Mathematical Modeling, Computation, and Experimental Imaging of ...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 635713 6 pageshttpdxdoiorg1011552013635713

Research ArticleMathematical Modeling Computation andExperimental Imaging of Thin-Layer Objects byMagnetic Resonance Imaging

Ivan Frollo

Institute of Measurement Science Slovak Academy of Sciences 84104 Bratislava Slovakia

Correspondence should be addressed to Ivan Frollo frollosavbask

Received 14 May 2013 Revised 25 September 2013 Accepted 11 November 2013

Academic Editor Eihab M Abdel-Rahman

Copyright copy 2013 Ivan Frollo This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Imaging of thin layers usingmagnetic resonance imaging (MRI)methods belongs to the special procedures that serve for imaging ofweakmagneticmaterials (weak ferromagnetic diamagnetic or paramagnetic)The objective of the paper is to presentmathematicalmodels appropriate for magnetic field calculations in the vicinity of thin organic or inorganic materials with defined magneticsusceptibility Computation is similar to the double layer theory Thin plane layers in their vicinity create a deformation of theneighboring magnetic field Calculations with results in the form of analytic functions were derived for rectangular circularand general shaped samples For experimental verification an MRI 02 Tesla esaote Opera imager was used For experimentsa homogeneous parallelepiped block (reference medium)mdasha container filled with doped watermdashwas used The resultant imagescorrespond to themagnetic field variations in the vicinity of the samples For data detection classical gradient-echo (GRE) imagingmethods susceptible to magnetic field inhomogeneities were used Experiments proved that the proposed method was effectivefor thin organic and soft magnetic materials testing using magnetic resonance imaging methods

1 Introduction

Imaging methods used for biological and physical structurebased on nuclear magnetic resonance (NMR) have becomea regular diagnostic procedure Special methods are neededwhen a thin-layer organic or inorganic object is inserted intoa static homogeneous magnetic field of the NMR tomographThis results in small variations of the static homogeneousmagnetic field near the sample It is possible to image themagnetic contours caused by the sample using a specialplastic holder filled with a water-containing substance nearthe sample The final image represents variations in themagnetic field resulting from the superpositioning of theimager magnetic field and fields produced by the sample

The basis for the mathematical modeling calculationand experimental verification based on magnetic resonanceimaging is the theory of ldquomagnetic thin-layerrdquoThe thin-layerfrom the point of view of physical properties is similar tothe magnetic double layer defined as a planar distribution

of magnetic dipoles characterized by surface density of themagnetic dipole momentsThe vector potential of a magneticdouble layer is equivalent to the vector potential of amagneticfield of the closed current loop [1 2]

Real physical layers in the presence of pure spin currentsin magnetic single and double layers in spin ballistic anddiffusive regimes were described in [3] Current-inducedmagnetization dynamics in single and double layer magneticnanopillars grown by molecular beam epitaxy were depictedin [4] Susceptibility MRI had been around for many yearsfor example it was discussed in [5] Indirect susceptibilitymapping of thin-layer samples using NMR imaging wasreported in [6]

First attempts of a direct measurement of the magneticfield variations utilizing the divergence in gradient strengththat occurs in the vicinity of a thin current-carrying copperwire were introduced in [7] A simple experiment with thinpulsed electrical current-carrying wire and imaging of amagnetic field using a plastic sphere filled with agarose

2 Mathematical Problems in Engineering

gel as phantom were published in [8] Single biogenic softmagnetite nanoparticle physical characteristics in biologicalobjects were introduced in [9] It was shown that thesusceptibility imaging needs to measure local magnetic fieldvariations superimposed on the field of the imager by thesample Such variations are representative of the sample [10]

In this paper an imaging method used for thin organicand soft magnetic materials detection was proposed Com-putation of the magnetic field variations based on thin-layermagnetic theory and a comparison of theoretical results withexperimental images were performed

2 Mathematical Model ofthe Magnetic Thin-layer

The goal of this section is to model the thin-layer structure(polarized in the main field) as a distribution of dipoles thefield of which is the superposition of fields due to all thedipoles representing the thin-layer structure

In our interpretation of the magnetic thin-layer model-ing we have started with a magnetic dipole moment of acurrent loop description (Figure 1) Let the current loop 119871with current 119868 be positioned in a plane of the rectangularcoordinate system [119909 119910 119911] The magnetic field at the point119861[1199090 1199100 1199110] determined by vector r can be expressed by a

vector potential using curve or surface integral form [11 12]

A (r) =1205830

4120587119868∮119871

119889lR=1205830

4120587119868int119878

119889S times R1198773 (1)

where R = r minus r1minus r2and 119878 is a plane limited by the loop 119871

Introducing the magnetic moment of the loop as m =

IS and magnetic induction B = curlA(r) we get the finalformula for magnetic field at the point 119861[119909

0 1199100 1199110] in the

form

B = 12058304120587

gradm sdot R1198773 (2)

The equivalency of the magnetic field generated by current 119868in a very small current loop to that generated by a magneticdouble layer is known from the literature [13] In the caseof a double layer the magnetic dipoles are continuouslydistributed on the surface 119878 The magnetic dipole is charac-terized by a surface density of the magnetic dipole momentM119878(R) The overall magnetic moment of the surface element

119878 is given by a surface integral as

m = int119878

M119878 (R) sdot 119889S (3)

For further analysis we suppose that the magnetic dipolemoment has a homogeneous density in the whole surfaceelement 119878

The final formula for magnetic induction of the magneticthin-layer which is also applicable for the closed current loopcalculation is expressed as follows [14 15]

B = minus1205830119868

4120587∮

R times 119889s1198773

=1205830119868

4120587int119878

(3R sdot R1198775

minus119868

1198773) 119889S (4)

Magnetic thin layer

0

L

I

r2

r1

B[x0 y0 z0]

S

R

+z

z0

+y

y0

+x

dl

x0

A

r

Figure 1 Theoretical configuration of the magnetic thin-layersample positioned in a plane of the rectangular coordinate system[119909 119910 119911]

Theoretical configuration of the thin-layer sample position-ing simulating the experimental laboratory arrangement isdepicted in Figure 2

We suppose that the thin-layer sample is positioned inthe 119909-119910 plane of the rectangular coordinate system [119909 119910 119911]and the thickness of the layer is neglected According toFigure 2(a) the layer is limited by lengths of 2a and 2b withthe left-right symmetry The magnetic field 119861

0of the NMR

imager is parallel with the +119911-axis The task is to calculatethe 119861119911(119909 119910 119911) component of the magnetic field in the point

119860[1199090 1199100 1199110]

Using general formula (4) for magnetic induction of themagnetic thin-layer calculation respecting the experimentalarrangement depicted in Figure 2 and assuming the positionvectors

r = r1minus r2 119903 = radic(119909

0minus 119909)2+ (1199100minus 119910)2+ (1199110minus 119911)2 (5)

where r is a position vector and 119868 is a current equivalent toplanar density of a dipole moment of themagnetic thin-layerwe can write the final formula in a double integral form asfollows

119861119911(119909 119910 119911)

=1205830119868

4120587

times int

119886

minus119886

[

[

int

119887

minus119887

3 minus 119868

[(1199090minus 119909)2+(1199100minus 119910)2+(1199110minus 119911)2]32119889119910]

]

119889119909

(6)

where limits for integration are [minus119886 119886] and [minus119887 119887]Numerical evaluation of (6) in an analytical form is rel-

atively complicated After integration one obtains relativelyhuge and problematic expressions To calculate the generalresultant expressions in analytical and numerical forms andto obtain the final graphical interpretation of the 119861

119911(119909 119910 119911)

components we used a simplified incremental calculationmodel using rectangular elements see Figure 2(b)

Mathematical Problems in Engineering 3

Imaging plane

RF field

Water solution

Oil slick

+y

+x+z

ds

B0

rminusa

x0

y0

a

minusb b

Bz

z0

A[x0 y0 z0]

r1

r2

(a)

minusan

an

+x

o

+y

dminustx tx

Oil slick image

(b)

Figure 2 (a) Theoretical configuration of the thin-layer sample positioned in the 119909-119910 plane In calculations the thickness of the layer andimaging planeswas neglected (b)Thin-layer sample (eg oil slick) positioned in the119909-119910 plane of the rectangular coordinate system Principlesof incremental elements integration are indicated

Imaging plane

0

10

20

30

4020

0

0

5

10

y0minus20

minus40

x0x0

y0

minus40 minus20 0

0

20

10

20

30

40

(a) (b) (c)

Figure 3 Calculatedmagnetic field variations near the organic slick sample positioned in the 119909-119910 plane of the rectangular coordinate systemrelative values (a) 3D plot (b) Contour plot (c) Density plot

For the final numerical calculation the following sim-plifying condition was assumed thickness of the layer beingnegligible and for the graphical interpretation of a rectangularmagnetic thin-layer sample we assume the following relativevalues 120583

01198684120587 = 1 position of the imaging plane 119911

0= 32

and dimensions of the magnetic thin-layer 119905119909and minus119905

119909were

assigned proportionally to real slick thin-layer dimensionsThe final simplified formula for incremental calculation

using rectangular elements is in the form

119861119911(119909 119910 119911)

=

+119899

sum

minus119899

int

+119905119909

minus119905119909

[

[

int

plusmn119886119899plusmn119889

plusmn119886119899

3minus119868

[(1199090minus 119909119905)2+ (1199100minus 119910119899)2+ (1199110)2]32119889119910]

]

119889119909

(7)

The resultant 3D and 2D plots of relative values of magneticfield for 119909

0 minus40 40 and 119910

0 minus4 36 are depicted in Figure 3

The experimental results confirmed that the density plotrepresentation corresponds best with the obtainedMR imagesee Figure 3(c)

3 Experimental Results

For experimental evaluation of the mathematical modelingwe have chosen a simple laboratory arrangement by applica-tion of the magnetic resonance imaging methods

For sample positioning a plastic vessel was used(Figure 4) A very thin isolating plasticmembrane separatingthe sample from the liquid was placed at its bottom Theheight of liquid level could be up to 10mm A special liquidsolution was used to shorten the measuring time of theimaging sequence gradient echo and to speed up the datacollection [8 9] The liquid contained 5mM NiCl

2+ 55mM

NaCl in distilled water This solution enabled to shorten


Plastic vessel

Imaging planeRF coilRF field

Liquid level

Sample position

+z

+y

B0

Figure 4 For sample positioning a plastic holder was constructedStatic magnetic field of the imager 119861

0is perpendicular and RF field

is parallel to the plane of the holder

the repetition time TR of the imaging sequence due to itsreduction of H

2O relaxation time 119879

1

A special solenoidal RF transducing coil was constructedand placed near the sample The plastic vessel and the RFcoil together with a sample were placed into the centreof a permanent magnet of the imager Imaging plane isperpendicular to the orientation of the static magnetic field(1198610)The next series of pictures shows theoretical and exper-

imental results with imaging of magnetic field variationsnear the very thin soft magnetic materials detected by themagnetic resonance imaging method using carefully tailoredGRE measuring sequences [10]

As a physical object oil slick (dimensions 75times60mm)wasused For comparison a softmagnetic sample cut from a datadisc thickness 80120583m and a thin copper wire formed in theshape of an oil slick were used Results bymagnetic resonanceimaging are visible in Figure 5

The second part of our experiments was aimed at thethin circularly shaped weak magnetic layers For theoreticalcalculations the formula (7) was applied where incrementalelements for integration respected the circular shapes Theresultant image was calculated as a difference of a largerand a smaller circle (Figure 6) Next experiments wereoriented to imaging of weak magnetic materialsmdashplasticcircular magnetic disks Figure 7(a) represents the originaldiskette used for data storing and separated circularmagneticdisc inserted to the holder used for imaging (Figure 7(b))Resultant images show the original diskette with visible datasectors and the same diskette after application of DC and ACmagnetic fields (Figure 8)

4 Discussion and Conclusion

A modified method for mapping and imaging of the planarorganic samples andweakmagnetic inorganic samples placedinto the homogenous magnetic field of an NMR imager wasproposed First experiments showed the suitability of themethod even in the low-field MRI (02 Tesla) The goal ofthis study was to propose an MRI method used for softmagnetic material detection Computation of the magneticfield variations based on thin-layer magnetic theory showedacceptable correspondence of theoretical results with experi-mental images

Mathematical analysis of an oil slick and weak magneticobject representing a shaped magnetic thin-layer showed

(a)

(b)

(c)

Figure 5 (a) Image of the oil slick GRE imaging sequence TR =800ms TE = 10ms and slice thickness 2mm (b) Image of the softmagnetic sample GRE TR = 400ms TE = 10ms and slice thickness2mm (c) Image of the thin coil wired up to the oil slick shape DCcurrent 20mA GRE TR = 500ms and TE = 10ms

theoretical possibilities to calculate magnetic field aroundany type of samples Calculated 3D images showed expectedshapes of the magnetic field in the vicinity of the thin-layer samples Density plot images showed magnetic fieldvariations caused by samples placed into the homogeneousmagnetic field of the NMR tomograph which is very similarto the images gained by magnetic resonance imaging using aGRE measuring sequence


00

0

55

10

5 minus5

minus5

x0

y0

(a)

minus2

minus2minus4

minus4

0

0 2

2

4

4

x0

y0

(b)

Figure 6 Calculated magnetic field of a small plastic circular magnetic disk used to store data or programs for a computermdashsample placedin the 119909-119910 plane Incremental calculation method was used (a) 3D plot (b) Density plot

(a)

y

x

empty1

empty2

(b)

Figure 7 (a) Verbatim DataLife MF 2HD 3510158401015840 Diskette to store data or programs for a computer in original packing dimensions 90 times 93 times32mm (b) Plastic circular magnetic disk separated from the diskette case diameters 0

1= 25mm 0

2= 85mm and thickness 008mm

(a) (b) (c)

Figure 8 Images of the diskette (a) original diskette with visible sectors (b) diskette with 2 larger and 4 smaller stains caused by 1-secondmagnetization bringing closer small circular neodymiummagnets 1 Tesla and (c) diskette demagnetized by 50Hz demagnetizer with 10mmlinear slot Experimental imaging parameters GRE TR = 400ms TE = 10ms and slice thickness 2mm


The experiments prove that it is possible to map themagnetic field variations and to image the specific structuresof thin samples using a special plastic holder The shapesof experimental images Figure 5 correspond to the realshapes of the samples Some of the resultant images areencircled by narrow stripes that optically extend the widthof the sample This phenomenon is typical for susceptibilityimaging when one needs to measure local magnetic fieldvariations representing sample properties [10 12]

The experimental results are in good correlation withthe mathematical simulations This validates the possiblesuitability of the proposed method for detection of selectedthin-layer organic and weak magnetic materials using theMRI methods Presented images of thin objects indicate thepotential of this methodology even in low-field MRI

The proposed method could be used in a variety ofimaging experiments for example on very silky samplestextile material treated by magnetic nanoparticles biologicalsamples documents equippedwith hiddenmagnetic domainmagnetic tapes credit cards and travel tickets with magneticstrips banknotes polymer fibers treated by a solution ofnanoparticles in the water used in surgery and more

Acknowledgments

This work was financially supported by the Grant Agencyof the SAS Project no VEGA 2009011 and by the projectCompetence Center for NewMaterials Advanced Technologiesand Energy ITMS 26240220073 and Operational Programfunded by the European Regional Development Fund Themathematical model and measured data were processedusing the program package Mathematica (Wolfram ResearchInc Champaign IL USA)

References

[1] G Wentzel ldquoComments on Diracrsquos theory of magneticmonopolesrdquo Progress of Theoretical Physics Supplement vol 37-38 pp 163ndash174 1966

[2] J A C Bland and B HeinrichUltrathinMagnetic Structures IIIFundamentals of Nanomagnetism Springer Berlin Germany2005

[3] O Mosendz G Woltersdorf B Kardasz B Heinrich and CH Back ldquoMagnetization dynamics in the presence of pure spincurrents in magnetic single and double layers in spin ballisticand diffusive regimesrdquo Physical Review B vol 79 no 22 ArticleID 224412 10 pages 2009

[4] N Musgens E Maynicke M Weidenbach et al ldquoCurrent-induced magnetization dynamics in single and double layermagnetic nanopillars grown by molecular beam epitaxyrdquo Jour-nal of Physics D vol 41 no 16 Article ID 164011 5 pages 2008

[5] E M Haacke M Makki Y Ge et al ldquoCharacterizing irondeposition in multiple sclerosis lesions using susceptibilityweighted imagingrdquo Journal of Magnetic Resonance Imaging vol29 no 3 pp 537ndash544 2009

[6] I Frollo P Andris J Pribil and V Juras ldquoIndirect susceptibilitymapping of thin-layer samples using nuclear magnetic reso-nance imagingrdquo IEEE Transactions on Magnetics vol 43 no 8pp 3363ndash3367 2007

[7] P T Callaghan and J Stepisnik ldquoSpatially-distributed pulsedgradient spin echo NMR using single-wire proximityrdquo PhysicalReview Letters vol 75 no 24 pp 4532ndash4535 1995

[8] M Sekino T Matsumoto K Yamaguchi N Iriguchi andS Ueno ldquoA method for NMR imaging of a magnetic fieldgenerated by electric currentrdquo IEEE Transactions on Magneticsvol 40 no 4 pp 2188ndash2190 2004

[9] O Strbak P Kopcansky M Timko and I Frollo ldquoSinglebiogenic magnetite nanoparticle physical characteristics Abiological impact studyrdquo IEEE Transactions on Magnetics vol49 pp 457ndash462 2013

[10] E M Haacke R W Brown M R Thompson and R Venkate-san ldquoMagnetic properties of tissues theory and measure-mentrdquo in Magnetic Resonance Imaging Physical Principles andSequence Design chapter 25 pp 741ndash779 Wiley-Liss JohnWiley and Sons New York NY USA 1st edition 1999

[11] J D Kraus andDA FleischElectromagnetics withApplicationschapter 1 2 McGraw-Hill 5th edition 1998

[12] J Jianming Electromagnetic Analysis and Design in MagneticResonance Imaging M R Neuman Ed Biomedical Engineer-ing Series CRC Press 1999

[13] G Retter Magnetische Felder und Kreise chapter 2 VEBDeutscher Verlag der Wissenschaften Berlin Germany 1961

[14] J D Jackson Classical Electrodynamicsedition John Wiley ampSons New York NY USA 3rd edition 1998

[15] Textbook of Physics Charles University in Prague Fac-ulty of Mathematics and Physics httpphysicsmffcuniczkfppskriptakurz fyziky pro DSwwwfyzika

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Algebra

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Stochastic AnalysisInternational Journal of


gel as phantom were published in [8] Single biogenic softmagnetite nanoparticle physical characteristics in biologicalobjects were introduced in [9] It was shown that thesusceptibility imaging needs to measure local magnetic fieldvariations superimposed on the field of the imager by thesample Such variations are representative of the sample [10]

In this paper an imaging method used for thin organicand soft magnetic materials detection was proposed Com-putation of the magnetic field variations based on thin-layermagnetic theory and a comparison of theoretical results withexperimental images were performed

2 Mathematical Model ofthe Magnetic Thin-layer

The goal of this section is to model the thin-layer structure(polarized in the main field) as a distribution of dipoles thefield of which is the superposition of fields due to all thedipoles representing the thin-layer structure

In our interpretation of the magnetic thin-layer model-ing we have started with a magnetic dipole moment of acurrent loop description (Figure 1) Let the current loop 119871with current 119868 be positioned in a plane of the rectangularcoordinate system [119909 119910 119911] The magnetic field at the point119861[1199090 1199100 1199110] determined by vector r can be expressed by a

vector potential using curve or surface integral form [11 12]

A (r) =1205830

4120587119868∮119871

119889lR=1205830

4120587119868int119878

119889S times R1198773 (1)

where R = r minus r1minus r2and 119878 is a plane limited by the loop 119871

Introducing the magnetic moment of the loop as m =

IS and magnetic induction B = curlA(r) we get the finalformula for magnetic field at the point 119861[119909

0 1199100 1199110] in the

form

B = 12058304120587

gradm sdot R1198773 (2)

The equivalency of the magnetic field generated by current 119868in a very small current loop to that generated by a magneticdouble layer is known from the literature [13] In the caseof a double layer the magnetic dipoles are continuouslydistributed on the surface 119878 The magnetic dipole is charac-terized by a surface density of the magnetic dipole momentM119878(R) The overall magnetic moment of the surface element

119878 is given by a surface integral as

m = int119878

M119878 (R) sdot 119889S (3)

For further analysis we suppose that the magnetic dipolemoment has a homogeneous density in the whole surfaceelement 119878

The final formula for magnetic induction of the magneticthin-layer which is also applicable for the closed current loopcalculation is expressed as follows [14 15]

B = minus1205830119868

4120587∮

R times 119889s1198773

=1205830119868

4120587int119878

(3R sdot R1198775

minus119868

1198773) 119889S (4)

Magnetic thin layer

0

L

I

r2

r1

B[x0 y0 z0]

S

R

+z

z0

+y

y0

+x

dl

x0

A

r

Figure 1 Theoretical configuration of the magnetic thin-layersample positioned in a plane of the rectangular coordinate system[119909 119910 119911]

Theoretical configuration of the thin-layer sample position-ing simulating the experimental laboratory arrangement isdepicted in Figure 2

We suppose that the thin-layer sample is positioned inthe 119909-119910 plane of the rectangular coordinate system [119909 119910 119911]and the thickness of the layer is neglected According toFigure 2(a) the layer is limited by lengths of 2a and 2b withthe left-right symmetry The magnetic field 119861

0of the NMR

imager is parallel with the +119911-axis The task is to calculatethe 119861119911(119909 119910 119911) component of the magnetic field in the point

119860[1199090 1199100 1199110]

Using general formula (4) for magnetic induction of themagnetic thin-layer calculation respecting the experimentalarrangement depicted in Figure 2 and assuming the positionvectors

r = r1minus r2 119903 = radic(119909

0minus 119909)2+ (1199100minus 119910)2+ (1199110minus 119911)2 (5)

where r is a position vector and 119868 is a current equivalent toplanar density of a dipole moment of themagnetic thin-layerwe can write the final formula in a double integral form asfollows

119861119911(119909 119910 119911)

=1205830119868

4120587

times int

119886

minus119886

[

[

int

119887

minus119887

3 minus 119868

[(1199090minus 119909)2+(1199100minus 119910)2+(1199110minus 119911)2]32119889119910]

]

119889119909

(6)

where limits for integration are [minus119886 119886] and [minus119887 119887]Numerical evaluation of (6) in an analytical form is rel-

atively complicated After integration one obtains relativelyhuge and problematic expressions To calculate the generalresultant expressions in analytical and numerical forms andto obtain the final graphical interpretation of the 119861

119911(119909 119910 119911)

components we used a simplified incremental calculationmodel using rectangular elements see Figure 2(b)


Imaging plane

RF field

Water solution

Oil slick

+y

+x+z

ds

B0

rminusa

x0

y0

a

minusb b

Bz

z0

A[x0 y0 z0]

r1

r2

(a)

minusan

an

+x

o

+y

dminustx tx

Oil slick image

(b)


Imaging plane

0

10

20

30

4020

0

0

5

10

y0minus20

minus40

x0x0

y0

minus40 minus20 0

0

20

10

20

30

40

(a) (b) (c)




0= 32


119909were



119861119911(119909 119910 119911)

=

+119899

sum

minus119899

int

+119905119909

minus119905119909

[

[

int


plusmn119886119899

3minus119868

[(1199090minus 119909119905)2+ (1199100minus 119910119899)2+ (1199110)2]32119889119910]

]

119889119909

(7)


0 minus40 40 and 119910






2+ 55mM



Plastic vessel


Liquid level

Sample position

+z

+y

B0






1








(a)

(b)

(c)




00

0

55

10

5 minus5

minus5

x0

y0

(a)

minus2

minus2minus4

minus4

0

0 2

2

4

4

x0

y0

(b)


(a)

y

x

empty1

empty2

(b)


1= 25mm 0


(a) (b) (c)






Acknowledgments


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Imaging plane

RF field

Water solution

Oil slick

+y

+x+z

ds

B0

rminusa

x0

y0

a

minusb b

Bz

z0

A[x0 y0 z0]

r1

r2

(a)

minusan

an

+x

o

+y

dminustx tx

Oil slick image

(b)


Imaging plane

0

10

20

30

4020

0

0

5

10

y0minus20

minus40

x0x0

y0

minus40 minus20 0

0

20

10

20

30

40

(a) (b) (c)




0= 32


119909were



119861119911(119909 119910 119911)

=

+119899

sum

minus119899

int

+119905119909

minus119905119909

[

[

int


plusmn119886119899

3minus119868

[(1199090minus 119909119905)2+ (1199100minus 119910119899)2+ (1199110)2]32119889119910]

]

119889119909

(7)


0 minus40 40 and 119910






2+ 55mM



Plastic vessel


Liquid level

Sample position

+z

+y

B0






1








(a)

(b)

(c)




00

0

55

10

5 minus5

minus5

x0

y0

(a)

minus2

minus2minus4

minus4

0

0 2

2

4

4

x0

y0

(b)


(a)

y

x

empty1

empty2

(b)


1= 25mm 0


(a) (b) (c)






Acknowledgments


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Plastic vessel


Liquid level

Sample position

+z

+y

B0






1








(a)

(b)

(c)




00

0

55

10

5 minus5

minus5

x0

y0

(a)

minus2

minus2minus4

minus4

0

0 2

2

4

4

x0

y0

(b)


(a)

y

x

empty1

empty2

(b)


1= 25mm 0


(a) (b) (c)






Acknowledgments


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00

0

55

10

5 minus5

minus5

x0

y0

(a)

minus2

minus2minus4

minus4

0

0 2

2

4

4

x0

y0

(b)


(a)

y

x

empty1

empty2

(b)


1= 25mm 0


(a) (b) (c)






Acknowledgments


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgments


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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