Content uploaded by Hessel Wijkstra

Author content

All content in this area was uploaded by Hessel Wijkstra

Content may be subject to copyright.

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 30, NO. 8, AUGUST 2011 1493

Contrast-Ultrasound Diffusion Imaging

for Localization of Prostate Cancer

Maarten P. J. Kuenen*, Massimo Mischi, and Hessel Wijkstra

Abstract—Prostate cancer is the most prevalent form of cancer

in western men. An accurate early localization of prostate cancer,

permitting efﬁcient use of modern focal therapies, is currently

hampered by a lack of imaging methods. Several methods have

aimed at detecting microvascular changes associated with prostate

cancer with limited success by quantitative imaging of blood

perfusion. Differently, we propose contrast-ultrasound diffu-

sion imaging, based on the hypothesis that the complexity of

microvascular changes is better reﬂected by diffusion than by

perfusion characteristics. Quantiﬁcation of local, intravascular

diffusion is performed after transrectal ultrasound imaging of an

intravenously injected ultrasound contrast agent bolus. Indicator

dilution curves are measured with the ultrasound scanner reso-

lution and ﬁtted by a modiﬁed local density random walk model,

which, being a solution of the convective diffusion equation,

enables the estimation of a local, diffusion-related parameter.

Diffusion parametric images obtained from ﬁve datasets of four

patients were compared with histology data on a pixel basis. The

resulting receiver operating characteristic (curve area )

was superior to that of any perfusion-related parameter proposed

in the literature. Contrast-ultrasound diffusion imaging seems

thereforetobeapromi

sing method for prostate cancer localiza-

tion, encouraging further research to assess the clinical reliability.

Index Terms—Biomedical imaging, blood vessels, cancer, pa-

rameter estimation, ultrasonography.

I. INTRODUCTION

PROSTATE cancer is the most prevalent form of cancer

in men in western countries. It accounts for 25% and

10% of all cancer diagnoses and deaths, respectively [1], [2].

Nowadays, a variety of focal therapies such as cryoablation,

brachytherapy, and high-intensity focused ultrasound, are avail-

able to efﬁciently treat early detected and localized prostate

cancer [3]. This may prevent a radical treatment as for example

radical prostatectomy, with the associated risks of the patient

Manuscript received January 12, 2011; revised February 24, 2011; accepted

February 27, 2011. Date of publication March 10, 2011; date of current ver-

sion August 03, 2011. This work was supported by the Dutch Organization for

Scientiﬁc Research (NWO) and the Technology Foundation (STW). Asterisk

indicates corresponding author.

*M. P. J. Kuenen is with the Department of Electrical Engineering, Eind-

hoven University of Technology, 5600 MB Eindhoven, The Netherlands and

also with the Department of Urology, AMC University Hospital, 1100 DD Am-

sterdam, The Netherlands (e-mail: m.p.j.kuenen@tue.nl).

M. Mischi is with the Department of Electrical Engineering, Eindhoven Uni-

versity of Technology, 5600 MB Eindhoven, The Netherlands.

H. Wijkstra is with the Department of Urology, AMC University Hospital,

1100 DD Amsterdam, The Netherlands, and also with the Department of Elec-

trical Engineering, Eindhoven University of Technology, 5600 MB Eindhoven,

The Netherlands.

Color versions of one or more of the ﬁgures in this paper are available online

at http://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/TMI.2011.2125981

becoming incontinent or impotent [4]. However, the limited

reliability of the available noninvasive diagnostic methods

hampers an efﬁcient use of focal therapies.

The main noninvasive diagnostic method, assessingthe

serum prostate-speciﬁc antigen (PSA) levelinblood,hasa

high false-positive rate (about 76%) [5]. Therefore, PSA does

not enable an efﬁcient mass screening [5] andisonlyusedfor

patient stratiﬁcation prior to biopsy investigation [6]. This inva-

sive and painful investigation commonly involves taking 6–12

spatially distributed samples of the prostatewithacoreneedle.

Although the biopsy investigation is currently the most reliable

diagnostic method, it is often repeated to achieve sufﬁcient

sensitivity [5], [6]. The limited cancer localization is another

drawback. Furthermore, a considerable fraction of all detected

carcinomas will not develop into a life-threatening disease [4].

Therefore, the risk of overdiagnosis and overtreatment, with a

related loss in quality of life, represents a major issue in current

prostate cancer care [4],[5].

These problems motivate the search for better noninvasive

methods for an early detection and localization of life-threat-

ening forms of prostate cancer. In particular, imaging methods

may reduce the number of biopsies by accurate targeting and

permit an efﬁcient application of focal treatments.

Several imaging modalities are being evaluated for early

prostate cancer detection. While computerized tomography

(CT) seems unsuitable for diagnostic prostate imaging [6],

cancer detection sensitivity (on a patient basis) with mag-

netic resonance imaging (MRI) techniques such as diffusion

weighted imaging (73%–89%), contrast-enhanced MRI

(69%–95%), and MR spectroscopy (59%–94%) is promising

[7]. Transrectal ultrasound (TRUS) techniques are however

equally promising, and they are more suitable than MRI in

terms of cost, time, resolution, and guidance of biopsies and

focal therapies [6], [8]. Therefore, TRUS improvements could

be of great value for early prostate cancer localization.

For imaging purposes, a key indicator for prostate cancer is

angiogenesis, i.e., the formation of a dense microvascular net-

work characterized by an increased microvessel density (MVD)

[9]–[12]. Angiogenesis, which is required for cancer growth

beyond 1 mm,correlates with prostate cancer aggressiveness

(i.e., risks of extracapsular growth and development of metas-

tases) [10]–[12]. Therefore, imaging methods based on angio-

genesis detection may help to identify life-threatening forms of

prostate cancer at an early stage.

Hypothesizing a correlation between MVD and perfusion,

i.e., blood ﬂow per tissue volume, the use of ultrasound contrast

agents (UCAs) for quantitative TRUS imaging of microvascular

perfusion has gained interest [6], [13], [14]. UCAs are dis-

persions of coated gas microbubbles that backscatter acoustic

0278-0062/$26.00 © 2011 IEEE

1494 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 30, NO. 8, AUGUST 2011

energy when hit by ultrasound waves [15], [16]. Despite an

improvement in biopsy targeting, contrast-enhanced TRUS

methods based on intermittent imaging and destruction-re-

plenishment techniques have however not proven sufﬁciently

reliable to replace systematic biopsies [17]–[19].

An alternative method involves dynamic TRUS imaging of

the passage of an intravenously injected UCA bolus [6], [8],

[14]. Up until now, only few quantitative studies have been car-

ried out. These studies quantify perfusion by extraction of time

and intensity features from the measured acoustic time-inten-

sity curves [20]–[23]. However, time features do not represent

the local hemodynamic characteristics since they generally de-

pend on the entire bolus history [24], whereas intensity features

are affected by scanner settings and nonlinear ultrasound prop-

agation [25].

The reasons that the developments in quantitative perfusion

imaginghavenotresultedinreliable prostate cancer localization

may be various. In addition to limitations in the ﬂow sensitivity,

important reasons may be linked to the complex and contradic-

tory effects of angiogenesis on perfusion [13], [22], [26], [27].

A lack of vasomotor control and the presence of arteriovenous

shunts cause a low ﬂow resistance [13], [27], but this can be

counterbalanced by the small microvessel diameter and an in-

creased interstitial pressure, due to extravascular leakage [13],

[27]. MVD characterization by quantiﬁcation of perfusion may

therefore be unreliable.

In this paper we propose contrast-ultrasound diffusion

imaging (CUDI) as an alternative noninvasive prostate cancer

localization method. CUDI is based on the hypothesis that

angiogenesis-induced changes in the microvascular architec-

ture correlate better with diffusion than with perfusion. In this

context, diffusion refers to the intravascular UCA spreading by

apparent diffusion, due to concentration gradient and ﬂow pro-

ﬁle, and by convective dispersion, due to multipath trajectories

through the microvasculature [28]–[30]. The microvascular

architecture of a solid tumor can be viewed as a distributed

network [31], in which ﬂow can be modeled as ﬂow through

a porous medium [32]. Structural characteristics of porous

media determine the diffusion [29], [30]. Therefore, we hy-

pothesize intravascular UCA diffusion to be correlated with the

microvascular structure and, therefore, with angiogenesis.

Based on the UCA bolus injection technique, CUDI is a new

method for quantiﬁcation of diffusion from time-density curves

(TDCs). These curves measure the image gray level versus time

at all pixels covering the prostate. By modeling the local in-

travascular UCA transport by the local density random walk

model, we provide a novel, theoretical framework to extract a

local, diffusion-related parameter from measured TDCs. CUDI

was evaluated in vivo by comparing ﬁve diffusion parametric

images from four patients, obtained by TDC ﬁtting at each pixel,

with histology data.

II. METHODOLOGY

A. Data Acquisition

The data acquisition was performed at the AMC University

Hospital (Amsterdam, The Netherlands), after approval was

Fig. 1. TRUS power modulation imaging of the prostate after intravenous in-

jection of a UCA bolus. The displayed frames are recorded before UCA appear-

ance (top left), at initial wash-in (top right), at peak concentration (bottom left),

and at wash-out (bottom right).

granted by the local ethics committee. Written informed con-

sent was obtained from all patients prior to their participating

in this study.

A 2.4 mL SonoVue (Bracco, Milan, Italy) UCA bolus was

injected intravenously in the patient arm. SonoVue is a dis-

persion of microbubbles coated by a phospholipid shell,

whose mean diameter is 2.5 [33]. TRUS imaging was per-

formed using an iU22 ultrasound scanner (Philips Healthcare,

Bothell, WA) equipped with a C8-4v probe. The adopted con-

trast-speciﬁc imaging mode was power modulation, at a fre-

quency of 3.5 MHz. The effective pulse length of two cycles

provided an axial resolution of 0.43 mm, while a low mechan-

ical index (MI) of 0.06 minimized microbubble disruption [15],

[16]. The compression was set to C38 and the gain was adjusted

to prevent truncation or saturation of the 8-bit gray level. All ac-

quired B-mode videos were stored in DICOM (Digital Imaging

and Communication in Medicine) format, which can be directly

input to the analysis software that we implemented in Matlab

(The MathWorks, Natick, MA). Four B-mode frames recorded

in power modulation mode are shown in Fig. 1.

B. Calibration

An accurate quantiﬁcation of the UCA diffusion dynamics

based on TRUS B-mode video data requires knowledge about

how the measured gray level relates to the UCA concentra-

tion. To this end, we investigated the relation between UCA

concentration and the backscattered acoustic intensity .We

also studied the measurement and conversion of into a gray

level .

For low concentrations and MI, a linear relation between

and has been reported [34]. We performed new measurements

at the Catharina hospital (Eindhoven, The Netherlands) to verify

this relationship for the current equipment, settings, and UCA.

The setup was similar to the static calibration setup reported

in [34]. The UCA dispersions were contained in polyurethane

bags that were submerged into a water-ﬁlled basin. The basin

walls were covered by acoustic absorbers to minimize the

KUENEN et al.: CONTRAST-ULTRASOUND DIFFUSION IMAGING FOR LOCALIZATION OF PROSTATE CANCER 1495

Fig. 2. In vitro measurement results. The gray error bars depict the acoustic

intensity measured in a ﬁxed ROI by their mean and standard deviation, whereas

the black line shows the linear approximation for SonoVue concentrations up to

1.0 mg/L.

acoustic reﬂections from the wall. To reproduce the clinical

conditions, the ultrasound probe was positioned about 1 cm

away from the UCA dispersion. For each concentration, three

measurements were performed, from three different SonoVue

vials. The mean acoustic intensity was evaluated in a ﬁxed

region of interest (ROI) of the recorded B-mode images, with

QLAB (Philips Healthcare) acoustic quantiﬁcation software.

The results are shown in Fig. 2. For SonoVue concentrations

up to 1.0 mg/L, and are linearly related ( )as

(1)

where deﬁnes the sensitivity and is the background inten-

sity due to backscatter from tissue and blood. The exact pa-

rameter values are not relevant for this study, since a linearly

related measure of is sufﬁcient for a complete description

of the UCA diffusion dynamics. With the injected dose, the in

vivo measurements are performed within the linear calibration

range that was estimated in vitro ( Fig. 2). In fact, by consid-

ering a simple system of two mixing chambers representing the

right and left ventricles (100 and 110 mL), the concentration

would not overtake 0.84% of the injected concentration (5.0

g/mL) [35]. Taking into account a blood volume fraction in the

prostate of 2% [36], the concentration in the prostate would re-

main below 0.84 mg/L, i.e., within the linear range.

The ultrasound transducer converts the backscattered ultra-

sound waves into an electrical voltage, proportional to .After

ampliﬁcation and demodulation, a compression of the signal dy-

namic range yields the quantization level ,andthe

gray mapping renders the displayed gray level [37].

This mapping is displayed on the B-mode image and can be

easily extracted and compensated for. The compression func-

tion, typically a logarithmic-like function [34], [37], is estimated

by comparing QLAB acoustic quantiﬁcation results to the quan-

tization level in single-pixel ROIs. The results

show a linear relationship ( ) between and

( Fig. 3). This implies a logarithmic compression function

,inwhich is determined by the dy-

namic range of the compression function as

(2)

Fig. 3. Quantization level versus QLAB normalized acoustic intensity ,

obtained from single-pixel ROIs. The black line represents the ﬁtted logarithmic

compression function.

For the estimated dynamic range (45.73 dB), equals 24.22.

This dynamic range is sufﬁciently large to enable an accurate

TDC quantiﬁcation [37].

Combining all relations, the function that maps UCA concen-

tration to gray level can be written as

(3)

In (3), the baseline equals the quantization level

for , i.e., before the UCA appearance in the prostate.

C. Diffusion Modeling

Physical modeling of the intravascular UCA transport is re-

quired to analyze diffusion. Our analysis is based on the local

density random walk (LDRW) model [38]–[40]. This model can

provide a physical interpretation of the diffusion process, and it

accurately ﬁts UCA indicator dilution curves (IDCs) [23], [34].

IDCs measure the UCA concentration in a ﬁxed sample volume

as function of time and can thus be obtained from TDCs via (3).

After a general introduction to the LDRW model, the local as-

pects of the diffusion process by this model are discussed.

The LDRW model characterizes the UCA transport in a

straight, inﬁnitely long tube of constant section ,inwhich

a carrier ﬂuid ﬂows with a constant velocity ,asshownin

Fig. 4. The model assumes a Brownian motion of microbub-

bles. The concentration dynamics is then given by the

mono-dimensional convective diffusion equation as

(4)

in which and represent the distance along the tube’s main axis

and the time variable, respectively. The diffusion coefﬁcient

is assumed constant. The boundary conditions, representing a

rapid bolus injection and the UCA mass conservation, are given

as

(5a)

(5b)

In (5), is the total injected UCA mass dose, and and

are the bolus injection time and site, respectively. The solution

is a normal distribution in space that translates at the

1496 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 30, NO. 8, AUGUST 2011

Fig. 4. UCA concentration dynamics by the LDRW model in an inﬁnitely-long

straight tube, with the lower curves describing the UCA concentration proﬁle

in space for increasing time.

carrier velocity and has a variance that increases linearly with

time, as shown in Fig. 4. The LDRW formulation for the IDC

is obtained by sampling at an arbitrary detection

site ()[39]–[41]

(6)

The parameters ,and are deﬁned as

(7)

where is the distance between the injection and

detection sites. The parameter equals the IDC integral, and

is the mean transit time (MTT), i.e., the average time a mi-

crobubble takes to travel the distance [39], [41]. The param-

eter is proportional to the Péclet number, which equals the

ratio of the diffusive time and the convective time [41].

In the appendix, we provide an analytical relation between and

the statistical skewness of the IDC.

By its relation to , the parameter is interesting for the char-

acterization of diffusion. However, also depends on the length

, which cannot be measured in our clinical application. As

a consequence, does not characterize diffusion locally. This

would require the deﬁnition of a local diffusion-related param-

eter that is independent of .

To describe local aspects of the UCA diffusion process, we

consider a short segment of the inﬁnitely-long tube without

making assumptions about the bolus injection. The boundary

condition (5a) representing the bolus injection is replaced by a

local boundary condition describing the spatial UCA concen-

tration proﬁle at time , just before the bolus passage at the

detection site . In line with the LDRW model, we assume a

normally distributed initial spatial concentration proﬁle given as

(8)

Fig. 5. Assumed UCA concentration proﬁle in space at , i.e., just before

the bolus passage at the detection site .

with mean and variance . By adopting

the boundary condition of (8) instead of (5a), we can obtain an

analytical solution for if we assume locally con-

stant hemodynamic parameters, i.e., and

for . This interval covers the tube segment

containing the bulk of the UCA bolus at (see Fig. 5). For

,and are not relevant and may have any value.

In particular, if for , the bolus injection

time can be estimated as

(9)

The estimate is a theoretical estimate that cannot be interpreted

as the true injection time, since only holds for

. However, (9) can be used to represent the IDC as

in (6), although with a different parametrization

(10a)

(10b)

(10c)

(10d)

Using the parametrization in (10), we deﬁne a new parameter ,

dependent on and only

(11)

After combining (11) and (6), the IDC can be expressed as

(12)

Being dependent on and only, is the local, diffusion-re-

lated parameter that we have adopted for characterization of the

microvascular structure. The parameter can be interpreted as

the local ratio between the diffusive time and the squared convec-

tive time. For low values of , the UCA concentration proﬁle

hardly spreads while passing . This leads to a symmetric IDC,

characterized by high values of . On the other hand, high values

KUENEN et al.: CONTRAST-ULTRASOUND DIFFUSION IMAGING FOR LOCALIZATION OF PROSTATE CANCER 1497

Fig. 6. IDC shape for various values of with .

Fig. 7. Signal conditioning and ﬁtting of a single-pixel TDC in the logarithmic

domain.

of lead to a skewed IDC, represented by small values of .

ThisisshowninFig.6.

D. Parameter Estimation

Local diffusion can be estimated from measured TDCs using

the modiﬁed LDRW IDC formalization in (12) and the relation

between UCA concentration and gray level in (3).

The accuracy of the parameter estimation is determined

by the temporal characteristics of IDC noise, i.e., all signals

that the model function (6) cannot explain. Typical ultrasound

noise sources such as speckle are less signiﬁcant here: sta-

tionary noise affects only the IDC baseline and signals from

moving linear scatterers (e.g., red blood cells) are effectively

suppressed by contrast-speciﬁc imaging techniques [16]. IDC

noise is therefore mainly related to microbubbles and might

be caused by random microbubble movement into and out of

the sample volume, represented by a pixel. Such movement

produces a noise component whose variance relates directly

to the microbubble concentration, satisfying the multiplicative

character of IDC noise that was previously measured [34].

We evaluated the multiplicative character of IDC noise by

analyzing over 50 000 curves measured in vivo. Signals at fre-

quencies above 0.5 Hz were considered as noise, because the

spectrum of (6) is restricted to frequencies lower than 0.5 Hz for

a realistic range of (0.1–1.5 )and (10–50 s). We com-

pared the high-frequency noise power with the low-frequency

signal magnitude using a short-time Fourier transform with a

Hamming window of 3.2 s. For IDCs in the real domain, the

average correlation coefﬁcient was , compared with

for log-domain TDCs. The relatively strong corre-

lation in the real domain indicates multiplicative noise. More-

over, frequencies above 0.5 Hz contained 54% of the total signal

power in the real domain, compared with 6% in the logarithmic

domain. For these reasons, parameter estimation is performed

in the logarithmic domain [42]. Although the logarithmic com-

pression affects the error metrics for TDC ﬁtting, the effects on

the estimation of and are negligible [34].

The modiﬁed LDRW TDC formalization is obtained by com-

bining (12) and (3). After compensating for the gray mapping,

and estimating and subtracting the baseline , this expression is

given as

(13)

In (13), the factor is included in , since only a relative

measure of the UCA concentration is required.

The accuracy of the parameter estimation is improved by

low-pass ﬁltering the TDCs both in space and time (see Fig. 7).

The spatial ﬁlter design is based on the size of the smallest

microvascular networks for which local diffusion must be es-

timated. As angiogenesis is required for cancer to grow beyond

1mm [11], a reliable analysis of image regions with a radius

as small as 0.62 mm is necessary. To maintain sufﬁcient res-

olution for accurate characterization at this scale, we adopted

a Gaussian ﬁlter with mm, whose 3-dB value is at

0.59 mm. The loss of information due to spatial ﬁltering is lim-

ited, because the axial scanner resolution (0.43 mm) is inferior

to the B-mode pixel resolution (0.15 mm). The nonuniform spa-

tial TRUS statistics, generally due to a lower lateral resolution

at larger depths, are compensated by spatial ﬁltering; after ﬁl-

tering, the average correlation coefﬁcient between neighboring

pixel IDCs is independent of the scanning depth, suggesting a

uniform spatial resolution. After spatial ﬁltering, downsampling

in both spatial dimensions by a factor three permits reducing the

computation time by 89%.

Low-pass ﬁltering may also be performed in time, given the

scanning frame rate (about 10 Hz) as compared to the maximum

TDC frequency of 0.5 Hz. We adopted a ﬁnite impulse response

ﬁlter of 100 coefﬁcients and a cutoff frequency of 0.5 Hz. A zero

phase shift is obtained by ﬁltering in both forward and backward

directions.

1498 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 30, NO. 8, AUGUST 2011

Fig. 8. In vivo CUDI parametric image obtained from the same data as shown in Fig. 1 with histology. On the left, the diffusion parametric image is overlaid on

the ultrasound power modulation image. The parameter is displayed as a color coding; uncolored pixels are associated with ﬁt failure. The manually selected

white contour represents the prostate boundary. The corresponding fundamental ultrasound image (middle) is also shown, in which the red and green polygons

represent the adopted ROIs for cancerous and healthy tissue, respectively. Three corresponding histology slices, all showing cancer in the right peripheral zone,

are shown on the right.

A well-known issue in IDC analysis is recirculation, i.e., the

subsequent bolus passages through a selected ROI that mask

the last segment of the ﬁrst-pass IDC tail [35], [39], [43]. In

cardiovascular applications, IDC parameter estimation is often

restricted to the interval where recirculation does not

occur. The recirculation time is commonly deﬁned as the time

when the IDC decays to 30% of its peak value [34], [43]. An ef-

fect similar to recirculation occurs also in the prostate, where

additional UCA passages may also be due to multiple feeding

arteries [44]. The effects on measured curves are, however, less

evident than in cardiovascular applications (see Fig. 7). There-

fore,weadoptamorespeciﬁc approach to deﬁne .TheIDC

decay during an interval is given as

(14)

where is assumed without loss of generality. For large

, the IDC decays approximately exponentially, which corre-

sponds to a linear TDC decay in the logarithmic domain. There-

fore, is determined such that the linear TDC approximation

has the highest on the interval between the peak time and

. The choice of is restricted to the interval where the IDC

amplitude, in the real domain, decays to between 20% and 50%

of the peak amplitude. The inﬂuence of TDC noise on the deter-

mination of is reduced by additional low-pass ﬁltering with

cutoff frequency at 0.1 Hz.

LDRW model ﬁtting can be performed with a linear regres-

sion algorithm as described in [34]. However, the computational

complexity of this algorithm is high as both the parameter

and the regression interval are iteratively determined. Given the

large number of TDCs (about for each dataset, after down-

sampling), we have adopted a different method, based on the

statistical IDC moments [45]. We have extended this method by

inclusion of the estimation of (see Appendix), which makes

the method non-iterative. However, the computation of the sta-

tistical IDC moments requires the complete ﬁrst-pass IDC. For

the interval , the IDC is approximated by an exponen-

tial decay, corresponding to the linear TDC approximation. As

aﬁnal step, the obtained parameter estimates are used as ini-

tialization to ﬁt (13) to the TDC for using the Leven-

berg–Marquardt iteration [46], to increase the ﬁtting accuracy.

Up to ﬁve iterations were used; a higher number of iterations

did not signiﬁcantly improve the results. On a Windows-based

workstation with an Intel Core2 Duo processor running at 3.16

GHzwith3.49GBofRAM,onevideoisprocessedinabout

5min.

A parametric image is produced by displaying the estimates

of for all pixels covering the prostate, as a color coding over-

laid on the B-mode image, see Fig. 8. The parametric image is

ﬁltered with the same Gaussian ﬁlter ( mm) as the one

adopted for spatial ﬁltering, to emphasize local diffusion at the

scale where microvascular networks can be associated with the

presence of cancer.

In some cases, shadowing effects or a lack of perfusion may

result in ﬁt failure. Then, the estimated parameters do not rep-

resent the UCA transport dynamics. If the determination coefﬁ-

cient of the TDC ﬁt in the logarithmic domain is lower than

0.75, the ﬁt is not accepted and no color is displayed.

E. Method Validation

To validate whether the diffusion-related parameter can be

estimated independently of the bolus history, the convective dif-

fusion (4) was simulated by a ﬁnite difference approximation.

The adopted boundary conditions were (5b) and (8). For various

, a large number of IDCs were obtained for increasing .We

analyzed the dependency of the diffusion-related parameters

and on both and ,byﬁtting the IDCs with the pro-

posed algorithm.

The performance of the parameter estimation algorithm was

then evaluated by ﬁtting simulated TDCs, for

and , with additive white noise sequences. By

adding these noise sequences to the TDCs, we reproduced the

multiplicative noise character in the real domain [34]. The noise

level was 12 dB was lower than the signal power, representing

the noise level encountered in vivo.

A preliminary clinical validation of the method was also per-

formed by analysis of ﬁve datasets registered from four pa-

tients referred for radical prostatectomy at the AMC University

Hospital (Amsterdam, the Netherlands). After radical prostate-

ctomy, the prostate was cut in slices of 4 mm thickness and a

pathologist marked the presence of cancer by histology anal-

ysis based on the level of cell differentiation [47]. We selected

KUENEN et al.: CONTRAST-ULTRASOUND DIFFUSION IMAGING FOR LOCALIZATION OF PROSTATE CANCER 1499

the histology slice(s) corresponding to the ultrasound imaging

plane and compared them with theCUDIresults.Fig.8shows

an example parametric diffusion image with the corresponding

histology. A quantitative comparison was performed for each

dataset by selecting two ROIs containing healthy and cancerous

tissue, based on the histology. Because the histology analysis

was not speciﬁcally aimed at a high-resolution validation, we

limited the ROI selection to areas larger than 50 mm that did

not show signiﬁcant variation across subsequent slices. We con-

sidered only the peripheral zone of the prostate [8], since about

70% to 80% of all cancers are found in this anatomical zone [8],

[14]. From the histogram of in each ROI, the mean value and

standard deviation of each speciﬁc class (healthy and cancerous

tissue) were used to determine the optimal tissue-classiﬁcation

threshold by Bayes inference [48]. This threshold (based on all

datasets) was used to derive the optimal sensitivity and speci-

ﬁcity for pixel classiﬁcation. In addition, we evaluated the re-

ceiver operating characteristic (ROC) curve on a pixel basis.

A comparison was performed with different IDC parameters

proposed in the literature [14], [20]–[23], by repeating the

same tissue classiﬁcation procedure. We extracted the peak

value (PV), the peak time (PT), the appearance time (AT, the

time at which the IDC achieves 5% of PV), the full-width

at half-maximum (FWHM, the time duration while the IDC

exceeds [22], [23]), the wash-in time (WIT, the time

it takes for the IDC to rise from 5% to 95% of its peak value

[23]), the area under the IDC (AUC, the LDRW parameter ),

the MTT (the LDRW parameter ) and the LDRW parameter

. The parameters PT and AT were computed with respect to

the estimated theoretical injection time . All parameters were

extracted from linearized ﬁts to ensure that the comparison is

not affected by differences in preprocessing.

III. RESULTS

The parameters and , estimated from IDCs obtained by

simulations of the convective diffusion (4), conﬁrmed the theo-

reticalresultsof(10d)and(11)withanestimationerrorbelow

1%. This result conﬁrms that ,incontrastto , is independent

of and , i.e., independent of the detection site and the his-

tory of the bolus, respectively.

The parameter estimation algorithm ﬁtted simulated TDCs

with an average and in the logarithmic

and real domains, respectively. If the complete TDC could be

used for ﬁtting, the mean relative error for was 4.33%. When

the IDC tail was excluded from the ﬁtting, as required for ﬁtting

of TDCs obtained in vivo , the mean relative error was 10.15%.

On the obtained B-mode image sequences that were com-

pared with histology, the algorithm showed an average

after ﬁltering in space and time, which was by 18% higher

than without ﬁltering. In these data, 89% of the pixel IDC ﬁts

were considered sufﬁciently accurate ( in the loga-

rithmic domain) and pixel TDC ﬁts were included in the

comparison with histology.

In all patients, we observed that the presence of cancer was

associated with higher values of . For each parameter, the mean

value and standard deviation in healthy and cancerous tissue

are reported in Table I. We used this information to derive the

sensitivity and speciﬁcity for pixel classiﬁcation, as well as the

TAB L E I

MEAN AND STANDARD DEVIATION OF IDC PARAMETERS

MEASURED IN HEALTHY AND CANCEROUS TISSUE

TAB L E I I

SENSITIVITY,SPECIFICITY,AND ROC CURVE AREA ON PIXEL BASIS OF

SEVERAL HEMODYNAMIC PARAMETERS EXTRACTED FROM FITTED IDCS

ROC curve area. For all the considered parameters, the results

are reported in Table II.

IV. DISCUSSION AND CONCLUSION

Contrast-ultrasound diffusion imaging (CUDI) is an inno-

vative noninvasive imaging method for prostate cancer local-

ization. The passage of an intravenously injected UCA bolus

through the prostate is measured by dynamic TRUS imaging.

The TDCs obtained from all pixels covering the prostate are an-

alyzed and a parametric image, based on intravascular diffusion,

is produced.

In a preliminary clinical validation, we have compared

the cancer localization accuracy of the proposed method,

CUDI, with several quantitative indicators of perfusion. More

precisely, we compared the correspondence between several

methods for MVD characterization and the level of cell differ-

entiation, evaluated by a histology analysis.

The results show that the diffusion-related parameter has a

ROC curve that is superior to that of any other IDC parameter.

Although the sensitivity of some other parameters is higher,

is the only parameter whose sensitivity and speciﬁcity both ex-

ceed 80%. While these results are obtained on a pixel basis, the

clinical diagnosis would be based on a larger scale; currently,

carcinomas are considered clinically signiﬁcant if their size is at

least 0.5 cm [47]. Lesion classiﬁcation is then based on large

amounts of pixels. A perfect pixel-based sensitivity is therefore

not strictly necessary. On the other hand, a high speciﬁcity is

essential to exclude healthy areas.

Although they are less speciﬁcthan , all IDC time parame-

ters are smaller in the cancerous areas than in healthy areas. This

1500 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 30, NO. 8, AUGUST 2011

is consistent with previous qualitative observations [6], [14].

Perhaps, the lower speciﬁcity of the time parameters compared

to is related to the difﬁculty of interpreting these time parame-

ters locally at the measurement site [24]. The amplitude-related

parameters PV and AUC show relatively large variations, in

both healthy and cancerous tissue. This may be due to their de-

pendency on the amount of UCA entering the prostate or to non-

linear ultrasound propagation through UCA dispersions [25].

Imaging of intravascular diffusion is a new concept, based

on modeling the intravascular UCA transport by the convec-

tive diffusion equation. Whereas other researchers have taken

a bottom-up approach, by analyzing ﬂow through individual

vessels and deducing the effects on vascular networks [49], we

have pursued a top-down approach, aimed at macroscopic mod-

eling of the microvascular network, similarly to [31]. In fact, we

characterize the microvascular network as similar to a porous

medium [32], whose structural characteristics are reﬂected in

diffusion [30].

The parameter measures the ratio between diffusion and

convection. High values of , with a relatively low diffusion

with respect to convection, seem to be associated with the

presence of cancer. This relative decrease in diffusion may be

caused by an increased microvessel tortuosity. This is however

the ﬁrst study that investigates the effects of changes in the

microvascular architecture on intravascular diffusion; a better

understanding requires additional research.

The presented method for estimation of diffusion has the ad-

vantage that a local diffusion-related parameter can be es-

timated independently for each pixel. This parameter depends

only on the local, hemodynamic parameters and and does

not depend on the entire dilution history between the injection

and detection site. This novel approach is based on the assump-

tion of a Normal UCA concentration distribution in space; an

assumption that is also included in the LDRW model [38]. The

width of this distribution before the bolus passage through the

detection site, given by , determines the resolution by which

we can estimate (see Fig. 5). We have not been able to verify

the use of this assumption in an experimental in vitro setup.

However, the spatial UCA distribution in the systemic arteries is

mainly determined by the transpulmonary circulation and can be

well described by the LDRW model [38]. Moreover, the LDRW

model is reported to be the most suitable for ﬁtting IDCs mea-

sured in the microcirculation of animal models [23]. These re-

sults support the validity of the LDRW model assumptions.

The proposed method focuses on the temporal characteris-

tics of the UCA diffusion dynamics. Alternative methods can

be based also on spatial diffusion characteristics and will be in-

vestigated in the future. In this study, relatively simple linear

ﬁlters were used to improve the robustness of parameter esti-

mation. More advanced ﬁltering methods can possibly provide

additional improvements. In this context, coherence-enhancing

diffusion ﬁltering seems an interesting method to improve the

signal quality given the anisotropy caused by the TRUS resolu-

tion and the microvascular characteristics.

An important issue concerns the validation of CUDI, as de-

termining the position of the imaging plane with respect to the

histology planes is difﬁcult. In fact, the imaging plane often

crosses several histology planes. The presented validation was

therefore restricted to patients whose histology did not show sig-

niﬁcant variation across subsequent slices. We are currently in-

vestigating new strategies to improve the comparison between

imaging and histology. The validation could be improved by

comparing CUDI results directly with the MVD, rather than

with the level of cell differentiation. This approach, requiring

the use of immunohistology [10]–[12], would be more accu-

rate as CUDI aims at characterizing the microvascular structure.

An additional step in the validation may also involve the zonal

anatomy of the prostate. Here, the validation was restricted to

the peripheral zone, where the majority of cancers are found

[8], [14]. As the microvascular structure varies among different

anatomical zones of the prostate, it may also be interesting to

investigate the intravascular diffusion in different zones.

In the future, three-dimensional ultrasound imaging may

offer great advantages for the proposed method. From a clinical

perspective, the entire prostate could be studied with a single

UCA bolus injection. This would resolve an important current

issue, i.e., the selection of proper TRUS imaging planes such

that any signiﬁcant carcinoma is covered. From a technical

perspective, the UCA transport could be observed in all spa-

tial dimensions, which would open up new possibilities for

spatio-temporal analysis of intravascular UCA diffusion. More-

over, the in vivo validation would be simpliﬁed as imaging and

histology results could be compared more accurately.

In conclusion, also given the additional possibilities offered

by three-dimensional ultrasound, imaging of intravascular dif-

fusion may be a promising alternative to perfusion imaging for

the localization of prostate cancer. The intravascular nature of

UCA microbubbles makes contrast-enhanced ultrasound an at-

tractive imaging modality to assess intravascular diffusion. Fur-

thermore, the use of CUDI should not be limited to prostate

cancer; the same diffusion principles also apply to many other

forms of cancer, such as breast cancer. Further clinical studies

are however required to evaluate the clinical reliability of CUDI.

APPENDIX

LDRW IDC MOMENTS AND SKEWNESS

For a random variable with probability density function

(PDF) , the moments for are given as

(15)

For , the central moments are given as

(16)

To interpret the LDRW IDC formalization in (6) as a PDF, we

deﬁne for and divide by its integral

[50]. The moments are then given as

(17)

The LDRW IDC moments, which for are denoted by ,

have been derived in [45] and [50]

(18a)

KUENEN et al.: CONTRAST-ULTRASOUND DIFFUSION IMAGING FOR LOCALIZATION OF PROSTATE CANCER 1501

(18b)

(18c)

In (18), equals the expectation of .If is known, the

moments and can be computed from measured IDCs.

Therefore, solving and from (18a) and (18b) provides a

noniterative method to estimate these parameters [45].

In the current study, is however unknown so we cannot

measure . We can only measure the moments , which de-

pend on . To estimate all LDRW parameters by measuring ,

we include in the moments analysis. The ﬁrst moment for

is by linearity of the expectation given as

(19)

This result can also be derived by substitution of in the

integrand of (17). Similarly, for can be derived. ,

and can then be solved from the obtained equations for ,

and . However, this system is very complicated and has

no analytical solution.

Alternatively, we can also measure the central moments

and derive expressions for in terms of ,and . By sub-

stituting , we observe that the central moments are

compensated for by the time shift by

(20)

In fact, the argument ensures that the resulting function

is independent of [see (6)]. Therefore, the moments

of are given by (18). By expanding ,the

central moments can be completely described in terms of

. In particular, and are given as

(21a)

(21b)

and can be solved from (21). To obtain the solution, we com-

pute the IDC skewness , i.e., the third standardized moment

of , which is a function of only

(22)

This result conﬁrms the relation between and the IDC skew-

ness [39], [41], [43]. Solving from (22) provides an estimate

of that is independent of and . Subsequently, can be es-

timated from (21a) as

(23)

Finally, we use (19) to estimate from as

(24)

The fourth parameter is given directly by the IDC integral.

Since the IDC integral as well as the moments ,,and

can directly be computed from measured IDCs, all LDRW IDC

parameters can be estimated by (22)–(24).

In summary, we have obtained a method to estimate all

LDRW IDC parameters, including . Being noniterative,

this method has low computational requirements compared to

various iterative methods.

ACKNOWLEDGMENT

The authors would like to thank the Department of Pathology

of the AMC University Hospital in Amsterdam for the histology

data, and I. Herold (M.D.) and Prof. Dr. H. Korsten (M.D.) of

the Department of Anesthesiology of the Catharina Hospital in

Eindhoven for providing equipment and assistance during the

in vitro measurements.

REFERENCES

[1] J. Ferlay, P. Autier, M. Boniol, M. Heanue, M. Colombet, and P. Boyle,

“Estimates of the cancer incidence and mortality in Europe in 2006,”

Ann. Oncol., vol. 18, pp. 581–592, 2007.

[2] A.Jemal,R.Siegel,E.Ward,Y.Hao,J.Xu,T.Murray,andM.J.Thun,

“Cancer statistics, 2008,” CA Cancer J. Clin., vol. 58, no. 2, pp. 71–96,

2008.

[3] T. J. Polascik and V. Mouraviev, “Focal therapy for prostate cancer,”

Curr. Opin. Urol., vol. 18, pp. 269–274, 2008.

[4] C. H. Bangma, S. Roemeling, and F. H. Schröder, “Overdiagnosis and

overtreatment of early detected prostate cancer,” WorldJ.Urol., vol.

25, pp. 3–9, 2007.

[5] F.H.Schröder,J.Hugosson,M.J.Roobol,T.L.J.Tammela,S.Ciatto,

V. Nelen, M. Kwiatkowski, M. Lujan, H. Lilja, M. Zappa, L. J. Denis,

F. Recker, A. Berenguer, L. Määttänen, C. H. Bangma, G. Aus, A.

Villers, X. Rebillard, T. van der Kwast, B. G. Blijenberg, S. M. Moss,

H. J. de Koning, and A. Auvinen, “Screening and prostate-cancer mor-

tality in a randomized European study,” N.Engl.J.Med., vol. 360, no.

13, pp. 1320–1328, 2009.

[6] M.H.Wink,J.J.M.C.H.delaRosette,C.A.Grimbergen,andH.

Wijkstra, “Transrectal contrast enhanced ultrasound for diagnosis of

prostate cancer,” WorldJ.Urol., vol. 25, pp. 367–373, 2007.

[7] M. Seitz, A. Shukla-Dave, A. Bjartell, K. Touijer, A. Sciarra, P. J. Bas-

tian, C. Stief, H. Hricak, and A. Graser, “Functional magnetic reso-

nance imaging in prostate cancer,” Eur. Urol., vol. 55, pp. 801–814,

2009.

[8] T. Loch, “Urologic imaging for localized prostate cancer in 2007,”

Wor l d J . U rol . , vol. 25, pp. 121–129, 2007.

[9] J. Folkman, K. Watson, D. Ingber, and D. Hanahan, “Induction of an-

giogenesis during the transition from hyperpla sia to neoplasia,” Nature,

vol. 339, no. 6219, pp. 58–61, 1989.

[10] N. Weidner, P. R. Carroll, J. Flax, W. Blumenfeld, and J. Folkman,

“Tumor angiogenesis correlates with metastasis in invasive prostate

carcinoma,” Am. J. Pathol., vol. 143, no. 2, pp. 401–409, 1993.

[11] M. K. Brawer, “Quantitative microvessel density. A staging and prog-

nostic marker for human prostatic carcinoma,” Cancer, vol. 78, no. 2,

pp. 345–349, 1996.

[12] M. Borre, B. V. Offersen, B. Nerstrøm, and J. Overgaard, “Microvessel

density predicts survival in prostate cancer patients subjected to

watchful waiting,” Br. J. Cancer, vol. 78, no. 7, pp. 940–944, 1998.

[13] D. Cosgrove, “Angiogenesis imaging—ultrasound,” Br.J.Radiol., vol.

76, pp. S43–S49, 2003.

[14] L. Pallwein, M. Mitterberger, A. Pelzer, G. Bartsch, H. Strasser, G.

M. Pinggera, F. Aigner, J. Gradl, D. zur Nedden, and F. Frauscher,

“Ultrasound of prostate cancer: Recent advances,” Eur. Radiol., vol.

18, pp. 707–715, 2008.

[15] S. B. Feinstein, “The powerful microbubble: From bench to bedside,

from intravascular indicator to therapeutic delivery system, and

beyond,” Am. J. Physiol. Heart Circ. Physiol., vol. 287, no. 2, pp.

H450–H457, 2004.

1502 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 30, NO. 8, AUGUST 2011

[16] N. de Jong, P. J. A. Frinking, A. Bouakaz, and F. J. ten Cate, “Detec-

tion procedures of ultrasound contrast agents,” Ultrasonics, vol. 38, pp.

87–92, 2000.

[17]A.Pelzer,J.Bektic,A.P.Berger,L.Pallwein,E.J.Halpern,W.

Horninger, G. Bartsch, and F. Frauscher, “Prostate cancer detection in

men with prostate speciﬁcantigen4to10mg/mLusingacombined

approach of contrast enhanced color doppler targeted and systematic

biopsy,” J. Urol., vol. 173, pp. 1926–1929, 2005.

[18] E. J. Halpern, J. R. Ramey, S. E. Strup, F. Frauscher, P. McCue, and L.

G. Gomella, “Detection of prostate carcinoma with contrast-enhanced

sonography using intermittent harmonic imaging,” Cancer, vol. 104,

pp. 2373–2383, 2005.

[19] R. A. Linden, E. J. Trabulsi, F. Forsberg, P. R. Gittens, L. G. Gomella,

and E. J. Halpern, “Contrast enhanced ultrasound ﬂash replenish-

ment method for directed prostate biopsies,” J. Urol., vol. 178, pp.

2354–2358, 2007.

[20] R. J. Eckersley, D. O. Cosgrove, M. J. Blomley, and H. Hashimoto,

“Functional imaging of tissue response to bolus injection of ultrasound

contrast agent,” in IEEE Ultrason. Symp. Proc., 1998, pp. 1779–1782.

[21] R.J.Eckersley,J.P.Sedelaar,M.J.K.Blomley,H.Wijkstra,N.M.

deSouza, D. O. Cosgrove, and J. J. M. C. H. de la Rosette, “Quanti-

tative microbubble enhanced transrectal ultrasound as a tool for mon-

itoring hormonal treatment of prostate carcinoma,” Prostate, vol. 51,

pp. 256–267, 2002.

[22] N.Elie,A.Kaliski,P.Péronneau,P.Opolon,A.Roche,andN.Lassau,

“Methodology for quantifying interactions between perfusion evalu-

ated by DCE-US and hypoxia throughout tumor growth,” Ultrasound

Med. Biol., vol. 33, no. 4, pp. 549–560, 2007.

[23] C. Strouthos, M. Lampask is,V .S boros, A. McNeilly, and M. Averkiou,

“Indicator dilution models for the quantiﬁcation of microvascular

blood ﬂow with bolus administration of ultrasound contrast agents,”

IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 57, no. 6, pp.

1296–1310, Jun. 2010.

[24] M. Mischi, J. A. den Boer, and H. H. M. Korsten, “On the physical

and stochastic representation of an indicator dilution curve as a gamma

variate,” Physiol. Meas., vol. 29, pp. 281–294, 2008.

[25] M.-X. Tang and R. J. Eckersley, “Nonlinear propagation of ultrasound

through microbubble contrast agents and implications for imaging,”

IEEE Trans. Ultrason., Ferroelect., Freq., Contr., vol. 53, no. 12, pp.

2406–2415, Dec. 2006.

[26] R. K. Jain, “Haemodynamic and transport barriers to the treatment of

solid tumours,” Int. J. Radiat. Biol., vol. 60, no. 1-2, pp. 85–100, 1991.

[27] S. Delorme and M. V. Knopp, “Non-invasive vascular imaging: As-

sessing tumour vascularity,” Eur. Radiol., vol. 8, pp. 517–527, 1998.

[28] G. Taylor, “Dispersion of soluble matter in solvent ﬂowing slowly

through a tube,” Proc.Roy.Soc.A, vol. 219, no. 1137, pp. 186–203,

1953.

[29] R. J. M. de Wiest, Flow Through Porous Media. London, U.K.: Aca-

demic, 1969.

[30] Handbook of Porous Media,K.E.Vafai,Ed.,2nded. NewYork:

Taylor & Francis, 2005.

[31] L. T. Baxter and R. K. Jain, “Transport of ﬂuid and macromolecules

in tumors: I. Role of interstitial pressure and convection,” Microvasc.

Res., vol. 37, pp. 77–104, 1989.

[32] S. J. Chapman, R. J. Shipley, and R. Jawad, “Multiscale modeling of

ﬂuid transport in tumors,” Bull. Math. Biol., vol. 70, pp. 2334–2357,

2008.

[33] M. Schneider, “Characteristics of SonoVue,” Echocardiography, vol.

16, no. 7, pp. 743–746, 1999.

[34] M.Mischi,A.A.C.M.Kalker,andH.H.M.Korsten,“Videodensito-

metric methods for cardiac output measurements,” EURASIP J. Appl.

Sign. Proc., vol. 5, pp. 479–489, 2003.

[35] X. Chen, K. Q. Schwarz, D. Phillips, S. D. Steinmetz, and R. Schlief,

“A mathematical model for the assessment of hemodynamic param-

eters using quantitative contrast echocardiography,” IEEE Trans.

Biomed. Eng., vol. 45, no. 6, pp. 754–765, Jun. 1998.

[36] L. Lüdemann, D. Prochnow, T. Rohlﬁng, T. Franiel, C. Warmuth,

M. Taupitz, H. Rehbein, and D. Beyersdorff, “Simultaneous quantiﬁ-

cation of perfusion and permeability in the prostate using dynamic

contrast-enhanced magnetic resonance imaging with an inversion-pre-

pared dual-contrast sequence,” Ann. Biomed. Eng., vol. 37, no. 4, pp.

749–762, 2009.

[37] N. G. Rognin, P. J. A. Frinking, M. Costa, and M. Arditi, “In-vivo

perfusion quantiﬁcation by contrast ultrasound: Validation of the use

of linearized video data vs. raw RF data,” in IEEE Ultrason. Symp.

Proc., 2008, pp. 1690–1693.

[38] C. W. Sheppard, Basic Principles of the Tracer Method.NewYork:

Wiley, 1962.

[39] M. E. Wise, “Tracer dilution curves in cardiology and random walk and

lognormal distributions,” Acta Physiol. Pharmacol. Neerl., vol. 14, pp.

175–204, 1966.

[40] K. H. Norwich and S. Zelin, “The dispersion of indicator in the cardio-

pulmonary system,” Bull. Math. Biophys., vol. 32, pp. 25–43, 1970.

[41] J. M. Bogaard, S. J. Smith, A. Versprille, M. E. Wise, and F. Hage-

meijer, “Physiological interpretation of the skewness of indicator-di-

lution curves; theoretical considerations and a practical application,”

Basic Res. Cardiol., vol. 79, pp. 479–493, 1984.

[42] A. Oppenheim, R. Schafer, and T. G. J. Stockham, “Nonlinear ﬁltering

of multiplied and convolved signals,” Proc. IEEE, vol. 56, no. 8, pp.

1264–1291, Aug. 1968.

[43] R. K. Millard, “Indicator-dilution dispersion models and cardiac output

computing methods,” Am. J. Physiol., vol. 272, no. Heart Circ. Physiol.

41, pp. H2004–H2012, 1997.

[44] A. Hendrikx, M. Klomp, J. Keyzer, B. Arends, and G. Zonneveld, “The

role of colour velocity imaging (CVI) in prostate ultrasound,” Eur. J.

Ultrasound,vol.3,no.1,pp.15–23,1996.

[45]M.Mischi,A.A.C.M.Kalker,andH.H.M.Korsten,“Moment

method for the local density random walk model interpolation of

ultrasound contrast agent dilution curves,” in Proc. 17th Int. EURASIP

Conf. BIOSIGNAL 2004, Brno, Czech Rep., 2004, pp. 33–35.

[46] G. A. F. Seber and C. J. Wild, Nonlinear Regression. : Wiley, 1989.

[47] R. Montironi, T. van der Kwast, L. Boccon-Gibod, A. V. Bono, and L.

Boccon-Gibod, “Handling and pathology reporting of radical prostate-

ctomy specimens,” Eur. Urol., vol. 44, pp. 626–636, 2003.

[48] C. M. Bishop, Pattern Recognition and Machine Learning.New

York: Springer, 2006.

[49] R. Karshaﬁan, P. N. Burns, and M. R. Henkelman, “Transit time ki-

netics in ordered and disordered vascular trees,” Phys. Med. Biol., vol.

48, pp. 3225–3237, 2003.

[50] M. Mischi, “Contrast echocardiography for cardiac quantiﬁcations,”

Ph.D. dissertation, Eindhoven Univ.Technol., Eindhoven, The Nether-

lands, 2004.