Empirical and Theoretical Characterization of the Diffusion...

Research ArticleEmpirical and Theoretical Characterization of the DiffusionProcess of Different Gadolinium-Based Nanoparticles within theBrain Tissue after Ultrasound-Induced Permeabilization of theBlood-Brain Barrier

Allegra Conti 12 Remi Magnin13 Matthieu Gerstenmayer1 Nicolas Tsapis 4

Erik Dumont3 Olivier Tillement5 Franccedilois Lux 5 Denis Le Bihan1

Sebastien Meriaux 1 Stefania Della Penna 26 and Benoit Larrat1

1NeuroSpin Institut des Sciences de La Vie Frederic Joliot Direction de La Recherche FondamentaleCommissariat a LrsquoEnergie Atomique et Aux Energies Alternatives Universite Paris Saclay Gif-sur-Yvette France2Department of Neuroscience Imaging and Clinical Sciences Institute for Advanced Biomedical TechniquesG DrsquoAnnunzio University Chieti Italy3Image Guided erapy Pessac France4Institut Galien Paris-Sud CNRS Univ Paris-Sud Universite Paris-Saclay Chatenay-Malabry France5University Lyon 1 Lyon France6Consiglio Nazionale Delle Ricerche Institute SPIN UOS LrsquoAquilaSite CNR-SPIN Co Universita di Chieti-Pescara ldquoG DrsquoAnnunziordquo Chieti Italy

Correspondence should be addressed to Allegra Conti contiallegragmailcom

Received 9 April 2019 Accepted 21 October 2019 Published 1 December 2019

Guest Editor Nicola Toschi

Copyright copy 2019 Allegra Conti et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Low-intensity focused ultrasound (FUS) combined with microbubbles is able to locally and noninvasively open the blood-brainbarrier (BBB) allowing nanoparticles to enter the brain We present here a study on the diusion process of gadolinium-basedMRI contrast agents within the brain extracellular space after ultrasound-induced BBB permeabilization ree compounds weretested (MultiHance Gadovist and Dotarem) We characterized their diusion through in vivo experimental tests supported bytheoretical models Specically by estimation of the free diusion coecients from in vitro studies and of apparent diusioncoecients from in vivo experiments we have assessed tortuosity in the right striatum of 9 Sprague Dawley rats through a modelcorrectly describing both vascular permeability as a function of time and diusion processes occurring in the brain tissue ismodel takes into account acoustic pressure particle size blood pharmacokinetics and diusion rates Our model is able to fullypredict the result of a FUS-induced BBB opening experiment at long space and time scales Recovered values of tortuosity are inagreement with the literature and demonstrate that our improved model allows us to assess that the chosen permeabilizationprotocol preserves the integrity of the brain tissue

1 Introduction

e in vivo characterization of gadolinium-(Gd-) based MRIcontrast agent (MR-CA) diusion within the brain tissue isof great interest for the understanding of drug transportmechanisms in the brain parenchyma in the framework ofthe recent pharmaceutical developments targeting entral

nervous system (CNS) diseases Despite increasing eortsand encouraging results drug delivery to the CNS remains achallenging task Indeed the blood-brain barrier (BBB) notonly prevents neurotoxic substances from entering the brainbut also limits the passage of therapeutic products to theCNS [1 2] Many strategies have been studied to overcomethis obstacle including direct injections [3 4] transient BBB

HindawiContrast Media amp Molecular ImagingVolume 2019 Article ID 6341545 13 pageshttpsdoiorg10115520196341545

disruption using chemical agents [5 6] or molecular en-gineering [7] More recently a promising technique has beenproposed allowing the delivery of various compounds to thebrain using low-intensity focused ultrasound combined withcirculating microbubbles [8]

However once the molecules have crossed the barrierthey have to diffuse in a highly constrained media theextracellular space (ECS) to reach their targets [9] More-over since the ECS architecture can change in case of pa-thologies [10 11] the characterization of the hindranceexperienced by molecules within the brain tissue is essentialwhen designing new therapeutic compounds or diagnosticmolecules for brain diseases Diffusion constraints can bestudied by estimating the ECS tortuosity (λ) is parametercompares the apparent diffusion coefficient (ADC) of amolecule within the complex architecture of the ECS to itsdiffusion coefficient in a free medium Dfree [12] Differentstrategies have been proposed to measure the ADC emost widely used method is real-time iontophoresis [9 13]using tetramethylammonium (TMA+) as a probe istechnique not only permits the in vivo characterization of theADC but also thanks to the small size of both detectionelectrodes and injection micropipette proves minimallyinvasive with consequent preservation of the integrity of thetissues Its main drawback consists in the measurementrelying on just one spatial point More recently diffusion-weighted magnetic resonance imaging (DW-MRI) has beenproposed to noninvasively measure the ADC of watermolecules in the brain [14 15] In comparison to the pre-vious techniques DW-MRI allows ADC measurements indeeper areas of the brain with a high (typically 2mm iso-tropic) spatial resolution [16] However contrary to TMA+

and other techniques using labelled molecules that diffuseonly across the ECS DW-MRI detects water which is alsopresent in the intracellular compartment To benefit fromthe advantages offered by MR in acquiring deep volumes ofthe brain a newmethod has been recently introduced by ourteam which allows us to detect molecular diffusion only inthe ECS structure [17] To do so MR-CAs are directly in-jected into the brain tissue and their diffusion is followed byacquisition of several longitudinal relaxation-time (T1)parametric maps MR-CA concentration maps at differentdiffusion times are then calculated and from these the ADCis estimated When compared to the typical diffusion-basedMRI techniques our method investigates larger areas of thebrain with a higher spatial resolution (about 02times 02mm2 inplane and 1mm in thickness) However a major issue raisedby this procedure consists in intracerebral injections in-ducing edema which modifies the diffusion properties ofbrain tissue

In the present study we have used two noninvasivemethods for the in vivo estimation of the ADC of differentGd chelates diffusing in the ECS after a FUS-induced BBBopening experiment In both cases contrast agent diffusionis recorded through dynamic acquisitions of MRI concen-tration maps In the first method the ADC evaluation isperformed as in [17] eg by fitting a 2D Gaussian curve tothe image intensity at different time points However dif-fusion of molecules delivered to the brain with the aid of

FUS-induced BBB permeabilization depends on many fac-tors such as tissue and particle properties as well as acousticparameters For this reason as a second approach to esti-mate contrast agent diffusion we introduce here the firstdiffusion model able to fully describe and predict at longspace and time scales the result of a FUS-induced BBBopening experiment is model takes into account acousticpressure particle size blood pharmacokinetics vascularpermeability as a function of time and diffusion processoccurring in brain tissue

Starting from ADC estimation performed with the helpof both methods and the evaluation of Dfree for all thecompounds bymeans of in vitro experiments it is possible tocalculate tortuosities in the target region of ratsrsquo brains toevaluate the effect of the selected BBB permeabilizationprotocol on the properties of brain tissue

2 Materials and Methods

21 Experimental Procedures All magnetic resonance ac-quisitions were performed by using a 7 T90mm Phar-mascan scanner (Bruker Ettlingen Germany) e in vitroacquisitions have been performed by using a 1H transmit-receiver volume coil (Bruker) e in vivo experiments havebeen conducted by using a dedicated ultrasound single-loopradiofrequency coil [18] whose diameter was wide enoughfor the ultrasound beam to pass through it and for extensivedisplacement of the transducer above the ratrsquos head A heaterdevice was used to keep temperature at the physiologicalvalue (37degC) as monitored by a temperature probe that wasinserted inside the magnet (see Figure 1)

ree different gadolinium (Gd) chelates were studiedDotaremreg (Gd-DOTA Guerbet France) Gadovistreg (Gd-DO3A-butrol Bayer Germany) and MultiHancereg (Gd-BOPTA Bracco Italy) First we assessed their longitudinalrelaxivities (r1) at 7 T and 37degC using phantoms made ofbundles of tubes containing different contrast agent (CA)concentrations in 03 ww agarose gel for each compoundFor these phantoms T1 values were measured by means ofan inversion-recovery fast gradient echo (IR-FGE) sequence[17 19] (echo time (TE)repetition time (TR1) 255ms 6segments 90 inversion times (TI) varying from 75ms to8975ms flip angle (FA) 5deg matrix 120times120times 5 withresolution 0250times 0250times125mm3 delay between theacquisitions of two segments (TR2) 15000ms and numberof averages (NA 6)) Resulting relaxivities are summarizedin Table 1 is table also includes the hydrodynamic di-ameter (dH) of each CA measured by dynamic light scat-tering (DLS) DLS experiments were performed using aNanoZS equipement (Malvern France) operating at anangle of 173deg For each Gd chelate the DLS acquisitions wereperformed at 25degC by using concentrations of 05M forMultiHance and Dotarem and of 10M for Gadovist egwithout diluting the samples We performed five differentDLS measurements for each sample e mean dH and thestandard deviation evaluated over the five measures arereported in Table 1

Evaluations of the Dfree of each compound were donewith amethod already presented in a previous work [17] For

2 Contrast Media amp Molecular Imaging

each product 10 μL of a 5mM solution was injected with aHamilton syringe (diameter 1mm) into a tube filled with03 ww agarose gel A stereotactic system was used tomake the injection central and vertical with respect to thetube e free diffusion of the CA was then dynamicallyfollowed by acquisition of five T1 parametric maps afterinjection (IR-FGE sequence with the following parametersTETR1 255ms 6 segments 60 TI from 88ms to 5100msFA 5deg matrix 128times104times14 with res 0225times 0225times 1mm3 TR2 9000ms NA 1 and total dura-tion 125min) A T1 map acquired before the injection wasused as a reference

e number of TI values has been chosen to ensure anaccurate estimation of T1 values for a large range of T1 Inparticular thanks to this sequence we are able to detect Gdconcentrations with a sensitivity threshold estimated around25 μM [17] e spatial and temporal resolutions of thismapping sequence were set in order to ensure a sufficientspace and time sampling of CA diffusion process Furtherdetails about the optimization of this MRI sequence can befound in [17] In all T1-parametric maps all voxels with a T1value larger than 5000ms that is much larger than both theT1 of gray and white matter at 7 T [20] have been maskedand considered as Not-a-Number

e measurements of the ADC were performed in vivoon 9 Sprague Dawley male rats (3 ratscompound 120ndash140 g Janvier Le Genest-Saint-Isle France) Animal testingcomplied with the recommendations of the EuropeanCommunity (86609EEC) and French legislation (decreeno 87848) e experimental setup is shown in Figure 1

e rats were anesthetized by means of 15ndash2 isoflurane ina mixture of air and oxygen and their heads were chemicallyshaved to ensure a proper coupling with the ultrasoundtransducer ey were then placed in prone position in acradle integrating a stereotactic frame and a dedicatedradiofrequency coil (Figure 1(b)) A custom build catheter(25G) was inserted into the caudal vein to perform injectionsfrom outside the MRI scanner Temperature monitoring andbreathing monitoring were performed using a rectal tem-perature probe and a respiration probe (Figure 1(c))respectively

A MR-compatible focalized transducer with 15MHzcentral frequency (diameter 25mm focal depth 20mm focalspot dimensions 11mm in-plane 6mm thickness ImasonicFrance) was coupled to any animalrsquos head via a balloon filledwith degassed water e transducer was mounted on amobile stage and its position could be tuned from outside themagnet by using MR-compatible motors (see Figure 1(b))emovement of themotors and ultrasound parameters werecontrolled by a dedicated software (ermoguidereg ImageGuided erapy France) (Figure 1(a)) All acoustic pressureswere estimated from previous calibration of the transducertaking a skull transmission factor varying with animalsrsquoweight [21]

In Figure 2 the experimental protocol is shown Afterrat installation an acoustic radiation force imaging (ARFI)sequence [22 23] was performed to localize the ultrasoundfocal point in ratsrsquo brains consisting in a standard mul-tislice multiecho sequence (MSME TETR 281080msmatrix 64times 64 times 5 and res 05 times 05 times 2mm3) modified

Motors

Water degassing

Motor control

Ultrasound generator

MRI

Intravenous injections

(a) (b)

Temperatureprobe

Respiration probe

(c)

MRI console

Figure 1 Experimental setup (a)eMRI console and the computer driving the electronic for ultrasound embedded in a tower composedby the water degassing system the motor control and the ultrasound generator (b) e transducer and its electronic compatible with theMRI scanner e transducer can move along the two perpendicular directions pictured by the green arrows (c) e respiration and thetemperature probes for real-time monitoring of the animalrsquos vital signs

Table 1 Table reporting the characteristics of the three contrast agents longitudinal relaxivity r1 (sminus 1mMminus 1) measured at 7 T and 37degChydrodynamic diameter found from both DLS measurements dH(DLS) and by using the StokesndashEinstein equation dH(S-E) free diffusion(Dfree) of the molecules Standard deviations (SD) are shown in bracket e SD of theDfree values has been calculated by averaging the errorestimated on both DfreeX and DfreeY components when fitting the Gaussian widths through equation (4)

Compound Number of phantoms r1 (sminus 1mMminus 1) dH (DLS) (nm) dH (S-E) (nm) Dfree (10minus 10m2s)Dotarem 1 47 (02) 16 (01) 15 (01) 45 (02)Gadovist 1 55 (03) 18 (01) 17 (01) 39 (02)MultiHance 1 69 (03) 23 (01) 23 (01) 28 (02)

Contrast Media amp Molecular Imaging 3

by the addition of two motion-sensitizing gradients (MSGsduration of one MSG 8ms and duration of the ultrasoundbursts 4ms) Knowing the current position of the focalspot the transducer was moved using the motors so as tofocalize ultrasound in the left striatum of the rats islocation has been chosen to ensure a high acoustictransmission through the skull as detailed in a recent workpublished by our team [21] A second ARFI image wasacquired to assess the good positioning of the ultrasoundfocal spot T1-weighted (T1w) anatomical images wereacquired before the BBB opening by using an MSME (TETR 83300ms matrix dimension 256 times 256times10 reso-lution of 0125times 0125 times1mm3 and 3 averages) is wasfollowed by a bolus injection of Sonovuereg microbubbles(Bracco Milan Italy 15 times108 bubblesmL 16mLkg 3 s)via tail vein catheter approximately 5 s before transcranialsonication (3ms burst every 100ms over a period of oneminute estimated focal acoustic pressure in thebrain 08MPa) 30 seconds after the end of the ultrasoundsession Gd chelates were intravenously injected via bolus(5 seconds 05M and 16mLkg for MultiHance andDotarem 1M and 08mLkg for Gadovist) T1-weighted(T1w) images were acquired 30 seconds after the CA in-jection to verify the BBB disruption Using the same IR-FGE sequence as the one used for in vitro diffusion T1parametric maps were acquired before and after sonicationin order to dynamically follow the diffusion of the Gdchelates in the brain At the end of each experimentalsession a T2-weighted (T2w) image was acquired to verifythe absence of any hemorrhage or edema due to ultra-sound-induced BBB disruption A rapid acquisition withrelaxation enhancement (RARE) sequence was used withthe following parameters TETR 103800ms RAREfactor 8 and matrix 128times128 times 32 with resolution

0225 times 0225times 05mm3

22 Data Analysis From T1 maps the correspondingconcentration maps were calculated using the followingrelationship between the longitudinal relaxation rates 1T1and the Gd-chelate concentrations [CA] [24]

1T1

1

T10+ r1 middot [CA] (1)

where 1T10 is the relaxation rate of the sample without CAie before the injection From this equation CA concen-trationmaps were then obtained All voxel values in the T1 orT10 maps larger than 5000ms were considered as Not-a-Number in the CA maps ese voxels were not consideredin the CA-diffusion analysis

In all cases (both for in vivo and in vitro acquisitions) wehave assigned to each CA map the time elapsed between theCA injection (in agarose gel or in the caudal vein for the invitro and the in vivo acquisitions respectively) and thebeginning of the CA-map acquisition sequence

To evaluate the Dfree value of injected molecules thefollowing bidimensional Gaussian function was fitted toconcentration-map data for each time point after the CAinjection [17 25 26]

[CA(x y)] Aeminus a(xminus x0)2minus 2b(xminus x0)(yminus y0)minus c(yminus y0)2( ) (2)

where A is the Gaussian amplitude and (x0 y0) are thecoordinates of its center along the absolute axes (x y) a band c are functions depending on the Gaussian widths (σXand σY) along its main axes (X and Y) and on the angle θbetween (X Y) and (x y)

a cos2(θ)

2σ2X+sin2(θ)

2σ2Y

b minussin(2θ)

4σ2X+sin(2θ)

4σ2Y

c sin2(θ)

2σ2X+cos2(θ)

2σ2Y

(3)

e regression algorithm used to fit the data withGaussian functions is the LevenbergndashMarquardt algorithm[27] available in the GSL GNU Scientific Library (httpswwwgnuorgsoftwaregsldochtmlnlshtml) In particularwe used the version of this algorithm implemented in thescaled LMDER routine in MINPACK written by Jorge J

Animalinstallation

T1-wMSME

ARFI T2-wRARE

T1-wMSME

T1-mappreinjection

T1-map T1-map

Ultrasoundopeningsession T1-map

[hellip]2prime20Prime 2prime00Prime 12prime30Prime 1prime00Prime 2prime00Prime 12prime30Prime 12prime30Prime 4prime00Prime

Gd chelateinjection

Dynamic diffusion (gt1 hour)

Figure 2 Experimental protocol for in vivomeasurements an ARFI sequence was used to detect the local acoustic intensity and choose theposition of the BBB opening indicated by the black arrowis 2-minute acquisition was followed by a T1-weightedMSME sequence and bythe first T1 map acquired just before opening One minute after the ultrasound opening session the MRI contrast agent was injected A T1-weighted image was acquired to evaluate the goodness of the opening procedure About two minutes after the CA injection the diffusionprocess was followed over more than 1 hour by acquiring several T1 maps At the end of each experimental session a T2-weighted RAREimage was acquired to evaluate damages such as hemorrhages and edema due to ultrasound All the images shown in this figure refer toacquisitions performed by using Gadovist


More Burton S Garbow and Kenneth E Hillstrom (httpspeoplescfsuedusimjburkardtf_srcminpackminpackhtml)

Defining σ2X and σ2Y as the molecular mean squaredisplacements along X and Y the diffusion coefficients alongthese axes DfreeX and DfreeY are given by Fickrsquos law

DfreeXY σ2XY

2t (4)

where t is the instant time after injection ie the diffusiontime Dfree values were then calculated as the mean value ofDfreeX and DfreeY components

Dfree DfreeX + DfreeY1113872 1113873

2 (5)

e first method used to evaluate the ADC consisted inplacing a mask surrounding the disruption site in CA mapsto which the same Gaussian fitting procedure was appliedFor any compound the ADC was estimated in any ratrsquosstriatum as the average

ADC ADCX + ADCY( 1113857

2 (6)

e second ADC estimation took into account how theBBB permeabilization changes after the ultrasound appli-cation together with CA pharmacokinetics after injection Ahomemade MATLAB code was used to simulate CA dif-fusion within the ECS after the BBB opening

e code comprises the following components

(i) A source function S (x y z t) describing the contrastagents that move from the blood to the brain wasmodeled as

S(x y z t) α middot QCA(x y z t) middot CAblood(t) (7)

where α is a proportionality constant requiring a firstguess on its value QCA(x y z t) is the amount of CAcrossing the BBB [28] and CAblood(t) describes CApharmacokinetics For a Gd chelate of hydrodynamicdiameter (dH) QCA(x y z t) is defined as [28 29]

QCA(x y z t) simσ20eminus 2kt

dH

middot

π2

1113970

1 minus erfdH

2radic

σ0(x y z)eminus kt1113888 11138891113888 11138891113888

+dH

σ0(x y z)eminus kte

minus d2H 2σ20(xyz)eminus 2kt( )( ) 1113889

(8)

where σ0 is the standard deviation of the distribution ofthe gap sizes generated in the BBB by ultrasound and kis the BBB closure rate (k 154eminus 5middotsminus 1) Since it hasbeen demonstrated that blood-brain barrier disruptionis characterized by a mechanical index (MI) which islinearly dependent on the effective acoustic pressure(Pex) [30] we considered the same dependence for σ0In particular according to the work published byMarty

et al in 2013 [28] we applied the relationshipσ0 21middotPex Starting from the simulated acousticpressure map we obtained the σ0(x y z) distributione kinetic term in equation (7) can be expressed by

CAblood(t) CAinj middot expminus t

b1113874 1113875 (9)

since our time resolution in the acquisition of CAmaps (125min) allows for just considering the washout of CAs in obedience to Toftsrsquo two-compartmentkinetic model [31] CAblood(t) depends on the in-jected CA concentration (CAinj) and on its clearancerate from the blood b CAinj was estimated for eachanimal by taking into account its weight and anaverage blood volume of 686mL100 g [32] while bwas fixed at 25 minutes [33]

(ii) Introducing the source term S(x y z t) into Fickrsquossecond law the evolution of CA-concentration longtime was found by simulating the equation

z[CA](x y z t)

zt ADCx middot

z2[CA](x y z t)

zx2

+ ADCy middotz2[CA](x y z t)

zy2

+ ADCz middotz2[CA](x y z t)

zz2

+ S(x y z t)

(10)

for a temporal and spatial resolution higher than thosecharacterizing [CA] maps Specifically an isotropicspatial resolution (dx dy dz) equal to 0125 μm wasselected while the temporal step dt was set at 5 s Whileα has been guessed the initial ADC values used forsimulations were chosen from the equation

λ

Dfree

ADC

1113970

(11)

starting from tortuosity values of the target region ofthe brain recovered from the literature [34] andmolecularDfree values retrieved from our experiments

(iii) e simulated CA volume was downsampled in spaceand time to the MRI acquisition resolution and thencoregistered to the experimental three-dimensional[CA] distribution in the CA-concentration maps

Due the large focal-spot length (sim6mm) CA concen-tration can be considered as constant along this direction(called z) for all the slices taken in account is makes theCA gradient negligible along z as well as the related dif-fusional process (see Figures S1 and S2 in the SupplementaryMaterials) For this reason the previous equation can beconsidered as reasonably describing the following bidi-mensional dynamics


z[CA](x y t)

zt ADCx middot

z2[CA](x y t)

zx2

+ ADCy middotz2[CA](x y t)

zy2 + S(x y t)

(12)

is last equation was integrated in order to estimate[CA](x y t) rough a cumulative fit including the exper-imental CA maps for the central slice the ADC componentsalong x and y and the proportionality constant α were foundDifferent ADCs around the value suggested by equation (11)were simulated until the fit algorithm converged

To evaluate the quality of the experimental approachchosen to mimic molecular free diffusion (ie the injectionof the compound in 03 ww of agarose gel) it is worthestimating the hydrodynamic diameter of the moleculesusing the StokesndashEinstein equation

dH kT

3πηDfree (13)

where k 138middot10minus 23 Pamiddotm3middotKminus 1 is the Boltzmann constant Tis the temperature in Kelvin degrees and η is the viscosity ofthe agar gel (692middot10minus 4 Pamiddots)

From the mean ADC recovered through the twoaforementioned methods the tortuosity values were esti-mated with the help of equation (11)

3 Results

Figure 3 shows an example of in vitro diffusion data and theiranalysis Concentration maps (Figure 3(a)) were acquired 4to 56minutes after the injection of MultiHance ese datawere fitted by means of the bidimensional Gaussian functionreported in equation (2) e simulated Gaussian distri-butions resulting from the fit are shown in Figure 3(b)Taking into account the voxel values in the central row of theGaussian spots pictured in Figures 3(a) and 3(b) it ispossible to assess the quality of the fit as illustrated inFigure 3(c) where the black dots represent the data and thered curve their Gaussian fit

Fickrsquos law (equation (4)) was used to fit the squares of thefitted Gaussian widths (σx and σy) as a function of thediffusion time in order to obtain an estimation ofDfreeX andDfreeY (Figure 3(d)) e Dfree values found for each com-pound are given by the average of the two components andare summarized in Table 1

e ADCs were estimated by analyzing in vivo con-centration maps as the ones shown in the upper panel ofFigure 4 Specifically these maps were acquired 2 to 84minutes after bolus injection of Dotarem Prior to computeGaussian fits on concentration maps a mask including onlythe BBB disruption site was applied (Figure 4(b)) e firstmethod for ADC evaluation consists in fitting 2D Gaussianfunctions to such maps e resulting distributions areshown in Figure 4(c)

As for the in vitro measurements the overlapping be-tween data and fit curve is shown (see Figure 4(d)) By

comparing through a two-sample KolmogorovndashSmirnovtest the data shown in Figure 4(c) with the respectiveGaussian profiles at each time point we obtained p valuesequal to 56e minus 4 0258 0258 0440 and 02581 meaningthat only at the first time point the Gaussian fit results to bedifferent from the data We also evaluated the ADC valueswithout taking into account the first time point Howeversince the values obtained with and without the first timepoint varied less than the error estimated by the respectivelinear fits and less than the variations inside the n 3 ratpools we also considered the first time point to estimate theADCs

e temporal evolution of the squared Gaussian widthsis shown in Figure 4(e) together with their fits by Fickrsquos LawStarting fromADCX and ADCY values the ADC in each ratrsquosstriatum was found By average over the entire set of rats themean ADCs reported in Table 2 were estimated as well asbrain tortuosity λI

e second method proposed to evaluate brain diffu-sional properties is based on a model taking in account boththe temporal changes in BBB permeabilization after ultra-sound application and CA blood pharmacokinetics

Figure 5 shows an example of CA distributions inside thebrain obtained by fitting this model to experimental con-centration maps obtained by diffusion measurements onMultihance

Once again Figure 5(a) reports the masked concen-tration maps used to evaluate brain tortuosity while themaps in Figure 5(b) are obtained via model e ADCsestimated by average of model results obtained for eachcompound are shown in Table 2 (ADCII) In the same tablethe values obtained for the proportionality constant α of thesource term are included Entering Dfree and ADCII valuesfound by this second approach in equation (11) braintortuosity is once again retrieved (λII in Table 2)

For the sake of comparison in Figure 6 the distributionprofiles extracted from the centers of [CA] maps are shownas previously done in Figure 4(d) is dataset refers to anexperiment on Gadovist with black dots representing ex-perimental data and [CA] red and blue profiles representingtheoretical data obtained from the first and second methodrespectively By comparing through a two-sampleKolmogorovndashSmirnov test the data with the simulated andthe Gaussian profiles we obtained at different time points pvalues equal to 0985 0374 0147 0047 0147 and 0047 formethod I and equal to 0675 0675 0736 0736 0736 and0736 for method II

ese results show that method II allows for obtainingdistribution shapes that are more similar to data at all thetime points than Gaussian fits in method I

4 Discussion

is work introduces two new methods suitable for the invivo characterization of molecular diffusion processes takingplace in the ECS after transient BBB permeabilization withlow-intensity focused ultrasound in order to deliver MR-contrast agents to the brain We used MRI to record MR-CAdiffusion By measuring DFree (free-medium diffusion) and


[CA

] (m

M)

t = 4 min t = 18 min t = 30 min t = 43 min t = 56 min01

005

0

(a)

[CA

] (m

M)


005

0

(b)

Voxel30 40 50 60

0

002

004

006

008

01

[CA

] (m

M)

Voxel30 40 50 60

Voxel30 40 50 60

Voxel30 40 50 60

Voxel30 40 50 60

DataMethod I

(c)

R2 = 0989

R2 = 0994

σY

σX

0

05

1

15

2

25

σ2 (10

ndash6m

2 )

15 30 45 600t (min)

(d)

Figure 3 In vitro diffusion ofMultiHance (a) Concentrationmaps acquired during 1 hour after the injection of 200 μL of the 5mM contrastagent in a phantom made of 03 ww agarose gel e time reported above each CA map refers to the time elapsed since the CA injection(b) Concentration maps obtained by fitting the maps shown in (a) through equation (2) for each time point (c) Shows a profile of the [CA]values (black dots) in the central rows on (a) and their corresponding fit (red line) from (b) ese curves are shown for each time point In(d) the trends of the square values of the Gaussian widths are shown as a function of the diffusion time In green and orange are pictured theexperimental data and the linear fits σ2XY DXYvitro middot 2t for σ2X and σ2Y respectively


[CA

] (m

M) 025

02015

01005

0

T = 2 min T = 15 min T = 28 min T = 41 min T = 53 min T = 66 min

(a)

[CA

] (m

M)

02502

01501

0050


(b)

[CA

] (m

M)

02502

01501

0050


(c)

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

[CA

] (m

M)

025

02

015

01

005

020 400

Voxel

DataMethod I

(d)

σY

R2 = 0898

R2 = 0996

σX

0

05

1

15

2

σ2 (10ndash6

mm

2 )

10 20 30 40 50 60 700T (min)

(e)

Figure 4 In vivo experiments performed with Dotarem where the BBB has been opened in the left striatum Figure (a) shows the concentrationmaps acquired between 2 and 66 minutes after the injection e masked maps used to perform the Gaussian fits are shown in row (b) while theGaussian surfaces obtainedwith the fit are pictured in (c) By comparing through a two-sampleKolmogorovndashSmirnov test the data shown in (c)withthe respective Gaussian profiles at each time point we obtained p values equal to 56e minus 4 0258 0258 0440 and 02581 (d) shows Gaussian profiles(red line) fitting the [CA] values (black dots) in the rows going through the centers of the spots in (b)e squares of the Gaussianwidths σX and σYare plotted over the diffusion time with the linear fit σ2XYDXYvivo middot 2t in (e) where the green and the orange colors refer to σX2 and σY2 respectively


ADC values within the ECS brain tissue tortuosity wascalculated in order to have information on brainarchitecture

To assess the quality of the experimental approachchosen to evaluate molecular free diffusion it is worthcomparing the hydrodynamic diameter of the moleculesdH(S-E) obtained through equation (13) to the ones foundby using DLS As can be noticed from Table 1 the hydro-dynamic diameter found through these two methods agreewhich means that the diffusion of the compounds in 03 w

w of agarose gel can be considered as free In addition Dfreevalues in Table 1 can be compared to the analogous onesalready published in the literature Specifically Marty et al[17] have found the sameDfree for Dotarem whereasorneand Nicholson [35] have estimated a free diffusion co-efficient equal to (222plusmn 016)middot10minus 10m2s for a molecule withhydrodynamic diameter of 295plusmn 002 nm which is com-parable to one that was found for a slightly smaller moleculeof MultiHance (dH 23plusmn 01 nm and Dfree (28plusmn 02) middot

10minus 10m2s)

Table 2e ADC and the λ values found with both methods are reported where the index I refers to the 2D gaussian fit methode ADCIIand the λII are the results obtained from the newmodel introduced in this work mimicking all the physiological processes occurring duringan experiment of FUS-induced blood-brain barrier opening for drug delivery (see Section 22) e parameter α is a proportionality factorused in the method II Standard deviations are shown in bracket

Compound Number of rats ADCI (10minus 10m2s) ADCII (10minus 10m2s) α (10minus 2 au) λI λIIDotarem 3 18 (06) 32 (04) 36 (05) 16 (02) 12 (01)Gadovist 3 15 (01) 29 (03) 22 (02) 15 (05) 12 (01)MultiHance 3 13 (03) 18 (05) 45 (35) 15 (02) 13 (02)

T = 1 min T = 16 min T = 29 min T = 44 min T = 57 min T = 70 min T = 83 min T = 96 min T = 108 min01

005

0[CA

] (m

M)

(a)

01

005

0[CA

] (m

M)

T = 1 min T = 16 min T = 29 min T = 44 min T = 57 min T = 70 min T = 83 min T = 96 min T = 108 min

(b)

Figure 5 Results obtained by using the second method of evaluation of the ADCs to investigate the delivery of MultiHance within one ratbrain In (a) we show the masked CA acquired for more than 1 hour after the BBB opening induced by ultrasound (b) shows the results ofour best fit simulation In particular the central slice showing the maximum CA concentration is pictured along the diffusion time etimes reported above each CA maps refer to the times elapsed after the injection of the compound

T = 1 min

0

005

010

015

02

[CA

] (m

M)

10 200Voxel

T = 16 min

0

005

010

015

02

10 200Voxel

T = 29 min

0

005

010

015

02

10 200Voxel

T = 44 min

0

005

010

015

02

10 200Voxel

T = 65 min

0

005

010

015

02

10 200Voxel

T = 78 min

0

005

010

015

02

10 200Voxel

Figure 6 Example of CA distributions over time after the CA injection ese data refer to the diffusion of Gadovist and are pictured withthe black dots In red the Gaussian fits are shown (method I of analysis) whereas in blue are shown the distributions profiles obtained withmethod II eg our mathematical model By comparing through a two-sample KolmogorovndashSmirnov test the data shown in figure with therespective Gaussian and simulated profiles at each time point we obtained p values equal to 0985 0374 0147 0047 0147 and 0047 formethod I and equal to 0675 0675 0736 0736 0736 and 0736 for method II


Table 2 shows that irrespective of the applied methodADC values scale correctly with molecular size decreasing atincreasing dH (ADCDotaremgtADCGadovistgtADCMultiHance)as expected from comparison to the literature [35] Fur-thermore all ADC values are smaller than their associatedDfree which confirms the hindrance experienced by diffusionacross the ECS

Tortuosities obtained by method I and II (λI and λII) arecompared to those appearing in the literature in order toassess the goodness of ADC estimation

λI and λII obtained for the different molecules turn outconstant which agrees with the literature Indeed all of ourtest molecules have a hydrodynamic diameter ten timessmaller than the intracellular gap d which is typicallycomprised between 20 and 64 nm in healthy ratsrsquo brains[35 36]

In this case the stationary wall-drag effect expected forlarger molecules by virtue of viscosity theory affects neithermolecular diffusion [36] nor tortuosity whose value onlydepends on the ECS structure and not on the size of thediffusion probes

41 Limitations and Future Perspectives In the presentwork both the methods used to estimate the molecularapparent diffusion coefficients are based on a protocolvalidated by our team in 2013 [17] eg the dynamic ac-quisitions of CA-concentration maps through an IR-FGEMRI sequence Although this sequence has been accuratelytuned to be sensitive to a large range of CA concentrationsand to have a sufficiently high temporal and spatial reso-lution to record molecular diffusion further work is neededto improve such resolutions For example a suitable way toincrease the speed of the MRI sequence currently used is byusing compressed sensing MRI techniques [37] Doing sowe expect to reduce the acquisition time and therefore toget access to diffusion data of MRI contrast agents at hightemporal resolution

e second limitation of our experimental approach isrelated to the possibility to evaluate CA diffusion only in twodimensions Indeed our method allows us to estimate thetransversal components (x and y) of the ADC but not toevaluate CA diffusion processes along z-axis is is due tothe gradient concentration and to the relatively low spatialresolution in this direction In order to improve our deliverymethod and to be more sensitive to Gd concentrationgradients along z-axis future experiments can be performedby using multielement transducer to produce a controlledsteering of the ultrasound beam in the z direction (seeFigure S3 in the Supplementary Materials) With thissteering approach it will be possible to permeabilize the BBBin a smaller region of the brain In addition by improvingthe spatial resolution in z of the concentration maps it willbe then possible to characterize the particle diffusion alsoalong this direction

Another limitation of our work concerns the capabilityof method II to fully predict the amount of particles gettingin the brain after a FUS-induced BBB opening experimentIndeed from a qualitative point of view one can expect the

inclusion of the source term to provide a better data de-scription when the blood-to-ECS flux is larger ie for CAsof smaller size since the QCA expression is a monotonicallydecreasing function with the molecular hydrodynamicdiameter dH However the amount of particles getting inthe brain after a FUS-induced BBB permeabilization isdependent from many factors some of them being difficultto precisely control For example if the coupling of thewater balloon between the transducer and the head or ifthe position of the transducer slightly changes betweentwo experiments the transmitted acoustic power couldvary inside the brain and consequently the amounts ofparticles delivered to brain tissue [21 38] McDannoldset al [39] have recently shown that even the level of oxygenused as a carrier gas for anesthesia during the experimentscan change microbubble activity and BBB disruption Allthese aspects varying among experiments change thevalue of the constant of proportionality α For this reasonin order to use our model to simulate an experimentaloutcome the simulations need to be performed by varyingα between 0 (eg the worst-case scenario corresponding toa failure of the experiment) and 007 (eg the maximumvalue of α found in this work)

42 Comparison between the Two ADC Estimation Methodse first method consists in fitting Gaussian distributionsto CA-map data in the brain region where diffusion occursFrom this fit the molecular square displacements and sotheir ADC can be evaluated is kind of postprocessing isalready accepted in the literature [16] although originallyapplied to CA diffusion patterns acquired after in-tracerebral injection of compounds However this methodpresents some limitations e first one concerns the ap-plication of this fit to CA maps with low signal-to-noiseratio (SNR)

In particular we define the SNR in each slice of the CAmaps as the ratio between themaximumCA delivered in theslice and the standard deviation in a region (20 voxelstimes 20voxels) located in the contralateral hemisphere eGaussian fit overestimates the distribution widths for SNRsmaller than 10 is is the case for example of the ac-quisition shown in Figure 6 e errors committed bymethod I on the estimation of the distributions widths areconfirmed by the p values obtained when comparing theGaussian profiles to the respective data points through atwo-sample KolmogorovndashSmirnov test Indeed the p valuesresulted to be smaller than 005 at two time points e sameissue does not affect ADC estimations when the compoundsare intracerebral injected as in [17] Indeed in this lattercase the SNR is higher than the one obtained through BBBopening since the CA concentration diffusing within theECS is 100 times larger than the CA delivered through BBB-opening

On the other hand when method II is applied to analyzethe same dataset it is possible to obtain particle distributionsmore similar to the experimental ones as confirmed by the p

values larger than 005 resulting from the same kind ofstatistical test


In addition to fit the data through the first method we usethe version of LevenbergndashMarquardt algorithm implementedin the scaled LMDER routine in MINPACK [27] is scaledLMDER routine makes use of both the function and itsderivative so it could explain why in some cases as the oneshown in Figure 6 the main differences between the data andthe respective Gaussian fit can be found near the peak

With respect to the first method the second ADC esti-mation method presented in this work is based on a diffusionmodel that includes a source term e source term describesthe flux from the blood to the ECS only which is appropriateif the two pools have a large concentration difference isapproximation can be quantitatively justified Indeed the CAconcentration injected in the blood system is around 3mMwhile as can be noticed from Figures 4ndash6 the maximum CAdelivered in the brain is estimated to be approximately 100times smaller In addition the CA concentration in blood ismuch higher than the ECS concentration during the durationof whole of the experiments (about 1 hour) (see Figure 4 inSupplementary Materials)

Another possible way to compare the two methods is tocompare the different tortuosity values λI and λII shown inTable 2 It has been recently shown with histology that low-intensity pulsed ultrasound could be used to transientlyenlarge the ECS width [40] In particular by estimating theoverall volume of distribution of different nanoparticlesFrenkel et al found an enhanced volume of 36 in averagee volume where particles diffuse in ECS is characterized bythe volume fraction υVECSVT defined as the ratio betweenthe volume of ECS (VECS) and the volume of the whole tissuemeasured in a small region of the brain (VT) (Sykova PhysiolRev 2008) In healthy brain tissue the ECS volume fraction υis estimated around 020 However by considering the studyproposed by Frenkel et al [40] the volume fraction enlargesof 36 after FUS application leading to a volume fraction ofυ 027 Since the relationship between the tortuosity value λand υ is the following as given by [41]

λ 2 minus υ

radic (14)

and the expected value of brain tortuosity after a FUS-in-duced BBB permeabilization experiment is equal to 132eg more similar to the values obtained through method IIthan the ones estimated through the Gaussian fit

5 Conclusions

In this study we used two methods to characterize thecontrast agent bidimensional diffusion within the brainafter ultrasound-induced BBB opening ese techniquesallow to investigate macromolecules biodistribution withinthe ECS with a slow time scale suitable for the study ofcellular uptake and transport as well as of the potentialclearance processes related to bulk flow or glymphaticpathway Although it is well known that focused ultra-sound combined with microbubbles permits to transientlyand noninvasively break tight junctions locally increasingthe BBB permeabilization and so promoting drug deliveryinto the brain [8 28 42ndash44] so far no study has beenperformed to fully characterize on a macroscopic space

and time scale the distribution of a compound when itenters the brain

By using a motorized and MR-compatible ultrasoundsystem we were able to target the right striatum of 9 rats in avery precise and reproducible manner in order to studydiffusion processes in a specific area of the brain reecommercially available MR-CAs were tested (DotaremregGd-DOTA Gadovistreg Gd-DO3A-butrol MultiHanceregGd-BOPTA) eir diffusion from the BBB-disruption sitewas followed by acquisition of several CA maps within1 hour from application of ultrasound e tested com-pounds are characterized by a similar hydrodynamic di-ameter (about 1ndash2 nm) which resulted in a similarhindering of diffusion in the ECS Since the CA distributiondepends on the diffusion properties of brain tissue we haveevaluated its tortuosity a parameter comparing molecularADC inside the tissue to its free-diffusion counterpart in amedia without obstacles e methods proposed here toestimate λ are both based on data processing of MR-CAmaps e first approach does not describe the dependenceof molecular diffusion neither on fundamental biologicalaspects nor on the specific protocol used to permeabilize theBBB

For this reason we have presented a mathematicalmodel able to fully predict time evolution of CA distri-butions within the brain after BBB permeabilization in-duced by FUS Our model takes into account differentbiological features concerning the BBB-opening mecha-nism such as the gap distribution between endothelialcells in turn depending on the effective acoustic pressuretransmitted through the skull and the shape of the focalspot the BBB closure rate and the CA concentration inblood after bolus injection and its physiological rate ofclearance e match with the experimental data allows usto introduce this approach as a new tool to successfullypredict and plan drug distribution after a BBB-openingexperiment for any particle size and acoustic parameter inall brain regions

Abbreviations

BBB Blood-brain barrierUS UltrasoundFUS Focused ultrasoundCA Contrast agentsCA map CA concentration mapADC Apparent diffusion coefficientECS Extracellular space

Data Availability

e MRI data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

Earlier results of the present work have been presented atIEEE International Ultrasonics Symposium (IUS) in 2016[26] and at the conference NeWS in 2017


Conflicts of Interest

e authors declare no potential conflicts of interest withrespect to the research authorship andor publication ofthis article

Authorsrsquo Contributions

AC SM and BL planned the experiments AC and RMperformed MRI acquisitions and FUS experiments ACperformed data analysis AC wrote the simulation code SMprovided MRI data analysis pipelines NT performed DLSexperiments SM BL and SDP managed the overall project

Acknowledgments

AC received an Enhanced Eurotalents postdoctoral fel-lowship (Grant Agreement no 600382) part of the MarieSklodowska-Curie Actions Program cofunded by the Eu-ropean Commission and managed by the French AtomicEnergy and Alternative Energies Commission (CEA)

Supplementary Materials

Supplementary Figure 1 (A) acoustic pressure field at15MHz e pressure field is normalized by the maximumpressure (obtained at the focal spot) e acoustic pressurefield shown on the right has been rescaled to the spatialresolution of the concentration maps (B) Axial and sagittalviews of the Gd-concentration maps it can be noticed thatthe focal spot dimension along z-axis is comparable to thethickness of the rat brain in the area where the BBB waspermeabilized Moreover the spatial resolution in this di-rection is much lower than the in-plane resolution (around44 times) is significantly lower resolution along z-axisdoes not allow to precisely quantify the variations of CAconcentration in this direction Supplementary Figure 2 leftsagittal views of the Gd-concentration maps right con-centration profiles extrapolated from the center of the BBBopening site ese profiles show that the Gd concentrationchanges only slightly with the z position is is due to thelow resolution of the Gd-concentration map along z and alsoto the dimensions of the focal spot along this directionSupplementary Figure 3 acoustic pressure fields at 15MHzfor the concave transducer (FD 08) without steering (left)and with a 25mm steering toward the transducer (right)Both pressure fields are normalized by the maximumpressure obtained at the focal spot in case of the absence ofsteering With a 25mm steering toward the transducer thevolume of the focal spot is decreased by 20 and themaximum pressure is increased by 10 compared to theexperiment without steering Supplementary Figure 4comparison between the CA concentration in blood (picturein blue) and the maximum CA delivered during a BBBopening experiment (in red) In particular experimentalpoints refer to the data shown in Figure 4 while the trend ofCAblood along time t has been derived through the equationCAblood(t)CAinjmiddotexp(minus tb) with b 25 minutes (Aime andCaravan JMRI 2009) (Supplementary Materials)

References

[1] W M Pardridge ldquoDrug transport across the bloodndashbrainbarrierrdquo Journal of Cerebral Blood FlowampMetabolism vol 32no 11 pp 1959ndash1972 2012

[2] D S Hersh A S Wadajkar N Roberts et al ldquoEvolving drugdelivery strategies to overcome the blood brain barrierrdquoCurrent Pharmaceutical Design vol 22 no 9 pp 1177ndash11932016 httpswwwingentaconnectcomcontentonebencpd20160000002200000009art00007

[3] C E Krewson M L Klarman and W M Saltzman ldquoDis-tribution of nerve growth factor following direct delivery tobrain interstitiumrdquo Brain Research vol 680 no 1-2pp 196ndash206 1995

[4] Q Yan C Matheson J Sun M J Radeke S C Feinstein andJ A Miller ldquoDistribution of intracerebral ventricularly ad-ministered neurotrophins in rat brain and its correlation withtrk receptor expressionrdquo Experimental Neurology vol 127no 1 pp 23ndash36 1994

[5] E A Neuwelt P A Barnett C I McCormick E P Frenkeland J D Minna ldquoOsmotic blood-brain barrier modificationmonoclonal antibody albumin and methotrexate delivery tocerebrospinal fluid and brainrdquo Neurosurgery vol 17 no 3pp 419ndash423 1985

[6] D J Guillaume N D Doolittle S GahramanovN A Hedrick J B Delashaw and E A Neuwelt ldquoIntra-arterial chemotherapy with osmotic blood-brain barrierdisruption for aggressive oligodendroglial tumors results of aphase I studyrdquo Neurosurgery vol 66 no 1 pp 48ndash58 2010

[7] D J Begley ldquoDelivery of therapeutic agents to the centralnervous system the problems and the possibilitiesrdquo Phar-macology amp 2erapeutics vol 104 no 1 pp 29ndash45 2004

[8] K Hynynen N McDannold N Vykhodtseva andF A Jolesz ldquoNoninvasive MR imagingndashguided focal openingof the blood-brain barrier in rabbitsrdquo Radiology vol 220no 3 pp 640ndash646 2001

[9] E Sykova and C Nicholson ldquoDiffusion in brain extracellularspacerdquo Physiological Reviews vol 88 no 4 pp 1277ndash13402008

[10] E Sykova ldquoDiffusion properties of the brain in health anddiseaserdquo Neurochemistry International vol 45 no 4pp 453ndash466 2004

[11] A Conti ldquoComparison of single spot and volume ultrasoundsonications for efficient nanoparticle delivery to glioblastomamodel in ratsrdquo in Proceedings of the 2017 IEEE InternationalUltrasonics Symposium (IUS) p 1 Washington DC USASeptember 2017

[12] C Nicholson and J M Phillips ldquoIon diffusion modified bytortuosity and volume fraction in the extracellular micro-environment of the rat cerebellumrdquo2e Journal of Physiologyvol 321 no 1 pp 225ndash257 1981

[13] C Nicholson J M Phillips and A R Gardner-MedwinldquoDiffusion from an iontophoretic point source in the brainrole of tortuosity and volume fractionrdquo Brain Researchvol 169 no 3 pp 580ndash584 1979

[14] E Sykova I Vorısek T Mazel T Antonova andM Schachner ldquoReduced extracellular space in the brain oftenascin-R- and HNK-1-sulphotransferase deficient micerdquoEuropean Journal of Neuroscience vol 22 no 8 pp 1873ndash1880 2005

[15] I Vorısek M Hajek J Tintera K Nicolay and E SykovaldquoWater ADC extracellular space volume and tortuosity in therat cortex after traumatic injuryrdquo Magnetic Resonance inMedicine vol 48 no 6 pp 994ndash1003 2002


[16] RW Brown Y-C N Cheng EM Haacke M Rompsonand R Venkatesan Magnetic Resonance Imaging PhysicalPrinciples and Sequence Design Wiley-Blackwell HobokenNJ USA 2014

[17] B Marty B Djemaı C Robic et al ldquoHindered diffusion ofMRI contrast agents in rat brain extracellular micro-envi-ronment assessed by acquisition of dynamic T1 and T2 mapsrdquoContrast Media amp Molecular Imaging vol 8 no 1 pp 12ndash192013

[18] R Magnin F Rabusseau F Salabartan et al ldquoMagneticresonance-guided motorized transcranial ultrasound systemfor blood-brain barrier permeabilization along arbitrarytrajectories in rodentsrdquo Journal of 2erapeutic Ultrasoundvol 3 no 1 2015

[19] R Deichmann D Hahn and A Haase ldquoFast T1mapping on awhole-body scannerrdquo Magnetic Resonance in Medicinevol 42 no 1 pp 206ndash209 1999

[20] P J Wright O E Mougin J J Totman et al ldquoWater protonT1 measurements in brain tissue at 7 3 and 15 T using IR-EPI IR-TSE and MPRAGE results and optimizationrdquoMagnetic Resonance Materials in Physics Biology and Medi-cin vol 21 no 1-2 pp 121ndash130 2008

[21] M Gerstenmayer B Fellah R Magnin E Selingue andB Larrat ldquoAcoustic transmission factor through the rat skullas a function of body mass frequency and positionrdquo Ultra-sound in Medicine amp Biology vol 44 no 11 pp 2336ndash23442018

[22] B Larrat M Pernot J-F Aubry et al ldquoMR-guided trans-cranial brain HIFU in small animal modelsrdquo Physics inMedicine and Biology vol 55 no 2 pp 365ndash388 2009

[23] N McDannold and S E Maier ldquoMagnetic resonance acousticradiation force imagingrdquo Medical Physics vol 35 no 8pp 3748ndash3758 2008

[24] T J Swift and R E Connick ldquoNMR-Relaxation mechanismsof O17 in aqueous solutions of paramagnetic cations and thelifetime of water molecules in the first coordination sphererdquo2e Journal of Chemical Physics vol 37 no 2 pp 307ndash3201962

[25] S Meriaux A Conti and B Larrat ldquoAssessing diffusion in theextra-cellular space of brain tissue by dynamic MRI mappingof contrast agent concentrationsrdquo Frontiers in Physics vol 62018

[26] A Conti R Magnin M Gerstenmayer et al ldquoCharacter-ization of the diffusion process of different gadolinium-basednanoparticles within the brain tissue after ultrasound inducedblood-brain barrier permeabilizationrdquo in Proceedings of the2016 IEEE International Ultrasonics Symposium (IUS)pp 1ndash4 Tours France September 2016

[27] J J More ldquoe Levenberg-Marquardt algorithm imple-mentation and theoryrdquo in Numerical Analysis pp 105ndash116Springer Berlin Germany 1978

[28] B Marty B Larrat M Van Landeghem et al ldquoDynamic studyof bloodndashbrain barrier closure after its disruption using ul-trasound a quantitative analysisrdquo Journal of Cerebral BloodFlow amp Metabolism vol 32 no 10 pp 1948ndash1958 2012

[29] A Conti S Meriaux and B Larrat ldquoAbout the Marty modelof blood-brain barrier closure after its disruption using fo-cused ultrasoundrdquo Physics in Medicine amp Biology vol 64no 14 Article ID 14NT02 2019

[30] N McDannold N Vykhodtseva and K Hynynen ldquoBlood-brain barrier disruption induced by focused ultrasound andcirculating preformed microbubbles appears to be charac-terized by the mechanical indexrdquo Ultrasound in Medicine ampBiology vol 34 no 5 pp 834ndash840 2008

[31] P S Tofts ldquoModeling tracer kinetics in dynamic Gd-DTPAMR imagingrdquo Journal of Magnetic Resonance Imaging vol 7no 1 pp 91ndash101 1997

[32] R J Probst J M Lim D N Bird G L Pole A K Sato andJ R Claybaugh ldquoGender differences in the blood volume ofconscious sprague-dawley ratsrdquo Journal of the AmericanAssociation for Laboratory Animal Science vol 45 no 2pp 49ndash52 httpswwwingentaconnectcomcontentaalasjaalas20060000004500000002art00009

[33] S Aime and P Caravan ldquoBiodistribution of gadolinium-based contrast agents including gadolinium depositionrdquoJournal of Magnetic Resonance Imaging vol 30 no 6pp 1259ndash1267 2009

[34] C Nicholson and E Sykova ldquoExtracellular space structurerevealed by diffusion analysisrdquo Trends in Neurosciencesvol 21 no 5 pp 207ndash215 1998

[35] R G orne and C Nicholson ldquoIn vivo diffusion analysiswith quantum dots and dextrans predicts the width of brainextracellular spacerdquo Proceedings of the National Academy ofSciences vol 103 no 14 pp 5567ndash5572 2006

[36] D A Rusakov and D M Kullmann ldquoGeometric and viscouscomponents of the tortuosity of the extracellular space in thebrainrdquo Proceedings of the National Academy of Sciencesvol 95 no 15 pp 8975ndash8980 1998

[37] M Doneva P Bornert H Eggers C Stehning J Senegas andA Mertins ldquoCompressed sensing reconstruction for magneticresonance parameter mappingrdquo Magnetic Resonance inMedicine vol 64 no 4 pp 1114ndash1120 2010

[38] S-YWu C Aurup C S Sanchez et al ldquoEfficient blood-brainbarrier opening in primates with neuronavigation-guidedultrasound and real-time acoustic mappingrdquo Scientific Re-ports vol 8 no 1 p 7978 2018

[39] N McDannold Y Zhang and N Vykhodtseva ldquoe effects ofoxygen on ultrasound-induced blood-brain barrier disruptionin micerdquo Ultrasound in Medicine amp Biology vol 43 no 2pp 469ndash475 2017

[40] V Frenkel D Hersh P Anastasiadis et al ldquoPulsed focusedultrasound effects on the brain interstitiumrdquo in Proceedings ofthe 2017 IEEE International Ultrasonics Symposium (IUS)p 1 Washington DC USA September 2017

[41] J Hrabe S Hrabĕtova and K Segeth ldquoA model of effectivediffusion and tortuosity in the extracellular space of thebrainrdquo Biophysical Journal vol 87 no 3 pp 1606ndash1617 2004

[42] R K Upadhyay ldquoDrug delivery systems CNS protection andthe blood brain barrierrdquo BioMed Research Internationalvol 2014 Article ID 869269 37 pages 2014

[43] C-H Fan and C-K Yeh ldquoMicrobubble-enhanced focusedultrasound-induced bloodndashbrain barrier opening for local andtransient drug delivery in central nervous system diseaserdquoJournal of Medical Ultrasound vol 22 no 4 pp 183ndash1932014

[44] A Burgess K Shah O Hough and K Hynynen ldquoFocusedultrasound-mediated drug delivery through the blood-brainbarrierrdquo Expert Review of Neurotherapeutics vol 15 no 5pp 477ndash491 2015


Stem Cells International

Hindawiwwwhindawicom Volume 2018


MEDIATORSINFLAMMATION

of

EndocrinologyInternational Journal of



Disease Markers


BioMed Research International

OncologyJournal of



Oxidative Medicine and Cellular Longevity


PPAR Research

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Immunology ResearchHindawiwwwhindawicom Volume 2018

Journal of

ObesityJournal of



Computational and Mathematical Methods in Medicine


Behavioural Neurology

OphthalmologyJournal of


Diabetes ResearchJournal of



Research and TreatmentAIDS


Gastroenterology Research and Practice


Parkinsonrsquos Disease

Evidence-Based Complementary andAlternative Medicine

Volume 2018Hindawiwwwhindawicom

Submit your manuscripts atwwwhindawicom

disruption using chemical agents [5 6] or molecular en-gineering [7] More recently a promising technique has beenproposed allowing the delivery of various compounds to thebrain using low-intensity focused ultrasound combined withcirculating microbubbles [8]

However once the molecules have crossed the barrierthey have to diffuse in a highly constrained media theextracellular space (ECS) to reach their targets [9] More-over since the ECS architecture can change in case of pa-thologies [10 11] the characterization of the hindranceexperienced by molecules within the brain tissue is essentialwhen designing new therapeutic compounds or diagnosticmolecules for brain diseases Diffusion constraints can bestudied by estimating the ECS tortuosity (λ) is parametercompares the apparent diffusion coefficient (ADC) of amolecule within the complex architecture of the ECS to itsdiffusion coefficient in a free medium Dfree [12] Differentstrategies have been proposed to measure the ADC emost widely used method is real-time iontophoresis [9 13]using tetramethylammonium (TMA+) as a probe istechnique not only permits the in vivo characterization of theADC but also thanks to the small size of both detectionelectrodes and injection micropipette proves minimallyinvasive with consequent preservation of the integrity of thetissues Its main drawback consists in the measurementrelying on just one spatial point More recently diffusion-weighted magnetic resonance imaging (DW-MRI) has beenproposed to noninvasively measure the ADC of watermolecules in the brain [14 15] In comparison to the pre-vious techniques DW-MRI allows ADC measurements indeeper areas of the brain with a high (typically 2mm iso-tropic) spatial resolution [16] However contrary to TMA+

and other techniques using labelled molecules that diffuseonly across the ECS DW-MRI detects water which is alsopresent in the intracellular compartment To benefit fromthe advantages offered by MR in acquiring deep volumes ofthe brain a newmethod has been recently introduced by ourteam which allows us to detect molecular diffusion only inthe ECS structure [17] To do so MR-CAs are directly in-jected into the brain tissue and their diffusion is followed byacquisition of several longitudinal relaxation-time (T1)parametric maps MR-CA concentration maps at differentdiffusion times are then calculated and from these the ADCis estimated When compared to the typical diffusion-basedMRI techniques our method investigates larger areas of thebrain with a higher spatial resolution (about 02times 02mm2 inplane and 1mm in thickness) However a major issue raisedby this procedure consists in intracerebral injections in-ducing edema which modifies the diffusion properties ofbrain tissue

In the present study we have used two noninvasivemethods for the in vivo estimation of the ADC of differentGd chelates diffusing in the ECS after a FUS-induced BBBopening experiment In both cases contrast agent diffusionis recorded through dynamic acquisitions of MRI concen-tration maps In the first method the ADC evaluation isperformed as in [17] eg by fitting a 2D Gaussian curve tothe image intensity at different time points However dif-fusion of molecules delivered to the brain with the aid of

FUS-induced BBB permeabilization depends on many fac-tors such as tissue and particle properties as well as acousticparameters For this reason as a second approach to esti-mate contrast agent diffusion we introduce here the firstdiffusion model able to fully describe and predict at longspace and time scales the result of a FUS-induced BBBopening experiment is model takes into account acousticpressure particle size blood pharmacokinetics vascularpermeability as a function of time and diffusion processoccurring in brain tissue

Starting from ADC estimation performed with the helpof both methods and the evaluation of Dfree for all thecompounds bymeans of in vitro experiments it is possible tocalculate tortuosities in the target region of ratsrsquo brains toevaluate the effect of the selected BBB permeabilizationprotocol on the properties of brain tissue

2 Materials and Methods

21 Experimental Procedures All magnetic resonance ac-quisitions were performed by using a 7 T90mm Phar-mascan scanner (Bruker Ettlingen Germany) e in vitroacquisitions have been performed by using a 1H transmit-receiver volume coil (Bruker) e in vivo experiments havebeen conducted by using a dedicated ultrasound single-loopradiofrequency coil [18] whose diameter was wide enoughfor the ultrasound beam to pass through it and for extensivedisplacement of the transducer above the ratrsquos head A heaterdevice was used to keep temperature at the physiologicalvalue (37degC) as monitored by a temperature probe that wasinserted inside the magnet (see Figure 1)

ree different gadolinium (Gd) chelates were studiedDotaremreg (Gd-DOTA Guerbet France) Gadovistreg (Gd-DO3A-butrol Bayer Germany) and MultiHancereg (Gd-BOPTA Bracco Italy) First we assessed their longitudinalrelaxivities (r1) at 7 T and 37degC using phantoms made ofbundles of tubes containing different contrast agent (CA)concentrations in 03 ww agarose gel for each compoundFor these phantoms T1 values were measured by means ofan inversion-recovery fast gradient echo (IR-FGE) sequence[17 19] (echo time (TE)repetition time (TR1) 255ms 6segments 90 inversion times (TI) varying from 75ms to8975ms flip angle (FA) 5deg matrix 120times120times 5 withresolution 0250times 0250times125mm3 delay between theacquisitions of two segments (TR2) 15000ms and numberof averages (NA 6)) Resulting relaxivities are summarizedin Table 1 is table also includes the hydrodynamic di-ameter (dH) of each CA measured by dynamic light scat-tering (DLS) DLS experiments were performed using aNanoZS equipement (Malvern France) operating at anangle of 173deg For each Gd chelate the DLS acquisitions wereperformed at 25degC by using concentrations of 05M forMultiHance and Dotarem and of 10M for Gadovist egwithout diluting the samples We performed five differentDLS measurements for each sample e mean dH and thestandard deviation evaluated over the five measures arereported in Table 1

Evaluations of the Dfree of each compound were donewith amethod already presented in a previous work [17] For








Motors

Water degassing

Motor control


MRI


(a) (b)

Temperatureprobe

Respiration probe

(c)

MRI console








1T1

1







a cos2(θ)

2σ2X+sin2(θ)

2σ2Y

b minussin(2θ)

4σ2X+sin(2θ)

4σ2Y

c sin2(θ)

2σ2X+cos2(θ)

2σ2Y

(3)


Animalinstallation

T1-wMSME

ARFI T2-wRARE

T1-wMSME

T1-mappreinjection

T1-map T1-map



Gd chelateinjection






DfreeXY σ2XY

2t (4)



2 (5)



2 (6)







dH

middot

π2

1113970

1 minus erfdH

2radic

σ0(x y z)eminus kt1113888 11138891113888 11138891113888

+dH



(8)




b1113874 1113875 (9)



z[CA](x y z t)

zt ADCx middot

z2[CA](x y z t)

zx2


zy2


zz2

+ S(x y z t)

(10)


λ

Dfree

ADC

1113970

(11)





z[CA](x y t)

zt ADCx middot

z2[CA](x y t)

zx2


zy2 + S(x y t)

(12)



dH kT

3πηDfree (13)



3 Results












4 Discussion



[CA

] (m

M)


005

0

(a)

[CA

] (m

M)


005

0

(b)

Voxel30 40 50 60

0

002

004

006

008

01

[CA

] (m

M)

Voxel30 40 50 60

Voxel30 40 50 60

Voxel30 40 50 60

Voxel30 40 50 60

DataMethod I

(c)

R2 = 0989

R2 = 0994

σY

σX

0

05

1

15

2

25

σ2 (10

ndash6m

2 )

15 30 45 600t (min)

(d)



[CA

] (m

M) 025

02015

01005

0


(a)

[CA

] (m

M)

02502

01501

0050


(b)

[CA

] (m

M)

02502

01501

0050


(c)

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

[CA

] (m

M)

025

02

015

01

005

020 400

Voxel

DataMethod I

(d)

σY

R2 = 0898

R2 = 0996

σX

0

05

1

15

2

σ2 (10ndash6

mm

2 )

10 20 30 40 50 60 700T (min)

(e)






10minus 10m2s)




005

0[CA

] (m

M)

(a)

01

005

0[CA

] (m

M)


(b)


T = 1 min

0

005

010

015

02

[CA

] (m

M)

10 200Voxel

T = 16 min

0

005

010

015

02

10 200Voxel

T = 29 min

0

005

010

015

02

10 200Voxel

T = 44 min

0

005

010

015

02

10 200Voxel

T = 65 min

0

005

010

015

02

10 200Voxel

T = 78 min

0

005

010

015

02

10 200Voxel



















λ 2 minus υ

radic (14)


5 Conclusions





Abbreviations


Data Availability


Disclosure







Acknowledgments




References



















































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of

























Motors

Water degassing

Motor control


MRI


(a) (b)

Temperatureprobe

Respiration probe

(c)

MRI console








1T1

1







a cos2(θ)

2σ2X+sin2(θ)

2σ2Y

b minussin(2θ)

4σ2X+sin(2θ)

4σ2Y

c sin2(θ)

2σ2X+cos2(θ)

2σ2Y

(3)


Animalinstallation

T1-wMSME

ARFI T2-wRARE

T1-wMSME

T1-mappreinjection

T1-map T1-map



Gd chelateinjection






DfreeXY σ2XY

2t (4)



2 (5)



2 (6)







dH

middot

π2

1113970

1 minus erfdH

2radic

σ0(x y z)eminus kt1113888 11138891113888 11138891113888

+dH



(8)




b1113874 1113875 (9)



z[CA](x y z t)

zt ADCx middot

z2[CA](x y z t)

zx2


zy2


zz2

+ S(x y z t)

(10)


λ

Dfree

ADC

1113970

(11)





z[CA](x y t)

zt ADCx middot

z2[CA](x y t)

zx2


zy2 + S(x y t)

(12)



dH kT

3πηDfree (13)



3 Results












4 Discussion



[CA

] (m

M)


005

0

(a)

[CA

] (m

M)


005

0

(b)

Voxel30 40 50 60

0

002

004

006

008

01

[CA

] (m

M)

Voxel30 40 50 60

Voxel30 40 50 60

Voxel30 40 50 60

Voxel30 40 50 60

DataMethod I

(c)

R2 = 0989

R2 = 0994

σY

σX

0

05

1

15

2

25

σ2 (10

ndash6m

2 )

15 30 45 600t (min)

(d)



[CA

] (m

M) 025

02015

01005

0


(a)

[CA

] (m

M)

02502

01501

0050


(b)

[CA

] (m

M)

02502

01501

0050


(c)

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

[CA

] (m

M)

025

02

015

01

005

020 400

Voxel

DataMethod I

(d)

σY

R2 = 0898

R2 = 0996

σX

0

05

1

15

2

σ2 (10ndash6

mm

2 )

10 20 30 40 50 60 700T (min)

(e)






10minus 10m2s)




005

0[CA

] (m

M)

(a)

01

005

0[CA

] (m

M)


(b)


T = 1 min

0

005

010

015

02

[CA

] (m

M)

10 200Voxel

T = 16 min

0

005

010

015

02

10 200Voxel

T = 29 min

0

005

010

015

02

10 200Voxel

T = 44 min

0

005

010

015

02

10 200Voxel

T = 65 min

0

005

010

015

02

10 200Voxel

T = 78 min

0

005

010

015

02

10 200Voxel



















λ 2 minus υ

radic (14)


5 Conclusions





Abbreviations


Data Availability


Disclosure







Acknowledgments




References



















































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of






















1T1

1







a cos2(θ)

2σ2X+sin2(θ)

2σ2Y

b minussin(2θ)

4σ2X+sin(2θ)

4σ2Y

c sin2(θ)

2σ2X+cos2(θ)

2σ2Y

(3)


Animalinstallation

T1-wMSME

ARFI T2-wRARE

T1-wMSME

T1-mappreinjection

T1-map T1-map



Gd chelateinjection






DfreeXY σ2XY

2t (4)



2 (5)



2 (6)







dH

middot

π2

1113970

1 minus erfdH

2radic

σ0(x y z)eminus kt1113888 11138891113888 11138891113888

+dH



(8)




b1113874 1113875 (9)



z[CA](x y z t)

zt ADCx middot

z2[CA](x y z t)

zx2


zy2


zz2

+ S(x y z t)

(10)


λ

Dfree

ADC

1113970

(11)





z[CA](x y t)

zt ADCx middot

z2[CA](x y t)

zx2


zy2 + S(x y t)

(12)



dH kT

3πηDfree (13)



3 Results












4 Discussion



[CA

] (m

M)


005

0

(a)

[CA

] (m

M)


005

0

(b)

Voxel30 40 50 60

0

002

004

006

008

01

[CA

] (m

M)

Voxel30 40 50 60

Voxel30 40 50 60

Voxel30 40 50 60

Voxel30 40 50 60

DataMethod I

(c)

R2 = 0989

R2 = 0994

σY

σX

0

05

1

15

2

25

σ2 (10

ndash6m

2 )

15 30 45 600t (min)

(d)



[CA

] (m

M) 025

02015

01005

0


(a)

[CA

] (m

M)

02502

01501

0050


(b)

[CA

] (m

M)

02502

01501

0050


(c)

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

[CA

] (m

M)

025

02

015

01

005

020 400

Voxel

DataMethod I

(d)

σY

R2 = 0898

R2 = 0996

σX

0

05

1

15

2

σ2 (10ndash6

mm

2 )

10 20 30 40 50 60 700T (min)

(e)






10minus 10m2s)




005

0[CA

] (m

M)

(a)

01

005

0[CA

] (m

M)


(b)


T = 1 min

0

005

010

015

02

[CA

] (m

M)

10 200Voxel

T = 16 min

0

005

010

015

02

10 200Voxel

T = 29 min

0

005

010

015

02

10 200Voxel

T = 44 min

0

005

010

015

02

10 200Voxel

T = 65 min

0

005

010

015

02

10 200Voxel

T = 78 min

0

005

010

015

02

10 200Voxel



















λ 2 minus υ

radic (14)


5 Conclusions





Abbreviations


Data Availability


Disclosure







Acknowledgments




References



















































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of





















DfreeXY σ2XY

2t (4)



2 (5)



2 (6)







dH

middot

π2

1113970

1 minus erfdH

2radic

σ0(x y z)eminus kt1113888 11138891113888 11138891113888

+dH



(8)




b1113874 1113875 (9)



z[CA](x y z t)

zt ADCx middot

z2[CA](x y z t)

zx2


zy2


zz2

+ S(x y z t)

(10)


λ

Dfree

ADC

1113970

(11)





z[CA](x y t)

zt ADCx middot

z2[CA](x y t)

zx2


zy2 + S(x y t)

(12)



dH kT

3πηDfree (13)



3 Results












4 Discussion



[CA

] (m

M)


005

0

(a)

[CA

] (m

M)


005

0

(b)

Voxel30 40 50 60

0

002

004

006

008

01

[CA

] (m

M)

Voxel30 40 50 60

Voxel30 40 50 60

Voxel30 40 50 60

Voxel30 40 50 60

DataMethod I

(c)

R2 = 0989

R2 = 0994

σY

σX

0

05

1

15

2

25

σ2 (10

ndash6m

2 )

15 30 45 600t (min)

(d)



[CA

] (m

M) 025

02015

01005

0


(a)

[CA

] (m

M)

02502

01501

0050


(b)

[CA

] (m

M)

02502

01501

0050


(c)

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

[CA

] (m

M)

025

02

015

01

005

020 400

Voxel

DataMethod I

(d)

σY

R2 = 0898

R2 = 0996

σX

0

05

1

15

2

σ2 (10ndash6

mm

2 )

10 20 30 40 50 60 700T (min)

(e)






10minus 10m2s)




005

0[CA

] (m

M)

(a)

01

005

0[CA

] (m

M)


(b)


T = 1 min

0

005

010

015

02

[CA

] (m

M)

10 200Voxel

T = 16 min

0

005

010

015

02

10 200Voxel

T = 29 min

0

005

010

015

02

10 200Voxel

T = 44 min

0

005

010

015

02

10 200Voxel

T = 65 min

0

005

010

015

02

10 200Voxel

T = 78 min

0

005

010

015

02

10 200Voxel



















λ 2 minus υ

radic (14)


5 Conclusions





Abbreviations


Data Availability


Disclosure







Acknowledgments




References



















































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















z[CA](x y t)

zt ADCx middot

z2[CA](x y t)

zx2


zy2 + S(x y t)

(12)



dH kT

3πηDfree (13)



3 Results












4 Discussion



[CA

] (m

M)


005

0

(a)

[CA

] (m

M)


005

0

(b)

Voxel30 40 50 60

0

002

004

006

008

01

[CA

] (m

M)

Voxel30 40 50 60

Voxel30 40 50 60

Voxel30 40 50 60

Voxel30 40 50 60

DataMethod I

(c)

R2 = 0989

R2 = 0994

σY

σX

0

05

1

15

2

25

σ2 (10

ndash6m

2 )

15 30 45 600t (min)

(d)



[CA

] (m

M) 025

02015

01005

0


(a)

[CA

] (m

M)

02502

01501

0050


(b)

[CA

] (m

M)

02502

01501

0050


(c)

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

[CA

] (m

M)

025

02

015

01

005

020 400

Voxel

DataMethod I

(d)

σY

R2 = 0898

R2 = 0996

σX

0

05

1

15

2

σ2 (10ndash6

mm

2 )

10 20 30 40 50 60 700T (min)

(e)






10minus 10m2s)




005

0[CA

] (m

M)

(a)

01

005

0[CA

] (m

M)


(b)


T = 1 min

0

005

010

015

02

[CA

] (m

M)

10 200Voxel

T = 16 min

0

005

010

015

02

10 200Voxel

T = 29 min

0

005

010

015

02

10 200Voxel

T = 44 min

0

005

010

015

02

10 200Voxel

T = 65 min

0

005

010

015

02

10 200Voxel

T = 78 min

0

005

010

015

02

10 200Voxel



















λ 2 minus υ

radic (14)


5 Conclusions





Abbreviations


Data Availability


Disclosure







Acknowledgments




References



















































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















[CA

] (m

M)


005

0

(a)

[CA

] (m

M)


005

0

(b)

Voxel30 40 50 60

0

002

004

006

008

01

[CA

] (m

M)

Voxel30 40 50 60

Voxel30 40 50 60

Voxel30 40 50 60

Voxel30 40 50 60

DataMethod I

(c)

R2 = 0989

R2 = 0994

σY

σX

0

05

1

15

2

25

σ2 (10

ndash6m

2 )

15 30 45 600t (min)

(d)



[CA

] (m

M) 025

02015

01005

0


(a)

[CA

] (m

M)

02502

01501

0050


(b)

[CA

] (m

M)

02502

01501

0050


(c)

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

[CA

] (m

M)

025

02

015

01

005

020 400

Voxel

DataMethod I

(d)

σY

R2 = 0898

R2 = 0996

σX

0

05

1

15

2

σ2 (10ndash6

mm

2 )

10 20 30 40 50 60 700T (min)

(e)






10minus 10m2s)




005

0[CA

] (m

M)

(a)

01

005

0[CA

] (m

M)


(b)


T = 1 min

0

005

010

015

02

[CA

] (m

M)

10 200Voxel

T = 16 min

0

005

010

015

02

10 200Voxel

T = 29 min

0

005

010

015

02

10 200Voxel

T = 44 min

0

005

010

015

02

10 200Voxel

T = 65 min

0

005

010

015

02

10 200Voxel

T = 78 min

0

005

010

015

02

10 200Voxel



















λ 2 minus υ

radic (14)


5 Conclusions





Abbreviations


Data Availability


Disclosure







Acknowledgments




References



















































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















[CA

] (m

M) 025

02015

01005

0


(a)

[CA

] (m

M)

02502

01501

0050


(b)

[CA

] (m

M)

02502

01501

0050


(c)

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

025

02

015

01

005

020 400

Voxel

[CA

] (m

M)

025

02

015

01

005

020 400

Voxel

DataMethod I

(d)

σY

R2 = 0898

R2 = 0996

σX

0

05

1

15

2

σ2 (10ndash6

mm

2 )

10 20 30 40 50 60 700T (min)

(e)






10minus 10m2s)




005

0[CA

] (m

M)

(a)

01

005

0[CA

] (m

M)


(b)


T = 1 min

0

005

010

015

02

[CA

] (m

M)

10 200Voxel

T = 16 min

0

005

010

015

02

10 200Voxel

T = 29 min

0

005

010

015

02

10 200Voxel

T = 44 min

0

005

010

015

02

10 200Voxel

T = 65 min

0

005

010

015

02

10 200Voxel

T = 78 min

0

005

010

015

02

10 200Voxel



















λ 2 minus υ

radic (14)


5 Conclusions





Abbreviations


Data Availability


Disclosure







Acknowledgments




References



















































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of






















10minus 10m2s)




005

0[CA

] (m

M)

(a)

01

005

0[CA

] (m

M)


(b)


T = 1 min

0

005

010

015

02

[CA

] (m

M)

10 200Voxel

T = 16 min

0

005

010

015

02

10 200Voxel

T = 29 min

0

005

010

015

02

10 200Voxel

T = 44 min

0

005

010

015

02

10 200Voxel

T = 65 min

0

005

010

015

02

10 200Voxel

T = 78 min

0

005

010

015

02

10 200Voxel



















λ 2 minus υ

radic (14)


5 Conclusions





Abbreviations


Data Availability


Disclosure







Acknowledgments




References



















































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



































λ 2 minus υ

radic (14)


5 Conclusions





Abbreviations


Data Availability


Disclosure







Acknowledgments




References



















































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of






















λ 2 minus υ

radic (14)


5 Conclusions





Abbreviations


Data Availability


Disclosure







Acknowledgments




References



















































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of























Acknowledgments




References



















































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of





















































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of























of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















Date post:	13-Nov-2020
Category:	Documents
Upload:	others
View:	1 times
Download:	0 times

Empirical and Theoretical Characterization of the Diffusion...

Documents