vp) is taken todescribe the particle core volume distribution

PP ðVP ;mP ; sP Þ ¼1ffiffiffiffiffi

2pp

sP VP

exp � ln2ðVP=mpÞ2s2

P

!ð7Þ

where μp is the median particle core volume and σp thestandard deviation of the core volume distribution. Sinceaccording to Eq. 4 the Néel relaxation time depends on thecore volume, the log-normal distribution of nanoparticlesizes also entails a log-normal distribution of relaxationtimes.

The distribution function Ph(Vh) of the hydrodynamicdiameters is more variable and typically more complicated.An obvious lower bound is that for a given particle Vh > Vp.

Typical cases that can easily be modeled:

a) Single (spherical) nanoparticles with a fixed shellthickness ds: Vh = π(dP + 2ds)

3/6. In this case, Vh isstill in the order of Vp.

b) Single nanoparticles where the shell thickness has anotherfunctional dependence on the particle diameter:Vh = f(VP).Also in this case, Vh is typically in the order of Vp.

c) Aggregates of colloidal compartments with a lognormaldistribution of aggregate volume characterized by itsmedian μh and distribution width σh.

Ph Vh;mh; shð Þ ¼ 1ffiffiffiffiffi2p

pshVh

exp � ln2 Vh=mhð Þ2sh

� �ð8Þ

In this case, Vh»Vp. The colloidal compartments mightbe magnetic, they might also consist of non-magneticmaterial. A special case of aggregates are so-calledmulticore particles, where a number of smaller mag-netic cores are assembled within one organic shell. Dueto the close distance between the different cores, themagnetic interaction between them can no longer beneglected for these multicore particles.

d) Immobilization of the nanoparticles to large or fixedobjects: Vh = ∞ for all MNPs in the sample.

A superposition of these cases occurs in real worldnanoparticle suspensions.

Due to the widely variable hydrodynamic diameter incolloidal dispersions, the effective relaxation time is only

1190 Wiekhorst, Steinhoff, Eberbeck and Trahms

loosely dependent on the particle core volume as shown inFig. 2.

For a magnetic nanoparticle suspension, the overalldistribution function of effective relaxation times ischaracterized by pairs of core volume and hydrody-namic volume:

P ¼ P Vp;Vh� � ð9Þ

Obviously, for a given particle formulation, the distri-bution of core diameters is independent of the bindingstate, whereas the hydrodynamic diameters and thus theeffective relaxation times will be altered by a bindingreaction or a cluster formation.

Magnetorelaxometry: Measurement Proceduresand Experimental Implementation

The phenomenon of delayed magnetic response of ferri- orferromagnetic materials to sudden changes of an externalapplied magnetic field has long been known and iscommonly denoted as magnetic relaxation, magnetic viscosity or

magnetic after effect (13). More recently, it was recognized thatthe measurement of this effect, i.e. in particular themeasurement of the time dependent response of super-paramagnetic nanoparticles to a sudden switch-off of themagnetic field, can be utilized to obtain specific informa-tion about the nanoparticles and their environment (14).Specific techniques and procedures of the measurement ofthese relaxation effects in magnetic nanoparticles aresubsumed under the concept magnetorelaxometry.

As depicted in Fig. 3, a magnetorelaxometric measure-ment generally consists of two phases: A magnetizing phasewhen a magnetizing field of defined strength and durationpartly aligns the MNP moments into the field direction, anda measurement phase, during which, shortly after theswitching-off of the magnetizing field, the decaying mag-netic field originating from the decay of the samples netmagnetic moment is detected by a sensitive magnetic fieldsensors like SQUID, fluxgate (15) or optical magnetometer(16). The latter two sensor types enable even the observa-tion of the magnetic nanoparticles during the magnetizingphase.

Fig. 1 Logarithmic 2D plot of theeffective relaxation time combiningNéel (abscissa, Eq. 4) and Brownianrelaxation (ordinate, Eq. 5) accord-ing to Eq. 6. Furthermore, thecorresponding core and hydrody-namic diameter are displayed formagnetite MNP with an effectiveanisotropy K~104 J/m3. Thestraight lines have been added toshow the influence of a fixed shellthickness of 5 nm (white line),10 nm (red line) or a fixed hydro-dynamic diameter of 150 nm (or-ange line) and 160 nm (black line)on the relaxation time with increas-ing core diameter.

Fig. 2 Effective relaxation time calculated by Eqs. (4–6) for different magneticcore diameters dp as a function of the hydrodynamic diameter dh forKeff=104 J/m3, T=290 K and η=10−3 Pa·s. The shaded area is accessibleby our MRX setup (10−4 s to 102 s), as determined by SQUID electronicsdead time, sampling frequency and relaxation measurement interval. Thegreen line indicates particles without a nonmagnetic shell layer where dh=dp.Immobilized particles are detectable only if they have a core diameter dp ofabout 16 nm to 22 nm. If both Brownian and Néel relaxation are present,particles with dp larger than 16 nm and at the same time dh in the range from50 nm to 5 μm are detectable by MRX. For particles with dp≥23 nm, onlyBrownian relaxation can be observed in the given time interval.

Magnetorelaxometry and Magnetic Nanoparticles 1191

The Magnetization of the Magnetic Nanoparticles

The net magnetization of a magnetic nanoparticle sampleachieved at the end of the MRX magnetizing phasecrucially depends on strength Hmag and duration tmag ofthe externally applied magnetic field. Since a magnetic fieldreduces the barrier to reach the energy minimum in theNéel relaxation process and acts as a torque in theBrownian relaxation mechanism, we have to take intoaccount the field influence on the magnetization time of theindividual particles. For the Néel relaxation we have (17)

tmag;NðHÞ ¼ tN 1� 0:82m0HMS

K

� �where τN is the Neel relaxation time as defined in Eq. 4.More sophisticated approaches for the field dependency ofthe Néel relaxation are discussed in (18) Accordingly, forBrownian relaxation the field dependency is found bysimulations (19)

tmag;BðHÞ ¼ tB 1þ 0:21mm0H

kBT

� �2 !�1=2

If both mechanisms are present, the resulting effectiverelaxation time τmag(H) is given by Eq. 6 again. Thus, theinitial magnetization amplitude M0 after a magnetizationtime tmag is then

M0 tmag� �¼Z

VP

ZVh

M Vp;Hmag ;T� �

1� exp �tmag=tmag Hmag ;Vp;Vh;T� ��

P VP ;Vhð ÞdVhdVP

Here, M(Vp,Hmag,T ) is the equilibrium value calculatedaccording to Eq. 2. The total magnetic moment of thesample is the superposition of the magnetic moments of allsingle particles that were aligned by this procedure. Thus,the measurement of their subsequent relaxation by magne-torelaxometry represents an integral view on the ensembleof the magnetic nanoparticles in the sample.

The Relaxation of the Magnetic Nanoparticles

For a magnetic nanoparticle sample the relaxation processis described by

MðtÞ ¼ZVp

ZVh

M0 Vp;Vh

� �exp �t=t eff Vp;Vh

� �� P Vp;Vh

� �dVh dVP

ð11ÞFrom the measured MRX relaxation curve two

parameters can be extracted that describe its mainfeatures, the relaxation amplitude ΔB and a relaxationdecay time t1/e. The MRX relaxation amplitude is themagnetic field change between two selected time pointswithin the recorded relaxation time interval. The MRXdecay time t1/e denotes the time after which the magneticfield value taken at a selected start point dropped by thefactor e≈2.718.

The appropriate number and type of magnetic fieldsensors for measuring the relaxation of the sample‘seffective magnetic moment is determined by the particularapplication. For several applications fluxgate and opticalmagnetometer having detection sensitivities down to10−12 T·Hz−1/2 with a maximum bandwidth of about10 kHz offer an acceptable way to apply magnetorelaxom-etry (15,16). At the cost of higher technical effort, the bestsensitivity so far is reached by low Tc superconductingquantum interference devices (SQUIDs). These sensorsenable the detection of magnetic fields as small as a few10−15 T·Hz−1/2, their bandwidth can reach several MHz.For measurement of samples at room temperature, theminimum distance between sample and SQUID is limitedby the requirements of thermal insulation.

A sample of volume V containing a magnetizationdistribution M(r,t) originating from magnetic nanoparticlesgenerates a magnetic flux density B(r,t) (hereafter simplytermed magnetic field) outside the sample

Bðr; tÞ ¼ m0

4p

ZV

3Mðr0; tÞ � ðr� r0Þðr� r0Þr� r0j j5 � Mðr0; tÞ

r� r0j j3 !

dV 0

ð12Þwith the integration carried out over the sample volume V andwith μ0 (= 4π×10−7 V·s/(A·m)) as the permeability of freespace. Due to the relaxation of the MNP after turning off the

Fig. 3 Magnetorelaxometry principle. The top line (from left to right)portrays a nanoparticle ensemble‘s behavior: initially in a disordered statewithout magnetization, partly rotated towards the field direction duringmagnetizing, and turning back into a randomly oriented distribution ofmagnetic moments leading to the detected magnetization relaxation.Typically a field of about 1.5 mT is applied for 1 s. After removal of thefield and a short interval the SQUID amplifier needs to recover, therelaxation signals are acquired for 0.5 s.

ð10Þ


magnetizing field, the magnetization M and hence themagnetic field B become time dependent. In cases where amagnetic nanoparticle accumulation can be assumed to bewithin a point-like region it can be described by a singlemagnetic moment m(r’,t) at r’. Eq. 12 simplifies then to thewell-known formula describing the field of a magneticdipole

Bðr; tÞ ¼ m0

4pmðr0; tÞ � ðr� r0Þðr� r0Þ

jr� r0j5 � mðr0; tÞjr� r0j3

!ð13Þ

Equation 12 reflects the integral character of MRX:after magnetizing the sample volume, all magnetic nano-particles in this volume contribute to the measuredmagnetic field signal at position r. As a consequence,MRX reflects all magnetic nanoparticles in the samplethat exhibit a relaxation within the observation timewindow.

The observation time window is limited by the deadtime of the SQUID sensor after the magnetizing field isswitched off and the ending of the recording interval ofthe MRX signal. Particles having an effective relaxationtime according to Eq. 6 that is much shorter than thedead time have decayed before they can be detected, thusthey do not contribute to the MRX signal just as the timeindependent remanent magnetization. Further limitingfactors of the magnetorelaxometric measurement tech-nique are the upper limit of the signal bandwidth fcutoff ofthe magnetic sensors and the sampling frequency fs of thedata acquisition system, only relaxation times with τeff >fs−1 and τeff > fcutoff

−1 can be resolved.The detection parameters limit the range of particle sizes

participating in the magnetorelaxometric measurement.The limits can be estimated by Eq. 3 as shown in Fig. 2.Assuming immobilized (Néel relaxation, solely) sphericaliron oxide particles with a typical effective anisotropy of104 J/m3, only particles having a core diameter of about16 nm to 22 nm are detectable in the time window of MRX(10−4 s to 102 s). If both Brownian and Néel relaxation arepresent, MNPs having core diameters larger than 16 nmand at the same time hydrodynamic diameters in the rangefrom 50 nm to 5 μm are detectable by MRX. For particleswith core diameters dp≥23 nm, only Brownian relaxationcan be observed in the given time interval.

Magnetorelaxometry Measurement Devices

In order to meet the technical requirements for the twophases of a magnetorelaxometric measurement, an MRXdevice is composed of a magnetizing unit and one or moremagnetic sensors that detect the relaxation. At present threedifferent MRX systems all based on low Tc superconduct-ing quantum interference device (SQUID) sensors are in

use in our lab at PTB, as shown in Fig. 4. Low Tc SQUIDShave to be operated at low temperatures that usually areaccomplished by housing the SQUIDs inside a Dewarcryostat filled with liquid Helium. Two of our SQUIDbased MRX devices are operated in a magnetically shieldedroom that suppresses environmental magnetic noise fromoutside, a third device has an integrated superconductingshield of its own. Due to the high performance of the PTBSQUID sensors and readout electronics, typically abandwidth from dc to several megahertz and a dynamicrange of over ±200 nT over a noise floor of 1.5 fT Hz−1/2

can be achieved.The single channel MRX system (1ch, Fig. 4a) used for

high resolution MRX combines two SQUID sensors toform a spatial gradiometer arrangement for the suppressionof magnetic far field distortions. At a distance of 12 mmbelow the nearest SQUID sensor located inside the tail of aliquid helium Dewar the samples of 150 μl volume(polyethylene microtiter vials) are placed inside the 10 mmbore of a twin coil (i.e. two parallel coils with oppositepolarity to reduce their stray field at the SQUID location).Alternatively, larger samples up to 8 ml volume can bemeasured replacing the magnetizing coil by a larger twincoil of 27.5 mm bore diameter. Both twin coil systems canbe energized by a magnetizing unit supplying magneticfields up to 2 kA/m (2.5 mT).

In the first step, the sample is magnetized for 1 s.Following a 200 μs dead time interval after switching offthe magnetic field that is required to recover the SQUIDsensor electronics, the relaxation is recorded for typically0.5 s, digitized at a 100 kHz sampling rate and 16 bit wordlength. The bandwidth of the MRX device is limited by thedata acquisition unit. The short duration of one measure-ment makes repetitive measurements feasible, which mayimprove the signal-to-noise (SNR) ratio by signal averaging.For this setting, relaxation amplitudes ΔB down to 1 pT canbe safely identified in an MRX measurement.

For MRX measurements of large sized samples up tohuman body dimensions, the PTB 304 SQUID vectormagnetometer (Fig. 4c) is used—a home-made biomagneticmeasurement system where the 304 SQUIDS are arrangedin a large Dewar vessel in various orientations measuringthe three magnetic field components that make up themagnetic field vector. For MRX applications this system iscombined with a magnetizing unit composed of a gradientamplifier (Bruker) and a Helmholtz coil of 84 cm diameter(Fig. 4d). A DC current of 80 A is fed into the coil toprovide a magnetizing field of 0.8 kA/m (1 mT). The largestray field of the Helmholtz coil requires the operation ofthe magnetizing coil outside the magnetically shielded roomand the subsequent transport of the magnetized sample intothe shielded room below the magnetometer. Since thistransport is currently done manually, a dead time of 5 s to


10 s to record the relaxation after switching off themagnetizing field has to be taken into account. Tocompensate for this inevitably large dead time, themagnetizing field is applied for 60 s. According to Eq. 10an extension of the magnetization time to 60 s gives also thelarge particles enough time to get aligned with the field sothat the magnetization has reached its maximum value (forthe given value of the magnetizing field). The dataacquisition at a sampling rate of 250 Hz starts 10 s priorto switching off the magnetizing field by which the sampletransport and start of the relaxation process are recorded.Thus the relaxation can be evaluated at the earliestmoment when the sample has come to rest. The relaxationsignals from the sample below the sensor are recorded forat least 75 s. During transport and measurements the doorof the shielded room is kept open. This avoids further timeloss and perturbations by the operation of the heavy slidingdoor, but admits the intrusion of low frequency magneticnoise signals of ±5 pT amplitude during the measurementphase, which limits the smallest detectable relaxationamplitude ΔB to about 10 pT for this setting.

In the third MRX device (Fig. 4b), 18 Helium cooledSQUIDs are arranged in a circle round a cylindricalhorizontal warm bore, with 11 cm diameter being largeenough to detect the relaxation signals from medium scale

accomplished by a cylindrical superconducting Niobiumshield which is integrated into this system. This magnetic

shield is assembled within the liquid Helium tank of theDewar vessel, but tightly aligned to the warm bore to keepenvironmental magnetic distortions from the SQUIDs.Sample magnetization can be performed either outside thewarm bore, where a large variety of magnetic field sourcescan be used, or inside the warm bore at the sensor locationby a Helmholtz coil with fields up to 4 mT. With this setup,the typical magnetizing duration amounts to 1 s followed bya 400 μs dead time interval, thus the relaxation data areacquired for 1 s at 100 kHz sampling rate and a 16 bit wordlength. For this device, a relaxation amplitude ΔB down toapproximately 1 pT becomes detectable in an MRXmeasurement.

APPLICATIONS

Exploiting the Relaxation Amplitude: Quantificationand Localization of Magnetic NanoparticlesAdministered to Organisms

The most relevant parameter for quantification andlocalization of MNP is the MRX relaxation amplitudeΔB. In addition, the relaxation curve should be assessed,because any changes in its shape may reflect changes of theparticle size distribution during the MNP application andmay add to the information of the quantification results.We classify the presented applications according to whethersingle channel or multi channel devices are used for MRXmeasurements.

Fig. 4 Magnetorelaxometry devices: (a) single channel device, the inset draft shows the centre part of the system with the 150 μl sample container in themagnetizing coil about 12 mm below the SQUID sensor in the Dewar flask. (b) 18 channel MRX scanner with integrated superconducting shield. Therelaxation of samples and animals up to rabbit size are measured by circumferentially arranged 18 SQUID sensors (inset shows the sensor support chain)located midway of the horizontal warm bore. (c) The PTB 304 channel vector magnetometer, the combination of this conventional biomagneticmeasurement system operated inside a magnetically shielded room with a (d) large Helmholtz coil system (d=84 cm, located in front of the shieldedroom) allows MRX measurements on samples up to human size.


samples or animals up to rabbit size. This system enablesMRX measurements in a typical laboratory environment, i.e. outside a shielded room. The necessary shielding is

Single Channel MRX: Quantification of MagneticNanoparticles in Tissue

Though of much less technical effort, single channel MRXprovides a useful tool for quantifying MNP in tissue. A crucialissue in single channel MRX applications is the control of thedistance between sample and detecting sensor, because the fieldof a magnetic dipole decays with the third power of thedistance. As a consequence small sample sizes are preferred,with an extent that is much less than the distance to the sensor,so that a point-like magnetic source can be assumed. For largersized samples a favourable relation between extent and distancecan be achieved by increasing the distance between sensor andsample, but at the expense of reduced relaxation amplitude. Togive a practical example, for measurements of tissue samples ina 8 ml container we usually increase this sample sensor distanceto about 40 mm.

Our single channel device magnetizes m = (0,0,mz) anddetects relaxation signals B = (0,0,Bz) solely in z-direction.With the origin of the coordinate system at the sensorposition and the sample position at r’ = (0,0,z’) the magneticfield of a magnetic moment according to Eq. 12 is given by

BzðzÞ ¼ m0

2pmz

jz � z0j3 ð14Þ

A most efficient quantification procedure using singlechannel MRX is to refer the sample relaxation amplitudeto the relaxation signal of a reference sample of knownMNP content measured under the same conditions. By thiswe can bypass all peculiarities concerning the relaxationmagnetization induced during the MRX magnetizing phaseas described by Eq. 11.

Identical and Homogenous Sample Volumes. Typical appli-cations are the quantification of the MNP content in bloodsamples, in small defined tissue samples supplied by abiopsy punch, or of MNP uptake by living cells (20). Herethe samples are of identical shape and with a uniformmagnetic nanoparticle distribution, so the nanoparticleamount of the sample XMNP,sample is determined by theratio of the relaxation curve amplitudes

XMNP ;sample ¼ ΔBz;sample

ΔBz;referenceXMNP ;reference ð15Þ

With XMNP we denote either the total number NMNP ofMNP or the total iron mass m(Fe) in magnetic materialdepending on our knowledge of the reference sample. Theiron amount of MNP reference samples is convenientlydetermined by photometry.

An important constraint noteworthy for any reliable MRXquantification is the congruence between sample and refer-ence relaxation curve shape which assures that the MNP in

sample and reference exhibit the same distribution ofrelaxation times. As an example Fig. 5a shows the relaxationcurves of a small tissue sample together with the relaxationcurves of two associated reference samples one with theMNP immobilized by freeze drying and the other in theoriginal fluid state. The relaxation curve of the immobilizedreference nearly perfectly matches the curve shape of thesample, thus being selected for the quantification.

Quantification of Small Irregularly Shaped Samples. In manypractical cases tissue samples show up in irregular shape. Thus,by simply applying the procedure described in "Identical andHomogenous Sample Volumes" quantification errors wouldemerge because the distance between MNP accumulationcentre of the sample and SQUID sensor may differ from thatof the homogeneous reference sample as illustrated in Fig. 5c.Examples are the quantification of uptake and clearance ofMNP by a small tumor or inadvertently by different organsof small animals like mice, rats or rabbits.

In these cases the location of the magnetic source(assumed to be point like) inside the sample volume a priori

is unknown, but can be determined by measuring thesample twice reversing the sample about the knowndistance d0 to the sensor, once in its normal posture andonce again placing the sample upside down into themagnetizing coils as depicted in Fig. 5b.

Let z0 mark the distance between the center of the samplecontainer to the sensor and thus the rotation axis for thesample reversion. Furthermore, d denotes the unknowndisplacement of the dipole location, depicted as a red circlein Fig. 5c, from position z0. So the dipole position is z’ = z0 + d

for the normal and z’ = z0-d for the reversed sample position.Then, d = |z-z’| can be extracted from Eq. 14 introducing theratio of the field amplitudes c = Bz,norm/Bz,rev

d ¼ z01� c1=3

1þ c1=3ð16Þ

Conveniently, the absolute distance z0 between rotationaxis and sensor is extracted from Eq. 16 by measuring themagnetic field of a small calibrated cylindrical coil withknown magnetic moment supplied by a square-wavealternating current, which is positioned at the rotation point.

In our example a displacement of d=9 mm wascalculated using a rotation point distance z0 of 37 mm asdetermined by the calibration coil measurement. Thequantification is then performed by means of Eq. 15with the measured relaxation amplitudes of sample andreference first normalized to the distance z0 by

Bz;corr ¼ Bz;normðz0 þ dÞ3=z30 ¼ Bz;revðz0 � dÞ3=z03.As an example of this scenario, the quantification of MNP

in cell samples is described. In a cell culture experimentcomplexes of MNP with lentiviruses, carrying the test gene


eGFP (for enhanced green fluoescent protein), were trans-duced to HUVECs (human umbilical vein endothelial cells) ina magnetic field gradient. For quantification of the MNPcontent within the cells after transduction, the HUVECs wereharvested, fixated, counted and transferred in vials for theMRXmeasurement. The cells stacked at the bottom of the vialby gravity forces, so that the spatial distribution of the sampleMNP distinguish from that of the reference sample, where theMNP were equally distributed. Thus, the cell samples weremeasured in up- and downward position and the center ofgravity of MNP was estimated by (16). We found a deviationfrom the geometric center of the sample, 5 mm in height, of(2.2±0.3) mm downwards. Accordingly, in the given setup, thequantification result was corrected by +35% compared to theresult which would be obtained by (15) being valid for ahomogeneous distribution of MNP in the measurementsample. The amount of MNP, quantified down to 1 pg/cell,correlated well with the gene expression, estimated bymeasurement of the green fluorescence light (20).

Reconstructing the MNP Distribution in a Sample of LargerExtension. The MRX approach based on "Quantificationof Small Irregularly Shaped Samples" can be utilized forthe spatially resolved quantitative reconstruction of a MNPdistribution throughout an extended tissue sample. Thiswas applied for ex-vivo quantification of MNP depositions inpig lungs after magnetic aerosol targeting (21). As shown inFig. 6a the extracted lung lobe was first dissected into about25 smaller pieces of about 2 cm3 volume. Each of these wasquantified by MRX as described in "Quantification of SmallIrregularly Shaped Samples" to obtain the absolute MNPcontent and then normalized to the corresponding tissue mass(leading to values for iron concentration in the tissue: μg Fe/gtissue). Finally, combining the single quantification results with

the individual locations within the whole sample (seephotography Fig. 6b), the MNP concentration distributionwas reconstructed over the whole lung lobe with a centimetrespatial resolution. For the example shown the MNP werefound nearly equally distributed over the whole lung lobewith a three-fold higher MNP uptake in the area of the mainsupplying bronchial tube. Additionally, the total absolutenanoparticle deposition within the whole lung and differ-ences between individual lung lobes were determined.

For the type of MNP used and the tissue pieces dissectedto fit into the 8 ml container, absolute iron amounts downto approximately m(Fe)=200 ng could safely be detected.By refining the sample dissection and using 150 ml samplecontainers at the closest possible sample sensor distanceeven a millimeter spatial resolution becomes feasible, withthe detection limit lowered to some 10 ng iron absolute.Note that this can only be regarded a rough estimationsince the particular detection limit is to a large extentdependent on type and size parameters of the individualMNP used in an application.

Multi Channel MRX: Localization and Quantificationof Magnetic Nanoparticle Accumulations in Tissue

The application of the dissection procedure described in"Reconstructing the MNP Distribution in a Sample ofLarger Extension" often becomes impractical for largersized samples or even impossible for in-vivo studies. In thissituation, often spatial information in terms of the magneticfield distribution is of interest, because these data may helpto reconstruct the distribution of the nanoparticle densitywithin the tissue. This information can be obtained byscanning a single channel SQUID system across the object,

Fig. 5 Quantifying the MNP amount of a small tissue sample by single channel MRX. (a) Relaxation curves of the sample piece together with immobilizedand fluid MNP reference signals. An arbitrary constant offset Bofs has been added to the relaxation curves solely for graphical representation. The bluedotted (fluid) and the green dashed (immobilized) curves are the reference signals normalized to match the sample curve. The normalization factor directlyis proportional to the iron amount in the sample. (b) Determining the centre of a MNP accumulation in an irregularly shaped tissue sample by doubleMRX measurement reversing the sample. (c) Individual relaxation curves of an irregularly shaped pig lung sample (inset picture) containing MNP. Togetherwith the relaxation curves of the sample measured in normal (dotted red curve) and upside down (dashed green) orientation the distance correctedrelaxation curve (straight black) is displayed.


or, more effectively, by using a multi-channel SQUIDsystem. In a multi-channel system a number of magneticfield sensors are located at different sites in a large Dewarcryostat, which is positioned close to the sample. Dependingon the distance and orientation of the sensor with respect tothe object, the relaxation amplitude is different for eachsensor, while the detected time dependent relaxation curvehas the same shape (Fig. 7a). For a fixed sample, a multi-channel system registers a steady magnetic field patternwhich decays monotonously in time (Fig. 7b).

A Single Point-Like MNP Accumulation. A single point-likeMNP accumulation is the simplest case of a nanoparticledistribution, which is relatively easy to reconstruct frommultichannel MRX data. Even if the real situation is morecomplicated, a point-like accumulation may be a usefulmodel that may help to obtain a first overview. Mathemat-

ically, the determination of location and amount of a point-like magnetic nanoparticle accumulation from a measuredrelaxation field pattern at a given time point t means theinversion of Eq. 13.

This can be performed analytically if the direction ofthe magnetic field during the magnetizing phasecoincides with the x-, y-, or z-direction, so that thelocalization of a point-like MNP accumulation can directlybe read off from a magnetic field pattern (22). With themagnetizing field along the z-direction (x-direction) thelateral x,y position is located at the maximum B+ of themagnetic field pattern. The depth or z location is determinedby the radius R of the zero line circle surrounding the fieldpattern, z = 2−1/2R, and the magnetic moment follows asmz=(2π/μ0)z

3B+ (mx=(55/2π/(12μ0))d

3B+). With the mag-netizing field applied along the x-direction the lateral x,y-position is located half way between the two extreme values

Fig. 6 Single channel MRX ondissected samples for quantitativereconstruction of a MNP distribu-tion after magnetic aerosol targetingin a pig lung lobe (part. caudalisdorsalis, numbers indicate μg iron/gtissue. (a) MRX quantificationresults of each tissue piecenormalized to tissue mass.(b) Reconstruction of the nano-particle distribution superimposedonto a picture of the (intact)lung lobe.

Fig. 7 Multi channel MRX of apoint-like MNP reference sample.Left) Relaxation curves (57 Bz-sensors) and corresponding mag-netic field patterns at t=5 s, 45 s,80 s. Right) Decay of magneticmoments (mx,my,mz) of the MNPdetermined by Levenberg-Marquardt fitting and analyticallyfor mz from the magnetic fieldpattern.


B+ and B−. The depth or z location is determined by thedistance between B+ and B−, while the magnetic momentfollows as mx = 25/12 51/2π/μ0d

3 B+.From four magnetic field patterns at t=5 s, 25 s, 45 s, and

80 s we determined a MNP accumulation localized atx=22 mm, y=44 mm, and z=−68 mm) with the magneticmoment decaying with time as displayed in Fig. 7b. Whenmagnetizing along z-direction the MNP accumulationcentred below the 304ch device should be closer than 7 cmto the sensor area (of radius 11 cm) since otherwise the zeroline of the field pattern would lie outside.

For an arbitrary magnetizing direction we use a non-linear Levenberg-Marquardt algorithm for fitting the threelocation parameters (x’,y’,z’) together with three magneticmoment (mx(t),my(t),mz(t)) parameters of the MNP accumu-lation describing best the corresponding magnetic fieldpattern of each relaxation time point according to Eq. 13.In this case, the location parameters x’,y’,z’ can be keptfixed in the fitting procedure, because the object does notmove with respect to the sensor system. These results aredisplayed for the moments in Fig. 7.

Similar to the single channel situation of "Identical andHomogenous Sample Volumes", the absolute MNPamount then follows by normalizing the magnetic momentto the moment of a reference sample of known MNPcontent measured and analyzed under same conditions.Though strongly depending on particular MNP type andsample properties MNP accumulations down to some μgiron can reliably be quantified and the localizationaccuracy often lies below one millimetre.

In-Vivo Localization and Quantification of a Magnetic Point-Like MNP Accumulation. For in-vivo MRX measurements onanimals (or even humans), additional signals from therespiration and muscle motion are superimposed to therelaxation signals even if the animal is kept fixed. An evengreater challenge is the recording of a free movingconscious animal, where the location of magnetic momentof the MNP accumulation cannot be assumed constant forthe fitting procedure.

As an example we present the in-vivo localization andquantification of a point like MNP accumulation in acarcinoma mouse model in a magnetic thermoablationtreatment investigation (23) In this study, MNP wereinjected into the tumor and subjected to an alternatingmagnetic field which is absorbed by the particles anddissipated by delivering heat to the surrounding tissue. Toassure an adequate heat production that kills the tumorcells, quantity and position of the MNP need to becontinuously controlled. To this end, the location andamount of the MNP accumulation were repeatedly deter-mined by 304ch MRX measurements of the consciousmouse, which was housed in a small cylindrically box.

Figure 8a shows the relaxation signal of a consciousmouse after MNP injection recorded by a central SQUIDof the 304ch system together with corresponding fieldpatterns (Fig. 8a top) at selected time points. Themovement of the mouse becomes nicely visible in the fieldpatterns, while the relaxation curves of the individualsensors detecting the magnetic field seem to be stronglydistorted (Fig. 8a). However, fitting the field patterns ateach time instant with the magnetic point dipole modelEq. 13 allows the identification of the spatial movement ofthe mouse (Fig. 8b) and the quantification of the magneticmoment relaxation of the nanoparticles (Fig. 8c).

Reconstruction of Extended MNP Accumulations. For thequantification and localization of magnetic nanoparticlesdistributed over larger extended tissue areas the magneticpoint dipole is an inadequate model. Nonetheless, aquantification and localization still can be accomplishedwith appropriate adopted forward models. So far we havetwo different approaches implemented for magnetorelax-ometry. As a straightforward expansion the multiple point

dipoles model uses a number of point dipoles at predefinedlocations within a sample for the description of themeasured magnetic relaxation field pattern (24). Addition-ally, adjusting the direction of the magnetizing field, onlyone parameter, the magnetic moment component pointinginto that direction, remains free for each of the dipoles.

Alternatively, at somewhat higher computational cost, themeasured magnetic relaxation field pattern can be analyzedby means of a multipole expansion (25). The resulting set ofmultipole coefficients in combination with an appropriatechoice of a magnetization distribution model enablesextracting important quantities like the total magneticmoment, the location and even the extension of anaccumulation.

Investigating the Relaxation Curve Shape: Interactionsof Magnetic Nanoparticles with Their BiologicalEnvironment

MNPs as Probes of the Local Hydrodynamic Environment

Magnetorelaxometry displays the superposition of signalsfrom all individual nanoparticles that are suspended in theliquid of the sample. Even though this is an integralapproach to monitor the behaviour of magnetic nano-particles, MNPs may serve as probes of the local hydrody-namic environment. As it was pointed out in theintroduction, the Brownian relaxation time of a givennanoparticle suspended in a fluid medium depends on itshydrodynamic volume, which includes the nonmagnetic (e.g.polymeric) shell surrounding the magnetic core as well as


possibly a hydration layer. The shape of the magneticrelaxation curve reflects the local hydrodynamic environmentof the MNP.

The organic molecules of the nanoparticle shell can becovalently or ionically bound to reactive biomolecules. Anyconformational change or chemical reaction of thesebiomolecules with analytes in the suspension will alter thehydrodynamic shape of the MNP. By this mechanism, thebinding reaction of a biomolecule and an analyte can bemonitored in terms of a change of the relaxation behaviour,so that MRX measurement data may even provide insightinto the microscopic behaviour of biochemical reactions.

A number of applications have been suggested that utilizethis mechanism, among them the so-called magnetic relaxationimmunoassay (MARIA) (26). In addition to the liquid phaseMRX immunoassay there are various assay layouts where theanalyte is not dissolved in the carrier medium (Fig. 9a) butimmobilized, e.g., by attaching it to a rigid surface (Fig. 9b) orto the surface of latex beads of much larger dimensions thanthat of the MNP, say, with diameters of around a micrometer.In such a setting, the Brownian relaxation is fully suppressedby the binding reaction, so that only the Néel mechanism isactive, leading to a slower effective relaxation. In Fig. 9c, abead based assay is depicted, where the analyte is fixed tolarge beads in the range 0.1.1 μm leading to a distinct changein hydrodynamic size (27).

Yet another type of assay (Fig. 9d) is the agglutination assay,which was recently suggested (28). In the agglutination assay,both the MNP and the analyte exhibit multiple binding sitesso that at a certain concentration ratio the analytes give rise to

multiple cross-linking between the particles. This results in theformation of large agglomerates, consisting of networks havingrelatively large hydrodynamic diameters.

In general, a series of MRX measurements is necessaryto interpret the relaxation measurements in terms ofchanges in the local hydrodynamic environment. The firststep is an MRX measurement that documents the initialstate of the MNP system under investigation. Then, thephysical parameters or the chemical composition of thesample is altered eventually leading to a change in thehydrodynamic behaviour of the MNPs, which is reflectedby changes of the MRX relaxation curve over subsequentmeasurements. Control measurements with removed orsaturated binding sites both on MNPs and on the analyteare in most cases appropriate to demonstrate the specificityof the hydrodynamic changes.

In the following, three practical approaches that quantifyMRX curve shape changes are discussed: 1) the comparisonto reference curves for the relaxation of bound andunbound nanoparticles, 2) the parameterization of therelaxation curve by a typical relaxation time t1/e and 3)modelling of the relaxation curve by assuming a specialdistribution function for the hydrodynamic diameters.

Characterization of the Relaxation Curve by Reference Curves

In the case of the solid phase assay layout where therotational diffusion of the bound particles is fully sup-pressed, the relaxation behavior of an individual particlechanges only upon the binding process, but remains

Fig. 8 304ch-MRX quantification of a magnetic nanoparticle accumulation in a conscious mouse: (a) the relaxation curve detection by the central SQUIDsensor starts some 10 s after switching off the magnetizing field and transporting the conscious mouse housed in a plastic tube beneath the MRX device.Magnetic field pattern snapshots (top row) at time points t=16 s, 19 s, 24 s, 71 s, 86 s display the point-like dipolar character of a nanoparticleaccumulation moving relatively to the sensor positions. Fitting the field patterns for each point in time with the magnetic point dipole model Eq. 13 allows(b) the identification of the spatial movement of the mouse and (c) the detection of magnetic moment relaxation of the nanoparticles. For comparison theresults for a fixed reference sample is added


unchanged as long as the nanoparticle is in its initial state.Thus, the task is to discriminate the signal contributions ofbound and unbound particles, where both fractions arecharacterized by a well defined relaxation behavior. Themost effective way to obtain reference data for thediscrimination task is by measuring separately MRX curvesof nanoparticles in the bound and in the unbound state.This is easily done by using samples of the nanoparticlepreparations in solution for the reference signal Bub(t) ofunbound particles and freeze-dried samples for the referencesignal Bb(t) of bound and immobilized particles.

The measured relaxation curve B(t) is modelled as the

superposition bBðtÞ of the reference curves Bb(t) for boundand Bub(t) for unbound MNP, complemented by anarbitrary constant offset Boffs that cannot be avoided inmeasurements using SQUIDs:

bBðtÞ ¼ a½bBbðtÞ þ ð1� bÞBubðtÞ� þ Boffs ð17ÞThe parameter a describes the iron content of the

measured sample in relation to that of the reference sample,and β describes the fraction of bound MNP. In matrixnotation, Eq. 17 is written as:

bB ¼ Pz with P ¼ Bb Bub 1� �

and z

ab

að1� bÞBoffs

264375 ð18Þ

The model is related to the vector of measured data Bby introducing a vector of model deviations u:

B ¼ bBþ u ¼ Pzþ u with u ¼B1 � bB1

..

.

BK � bBK

2666437775 ð19Þ

The unknown parameter vector z is found by minimizingthe variance of u which is satisfied for the ordinary leastsquare estimator zOLS:

zOLS ¼ab

a 1� bð ÞBoffs

264375 �

k1

k2

Boffs

264375 ¼ PþB ¼ PTP

� ��1PTB ð20Þ

Here, P+ denotes the Moore-Penrose pseudoinverse ofP. The fraction of bound magnetic nanoparticles is theneasily calculated as β ¼ k1= k1 þ k2ð Þ and the relative ironcontent follows as a ¼ k1 þ k2.

Since an MRX measurement can be repeated inintervals of a few seconds and washing steps are notnecessary, β can be updated with a temporal resolution of afew seconds. Thus, the method may be used as a tool tomonitor the binding kinetics of a reaction that proceedswithin a corresponding time interval.

Fig. 9 Different kinds of MNPbased binding-assays for biomole-cule detection by MRX: (a)solid phase, (b) liquid phase, (c)bead based, (d) agglutination assay


Characterization of the Relaxation Curves by t1/e

In some assay types, e.g. in the agglutination assay (Fig. 9d), thedistribution of hydrodynamic diameters of the colloidalentities containing nanoparticles is permanently changingthroughout the binding process. Consequently, while thedistribution of magnetic core diameters will remain constant,the distribution P(dh) of hydrodynamic diameters at a specifictime instant t is not known. In this case, it is impossible to usereference curves for quantification of the binding process as itwas proposed in section "Characterization of the RelaxationCurve by Reference Curves". A pragmatic way to documenta change in curve shape is given by computing t1/e, i.e. thetime when the amplitude of the relaxation signal has reached1/e of its initial value. The total signal is the superposition ofall relaxation processes occurring with τeff = τeff(dp,dh) in thesuspension. Especially for larger so-called “blocked” particles,where τN >>tmeas, the effective relaxation time τeff is directproportional to the hydrodynamic volume Vh and all changesin the hydrodynamic volume Vh are reflected in t1/e.

A significant rise in t1/e means increased hydrodynamicdiameters of the colloidal compartments and may indicateaggregation processes. Thus, this method can provide anefficient quality check of MNP suspensions.

Parameter Estimation for the Distribution of HydrodynamicDiameters

Under the assumption of a special distribution function forthe hydrodynamic diameters, the parameters of thisdistribution can be estimated from the relaxation data. Atypical example is the assumption of a lognormal distribu-tion of hydrodynamic diameters dh, as in Eq. 8. By fittingthis model to the measurement data via a nonlinear fittingroutine, the mean hydrodynamic diameter μh and thedistribution width σh can be estimated.

CONCLUSION

Magnetic nanoparticles have a wide range of applications inmedicine and biomedical research. Their physical propertiesenable the measurement of a magnetic relaxation signal aftera rapid change of an external magnetic field. The relaxationsignal depends on core sizes and hydrodynamic volumes of theMNPs. Because neither the surrounding biological environ-ment nor any remanent magnetic material in the vicinity willgenerate such relaxation signals, magnetorelaxometry ishighly specific for MNPs.

Thus the quantification ofMNP in living conscious animalsis feasible. The temporal decoupling from the large change inexcitation field allows one the use of ultrasensitive SQUIDsensors for readout of the magnetic relaxation signals. This

technique together with the high specificity of MRXmeasure-ments make the quantitative determination of the MNPcontent in large tissue samples down to a few picogrampossible. The analysis of the MNP relaxation curves enablesthe observation of colloidal structure changes.

ACKNOWLEDGMENTS & DISCLOSURES

The authors gratefully acknowledge the funding by theGerman Research Foundation (Grants TR408/5-1 andTR408/6-1).

Open Access This article is distributed under the terms of theCreative Commons Attribution Noncommercial Licensewhich permits any noncommercial use, distribution, andreproduction in any medium, provided the original author(s)and source are credited.

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