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Chapter 9

STR TEGIES FOR BSOLUTE C LIBR TION

OF NE R INFR RED TOMOGR PHIC TISSUE

IM GING

Troy

O.

McBride, Brian

W.

Pogue, UlfL. Osterberg, Keith

D.

Paulsen

Thayer School

o

Engineering. 8000 Cummings

Hall

Dartmouth College. Hanover. New

Hampshire 03755

Abstract: Quantitative near infrared NIR) imaging of tissue requires the

use

o a diffusion

model-based

reconstruction algorithm,

which solves for the

absorption

and

scattering coefficients of a tissue volume by matching transmission

measurements

o light to

the

predictive diffusion

equation

solution. Calibration

problems

as well as other practical considerations arise for an

imaging

system

when using

a

model-based method

for a real system.

For

example, systematic

noise in

the data acquisition

hardware

and source/detector

fibers must

be

removed

to

prevent spurious results

in the

reconstructed image. Practical

considerations for a NIR diffuse tomographic imaging system

include: I)

calibration with a homogeneous phantom, 2)

use

of a homogeneous fitting

algorithm

to arrive

at an initial

optical property

estimate for image

reconstruction o a heterogeneous

medium.

and

3) correction

for fluctuations

in

source strength and initial phase offset during data acquisition. These practical

considerations, which rely on an accurate

homogeneous fitting

algorithm are

described.

They have allowed demonstration of a prototype imaging system

that has the ability to quantitatively

reconstruct

heterogeneous

images

o

hemoglobin

concentrations

within

a

highly scattering medium with

no

a priori

information.

Key

words:

blood,

calibration, hemoglobin, photon migration, reconstruction,

tomography

Oxygen Transport to Tissue XXIV edited by

Dunn

and

Swartz, Kluwer AcademiclPlenum Publishers, 2003

85

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86 Mc ride et

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1

INTRODUCTION

Tissue

is

highly scattering at visible and near-infrared

NIR.)

wavelengths, thus a simplistic back-projection image is highly blurred and

difficult to interpret

as

the effects

o

scattering and absorption can not be

distinguished. However, by using model-based imaging and computational

methods in combination with frequency domain measurements, moderate

resolution, quantitatively accurate images o absorption and scattering can be

obtained within highly scattering media, such

as

tissue [1,2]. This method

may lead to a potentially powerful medical imaging modality for non

invasive hemoglobin tomography.

Practical calibration issues arise when using a model-based image

reconstruction approach which may not be present in modalities such as x

ray tomography due to the necessity o fitting the data to a partial differential

equation solution. In addition, while many system calibrations can be

ignored in qualitative imaging, more care is required for quantitative

imaging o hemoglobin concentrations on an absolute scale. Issues

encountered in the development o the NIR imaging system addressed in this

paper include: 1) elimination o system-based offsets, 2) choice o

heterogeneous starting value, and 3) offsets due to long-term drift. These

issues are addressed through a calibration protocol which involves a

homogeneous phantom and the use

o

a precise homogeneous fitting

algorithm that is resistant to measurement noise.

A 4: 1 increase in hemoglobin concentration

[3]

and a 1.4 - 4.4 times

lower oxygen pressure [4] have been observed in breast cancer. By using

NIR. frequency domain measurements which have been properly calibrated

at multiple wavelengths, quantitative absolute images o hemoglobin related

parameters can be obtained [5]. Once calibrated imaging is implemented in

practice it should be possible to determine whether NIR hemoglobin

concentration and oxygen saturation information is useful in the diagnosis

o

breast tumors.

2 METHODS

2 1 maging system

The frequency-domain near-infrared imaging system consists o three

main components: 1) data acquisition system, 2) image reconstruction

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Data

Acquisition

At a single

wavelength .

acquire

256

measurements

Inverse Image

Reconstruction

Reconstruct into

quantitative image

o absotption and

scattering

Repeat

for

all

wavelengths

Tissue Functional

Parameters

Combine I east squares regression)

absotption images at multiple wavelengths

to generate images ofhemoglohin

concentration and oxygen saturation.

Figure 1. Schematic o near-infrared tomographic imaging

system

87

algorithm, and (3) spectroscopic determination

o

functional properties. A

schematic

o

the imaging process

is

shown in Figure 1.

2 1 1 Data acquisition

system

The data acquisition system (shown in Figure 2) uses an amplitude

modulated 100 MHz) wavelength-tunable 700-850 nm) TiSapphire laser as

the light source. Sixteen source and sixteen detector optical fibers are

arranged in a circular geometry to analyze a single tomographic plane

o

the

measured tissue or tissue-simulating phantom. The detector for the system is

a photomultiplier tube with built-in heterodyning circuitry. The heterodyne

signal 1 kHz) amplitude and phase shift are read into a computer using a

commercial-grade data acquisition board. The source and detector are

multiplexed to acquire the 256 data points using linear translation stages.

[5,6]

(a)

(b)

....----., loo.

oonrnz

h

tll

M z

I

Figure 2. a)

Diagram

and

b) photograph of

the

data acquisition system.

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2.1.2 Image reconstruction

A finite element based solution to the frequency-dependent diffusion

equation is used to calculate a fit to the measured projection data o

amplitude and phase shift. Absorption and scattering coefficient images are

generated using a Newton iterative scheme [1,7] with update vectors

determined from a full matrix inversion during each iteration. A schematic

o the image reconstruction algorithm is illustrated in Figure

3.

Measured Phase and Amplitude

data, ~ . a S J r e d h.t.ro)

,

at

256

points

IOriginal

estimate

of

I

olve forward diffusion equation using latest

optical properties

.

estimate of absorption and scattering at each

I

finite element node to determine calculated

phase and amplitude at 256

points.

Compare calculated and measured

IReturn New I

phase and amplitude

data at 256

points.

Estimate

Error below

I

rror above

tolerance

tolerance

r

iniS"ied

I

pdate vector based on full matrix

inversion generates new absorption

and

scattering at

each

me& node

Figure 3. Schematic

o

the finite element based reconstruction algorithm based on the non

linear estimation problem

o

inferring optical property maps using the frequency-domain

diffusion equation.

2.1.3 Determination

o

functional properties

Multiple wavelength images o absorption coefficient can be used to

determine maps o hemoglobin related parameters by incorporating the

previously measured absorption spectra o pure oxygenated hemoglobin, de

oxygenated hemoglobin [8], and other tissue chromophores such as lipids [9]

and water [10]. Because the image reconstruction algorithm recovers

quantitative values for the absorption coefficient, a least squares fit can be

used to determine the metabolic chromophore concentrations from that data

as shown schematically in Figure 4.

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Known

mol ar

absorption

a) at

each

wavelength A) from measured values for

the unknowns (e.g. HbR and Hb02)

a

Hb

-

R

=

1 l 1 i 2 ~ n

[ m m ~ m M

a

Hb

-

=

1l,1i2,b

[mm.

1

m ]

+

Measured absorption W at n

wavelengths at each point in the image

from

finite element reconstruction algorithm

Ji ,.tI,.t2,An =

~ m

]

t

Calculate

concentrations in. this case [Hb

02] and [HbR]) o unknowns for each

point using least squares

fit.

1i.,AI

[Hb

-

2 l

a':-02 +[Hb -

R]

.a':-R

l i

a

A2

= [ H b - 0 2 l a ~ - 0 2

[ H b - R ] . a ~ - R

1i.,hI [Hb 02laJ b-02 [Hb R]a: '-R

Figure 4. Schematic

o methodology

for

determining

hemoglobin related parameters from

multiple wavelength

images

o absorption coefficient.

2.2 Practical considerations

Calibration and other practical considerations arise when using a model

based image reconstruction method with data acquired from a real imaging

system. In the NIR diffuse tomography context these include: (1) system

based offsets, (2) initial optical property estimate for a heterogeneous

medium, and (3) long term fluctuations o source strength and initial phase

offset.

2.2.1 System-based offsets

Systematic offset in the measured data occurs at various source and

detector locations due to optical fiber differences, multiplexing imprecision,

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and other systematic inconsistencies. Obviously, these offsets need to be

removed from the' data prior to image reconstruction in order to prevent the

appearance

of

artifacts in the resultant tissue property profiles. The method

adopted here measures a homogeneous phantom and subtracts differences

between measured and calculated values:

:alibrated heterO) = ~ e a S U r e d h e t e r O ) - ~ e a s u r e d h O m O ) - :alculated hOmO) (1)

where ~ e s u r e d is the measured amplitude and phase at each

of

the 256

measurement locations for the calibration phantom

homo)

and the actual

heterogeneous object hetero), while ;alculated is the corresponding

calculation for the homogeneous

homo)

case. This requires either exact

knowledge

of

the scattering and absorption coefficients for the measured

homogeneous phantom or a homogeneous fitting algorithm which

determines these properties for the measured data on the homogeneous

calibration phantom. The homogeneous fitting algorithm (described in

Section 2.3) is an important component

of

the practical reconstruction

algorithm and is, therefore, also used in the calibration procedure.

The calibration procedure for eliminating systematic offset at different

source and detector locations involves: (1) measurement of a homogeneous

phantom (each day and after equipment changes), (2) determination of the

scattering and absorption coefficient for the phantom (from the measured

data using the homogeneous fitting algorithm), and (3) use of the difference

between calculated and measured values as the calibration factor.

2 2 2

Heterogeneous starting values

The image reconstruction algorithm starts from an estimated set of

optical properties and then calculates an update vector based on

l

(the

squared difference between the calculated and measured values).

f

the

initially estimated set

of

optical properties is far from the actual optical

properties

of

the heterogeneous object being imaged, this inaccurate starting

point can slow convergence and even lead to erroneous answers.

n

order to

determine the initial optical property estimate for patient data or a

heterogeneous phantom, the measurements are averaged for each of the

sixteen sources and the homogeneous fitting algorithm is used to determine a

homogeneous estimate of the properties that are consistent with a diffusion

equation solution which matches the averaged data. n this manner, the

initial estimate is determined solely from the heterogeneous phantom or

patient measurements, yet, is close enough to the unknown actual

heterogeneous parameter distribution that the algorithm will converge.

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2.2.3 ong term drift

Differences between the source strength and initial phase at the time

o

the measurement o the homogeneous calibration phantom and the actual

source strength and initial phase observed at the time o the heterogeneous

measurement can lead to an overall offset mainly due to long term drift)

between the calibrated measured and calculated data. This difference in

overall offset needs to be removed in order for the reconstruction algorithm,

which fits to the absolute amplitude and phase data, to be effective. To

account for long term drift, the offset for both the homogeneous phantom

measurements and the actual heterogeneous measurement are calculated

based on the homogeneous fitting algorithm. The homogeneous fitting

algorithm responds to the slope o the data and is therefore independent o

initial source strength and phase shift.) The offset is estimated as the

average difference between the measured and calculated data.

N

t P ~ e a s U r e d h O m O ) - tP:alculated hOmO)

tPojfset homo) =...: i=..:. l N

N

L

t P ~ e a s U r e d h e t e r O )

- tP:alculated heterO)

ffojfset hetero) =...: i=::.: l N

tPojJset net) = Pojfset heterO) -

tPojJset homO)

2)

3)

4)

where tPojfset nel) is the long term drift correction for initial phase and

source strength.

2 3 Homogeneous fitting algorithm

An important part o practical imaging is the homogeneous fitting

algorithm. The homogeneous fitting algorithm is critical for both calibration

o the system and for providing the initial optical property estimate for the

image reconstruction process. Assuming a homogeneous medium, only one

source location is needed to determine the absorption and scattering

coefficient

o

the material from measurements around its periphery. The

absorption and scattering coefficients which result in the best fit to the

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measured data can be determined by using a Newton-Raphson iterative

scheme applied to the finite element forward solution of the diffusion

equation

for

the relevant geometry. To reduce the effect

of

noise on the

fitting algorithm, the data acquired is averaged together

for

all sources based

on detector distance from the source location. The Newton-Raphson

solution is simplified by reducing the fit to two parameters: slope of the

phase with respect to distance from the source location and slope of the log

of intensity times distance with respect to distance. These two parameters

were chosen because they are nearly constant and can be obtained from the

data through linear regression. They are constant

for the analytic infinite

medium diffusion equation solution [11]. This method

is

insensitive to noise

due to the large amount

of

averaging which occurs (256 measurements are

used

to

find two parameters).

Figure 5 contrasts two versions

of

the homogeneous fitting algorithm.

Method A performs a Newton-Raphson iterative scheme based on an

estimate of the first derivative for the averaged data (number of detectors

minus one (15) parameters), while Method B uses the slope of the phase

with respect to distance from the source location and slope of the log of

intensity times distance with respect to distance determined from linear

regression (2 parameters).

I

Measured data

--

ZS (16 source x 16

detector) JMllnts

for

phase and Intensity

I

I

verage

data

to

I

I

source x 16 detectors

Iteratively

solve for

optical

Method A: Newton-

Method

B:

Newton-Raphson

Raphson Iterative method

properties

by

comparing

Iterative

method

based on

slope

of

(minimization of

I )

based

wtth

nnlte element

fOl ward

de

On

approximation

or

nrst

solution

of dl/fuslon equation

the phase

shirt versus distance

7,)

for drcular geometry

and

a

and slope o

In(dlstance intensity)

derivative of the measured

homogeneous medium

d(ln(d

AC

data

versus

distance

- d r

z

= Qin(I '1 l-ln(I, lHn(I :1l-ln(I,mlY

X = d ( I n ~ I _ d ) ) d ( i n ~ I ' ' I ~ ' ) ) J

P 1

dr

dr

+ cro. 1

-

11, ]- [0 :1

_

1,m

Y

[

dfl' '''- _dr I

dr dr

where N =

16detectors,

In(J')am f are

the

[d(ln(>-i) del

where all slope calculations -----;;:-.b

caJculaJed

In(Iniensity) and

phase

shift

from

are perfonnedwith a linear

regression

to

finite element

solution,

m

In(i ) and

r are

the measured

daJa

or the calculated

daJa

the measured

In(Intensity)

andphllJie.

from

the

finite element solution

Figure 5. Schematic describing two homogeneous fitting algorithms.

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3.

RESULTS

3.1

Practical considerations

3.1.1 Homogeneous calibration

Figure 6 shows systematic offset from a slight difference in alignment of

two optical fibers with the laser source fiber during multiplexing. This offset

is consistent over a period of weeks and only changes during system

modifications/maintenance such as re-alignment

of

the laser. The resultant

difference data (Fig.6.c.) between the measured (Fig.6.a.) and the calculated

data (Fig.6.b.) is used as a calibration factor in the reconstruction algorithm.

a)

(b) c)

6,00

.

6,00

6.00

5,00

5,00

5,00

' .00

' ,00

' ,00

3.00

3,00

3.00

E

~ 2 . 0 0

1

00

E

.

00

0.00

,00

1.00

1.00

1 ,00

2 ,00

2,00

2,00

3,00

3,00 -

-

_

-3 .00

Figure 6. Plot of (a) In(Intensity) data at 16 detector sites showing offset due

to

alignment

differences, (b) calculated data from finite element solution, and (c) difference between (a)

and (b). Difference data (c) is used

as

the 'calibration' factor.

3.1.2 Heterogeneous starting value

Simulations were performed to demonstrate the effect

of

different

starting values during reconstruction of a heterogeneous object. Images for

three different initial optical property estimates ( 0 , 10 , and 50 error

from the actual average optical properties) are compared in Figure 7 after

one iteration. The algorithm converges to the correct image

2:

I absorbing

object with Gaussian profile) during the first iteration for an initial estimate

which is equal to the average optical properties of the heterogeneous test

object (Fig.7.a,d). When the initial estimate is significantly removed from

the actual average optical properties, the algorithm takes longer to converge.

This slowdown is evident in the other images in Figure

7.

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(a)

r:

::,,

" I

n

; 1

.1

~ i I U

(d)

, [7\

I

l n .

n

t lO

II iIP .

n

llO lilll

II lIl J

II 4lD '0.

II UI IU .

I I I I I I

I

.

10

II

)0

(b)

r-;

"

"

I , I I I I

II

: J I. ,. )

(e)

.

[1\

.

uuHIO

O.UtIH

lll

U Ul3110

.H'. , : :

I

111

I'll

)11

4 U

JtI

McBride et al

(t)

[

lIHt 4 UO

(lOUHCI

tI

Oll j lUI

(J

OOl$C1

(J OO}OO

I() OtHSCl

o OIHUCJ

I I J I I I

I 10 i l l

;lO 40

J O

Figure 7. Image after first iteration

of

reconstruction

of

absorbing object with Gaussian

profile with (a) initial estimate equal to average optical properties, (b) \0% error in initial

estimate, and (c) 50% error in initial estimate. The vertical axis units are

in

mm

t

(d), (e),

and (t) are profile plots along a vertical line through the center

of

images (a), (b), and (c)

respectively.

3.1.3 ()ffset

Offset between the natural log

of

intensity profile

of

a calculated and

measured phantom due to long term drift in the Ti:Sapphire laser intensity is

shown in Figure 8. The offset is accounted for by comparison of the changes

in offset between the homogeneous calibration phantom and the measured

heterogeneous object.

In

the extreme case shown in Figure

8,

the offset in

intensity is almost two natural log units.

10.00

8.00

6.00

4.00 .

2.00

'iii

0.00

c

-4 .00 .

-6.00

8

.

00

10.00

I calibrated Measured d1

* Calculated

Data

_ .. J

Figure 8. Plot demonstrating measured data with overall offset due to long term drift

of

laser

intensity. The average difference

is

used to correct this offset.

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3.2 Testing

o

homogeneous fitting algorithm

Using repeated measurements

of

the same phantom, the results

of

the

homogeneous fitting algorithm (Method B in Figure

5)

have been shown to

have an average deviation of 0.5 even in the presence

of 5%

measurement

noise. This compares very favorably with our previous approach (Method A

in Figure 5), which did not take full advantage

of

the data averaging, where

5%

measurement noise translated to 5 average deviation in the

homogeneous fit. This decrease in deviation is demonstrated in Figure 9

which shows data for a phantom study involving triplicate measurement of

objects with increasing amounts

of

blood. The absorption coefficient

increases while the scattering coefficient remains constant. The study is

performed at three wavelengths and the slope of the line of increasing

absorption with added blood is compared with expected values

of

the molar

extinction coefficient from Wray et

al. [8]

in Table

1.

(a)

0.8

0.7

_ 6

0.5

I

I

I

1 0.4

'

0.3

E

0.2

0.1

0.0

a

3

ml blood I liter 0.5 Intrallpld

(c)

0.007

--

_ 0.006

1 0.005

.

0.

004

,

.

0.

003 _ - -

a

3

4

5

mI blood 0.5 Intrallpld

5

(b)

0 8

r ------------

0.7 1

:

0.6 t

t

n

E 0.2

J

0.1

I

0.0 - - - ~ ~ - _ _ . ___I

a

1

3

0.

007

-

0.006

I

i o.005 '

.

0.004 '

ml blood I

l i ter O.soh

ntrallpld

(d)

--

_ 1

0.003 . - . - -

;

o

2 5

ml blood I liter 0.5 Intrallpld

5

Figure

9. Homogeneous fits to measured data for

the

two

methods

described

in

Figure 5: (a)

Method A scattering coefficient

data, (b) Method

8 scattering coefficient

data,

(c)

Method

A

absorption coefficient

data, (d) Method

B absorption coefficient

data. Each

concentration

was

measured

three

times

at

each

of three wavelengths.

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Molar Extinction Coefficient

(mm

1

mM

1

)

Wavelength

Wrall et al.

data Method B

Percent

difference

750 nm 3.10E-04 2.49E-04 -19.68

800 nm

4.50E-04 5.06E-04 12.44

830 nm

5.30E-04

4.79E-04 -9.62

Table

1 Comparison

of molar extinction

coefficient

for

oxygenated

hemoglobin

measured

by Wray et al.

[8] with

results from use of homogeneous fitting algorithm (Method B.) based

on data from phantoms increasing hemoglobin concentration

(Figure

9.d).

3.3 Modified imaging algorithm

The modified image reconstruction algorithm is shown schematically in

Figure 10. This modified algorithm uses

the

three practical imaging

considerations discussed in this paper. These modifications are denoted in

the figure by the thicker border lines.

Once a day MellllUfe

Mea llred Phase and Amplilude Avet'l8e over

16

sources and

obtain

data, '-..ur.d

(Iw

. , )

, 81256

estimate

for

qtical properties

by

Iterative

known hcmogenoous

Least Squares Fit to data as llming

phantom;-

:

..,.d

(homo)

p.xnts for objec:t ofinterest

bomogeoeous

Iis II

"

y

Use calilnled data:?ca.bralod(Iw' ,) - . . . . vod(Iw,. ') -

. . .

urwd(ho lfID) -

caJaJaIod(ho fOO

- .ffi.

.

)

olve forward diffusion e"",l ion using latest

,

--+

eslimllle of absorplion and scaltering at

each

J Original

estimate

of

finite .lemeDl node to determin. calculllled

I

qtical properties

phase and

8IIlplilude

at

256

points.

~

Compare calculilled and measured

I dum ew I

pbase and amplilude data

at 256

points.

Estimate

Error

below

I

rror

above

tolerance tolOIllD O

~ F i n i h e d

I

~

Update vector baaed

]II full

mabix

I

nveni

]II

generatea

new absorpti(]ll

and scattering III each melb node

Figure

10.

Schematic of revised image reconstruction algorithm (compare with Figure

3)

which

includes calibration

and

other practical

imaging

considerations added in bold

boxes.

3.4 Phantom imaging results

The final calibrated system was used to measure a tissue-simulating

phantom. A representative image is shown in Figure

11.

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a)

b)

c)

I

-0.015

I

-0.010

-0.005

-0.000

d)

-60.000

-100.000

-50.000

-80.000

-40.000

-30.000

-60.000

-20.000

I

I

10.000

-0.000

-40.000

-20.000

-0.000

D

60.0

40.0

20.0

0.0-

Figure

11.

Representative phantom

images from the

calibrated

imaging system. The 90 mm

diameter phantom consists of a mixture o

0.5

Intralipid and 20 microMolar hemoglobin

concentration in water. The hemoglobin in

the

phantom is

fully

oxygenated except for

a 24

mm

diameter object

to

the

right

of center which

was

de-oxygenated by bubbling

with

Nitrogen

gas. Absorption

coefficient images

(scale

is

rom-I)

of

the

phantom are shown

(a)

at

720 nm, (b)

at

750

nm,

and (c) 800 nm.

These

three

images

were

combined

to form images

of

(d)

hemoglobin concentration

(units are microMolar) and (e)

hemoglobin oxygen saturation

(in percent

oxygenation).

t)

and

(g) are

horizontal profile plots

through

the center of (d) and

(e)

respectively.

4 DIS USSION

4 1 Practical considerations

Practical considerations for a NIR diffuse imaging system include: 1)

system calibration with a homogeneous phantom, 2) use

o

a homogeneous

fitting algorithm to arrive at an initial optical property estimate for image

reconstruction o a heterogeneous medium, and 3) correction for long term

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drift by subtraction o source strength and initial phase offsets. These

practical considerations necessary when using data from a real imaging

system with a model-based reconstruction scheme in order

to

reduce data

model mismatch errors associated with system imperfections which are not

idealized in the model. These modifications allow for the realization o a

prototype imaging system that has the potential to quantitatively reconstruct

heterogeneous images

o hemoglobin-related parameters o highly scattering

tissue in vivo

4 2 Testing

o

homogeneous fitting algorithm

n

important component

o

our practical imaging system is a robust,

stable homogeneous fitting algorithm. One method which is insensitive to

noise due

to

the large amount

o

data averaging which is possible (256

measurements are used to find two parameters) is a Newton-Raphson

minimization based on two parameters: slope o the phase with respect to

distance from the source location and slope o the log o intensity times

distance with respect to distance. This method produces only 0.5% deviation

in determined optical properties in the presence o over

5

measurement

noise

in

repeated phantom measurements.

5 CONCLUSION

Quantitatively accurate images o hemoglobin related parameters

recovered

on

an absolute scale have been demonstrated with no a priori

information. Production o these quantitative images o hemoglobin

concentration and oxygen saturation in a tissue-simulating phantom were

critically dependent on a prototype imaging system implementation which

includes the practical considerations o source/detector calibration, an

accurate first estimate o the optical property distribution, and a correction

for long term source/sensor drift. Once properly calibrated, this near

infrared tomographic imaging system has the potential to provide

in vivo

images o hemoglobin related parameters which may make it useful in breast

cancer imaging and therapy monitoring.

ACKNOWLEDGEMENTS

This work has been supported by the NIH through grants ROICA69544 and

POICA80139 awarded by the National Cancer Institute. Authors gratefully acknowledge

previous development work by Huabei Jiang and David Rinehart.

7/23/2019 Mcbride 2003

15/15

Strategies For

NIR

alibration

99

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