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IOP PUBLISHING

PHYSICS IN MEDICINE AND BIOLOGY

Phys. Med. Biol. 53 (2008) 837–859

doi:10.1088/0031-9155/53/4/002

Pharmacokinetic-rate images of indocyanine green for

breast tumors using near-infrared optical methods

Burak Alacam1, Birsen Yazici1,2, Xavier Intes2, Shoko Nioka3and

Britton Chance3

1Department of Electrical, Computer, and Systems Engineering, Rensselaer Polytechnic

Institute, Troy, NY, USA

2Department of Biomedical Engineering, Rensselaer Polytechnic Institute, Troy, NY, USA

3Department of Biophysics and Biochemistry, University of Pennsylvania, PA, USA

E-mail: yazici@ecse.rpi.edu

Received 12 October 2007, in final form 30 November 2007

Published 15 January 2008

Online at stacks.iop.org/PMB/53/837

Abstract

In this paper, we develop a method of forming pharmacokinetic-rate images

of indocyanine green (ICG) and apply our method to in vivo data obtained

from three patients with breast tumors. To form pharmacokinetic-rate images,

we first obtain a sequence of ICG concentration images using the differential

diffuse optical tomography technique. We next employ a two-compartment

model composed of plasma, and extracellular–extravascular space (EES), and

estimate the pharmacokinetic rates and concentrations in each compartment

using the extended Kalman filtering framework. The pharmacokinetic-rate

images of the three patient show that the rates from the tumor region and

outside the tumor region are statistically different.

concentrations in plasma, and the EES compartments are higher around the

tumor region agreeing with the hypothesis that around the tumor region ICG

may act as a diffusible extravascular flow in compromised capillary of cancer

vessels. Our study indicates that the pharmacokinetic-rate images may provide

superior information than single set of pharmacokinetic rates estimated from

the entire breast tissue for breast cancer diagnosis.

Additionally, the ICG

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Near-infrared (NIR) diffuse optical imaging offers several advantages over other imaging

modalities(Arridge1999,Boppartetal2004,Guetal2004,IntesandChance2005,Mahmood

et al 1999, Sevick-Muraca et al 1997, Yodh and Chance 1995). NIR techniques are minimally

invasive, easy to use, relatively inexpensive and can be made portable. Moreover, optical

0031-9155/08/040837+23$30.00 © 2008 Institute of Physics and Engineering in MedicinePrinted in the UK837

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838 B Alacam et al

techniques,whencoupledwithcontrastagents,havethepotentialtoprovidemolecular/cellular

level information, which can improve cancer detection, staging and treatment monitoring

(Alacam et al 2006, Cuccia et al 2003, Intes et al 2003, Mahmood et al 1999, Sevick-Muraca

et al 1997).

Among many commercially available optical contrast agents, only indocyanine green

(ICG) is approved for use in humans by the Food and Drug Administration (ElDeosky et al

1999, Hansenetal1993, Shinoharaetal1996). ICGisabloodpoolingagentthathasdifferent

delivery behavior between normal and cancer vasculature. In normal tissue, ICG acts as a

blood flow indicator in tight capillaries of normal vessels. However in tumors, ICG may act

as a diffusible (extravascular) flow in leaky capillary of vessels (Alacam et al 2006, Cuccia

et al 2003, Ntziachristos et al 2000, Vaupel et al 1991). Therefore, pharmacokinetics of ICG

has the potential to provide new tools for tumor detection, diagnosis and staging.

One approach to analyze pharmacokinetics of contrast agents is the compartmental

modeling (Anderson 1983, Jacquez 1972, Tornoe 2002).

compartmental modeling were reported to show the feasibility of ICG pharmacokinetics in

tumor characterization (Alacam et al 2006, Cuccia et al 2003, Intes et al 2003). Cuccia et al

(2003) presented a study of the dynamics of ICG in an adenocarcinoma rat tumor model using

a two-compartment model. Intes et al (2003) presented the uptake of ICG in breast tumors

using a continuous wave diffuse optical tomography apparatus using a two-compartment

model. We recently introduced the extended Kalman filtering (EKF) framework to model and

estimate the ICG pharmacokinetics and tested three different compartmental models for the

ICG pharmacokinetics using the in vivo NIR data collected from Fischer rats with cancerous

tumors (Alacam et al 2006). Our study suggests that the pharmacokinetic rates out of the

vasculature are higher in edematous tumors as compared to necrotic tumors.

In all the studies described above, the pharmacokinetic rates are assumed to be constant

over a tissue volume that may be as large as the entire imaging domain.

pharmacokinetic rates are expected to be different in healthy and tumor tissue as reported

in positron emission tomography (PET) and magnetic resonance imaging (MRI) literature.

It was shown that the spatially resolved pharmacokinetic-rate analysis provides increased

sensitivity and specificity for breast cancer diagnosis (Mussurakis et al 1997, Su et al

2005, Sun et al 2006). For example, Sun et al (2006) showed that FAU (1-2?-deoxy-2?-

fluoro- β-D-arabinfuranosyl urasil, a PET contrast agent) accumulation in tumor regions is

significantly higher when compared to normal breast tissue based on pharmacokinetic-rate

images. Mussurakis et al (1997) showed that the pharmacokinetics of gadolinium-DTPA (an

MRI contrast agent) can be used to differentiate between malignant and benign breast tumors

with a high accuracy. It has also been shown that the spatially resolved image interpretation

is superior to the isolated use of quantitative pharmacokinetic rates.

In the area of diffuse NIR spectroscopy and imaging, a number of studies on spatially

resolved pharmacokinetic rates has been reported (Gurfinkel et al 2000, Milstein et al 2005).

Gurfinkel et al (2000) presented in vivo NIR reflectance images of ICG pharmacokinetics to

discriminatecanineadenocarcinoma(locatedat0.5–1cmdepth)fromnormalmammarytissue.

These images were generated by a non-tomographic technique using a CCD camera that is

suitable only for imaging tumors close to surface. Milstein et al (2005) presented a Bayesian

tomographic image reconstruction method to form pharmacokinetic-rate images of optical

fluorophores based on fluorescence diffuse optical tomography. Numerical simulations show

that the method provides good contrast. However, no real data experiments were presented to

study the diagnostic value of spatially resolved pharmacokinetic rates.

In this paper, we present a method of forming pharmacokinetic-rate images and report

spatially resolved pharmacokinetic rates of ICG using in vivo NIR data acquired from three

A number of studies using

However,

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Pharmacokinetic-rate images of indocyanine green for breast tumors839

patients with breast tumors.

presenting the pharmacokinetic-rate images of an optical contrast agent using in vivo breast

data based on tomographic techniques. We first develop a set of spatio-temporally resolved

ICG concentration images based on differential diffuse optical tomography. We model the

ICG pharmacokinetics by a two-compartment model composed of plasma and extracellular–

extravascular space (EES) compartments. We then estimate the ICG pharmacokinetic rates

and the concentrations in different compartments based on the EKF framework (Alacam

et al 2006). We show that the pharmacokinetic rates from the tumor region and outside the

tumor region are statistically different. We also estimate a single set of pharmacokinetic rates

(bulk pharmacokinetic rates) for the entire breast tissue. Our study indicates that spatially

resolved pharmacokinetic rates provide more consistent and superior diagnostic information

as compared to the bulk pharmacokinetic rates.

The rest of the paper is organized as follows. In section 2, we present the reconstruction

of ICG concentration images. In section 3, we present modeling and estimation of ICG

pharmacokinetic-rate images using the EKF framework. In section 4, we present the spatially

resolved ICG pharmacokinetic-rate analysis of in vivo breast data. Section 5 summarizes our

results and conclusion.

To the best of our knowledge, our work is the first study

2. Reconstruction of bulk ICG concentration images

Inourdatacollectionprocess,asequenceofboundarymeasurementsarecollectedoveraperiod

of time. Each set of measurements are used to form a frame of the ICG concentration images.

The resulting sequence of ICG concentration images are then used to form pharmacokinetic-

rate images. To reconstruct each frame of the ICG concentration images, we follow a static

reconstruction approach and use differential diffuse optical tomography (DDOT) technique

(Intes et al 2003, Ntziachristos et al 1999).

In DDOT, two sets of excitation measurements are collected corresponding to before and

after the ICG injection, and the ICG concentration is determined by the perturbation method

(Intes et al 2003, Ntziachristos et al 1999). The photon propagation before and after the

injection is modeled by the following diffusion equations:

∇ · Dx(r)∇?±

with Robin-type boundary conditions:

2Dx(r)∂?±

∂ν

where x stands for the excitation, c is the speed of light inside the medium ?; ω denotes the

modulation frequency of the source, µ−

and after the ICG injection, Dxis the diffusion coefficient which is assumed independent of

µ±

and after the ICG injection. Here, ν denotes the outward normal to the boundary ∂? of ?, ρ

is a constant representing the refractive index mismatch between the two regions separated by

∂? and S(r,ω) is the excitation source on the boundary.

The absorption coefficients after the injection µ+

coefficient of the medium before the ICG injection µ−

ICG ?µax(r):

?µax(r) = µ+

Intheforwardmodel,theanalyticalsolutionsoftheheterogonousdiffusionequationgiven

in (1) is derived using first-order Rytov approximation (Intes et al 2003). The sample volume

x(r,ω) −?µ±

ax(r) + jω/c??±

x(r,ω) = 0,r ∈ ? ⊂ R3,

(1)

x(r,ω)

+ ρ?±

x(r,ω) = −S(r,ω),r ∈ ∂?,

(2)

ax(r) and µ+

ax(r) are the absorption coefficients before

ax, known but not necessarily constant, ?±

x(r,ω) denotes optical field at location r before

axare modeled as a sum of the absorption

axand the perturbation caused by the

ax(r) − µ−

ax(r),r ∈ ? ⊂ R3.

(3)

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is divided into a set of voxels and the measurements are related to the relative absorption

coefficients of each voxel by a system of linear equations. The shape of the breast was

approximated as a cylinder and the Kirchhoff approximation (Ripoll et al 2001a, 2001b) for

diffuse waves was used to model the interaction of light with boundaries. In order to minimize

optode-tissue coupling mismatch due to breathing motion, the forward model was augmented

with the coupling coefficient technique as described in Boas et al (2001).

Here, the Rytov-type measurements, which are defined by the natural logarithm of the

ratio of the post-ICG measurements to the pre-ICG measurements were used (Ntziachristos

et al 1999). Let ?x(ω;rd;rs) denote the Rytov-type measurements at location rd due to

source at rs. The linearized relationship between the differential absorption coefficient and

measurements is given by O’Leary (1996)

?

where ?−

injection, Ix(r) = c?µax(r)/Dx, and G−

at rsbefore the injection.

We address the inverse problem of recovering ?µaxfrom Rytov measurements ?xbased

on the forward model (4) using the singular-value decomposition of the Moore–Penrose

generalized system. We use a zeroth-order Tikhonov regularization to stabilize the inversion

procedure. The regularization parameter was determined by L-curve analysis (Hansen and

O’Leary 1993) using the data obtained from a phantom study previously employed to validate

theapparatus(Intesetal2003). Theoptimalregularizationparameterwasfoundtobe6×10−4

and set to be the same for all patient images and time instances. A detailed discussion of the

forward and inverse models used for the reconstruction of differential absorption coefficients

(?µax) can be found in Intes et al (2003).

To construct a set of ICG concentration images, we use the linear relationship between

the differential absorption coefficients and ICG concentrations (Landsman et al 1976),

?x(ω;rd;rs) = −

1

?−

x(ω;rd;rs)

?

G−

x(r,ω;rd)Ix(r)?−

x(r,ω;rs)d3r,

(4)

x(r,ω;rs) is the photon density obtained at the excitation wavelength before ICG

x(r,ω;rd) is the Green’s function of (1) for a source

?µa(r) = ln10?λm(r),

(5)

where ?λis the extinction coefficient of ICG at the wavelength 805 nm, m(r) is the bulk ICG

concentration in the tissue and ?µa(r) is as defined in (3).

Note that the method described here is applicable for frequency domain case but for

simplicity we set the frequency to zero, i.e. ω = 0.

3. Modeling and estimation of ICG pharmacokinetics

3.1. Two-compartment model of ICG pharmacokinetics

Using the method outlined in section 2, we reconstruct a sequence of ICG concentration

images. As an example, figures 1–3 show a set of images reconstructed from in vivo breast

data.

OurobjectiveistomodelthepharmacokineticsofICGateachvoxelofICGconcentration

images using compartmental modeling. To do so, we first extracted the time varying ICG

concentration curves for each voxel from the sequence of ICG concentration images. An

example of such a curve is shown in figure 4. We next fit a two-compartment model to each

ICG concentration curve (Alacam et al 2006, Gurfinkel et al 2000).

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ICG concentration

µM

t1

t ces 4 4 =

2

t ces 8 8 =

3 = 132 sec

t4 = 176 sec

t5 = 220 sec

t6 = 264 sec

t8 = 352 sec

t9 = 396 sec

t7 = 308 sec

Figure 1. ICG concentration images for a set of time instants for case 1.

Using the two-compartment model introduced by Alacam et al (2006), ICG transition

between plasma and extracellular–extravascular space (EES) can be modeled as follows:

?˙Ce(t)

where Cp(t) and Ce(t) represent the ICG concentrations in plasma and EES at time t,

respectively.The rates kin,kout and kelm have a unit of sec−1.

permeability surface area products given by PSγ, where P is the capillary permeability

constant, S is the capillary surface area and γ is the tissue density. kin and kout govern

the leakage into and the drainage out of the EES. The parameter kelm describes the ICG

elimination from the body through kidneys and liver. Here, ω(t) is uncorrelated zero-mean

Gaussian process with covariance matrix Q representing the model mismatch.

The actual total ICG concentration in the tissue is a linear combination of plasma and the

EES ICG concentrations, and modeled as

m(t) =?ve

where m(t),Ce(t) and Cp(t) are defined in (5) and (6), vpand veare plasma and the EES

volume fractions, respectively, and η(t) is uncorrelated zero-mean Gaussian process with

covariance matrix R, representing the measurement noise.

˙Cp(t)

?

=

?−kout

kin

kout

−(kin+ kelm)

??Ce(t)

Cp(t)

?

+ ω(t),t ∈ [T0,T1],

(6)

They are defined as the

vp

??Ce(t)

Cp(t)

?

+ η(t),t ∈ [T0,T1],

(7)

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t1 = 44 sec

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t9 = 396 sec

X [cm]

X [cm]

Y [cm]

Y [cm]

ICG concentration

µM

ICG concentration

µM

ICG concentration

µM

Figure 2. ICG concentration images for a set of time instants for case 2.

3.2. Estimation of ICG pharmacokinetics using extended Kalman filtering

In matrix–vector notation, (6) and (7) can be expressed as

˙C(t) = K(α)C(t) + ω(t),

where C(t) denotes the concentration vector with elements Ce(t), and Cp(t); K(α) is the

system matrix, V(α) is the measurement matrix as defined in equation (7) and α is the

parameter vector given by

α = [kout kin kelm ve vp]T.

The ICG measurements in (8) are collected at discrete time instances, t = kT,k =

0,1,..., where T is the sampling period. Therefore, the continuous model described in (8) is

discretized. We can express the discrete compartmental model as follows:

Cd(k + 1) = Kd(θ)Cd(k) + ωd(k),

where Kd(θ) = eK(α)is the discrete time system matrix; Vd(θ) = V(α) is the discrete

measurement matrix; ωd(k) and ηd(k) are zero-mean Gaussian white noise processes with

covariances matrix Qdand variance Rd, respectively. The vector θ is composed of parameters

τijwhich are functions the pharmacokinetic rates and volume fractions:

θ = [τ11

m(t) = V(α)C(t) + η(t),

(8)

(9)

m(k) = Vd(θ)Cd(k) + ηd(k),

(10)

τ12

τ21

τ22

ve

vp]T.

(11)

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t1 = 44 sec

t2 = 88 sec t3 = 132 sec

t4 = 176 sec

t5 = 220 sec

t6 = 264 sec

t7 = 308sec

t8 = 352 sec

t9 = 396 sec

Figure 3. ICG concentration images for a set of time instants for case 3.

0100200300 400 500600

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time (sec)

ICG Concentration (µ M)

Case 1

Case 2

Case 3

Figure 4. Time course of ICG concentration curves for a specific voxel, 65th, 276th, 188th voxel

for cases 1, 2 and 3, respectively.

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Figure 5. (Left) Schematic diagram. (Right) The cut section of the CW NIR imaging apparatus

with 16 sources and detectors.

We first estimate τij, i,j = 1,2 and then compute the pharmacokinetic rates kin,koutand kelm

(Alacam et al 2006, Chen 1999). The explicit form of the discrete state-space model is given

as follows:

?Ce(k + 1)

?Ce(k)

We estimate the parameter vector θ and concentration vector Cd by using the EKF

framework.The EKF is a recursive modeling and estimation method with numerous

advantages in ICG pharmacokinetic modeling (Alacam et al 2006). These include effective

modeling of multiple compartments, and multiple measurement systems in the presence of

measurement noise and uncertainties in the compartmental model dynamics, simultaneous

estimation of model parameters and ICG concentrations in each compartment, statistical

validationofestimatedconcentrationsanderrorboundsonthemodelparameterestimates, and

incorporation of available a priori information about the initial conditions of the permeability

rates into the estimation procedure.

When both states (ICG concentrations) and model parameters (pharmacokinetic rates

and volume fractions) are unknown, a linear state-space model can be regarded as a nonlinear

model; thelinear systemparameters and states combine toformthenew states of thenonlinear

model. This system is then linearized and the new unknown states are found using the EKF

estimator (Alacam et al 2006, Ljung 1979, Togneri and Deng 2003, Nelson and Stear 1976).

In EKF framework, θ can be treated as a random process with the following model:

Cp(k + 1)

?

=

?τ11

τ12

τ22

τ21

??Ce(k)

?

Cp(k)

?

+ ωd(k)

(12)

m(k) = [ve

vp]

Cp(k)

+ ηd(k).

θ(k + 1) = θ(k) + ςd(k),

(13)

where ςd(k) is a zero-mean Gaussian process with covariance matrix Sd.

Table 1 summarizes the joint estimation of pharmacokinetic rates and ICG concentration

in different compartments. In table 1,ˆCd(k|k − 1) is the state estimate propagation at step

k given all the measurements up to step k − 1,ˆCd(k) is the state estimate update at step

k,Pk,k−1denotes the error covariance propagation at step k given all the measurements up to

step k−1,Pk,kis the error covariance update at step k,Sdis the preassigned covariance matrix

of ςd(k),Jkis the Jacobian matrix due to iterative linearization of the state equation at step

k,Gkis the recursive Kalman gain at step k,Rdis the covariance matrix of the measurements,

Qdis the covariance matrix of the concentration vector and I is the identity matrix. A detailed

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Figure 6. Pharmacokinetic rate images, (a) kinand (b) koutfor case 1. The kinimages are shown

with approximate tumor location and size.

Table 1. EKF algorithm for simultaneous estimation of states and parameters.

?ˆCd(0)

ˆθ(0)

?ˆCd(k|k − 1)

ˆθ(k|k − 1)

Initial conditions

?

=

?E(Cd(0))

?

ˆθ(0)

?Kd(ˆθ(k − 1))ˆCd(k − 1)

ˆθ(k − 1)

?

,P0,0=

?Var(Cd(0))

0

0

Sd

?

State estimate propagation

=

?

Error covariance propagation

Pk,k−1= Jk−1Pk−1,k−1JT

?ˆCd(k)

ˆθ(k)

+Gk(m(k) − Vd(ˆθ(k|k − 1))ˆCd(k|k − 1))

Pk,k= [I − Gk?k|k−1]Pk,k−1

Gk= Pk,k−1?T

Jk=

0

?Vd(ˆθ(k|k − 1))

0

k−1+

?

?Qd

0

0

Sd

?

State estimate update

?

=

?ˆCd(k|k − 1)

ˆθ(k|k − 1)

Error covariance update

Kalman gain

k|k−1[?k|k−1Pk,k−1?T

?Kd(ˆθ(k))

k|k−1+ Rd]−1

?

Definitions

∂

∂θ[Kd(ˆθ(k))ˆCd(k)]

I

?T

?k|k−1=

discussion of the extended Kalman filtering algorithm, and the initialization of the parameters,

concentrations and covariance matrices can be found in Alacam et al (2006).

4. Spatially resolved ICG pharmacokinetic rate analysis of in vivo breast data

4.1. Apparatus

In this work, we use the data collected with a continuous wave (CW) NIR imaging apparatus.

The apparatus has 16 light sources, which are tungsten bulbs with less than 1 W of output

power. They are located on a circular holder at an equal distance from each other with 22.5◦

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Figure 7. Pharmacokinetic-rate images, (a) kinand (b) koutfor case 2. The kinimages are shown

with approximate tumor location and size.

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Figure 8. Pharmacokinetic-rate images, (a) kinand (b) koutfor case 3. The kinimages are shown

with approximate tumor location and size.

apart. Sixteen detectors, namely silicon photodiodes, are situated in the same plane. The

breast is arranged in a pendular geometry with the source-detector probes gently touching its

surface. Figure 5 illustrates the configuration of the apparatus and the configuration of the

detectors and the sources in a circular plane. Note that sources and detectors are co-located.

The detectors use the same positions as the sources to collect the light originating from one

source at a time. Only the signals from the farthest 11 detectors are used in the analysis.

For example, when source 1 is on, the data are collected using detectors 4–14. This provides

sufficient number of source-detector readings (176 readings) to reconstruct ?µaimages at

each time instant. A band pass filter at 805 nm, the absorption peak of ICG, is placed in front

of the sources to select the desired wavelength. A set of data for one source is collected every

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Figure 9.

(c) 422.4th s.

ICG concentration images in plasma for case 1 for (a) 246.4th, (b) 334.4th and

∼500 ms. The total time for a whole scan of the breast including 16 sources and 16 detectors

is ∼8.8 s. A more detailed explanation of the apparatus and the data collection procedure can

be found in (Nioka et al 1997).

4.2. Tumor information and protocol

Three different patients with different tumor types are included in this study. Measurements

are made before the biopsy to avoid modification of the blood volume and flow in the tumor

region. First case (case 1) is fibroadenoma, which corresponds to a mass estimated to be

1–2 cm in diameter within a breast of 9 cm diameter located at 6–7 o’clock. Second case

(case 2) is adenocarcinoma corresponding to a tumor estimated to be 2–3 cm in diameter

within a breast of 7.7 cm diameter located at 4–5 o’clock. The third case (case 3) is invasive

ductal carcinoma, which corresponds to a mass estimated to be 4 cm by 3 cm located at

6 o’clock. Table 2 describes the tumor information for each patient. A priori information on

the location and size of the tumor was obtained by palpation and the diagnostic information

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Figure 10. ICG concentration images in the EES for case 1 for (a) 246.4th, (b) 334.4th and

(c) 422.4th s.

Table 2. Tumor information for each patient.

Tumor type Tumor size Tumor location

Case 1

Case 2

Case 3

Fibroadenoma

Adenocarcinoma

Invasive ductal carcinoma

1–2 cm

2–3 cm

4 cm by 3 cm

6–7 o‘clock

4–5 o‘clock

6 o‘clock

was derived a posteriori from biopsy and surgery. ICG is injected intravenously by bolus with

a concentration of 0.25 mg kg−1of body weight. Data acquisition started before the injection

of ICG and continued for 10 min.

4.3. Results and discussion

UsingtheCWimagerdescribedabove, source–detectorreadingswerecollectedfromdifferent

angles for each patient. Differential absorption coefficient images were reconstructed based

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Figure 11. ICG concentration images in plasma for case 2 for (a) 228.8th, (b) 316.8th and

(c) 404.8th s.

on DDOT forward model given in equations (1)–(4) with ω set to zero. Using the linear

relationship (5) between ICG concentration and absorption coefficient, ICG concentration

images were obtained for each case. A sample set of ICG concentration images for the

selected time instants are shown in figures 1–3 for cases 1–3, respectively. Although only nine

images are displayed, there are approximately 50 images for each case, each corresponding

to a different time instant. Each image is composed of 649 voxels. Note that the ICG

concentration images in figures 1–3 represent the bulk ICG concentrations in the tissue, not

the ICG concentrations in plasma or the EES compartments.

We next extracted the time course of ICG concentration for each voxel. As an example,

figure 4 shows the time course of ICG concentrations for all three cases for a specific voxel

in the tumor region (65th, 276th, 188th voxel for cases 1, 2 and 3, respectively). We then

fit the two-compartment model to each time course data using the EKF framework; and

estimated kin,kout,kelm, and the ICG concentrations in plasma and the EES. We chose initial

values within the biological limits that lead to minimum norm error covariance matrix. The

images of kinand koutfor each case are shown in figures 6(a)–(b), and 7(a)–(b), 8(a)–(b),

respectively. Additionally, we constructed the ICG concentration images for plasma and

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Figure 12. ICG concentration images in the EES for case 2 for (a) 228.8th, (b) 316.8th and

(c) 404.8th s.

the EES compartments. Figures 9–14 show the ICG concentration in plasma and the EES

for three different time instants for cases 1, 2 and 3, respectively. Our results show that

the pharmacokinetic rates are higher around the tumor region agreeing with the fact that

permeability increases around the tumor region due to compromised capillaries of tumor

vessels. We also observed that ICG concentrations in plasma and the EES compartments are

higher around the tumors agreeing with the hypothesis that around the tumor region ICG may

act as a diffusible extravascular flow in leaky capillary of tumor vessels.

Using the a priori and a posteriori information on the location, and the size of the tumors,

we plotted an ellipse (or a circle) to identify the approximate location and size of the tumor

in the pharmacokinetic-rate images. We note that the radii of the ellipses were chosen large

enough to include the tumor boundaries. Figures 6(a), 7(a) and 8(a) present the kinimages

with approximate tumor location and size for cases 1, 2 and 3, respectively. The consistency

of the bright regions in the kinimages, and circular/elliptical regions drawn based on the a

priori and a posteriori information shows that the pharmacokinetic-rate images may provide

good localization of tumors.

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Figure 13. ICG concentration images in plasma for case 3 for (a) 246.4th, (b) 378.4th and

(c) 510.4th s.

Table 3. Mean and standard deviation of pharmacokinetic rates for the tumor region and outside

the tumor region.

kin(sec−110−2)kout(sec−110−2)kelm(sec−110−3)

InsideOutsideInsideOutside InsideOutside

Case 1

Case 2

Case 3

2.14 ± 0.018

2.92 ± 0.076

6.87 ± 0.093

0.73 ± 0.011

1.14 ± 0.052

3.06 ± 0.015

1.24 ± 0.069

1.58 ± 0.051

4.96 ± 0.048

0.43 ± 0.013

0.65 ± 0.036

1.66 ± 0.072

4.11 ± 0.057

3.94 ± 0.081

4.49 ± 0.056

3.87 ± 0.012

4.12 ± 0.047

4.46 ± 0.081

The histograms of kin and kout images for the tumor region (as indicated by

circular/elliptical regions) and outside the tumor region are shown in figures 15(a)–(c) and

figures 16(a)–(c), respectively. Note that all nonzero voxels outside the elliptical region

constitute ‘outside the tumor region’. The solid curves in figures 15 and 16 show the Gaussian

fit. The histograms and their Gaussian fits in figures 15 and 16 show that the mean and the

standarddeviationofkinandkoutvaluesaredifferentforthetumorandoutsidethetumorregion.

Table 3 tabulates the mean values (± spatial standard deviation) of the pharmacokinetic rates