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Direct-current-based image reconstruction

versus direct-current included or excluded

frequency-domain reconstruction

in diffuse optical tomography

Guan Xu,1Daqing Piao,1,* Charles F. Bunting,1and Hamid Dehghani2

1School of Electrical and Computer Engineering, Oklahoma State University, Stillwater, Oklahoma, USA 74078

2University of Birmingham, Birmingham B15 2TT, UK

*Corresponding author: daqing.piao@okstate.edu

Received 25 September 2009; revised 16 March 2010; accepted 29 March 2010;

posted 8 April 2010 (Doc. ID 117721); published 25 May 2010

We study the level of image artifacts in optical tomography associated with measurement uncertainty

under three reconstruction configurations, namely, by using only direct-current (DC), DC-excluded fre-

quency-domain, and DC-included frequency-domain data. Analytic and synthetic studies demonstrate

that, at the same level of measurement uncertainty typical to optical tomography, the ratio of the stan-

dard deviation of μaover μareconstructed by DC only is at least 1.4 times lower than that by frequency-

domain methods. The ratio of standard deviations of D(or μ0

s) over D(or μ0

s) reconstructed by DC only are

slightly lower than those by frequency-domain methods. Frequency-domain reconstruction including DC

generally outperforms that excluding DC, but as the amount of measurements increases, the difference

between the two diminishes. Under the condition of a priori structural information, the performances of

OCIS codes: 170.3880, 170.3010, 170.6960, 170.5270.

1. Introduction

Diffuse optical tomography (DOT) based on measure-

ment of near-infrared (NIR) light diffused through

thick biological tissue aims to quantify the heteroge-

neities of NIR-absorbing chromophors and scattering

particles [1]. There are generally three categories

of DOT measurements: (1) continuous wave (CW),

wherein only steady-state or direct-current (DC)

detection is carried out, (2) time domain, wherein

the attenuation and pulse-width broadening of the

excitation light are the measurands [2–5], and (3)

frequency domain, which is mathematically the Four-

ier-transform equivalent of the time-domain method

[6–17] but is considerably less complicated in instru-

mentation. Frequency-domain detection ideally

renders three types of information: the DC attenua-

tion, the modulation intensity change (AC), and the

modulation phase shift (PHS). Some frequency-

domain DOT works, however, have utilized AC and

PHS [6–12,14,15], rather than the complete measur-

ands of DC, AC, and PHS. Excluding the DC in fre-

quency-domain DOT reconstruction implied that

the DC information was considered unlikely to im-

prove the outcome of reconstruction when the AC

and PHS are available. Such consideration could have

been prompted if the DC information had been redun-

dant in frequency-domain reconstruction, but indeed

it has not been either justified or negated.

On the other hand, many works in DOT have relied

on only the DC measurements [18–26]. Although

lacking phase information will certainly reduce the

accuracy or confidence of quantitative reconstruction,

almost all these studies have demonstrated that the

absorption and reduced scattering characteristics can

0003-6935/10/163059-12$15.00/0

1 June 2010 / Vol. 49, No. 16 / APPLIED OPTICS 3059

be separately and absolutely reconstructed by use of

DC information only. But all these works lack a direct

comparison of the outcome of DC-based reconstruc-

tion with that of frequency-domain reconstruction,

which is needed to provide a basis to assess the

compromise for reconstruction based solely upon

DC information. Out of these DC-based DOT recon-

structions, there also exists a common but not widely

stated feature in the images—the recovered back-

ground is usually more homogeneous than the gener-

al level of background artifacts seen in images

reconstructed in the frequency domain. Fewer image

artifacts in the background may be beneficial for iden-

tifying the target of interest over a relatively hetero-

geneous background, but what contributes to fewer

image artifacts in the background has not been well

understood.

This work studies the level of artifacts associated

with measurement uncertainties in three modes of

image reconstruction, namely DC, AC þPHS, and

DC þAC þPHS. The studies are conducted both

analytically and by synthetic measurements, to ad-

dress why the DC-based reconstruction results in

fewer background artifacts and to demonstrate that

including DC information in frequency domain gen-

erally improves the reconstruction outcome. Clearly,

the analysis of this study shall be based upon the pro-

pagation of measurement noises to the image. Con-

tributing to the image artifacts are a number of noise

sources, among which is an error due to coupling loss,

as studied by Schweiger et al. [27]. That study trea-

ted coupling errors as coupling coefficients appended

to the solution space, and demonstrated reconstruc-

tion of frequency-domain data contaminated with

synthetic coupling errors. Similar studies are neces-

sary to understanding reconstruction with contami-

nated DC data.

The level of artifacts is a critical indicator of the

capability of reliably recovering the optical heteroge-

neity. Ntziachristos et al. [28] demonstrated that the

reconstruction of localized lesions deteriorated as a

function of background heterogeneity. They also

found that increasing the dataset size, specifically

the number of detectors used, improves the recon-

struction of the lesion structure, but does not remove

the artifacts. Those results, performed on frequency-

domain synthetic and experimental data, indicate

that certain artifacts are inherent to image forma-

tion and, thereby, cannot be removed completely. The

cause of such artifacts must also be inherent to DC-

based reconstruction, wherein the outcome relative

to frequency-domain reconstruction is unknown.

The analytic approach of this study is based pri-

marily upon a method introduced by Fantini et al.

[29] to model the accuracies or, equivalently, the er-

rors associated with a two-distance measurement

technique for quantifying the optical properties of

a bulk homogeneous medium. Reconstructing optical

properties in a homogeneous medium is essentially a

process of fitting the slopes of measurements with

respect to different source–detector distances, for

which Fantini et al. introduced their models of the

“relative error”of absorption and reduced scattering

coefficients using the intensity exponential factor,

the AC exponential factor, and the phase factor be-

tween the measurements made at two different

source–detector distances. The tomography of optical

heterogeneity relies on multiple measurements

among spatially resolved sources and detectors, and

image reconstruction is a process of optimizing the

local optical properties to minimize the difference

of model prediction for these source–detector pairs

with respect to the measured values. The accuracy

of reconstruction is thereby dependent upon the cap-

ability of distinguishing the signal variations for a

single source–detector pair due to all types of mea-

surement fluctuations, as well as local changes of

tissue optical properties, such variations among dif-

ferent source–detector pairs, and mapping such var-

iations to the image space. Hence, the “relative error”

initially discussed in [29] equally applies to tomogra-

phy of optical heterogeneity, because the “relative er-

ror”of measurement determines the upper limit of

reconstruction accuracy; in other words, it sets the

“parameter-recovery-uncertainty level”(PRUL) in

the tomography images.

This study analyzes the PRULs of the absorption

coefficient, the reduced scattering coefficient, and the

diffusion coefficient, for the measurement conditions

of DC, AC + PHS, and DC + AC + PHS and examines

their representations as image artifacts in synthetic

models. Much of the analytic approach of this study

is based upon the method established in [29]; how-

ever, there are substantial differences in the mea-

surement configurations investigated, and also, in

this novel study, the analytic results partially sug-

gested by [29] are quantitatively evaluated to com-

pare the PRULs among these configurations. It is

also noted that [29] considered the measurement

configurations of DC þAC, AC þPHS, and DCþ

PHS. When frequency-domain (FD) information is

available, it is straightforward to apply AC þPHS,

as employed by many works [6–12,14,15], to image

reconstruction. The utilization of DC þAC and DC þ

PHS are mathematically valid; however, those con-

figurations have seldom been used for image recon-

struction. This study investigates the level of

artifacts in the DC, AC þPHS, and DC þAC þPHS

configurations, as they are the most likely implemen-

ted approaches toward image reconstruction. There-

fore, among the results previously stated in [29], only

those related to AC þPHS have been included in this

study when appropriate. The AC þPHS result for

the absorption coefficient in [29] is cited directly,

but the AC þPHS result in [29] for reduced scatter-

ing is revised to a more generalized form that is con-

sistent with the result for the absorption coefficient.

Table 1in Subsection 2.A is introduced to make clear

these distinctions. This study also investigates re-

construction of the diffusion coefficient, because,

not only are the absorption and reduced scattering

coefficients coupled, but also generally the diffusion

3060 APPLIED OPTICS / Vol. 49, No. 16 / 1 June 2010

coefficient is involved in the reconstruction process

prior to formulating the reduced scattering coeffi-

cient. The diffusion coefficient image may provide

new insights to the study even though its artifacts

are expected to be close to those seen in reduced scat-

tering image.

The rest of the paper is organized in the following

sections. Section 2analyses the PRUL for three ca-

tegories: (1) D.C. only, (2) AC þPHS, and (3) DCþ

AC þPHS. Tissue and measurement parameters

typical to optical tomography applications are imple-

mented to evaluate quantitatively the PRULs ex-

pected in the images. Section 3uses synthetic data

to examine the uncertainty of the parameters recov-

ered for homogeneous medium, single inclusion with

different types of optical contrast, and multiple in-

clusions with specific optical contrasts. These syn-

thetic models are also evaluated selectively for the

condition of having spatial a priori information in

the image reconstruction. Section 4discusses the im-

plications of the results.

2. Theory

The reconstruction accuracy of optical tomography is

determined by many factors, including the accuracy

of the forward model, the determinacy of inverse for-

mulation, and the characteristics of instrument noise

[30]. An analytic approach has been introduced in

[29] to demonstrate that the uncertainty (or error)

in the measurement maps to the uncertainty of reco-

vering the assembled optical properties of bulk

tissue. The same uncertainty (or error) of the mea-

surement, when involved in tomographic reconstruc-

tion to recover spatially resolved tissue optical

properties, will translate to spatially varying arti-

facts that reduce the contrast-to-noise ratio (CNR)

of the target of interest. This effect may seem ob-

vious; however, the extent of it is not well understood.

This work closes this gap of knowledge in three con-

ditions of DOT measurements, namely DC, ACþ

PHS, and complete frequency-domain information

by DC þAC þPHS.

A. Parameter-Recovery-Uncertainty Level

The variation of the recovered optical properties is

modeled as PRUL, which for AC þPHS has been

derived in [29] in terms of the attenuation of the

AC amplitude and phase shift versus a change of

source–detector distances. We implement the ap-

proach in [29], but extend it to DC-only and DC þ

AC þPHS configurations, and apply it to diffusion

coefficients in addition to absorption and reduced

scattering coefficients.

The frequency-domain measurement of photon den-

sity consists of a steady state and time-varying com-

ponents as UFDð~

r;ωÞ¼UDCð~

rÞþUACð~

r;ωÞ, where~

ris

the position vector and ωis the angular modulation

frequency of the light source. The UFDð~

r;ωÞsatisfies

the photon diffusion equation of

−μað~

rÞ

Dð~

rÞþiω

vDð~

rÞUFDð~

r;ωÞþ∇2UFDð~

r;ωÞ

¼−Sð~

r;ωÞ

Dð~

rÞ;ð1Þ

where vis the speed of light in the medium, μais the

absorption coefficient, D¼½3ðμaþμ0

sÞ−1is the diffu-

sion coefficient, μ0

sis the reduced scattering coefficient,

and the source term Sð~

r;ωÞhas a DC component

SDCð~

rÞand a time-varying component SACð~

r;ωÞ.For

a homogeneous infinite medium with a detector at ~

and a source at~

r0, thereby a source–detector distance

of d¼j

r0−~

rj,wehave

UFDðr;ωÞ¼UDCðrÞþjUACðr;ωÞjexpðiΦACÞ

¼SDCðr0Þ

4πDd expð−kDCdÞ

þSACðr0;ωÞ

4πDd expð−kACdÞ· expðikPHSdÞ;

ð2Þ

where

kDC ¼ﬃﬃﬃﬃﬃ

μa

r;kAC ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

μa

2Dﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1þω2

v2μ2

sþ1

kPHS ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

μa

2Dﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1þω2

v2μ2

s−1

ð3Þ

It is noted that kAC >kDC and kAC is correlated with,

but not linearly dependent upon, kDC. The attenuation

of the DC component of the photon density is thus not

equal to or linearly dependent upon that of the AC

component, which is an indication that the DC infor-

mation would not be a duplication of any of AC or PHS.

Denoting d2>d1and ρ¼jd1−d2jas the differ-

ence of source–detector distance between two mea-

Table 1. Comparison of the Analytic Derivations in This Work with That in [29]

Measurements

DC DC þAC AC þPHS DC þPHS DC þAC þPHS

Δμa

μaðσμa

μaÞThis study [29][29][29] This study

Δμ0

μ0

sðσμ0

μ0

sÞThis study [29][29]a[29] This study

ΔD

DðσD

DÞThis study This study This study

aThe derivation was revised to a more generalized form.

1 June 2010 / Vol. 49, No. 16 / APPLIED OPTICS 3061

surements corresponding to the same source, one has

[29] (reproduced here for convenience)

δ¼lnd2

UDCðd2Þ

UDCðd1Þ¼−ρ·kDC ¼−ρ·ﬃﬃﬃﬃﬃ

μa

α¼lnd2

UACðd2Þ

UACðd1Þ¼−ρ·kAC

¼−ρ·ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

μa

2Dﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1þω2

v2μ2

sþ1

ϕ¼Φðd2Þ−Φðd1Þ¼ρ·kPHS

¼ρ·ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

μa

2Dﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1þω2

v2μ2

s−1

t:ð4Þ

Table 1lists the PRUL of five different measurement

configurations, among which three were investigated

in [29]. As stated previously, the configuration of

DC þAC and DC þPHS were seldom used for image

reconstruction, therefore, only the AC þPHS results

of [29] are cited for this comparative study.

In CW measurement, we have

μajDC ¼D·δ

ρ2

:ð5Þ

References [31,32] suggest that, for steady-state sur-

face measurements, μaand Dcollectively determine

the diffuse reflectance, denoted as R∞, by the rela-

tionship ½μa·D¼KðR∞Þ. It is noted that the diffuse

reflectance is not UDCð~

rÞ, which implies treating

KðR∞Þas not significantly dependent upon UDCð~

rÞ,

thereby Eq. (5) may be converted to

μajDC ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

KðR∞Þ

pρ·δ;ð6Þ

and estimating the PRUL of μafor DC by

σμa

μaDC¼1

μa∂μa

∂δσδ¼σδ

δor σ2

δ21=2

:ð7Þ

We have, for AC þPHS [29],

μajACþPHS ¼ω

2vϕ

α−α

ϕ;ð8Þ

and a PRUL of [29]

σμa

μajACþPHS ¼1

μa∂μa

∂α2

σ2

αþ∂μa

∂ϕ2

σ2

ϕ1=2

¼α2þϕ2

α2−ϕ2σ2

α2þσ2

ϕ21=2

:ð9Þ

For DC þAC þPHS measurement, we have

μajDCþACþPHS ¼−ω

v·δ2

2αϕ ð10Þ

and, accordingly, a PRUL of

σμa

μaDCþACþPHS

¼1

μa∂μa

∂δ2

σ2

δþ∂μa

∂α2

σ2

αþ∂μa

∂ϕ2

σ2

ϕ1=2

¼4σ2

δ2þσ2

α2þσ2

ϕ21=2

:ð11Þ

The PRULs in Eqs. (7), (9), and (11) all have the

shape of

σμ

μ¼η·ðξÞ1=2;ð12Þ

which contains a multiplication factor ηand a

square-root term ﬃﬃﬃ

p. The relative levels of these

PRULs become comparable as σ2

ϕ2,σ2

α2, and σ2

δ2are prac-

tically the same [29]. It is indicated in Table 2that

the PRUL of μawill be the lowest in DC-based recon-

struction, but whether the PRUL of μais lower in

AC þPHS or in DC þAC þPHS depends upon the

difference in αand ϕ.

Because the image reconstruction recovers Dto

formulate μ0

s, it is imperative to analyze the PRUL

of D. For the case of DC, similar to the derivation

for μa, we have

DjDC ¼KðR∞Þ·ρ

δ;ð13Þ

Table 2. Comparison on PRUL of μaðσμa=μaÞ

Eq. Condition

ηﬃﬃﬃ

Normalized η·ﬃﬃﬃ

Expression Value Expression Normalized value

(7)DC 1 σ2

δ21=211

(9)ACþPHS α2þϕ2

α2−ϕ2>1σ2

α2þσ2

ϕ21=21.41 >1:41

(11)DCþAC þPHS 1 4σ2

δ2þσ2

α2þσ2

ϕ21=22.45 2.45

3062 APPLIED OPTICS / Vol. 49, No. 16 / 1 June 2010

σD

DDC¼1

D∂D

∂δσδ¼σδ

δorσ2

δ21=2

:ð14Þ

For AC þPHS and DC þAC þPHS, the expressions

are the same:

DjACþPHS ¼DjDCþACþPHS ¼−ωρ2

2v·1

αϕ :ð15Þ

Therefore,

σD

DjACþPHS ¼σD

DjDCþACþPHS

¼1

D∂D

∂α2

σ2

αþ∂D

∂ϕ2

σ2

ϕ1=2

¼σ2

α2þσ2

ϕ21=2

:ð16Þ

The PRULs of Din Eqs. (14) and (16) are compared in

Table 3. Apparently, when AC and phase are em-

ployed, the DC component is redundant for the recov-

ery of D.

The PRUL of μ0

sis derived by

σμ0

μ0

s¼1

μ0

s∂μ0

∂D2

σ2

Dþ∂μ0

∂μa2

σ2

μa1=2

¼1

3D−μa−1

· 1

3D2σD

D2

þðμaÞ2σμa

μa21=2

;

ð17Þ

so the PRUL of μ0

sfor DC is

σμ0

μ0

sDC¼1

3D−μa−1 1

3D2

·σ2

δ2þμ2

a·σ2

δ21=2

ð18Þ

For AC þPHS, it is [29,33]

σμ0

μ0

sACþPhs ¼1

3D−μa−1

· 1

3D2σ2

α2þσ2

ϕ2

þμ2

a·α2þϕ2

α2−ϕ22

·σ2

α2þσ2

ϕ21=2

;ð19Þ

and for DC þAC þPHS, it is

σμ0

μ0

sjDCþACþPhs ¼1

3D−μa−1

· 1

3D2σ2

α2þσ2

ϕ2þμ2

·4σ2

δ2þσ2

α2þσ2

ϕ21=2

:ð20Þ

Based on the estimation leading to Table 2, the

PRULs in Eqs. (19)and(20) can be normalized with

respect to Eq. (18). The results are given in Table 4.

Again, the PRUL of μ0

swill be the lowest for DC.

Whether the PRUL of μ0

sis lower in AC þPHS or

in DC þAC þPHS depends also upon the difference

in αand ϕas for the PRUL of μa, but, because of the

dominance of 1=3Dover μa, the difference between

AC þPHS and DC þAC þPHS will be less than that

observed for PRUL of μain Table 2.

B. Summary of the PRUL Analyses

The DC-only reconstruction seems to give the least

level of relative uncertainty of the parameter in

the reconstruction. The AC þPHS configuration

seems to be equivalent to DC þAC þPHS in the le-

vel of PRULs of reduced scattering and diffusion coef-

ficient, but it is unclear for the absorption coefficient.

These analyses have been conducted for an infinite

homogeneous medium, but the results will be readily

translatable to a medium with boundaries and with

inclusions.

3. Synthetic Studies

Simulations are carried out to study the practical

issues of PRUL, such as background noise, the accu-

racy of optical property recovery, and the interpara-

meter cross coupling, of the three measurements

setups.

A. Synthetic Model

The forward model is carried out by the finite-

element method (FEM) solution of Eq. (1) using the

Robin-type boundary condition [34]:

Table 3. Comparison on PRUL of D

Eq. Condition Expression

Normalized

Value

(14)DC σ2

δ21=21

(16)ACþPHS & DC

þAC þPHS σ2

α2þσ2

ϕ21=2∼1:41

Table 4. Comparison on PRUL of μ0

Eq. Condition Expression Normalized as

(18)DC h1

3D2σ2

δ2þμ2

a·σ2

δ2i1=21

(19)ACþPHS h1

3D2σ2

α2þσ2

ϕ2þμ2

a·α2þϕ2

α2−ϕ22·σ2

α2þσ2

ϕ2i1=2>1:41

(20)DCþAC þPHS h1

3D2σ2

α2þσ2

ϕ2þμ2

a·4σ2

δ2þσ2

α2þσ2

ϕ2i1=2>1:41

1 June 2010 / Vol. 49, No. 16 / APPLIED OPTICS 3063

Uð~

r0;ωÞ−2DA^

n0·∇Uð~

r0;ωÞ¼0;ð21Þ

where Ais related to refractive index mismatch and

n0is an outgoing normal vector. The Jacobian is

structured to the form of

J¼2

PHS 3

5¼2

∂ln UDC

∂μa

∂ln UDC

∂D

∂ln jUACj

∂μa

∂ln jUACj

∂D

∂ΦAC

∂μa

∂ΦAC

∂D

;ð22Þ

where the indices of each block of the Jacobian could

be node based for pixelwise reconstruction or region

based for prior-guided regionwise reconstruction.

Utilizing only the first row leads to CW, utilizing

the second and third rows renders AC þPHS, and

utilizing all three rows gives DC þAC þPHS.

The inverse solver implements the Levernberg–

Marquardt algorithm as

xkþ1¼xkþα·½JTðxkÞJðxkÞþλI−1JTðxkÞΔvðxkÞ;

ð23Þ

where xis the array of unknown parameters, Δν is

the forward projection error and λis a penalty or reg-

ularization term. The value of λis initially set as 100,

and is reduced to its fourth root with each continued

iteration. The damping factor, α, in the range of (0, 1),

is introduced when only regionwise reconstruction is

performed to facilitate stable convergence [35] and is

set at 0.5 in this study when included. For pixelwise

reconstructions using NIRFAST [36,37], αis set to 1.

B. Simulation Results

Synthetic data are generated for a homogeneous

medium, a medium with a single inclusion, and a

medium with multiple inclusions with mixed types

of optical heterogeneities.

1. PRULs in a Homogeneous Medium

A cylinder-applicator geometry [38]of60 mm in

height and 86 mm in diameter with 16 optodes is

adopted, like the one shown in Fig. 1. The optodes are

turned on sequentially for the measurements being

taken by all other optodes, generating a total of

240 measurements for each dataset.

The volume is discretized into a FEM mesh of

12,695 nodes for forward computation, while a smal-

ler FEM mesh of 600 nodes is used in the recon-

struction. Because this synthetic study specifically

investigates the level of artifacts reconstructed to

the same level of recovered parameters in an other-

wise homogeneous medium, the same optical proper-

ties of μa¼0:01 mm−1and μ0

s¼0:01 mm−1are used

for both forward computation and as the initial

values of the inverse routine, with 1% noise added

to the forward simulation data to maintain the same

measurement error. In addition, all controlling

parameters of the inverse model are maintained

the same for DC, AC þPHS, and DC þAC þPHS

configurations.

Table 5demonstrates that the variations recov-

ered to the parameters of a homogeneous medium

are lowest in DC, as expected from the analytic anal-

ysis. The DC þAC þPHS slightly outperforms AC þ

PHS in μarecovery, but AC þPHS slightly outper-

forms DC þAC þPHS in μ0

s=Drecovery.

The normalized numbers (1:45–1:64) for μ0

s=D

recovery are considerably close to those in the analy-

tical derivation—with the same average optical prop-

erties, the background standard deviation of the

images reconstructed by FD system measurements

is at least 1.41 times larger than those reconstructed

by the CW system. However, in μareconstruction, the

variations in FD configurations are about twice those

predicted in Table 2. It is noted that the analytic

results in this study are based upon perturbation

analysis. It is well known that DOT is a nonlinear

process, wherein the absorption perturbation is more

Fig. 1. (Color online) Imaging geometry for a homogeneous

medium.

Table 5. Mean Value and Standard Deviation Reconstructed for Homogeneous Medium`

μa

σμaðmm−1Þ

μ0

σμ0

sðmm−1Þ

σDðmmÞ

Abs. Norm. Abs. Norm. Abs. Norm.

DC 0.01 0:69 ×10−61 1.00 0:80 ×10−41 0.33 2:64 ×10−31

AC þPHS 0.01 3:13 ×10−64.50 1.00 1:18 ×10−41.47 0.33 3:83 ×10−31.45

DC þAC þPHS 0.01 2:98 ×10−64.29 1.00 1:31 ×10−41.64 0.33 4:24 ×10−31.60

a“Abs.”denotes the absolute value of the standard deviation. “Norm.”denotes the standard deviation normalized with respect to the

standard deviation of DC. The same notations apply to Tables 6and 8.

3064 APPLIED OPTICS / Vol. 49, No. 16 / 1 June 2010

pronounced than scattering perturbation. In this

specific model of homogeneous medium, the signal

perturbation is evenly distributed to the entire vo-

lume of the homogeneous medium instead of mostly

confined to smaller lesions with higher optical prop-

erty contrast, as in the later examinations. There-

fore, the perturbations from AC and PHS could

have been coupled to and nonlinearly amplified as

the variation of absorptions.

2. Contrast-to-Noise Ratio Analysis for Single

Target

The results in Subsection 3.B.1 indicate that, for 1%

noise in the measurement of homogeneous medium,

DC-only reconstruction clearly maintains a lower

artifact level compared to DC þAC þPHS and

AC þPHS. This study examines the contrast of a tar-

get inclusion in an otherwise homogeneous medium

at different measurement noise levels when recon-

structed by DC, AC þPHS, and DC þAC þPHS con-

figurations. The synthetic model is similar to that in

Subsection 3.B.1, but with a spherical heterogeneity

added at (x¼0mm. y¼−20 mm, z¼0mm), with

μa¼0:025 mm−1and μ0

s¼1:75 mm−1. The recon-

struction basis of 2760 nodes is larger than the

one used for Subsection 3.B.1. Varying noise levels,

of 0% to 10%, are integrated into the forward data

to examine the CNR of the target (CNR ¼½max

ðtarget-region-valueÞ−meanðbackground-valueÞ=

background-standard-deviation) with respect to

the background artifacts. The background deviation

is calculated by excluding the areas within a distance

of 1.5 times the target radius away from its center

[39]. The calculated CNRs are given in Fig. 2for

the three types of target contrast. It is observed in

Fig. 2that the CNR levels of μaand Dlook similar

when compared to that of μ0

s, which supports the

Fig. 2. (Color online) Contrast-to-noise-ratio (CNR) with respect to the measurement noise levels. (a), (b), (c) μa=μ0

s=Ddistribution in the

z¼0plane of forward model; (d) μaCNR comparison; (e) μ0

sCNR comparison; (f) DCNR comparison.

Fig. 3. (Color online) Simulation studies for reconstructing multiple targets in a three-dimensional cylindrical geometry with the optodes

and targets located on one plane.

1 June 2010 / Vol. 49, No. 16 / APPLIED OPTICS 3065

assumptions made for deriving PRULs of μaand Din

Eqs. (7) and (14). In Fig. 2, the CNR levels of μaare

found to be lower than that of μ0

s, which may be due to

underestimation of μaand overestimation of μ0

sin

such a pixelwise image reconstruction [24]. Despite

this, several features can be observed in Fig. 2. (1)

At a zero noise level, the three methods are compar-

able in the CNR. (2) When the noise becomes higher,

the D.C. clearly outperforms the other two in CNR,

while DC þAC þPHS slightly outperforms ACþ

PHS. (3) At a 10% noise level, the CNRs of all meth-

ods are similar for μ0

sand Drecovery, but DC still

outperforms the other two in μareconstruction.

3. Multiple Target Case

The geometry for having multiple inclusions is shown

in Fig. 3, where three spherical targets with radii of

7:5mm are located in the longitudinal middle plane

(z¼0) of the cylindrical imaging volume and are all

20 mm away from the center of the circular cross sec-

tion, ensuring the same spatial sensitivity at their po-

sitions. Target 1, at the upper left (x¼−14:14 mm,

y¼14:14 mm, z¼0mm), has only absorption con-

trast (μa¼0:025 mm−1,μ0

s¼1mm−1), target 2, at up-

per right (x¼14:14 mm, y¼14:14 mm, z¼0mm),

has only scattering contrast (μa¼0:01 mm−1,μ0

s¼

1:75 mm−1), and target 3, at lower side (x¼0mm,

y¼−20 mm, z¼0mm), has contrasts of both absorp-

tion and reduced scattering (μa¼0:025 mm−1,

μ0

s¼1:75 mm−1). The dashed line in the figure marks

the position of the target when it presents no contrast

in that category. Table 6lists the deviation of the

background optical property in the reconstructed

images. Standard deviation values in Table 6are nor-

malized along each column versus those of DC-only

reconstruction.

For background homogeneity, comparison in

Table 6indicates that DC only demonstrates the low-

est artifact level in the image background, while the

background artifact levels of DC þAC þPHS and

AC þPHS are approximately 1 to 2 times higher.

Although the numerical simulative result does not

exactly match the values in Tables 2–4, it qualita-

tively agrees with the analytical derivations. The

analytical derivations given in Tables 2–4indicate

that DC þAC þPHS and AC þPHS produce similar

background homogeneities, but the simulation re-

sults all indicated a slightly lower background arti-

fact level in DC þAC þPHS reconstruction. For

target accuracy, the reconstructed images in Fig. 3

and the data comparison in Table 7are seen with

DC þAC þPHS as superior to AC þPHS, which,

along with the comparison on the background

homogeneity, indicates that including DC generally

improves the FD reconstruction. In terms of inter-

parameter cross coupling, DC has more coupling

than FD, which is well known. The cross coupling

Table 6. Standard Deviation of Background Optical Properties in Fig. 3

σμaðmm−1Þσμ0

sðmm−1ÞσDðmmÞ

Abs. Norm. Abs. Norm. Abs. Norm.

DC 1:92 ×10−412:46 ×10−217:25 ×10−31

AC þPHS 3:63 ×10−41.89 2:88 ×10−21.17 9:01 ×10−31.24

DC þAC þPHS 3:45 ×10−41.79 2:49 ×10−21.01 7:75 ×10−31.07

Table 7. Comparison of the Accuracy of Recovered Optical Properties in Fig. 3

μa1ðmm−1Þμ0

s1ðmm−1ÞD1ðmmÞ

Value Error Value Error Value Error

Set 0.025 1 0.325

DC 0.0125 −50:16%1.398 39.84% 0.236 −27:35%

AC þPHS 0.0146 −41:62%1.293 29.27% 0.255 −21:59%

DC þAC þPHS 0.0149 −40:30%1.201 20.06% 0.274 −15:67%

μa2ðmm−1Þμ0

s2ðmm−1ÞD2ðmmÞ

Value Error Value Error Value Error

Set 0.01 1.75 1.75

DC 0.0114 13.68% 1.238 −29:25%1.639 −6:34%

AC þPHS 0.0107 6.95% 1.250 −28:56%1.619 −7:47%

DC þAC þPHS 0.0104 3.81% 1.375 −21:45%1.635 −6:55%

μa3ðmm−1Þμ0

s3ðmm−1ÞD3ðmmÞ

Value Error Value Error Value Error

Set 0.025 1.75 0.188

DC 0.0141 −43:48%1.639 −6:34%0.2012 7.37%

AC þPHS 0.0139 −44:31%1.619 −7:47%0.204 8.70%

DC þAC þPHS 0.0137 −45:24%1.635 −6:55%0.202 7.64%

3066 APPLIED OPTICS / Vol. 49, No. 16 / 1 June 2010

in DC þAC þPHS is slightly less severe than that

in AC þPHS.

A similar study is conducted for the same targets

in a three-ring setup [38] in Fig. 4, which has three

identical rings of optodes at the azimuthal planes of

z¼−10 mm, z¼0mm, and z¼10 mm. Each set of

data contains a total of 2256 measurements by

turning on one source and detecting at all other op-

todes. The key values are compared in Tables 8and 9.

Most features of the three aspects discussed for the

single-ring case can be reconfirmed, except that the

target contours recovered by FD reconstructions are

more accurately defined, but, nonetheless, the differ-

ence between DC þAC þPHS and AC þPHS is

insignificant.

Prior-guided region-based reconstructions are also

performed on both of the imaging geometries of

Figs. 3and 4to examine if including accurate a priori

structural information of the target affects the out-

come of the three reconstruction configurations. As

is shown in Figs. 5and 6, with forward models the

same as those in Figs. 3and 4, the inverse model

has integrated spatial a priori information by assum-

ing a homogeneous target of the accurate size in a

homogeneous background. Results of both cases indi-

cate that, with the structural a priori information,

the performances of the three configurations are es-

sentially equivalent.

4. Discussions

Using only the DC information to simultaneously re-

cover the absorption and diffusion (or the reduced

scattering) distributions has been controversial.

The nonuniqueness that may be inherent to DC-only

measurements was described in a seminal study [40].

However, despite the negative predictions in [40]

that there could be an infinite number of diffusion

and absorption pairs leading to the same surface

measurements, Harrach [41] proved that, at most,

one of them consists of a piecewise constant diffusion

and piecewise analytic absorption, and if the true

medium has these properties, as in virtually any

practical condition, a reconstruction algorithm favor-

ing these properties will pick the right combination

of profiles. Harrach’s study theoretically justified the

experiences in many works that the absorption and

scattering distributions have been separately and

uniquely recovered by surface measurement of DC

only [18–26].

The primary aim of this work is to understand the

expectation for DC-based reconstruction in a more

systematic approach, thereby establishing a certain

level of confidence for the recovered information

when only DC information can be relied upon. This

work, conveyed by a side-by-side comparison of

the reconstructions based on DC, AC þPHS, and

DC þAC þPHS, does provide direct evidence that

DC-based reconstruction is much less accurate in re-

covering the absolute optical properties of the target

of interest when no additional spatial information is

available to confine the reconstruction, as having

been universally recognized by the DOT community.

However, apart from these well-expected shortcom-

ings, it seems that DC-based reconstruction may not

be completely unfavorable. This study generalized

the analytical approach initially proposed in [29]

Table 8. Standard Deviation of Background Optical Properties in Fig. 4

σμaðmm−1Þσμ0

sðmm−1ÞσDðmmÞ

Abs. Norm. Abs. Norm. Abs. Norm.

DC 2:26 ×10−413:00 ×10−218:47 ×10−31

AC þPHS 4:07 ×10−41.80 3:26 ×10−21.09 9:78 ×10−31.15

DC þAC þPHS 3:95 ×10−41.75 3:18 ×10−21.06 9:51 ×10−31.12

Fig. 4. (Color online) Simulation studies for reconstructing multiple targets in a three-dimensional cylindrical geometry with the optodes

located on three different planes and targets located on the middle plane.

1 June 2010 / Vol. 49, No. 16 / APPLIED OPTICS 3067

to quantify the level of image artifacts that is ex-

pressed by the standard deviation of a parameter

over the parameter itself. Parameters representative

of tissue measurements are used to evaluate the ana-

lytic results and conduct the synthetic studies, in

both of which the DC reconstruction produced a low-

er level of relative variation in the optical para-

meters recovered, and some advantages in the CNR.

It may be argued that DC flattens images, leading to

a lower standard deviation in the background and,

because the background standard deviation is the de-

nominator of CNR, the CNR of DC could become bet-

ter. But, if there were flattening of the image, then

the numerator of the CNR would also be flattened,

and perhaps flattened more strongly owing to the

nonlinearity of DOT and, thereby, underestimated

at a higher level, which collectively might reduce

the CNR rather than increase the CNR. The slight

but notable CNR advantage of DC-based over FD-

based reconstruction demonstrated in this study

strongly suggests some inherent advantages of DC,

but it could be just because DC has lower information

content, similar to what one could expect by reducing

the amount of data available or increasing the regu-

larization in FD-based reconstructions.

It is worthwhile to note that this study (as well as

most other synthetic studies) assumes a step change

of the optical properties of the target of interest with

respect to the background. This is not a faithful

representation of actual tissue-imaging applica-

tions, wherein the target of interest frequently has

a tapered or smooth change of contrast over the back-

ground. The stronger cross talk between absorption

and scattering seen for DC-only reconstruction in

this study, as well as many other studies, could have

been the outcome of the nonuniqueness, revealed by

[40], which is pronounced when the target of interest

has a step contrast over the background. In fact, the

DC-based reconstruction of in vivo measurements

has encountered notably different absorption and

scattering patterns of a target of interest [42], which

may indicate a weaker cross talk for smoother con-

trast of the target of interest. It is also noted that this

study, as well as most other synthetic studies, as-

sumes a globally homogenous yet locally heteroge-

neous background. An actual tissue environment

could be locally homogenous but globally strongly

heterogeneous, such as is found in the prostate [26].

In such conditions, a balance or trade-off may exist

between the ability of suppressing the background

heterogeneity and the likelihood of identifying a

Table 9. Comparison of the Accuracy of Recovered Optical Properties in Fig. 4

μa1ðmm−1Þμ0

s1ðmm−1ÞD1ðmmÞ

Value Error Value Error Value Error

Set 0.025 1 0.325

DC 0.0133 −46:93%1.528 52.81% 0.216 −33:49%

AC þPHS 0.0169 −32:39%1.288 28.77% 0.256 −21:43%

DC þAC þPHS 0.0171 −31:42%1.292 29.21% 0.255 −21:70%

μa2ðmm−1Þμ0

s2ðmm−1ÞD2ðmmÞ

Value Error Value Error Value Error

Set 0.01 1.75 0.189

DC 0.0117 16.95% 1.319 −24:63%0.251 32.26%

AC þPHS 0.0104 3.73% 1.427 −18:45%0.232 22.46%

DC þAC þPHS 0.0103 3.09% 1.441 −17:66%0.230 21.31%

μa3ðmm−1Þμ0

s3ðmm−1ÞD3ðmmÞ

Value Error Value Error Value Error

Set 0.025 1.75 0.188

DC 0.0156 −37:57%1.847 5.57% 0.178 −4:73%

AC þPHS 0.0163 −35:02%1.731 −1:10%0.191 1.61%

DC þAC þPHS 0.0163 −34:77%1.726 −1:38%0.191 1.88%

Fig. 5. (Color online) Region-based reconstruction for multiple targets in a three-dimensional cylindrical geometry with the optodes and

targets located on one plane. (a) Imaging geometry and the regions of interest; (b) comparison of the results for DC, AC þPHS, and

DC þAC þPHS.

3068 APPLIED OPTICS / Vol. 49, No. 16 / 1 June 2010

target of interest in which the contrast is strong

locally but weak globally.

This study has also indicated that including DC in-

formation in FD reconstruction can sometimes lead

to better images than those obtained by ignoring

it. The expressions of δand αin Eq. (4) demonstrate

that the DC attenuation is not linearly dependent

upon the AC attenuation, and the difference between

the two attenuation values increases as the modula-

tion frequency increases. The necessity of including

DC in order to optimize the FD reconstruction is

made evident by the results in Subsections 3.B.2

and 3.B.3, wherein the DC þAC þPHS results have

always been slightly better than the AC þPHS

results on the background artifacts, the target prop-

erties, and the cross coupling between μaand μ0

s=D.

However, the slightly better performance of DC þ

AC þPHS over AC þPHS diminishes as the total

number of measurements goes up, as is shown in

the three-ring case in Subsection 3.B.3. When fewer

measurements are available in application situa-

tions, including the DC information in the limited

FD measurements likely will improve the overall re-

construction outcome.

This study is carried out for the measurements at a

single wavelength. Investigating the PRUL issues in

the context of multiband FD measurements will be a

natural and more practical extension of this work be-

cause most optical tomography measurements are

conducted with some kind of spectral information.

Besides, similar approaches may be extended to

other applications wherein the measurement data

contains multiple aspects of information, from which

the data usage may be optimized for the specific sys-

tem configuration.

5. Conclusions

The level of variations of recovered optical properties

in optical tomography associated with the measure-

ment uncertainty under three reconstruction config-

urations of DC-only, the DC-excluded FD, and the

DC-included FD is studied by analytic and synthetic

means. It is demonstrated that, at the same level of

measurement uncertainty typical to optical tomogra-

phy and under pixelwise reconstruction without spa-

tial a priori information, the standard deviations of

μaover μareconstructed by DC only are at least 1.4

times lower than those obtained by FD methods. The

standard deviations of D(or μ0

s) over D(or μ0

s) recon-

structed by DC only are slightly lower than those by

FD methods. Frequency-domain reconstruction in-

cluding DC generally outperforms reconstruction ex-

cluding DC, but the difference between the two

becomes less significant when the total amount of

measurements becomes larger. For FD reconstruc-

tion with no spatial a priori information and a smal-

ler number of measurements, including DC is

recommended. When a priori structural information

is available, the three reconstruction configurations

investigated in this study perform equally well.

This work has been supported in part by the Pros-

tate Cancer Research Program of the U.S. Army Med-

ical Research Acquisition Activity (USAMRAA)

through grant W81XWH-07-1-0247, and the Health

Research Program of Oklahoma Center for the Ad-

vancement of Science and Technology (OCAST)

through grant HR06-171. We are grateful to the

anonymous reviewers for their constructive com-

ments that enriched the discussions of this work.

References and Notes

1. S. R. Arridge, “Optical tomography in medical imaging,”

Inverse Probl. 15, R41–R93 (1999).

2. F. Gao, H. Zhao, Y. Tanikawa, and Y. Yamada, “Optical tomo-

graphic mapping of cerebral haemodynamics by means of

time-domain detection: methodology and phantom valida-

tion,”Phys. Med. Biol. 49, 1055–1078 (2004).

3. H. Rinneberg, D. Grosenick, K. T. Moesta, J. Mucke,

B. Gebauer, C. Stroszczynski, H. Wabnitz, M. Moeller,

B. Wassermann, and P. M. Schlag, “Scanning time-domain op-

tical mammography: detection and characterization of breast

tumors in vivo,”Technol. Cancer Res. Treat. 4, 483–496 (2005).

4. T. Tanifuji and M. Hijikata, “Finite difference time domain

(FDTD) analysis of optical pulse responses in biological tis-

sues for spectroscopic diffused optical tomography,”IEEE

Trans. Med. Imaging 21, 181–184 (2002).

5. W. Mo and N. Chen, “Fast time-domain diffuse optical tomo-

graphy using pseudorandom bit sequences,”Opt. Express 16,

13643–13650 (2008).

6. B. W. Pogue, M. S. Patterson, H. Jiang, and K. D. Paulsen, “In-

itial assessment of a simple system for frequency domain dif-

fuse optical tomography,”Phys. Med. Biol. 40, 1709–1729

(1995).

7. B. W. Pogue, T. O. McBride, J. Prewitt, U. L. Osterberg, and

K. D. Paulsen, “Spatially variant regularization improves

diffuse optical tomography,”Appl. Opt. 38, 2950–2961 (1999).

8. J. Wang, S. Jiang, K. D. Paulsen, and B. W. Pogue, “Broadband

frequency-domain near-infrared spectral tomography using a

mode-locked Ti:sapphire laser,”Appl. Opt. 48, D198–D207

(2009).

9. D. M. Hueber, M. A. Franceschini, H. Y. Ma, Q. Zhang,

J. R. Ballesteros, S. Fantini, D. Wallace, V. Ntziachristos,

Fig. 6. (Color online) Region-based reconstruction for multiple targets in a three-dimensional cylindrical geometry with the optodes

located on three different planes and targets located on the middle plane. (a) Imaging geometry and the regions of interest; (b) comparison

of the results for DC, AC þPHS, and DC þAC þPHS.

1 June 2010 / Vol. 49, No. 16 / APPLIED OPTICS 3069

and B. Chance, “Non-invasive and quantitative near-infrared

hemoglobin spectrometry in the piglet brain during hypoxic

stress, using a frequency-domain multidistance instrument,”

Phys. Med. Biol. 46,41–62 (2001).

10. M. A. Franceschini, K. T. Moesta, S. Fantini, G. Gaida,

E. Gratton, H. Jess, W. W. Mantulin, M. Seeber, P. M. Schlag,

and M. Kaschke, “Frequency-domain techniques enhance op-

tical mammography: initial clinical results,”Proc. Natl. Acad.

Sci. USA 94, 6468–6473 (1997).

11. N. G. Chen, M. Huang, H. Xia, D. Piao, E. Cronin, and

Q. Zhu, “Portable near-infrared diffusive light imager for

breast cancer detection,”J. Biomed. Opt. 9, 504–510 (2004).

12. M. J. Holboke, B. J. Tromberg, X. Li, N. Shah, J. Fishkin,

D. Kidney, J. Butler, B. Chance, and A. G. Yodh, “Three-

dimensional diffuse optical mammography with ultrasound

localization in a human subject,”J. Biomed. Opt. 5, 237–247

(2000).

13. B. J. Tromberg, B. W. Pogue, K. D. Paulsen, A. G. Yodh,

D. A. Boas, and A. E. Cerussi, “Assessing the future of diffuse

optical imaging technologies for breast cancer management,”

Med. Phys. 35, 2443–2451 (2008).

14.E.S.Papazoglou,M.S.Weingarten,L.Zubkov,L.Zhu,

S. Tyagi, and K. Pourezaei, “Near infrared diffuse optical tomogra-

phy: improving the quality of care in chronic wounds of patients

with diabetes,”Biomed. Instrum. Technol. 41,83–87 (2007).

15. G. Gulsen, O. Birgul, M. B. Unlu, R. Shafiiha, and O. Nalcioglu,

“Combined diffuse optical tomography (DOT) and MRI system

for cancer imaging in small animals,”Technol. Cancer Res.

Treat. 5, 351–363 (2006).

16. J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran,

A. Slemp, V. Ntziachristos, B. Chance, and A. G. Yodh, “Three-

dimensional diffuse optical tomography in the parallel plane

transmission geometry: evaluation of a hybrid frequency do-

main/continuous wave clinical system for breast imaging,”

Med. Phys. 30, 235–247 (2003).

17. H. K. Kim, U. J. Netz, J. Beuthan, and A. H. Hielscher, “Opti-

mal source-modulation frequencies for transport-theory-

based optical tomography of small-tissue volumes,”Opt.

Express 16, 18082–18101 (2008).

18. N. Iftimia and H. Jiang, “Quantitative optical image recon-

struction of turbid media by use of direct-current measure-

ments,”Appl. Opt. 39, 5256–5261 (2000).

19. A. M. Siegel, J. J. A. Marota, and D. A. Boas, “Design and

evaluation of a continuous-wave diffuse optical tomography

system,”Opt. Express 4, 287–298 (1999).

20. Y. Xu, X. Gu, T. Khan, and H. Jiang, “Absorption and scatter-

ing images of heterogeneous scattering media can be simulta-

neously reconstructed by use of dc data,”Appl. Opt. 41,

5427–5437 (2002).

21. Z. Yuan and H. Jiang, “Image reconstruction scheme that com-

bines modified Newton method and efficient initial guess es-

timation for optical tomography of finger joints,”Appl. Opt.

46, 2757–2768 (2007).

22. H. Jiang, K. D. Paulsen, U. L. Osterbergy, and M. S. Patterson,

“Improved continuous light diffusion imaging in single- and

multi-target tissue-like phantoms,”Phys. Med. Biol. 43,

675–693 (1998).

23. Y. Pei, H. L. Graber, and R. L. Barbour, “Normalized-

constraint algorithm for minimizing inter-parameter cross-

talk in DC optical tomography,”Opt. Express 9,97–109 (2001).

24. Z. Jiang, D. Piao, G. Xu, J. W. Ritchey, G. R. Holyoak,

K. E. Bartels, C. F. Bunting, G. Slobodov, and J. S. Krasinski,

“Trans-rectal ultrasound-coupled near-infrared optical tomo-

graphy of the prostate Part II: Experimental demonstration,”

Opt. Express 16, 17505–17520 (2008).

25. A. H. Hielscher, A. D. Klose, A. K. Scheel, B. Moa-Anderson,

M. Backhaus, U. Netz, and J. Beuthan, “Sagittal laser optical

tomography for imaging of rheumatoid finger joints,”Phys.

Med. Biol. 49, 1147–1163 (2004).

26. K. K. Wang and T. C. Zhu, “Reconstruction of in-vivo optical

properties for human prostate using interstitial diffuse optical

tomography,”Opt. Express 17, 11665–11672 (2009).

27. M. Schweiger, I. Nissilä, D. A. Boas, and S. R. Arridge, “Image

reconstruction in optical tomography in the presence of cou-

pling errors,”Appl. Opt. 46, 2743–2756 (2007).

28. V. Ntziachristos, A. H. Hielscher, A. G. Yodh, and B. Chance,

“Diffuse optical tomography of highly heterogeneous media,”

IEEE Trans. Med. Imaging 20, 470–478 (2001).

29. S. Fantini, M. A. Franceschini, J. B. Fishkin, B. Barbieriand,

and E. Gratton, “Quantitative determination of the absorption

spectra of chromophores in strongly scattering media: a light-

emitting-diode based technique,”Appl. Opt. 33, 5204–5213

(1994).

30. D. Boas, T. Gaudette, and S. R. Arridge, “Simultaneous

imaging and optode calibration with diffuse optical tomogra-

phy,”Opt. Express 8, 263–270 (2001).

31. S. L. Jacques, “Reflectance spectroscopy with optical fiber de-

vices and transcutaneous bilirubinometers,”in Biomedical

Optical Instrumentation and Laser-Assisted Biotechnology,

A. M. Verga Scheggi, S. Martellucci, A. N. Chester, and R.

Pratesi, eds. (Kluwer Academic, 1996), pp. 83–94.

32. F. Fabbri, M. A. Franceschini, and S. Fantini, “Characteriza-

tion of spatial and temporal variations in the optical proper-

ties of tissuelike media with diffuse reflectance imaging,”

Appl. Opt. 42, 3063–3072 (2003).

33. The original derivation in [29] for σμ0

s=μ0

shas ρin the equation,

which is inconsistent with that obtained for σμa=μa. Equation

(19) corrected this inconsistency.

34. B. W. Pogue, S. Geimer, T. O. McBride, S. Jiang, U. L. Osterberg,

and K. D. Paulsen, “Three-dimensional simulation of near-

infrared diffusion in tissue: boundary condition and geometry

analysis for finite-element image reconstruction,”Appl. Opt.

40, 588–600 (2001).

35. G. Xu, D. Piao, C. H. Musgrove, C. F. Bunting, and H. Dehghani,

“Trans-rectal ultrasound-coupled near-infrared optical tomo-

graphy of the prostate, part I: simulation,”Opt. Express 16,

17484–17504 (2008).

36. H. Dehghani, B. W. Pogue, S. Jiang, B. Brooksby, and

K. D. Paulsen, “Three-dimensional optical tomography: reso-

lution in small-object imaging,”Appl. Opt. 42, 3117–3128

(2003).

37. H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis,

S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paul-

sen, “Near infrared optical tomography using NIRFAST: algo-

rithms for numerical model and image reconstruction

algorithms,”Commun. Numer. Meth. Eng. 25, 711–732 (2009).

38. P. K. Yalavarthy, H. Dehghani, B. W. Pogue, and K. D. Paulsen,

“Critical computational aspects of near infrared circular tomo-

graphic imaging: Analysis of measurement number, mesh

resolution and reconstruction basis,”Opt. Express 14,

6113–6127 (2006).

39. M. Huang and Q. Zhu, “A dual-mesh optical tomography

reconstruction method with depth correction using a priori

ultrasound information,”Appl. Opt. 43, 1654–1662 (2004).

40. S. R. Arridge and W. R. B. Lionheart, “Nonuniqueness in dif-

fusion-based optical tomography,”Opt. Lett. 23, 882–884

(1998).

41. B. Harrach, “On uniqueness in diffuse optical tomography,”

Inverse Probl. 25, 055010 (2009).

42. Z. Jiang, G. R. Holyoak, K. E. Bartels, J. W. Ritchey, G. Xu,

C. F. Bunting, G. Slobodov, and D. Piao, “In vivo trans-rectal

ultrasound coupled near-infrared optical tomography of a

transmissible venereal tumor model in the canine pelvic ca-

nal,”J. Biomed. Opt. 14, 030506 (2009).

3070 APPLIED OPTICS / Vol. 49, No. 16 / 1 June 2010

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