Quantitative pharmacokinetic analysis of DCE-MRI data without...

Magnetic Resonance Im

Original contributions

Quantitative pharmacokinetic analysis of DCE-MRI data without

an arterial input function: a reference region model

Thomas E. Yankeelova,b,T, Jeffrey J. Lucia,b, Martin Lepagea,b, Rui Lib,c,

Laura Debuskd, P. Charles Lind,e, Ronald R. Pricea,b, John C. Gorea,b

aInstitute of Imaging Science, Vanderbilt University, Nashville, TN 37232-2675, USAbDepartment of Radiology and Radiological Sciences, Vanderbilt University, Nashville, TN 37232-2675, USA

cDepartment of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, TN 37232-2675, USAdDepartment of Cancer Biology, Vanderbilt University, Nashville, TN 37232-2675, USA

eDepartment of Radiation Oncology, Vanderbilt University, Nashville, TN 37232-2675, USA

Received 17 February 2005; accepted 17 February 2005

Abstract

Dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) can assess tumor perfusion, microvascular vessel wall permeability

and extravascular–extracellular volume fraction. Analysis of DCE-MRI data is usually based on indicator dilution theory that requires

knowledge of the concentration of the contrast agent in the blood plasma, the arterial input function (AIF). A method is presented that

compares the tissues of interest (TOI) curve shape to that of a reference region (RR), thereby eliminating the need for direct AIF measurement.

By assigning literature values for Ktrans (the blood perfusion-vessel permeability product) and ve (extravascular–extracellular volume fraction)

in a reference tissue, it is possible to extract the Ktrans and ve values for a TOI without knowledge of the AIF. The operational RR equation for

DCE-MRI analysis is derived, and its sensitivity to noise and incorrect assignment of the RR parameters is tested via simulations. The method

is robust at noise levels of 10%, returning accurate (F20% in the worst case) and precise (F15% in the worst case) values. Errors in the TOI

Ktrans and ve values scale approximately linearly with the errors in the assigned RR K trans and ve values. The methodology is then applied to a

Lewis Lung Carcinoma mouse tumor model. A slowly enhancing TOI yielded K trans=0.039F0.002 min�1 and ve=0.46F0.01, while a rapidly

enhancing region yielded Ktrans=0.35F0.05 min�1 and ve=0.31F0.01. Parametric K trans and ve mappings manifested a tumor periphery with

elevated Ktrans (N0.30 min�1) and ve (N0.30) values. The main advantage of the RR approach is that it allows for quantitative assessment of

tissue properties without having to obtain high temporal resolution images to characterize an AIF. This allows for acquiring images with higher

spatial resolution and/or SNR, and therefore, increased ability to probe tissue heterogeneity.

D 2005 Elsevier Inc. All rights reserved.

Keywords: DCE-MRI; Arterial input function; Pharmacokinetics; Reference region model

1. Introduction

Dynamic contrast-enhanced magnetic resonance imaging

(DCE-MRI) involves the serial acquisition ofMR images of a

tissue of interest (TOI) (e.g., a tumor locus) before, during

and after an intravenous injection of contrast agent (CA). As

the CA perfuses into the tissue under investigation, the T1 and

T2 values of tissue water decrease to an extent that is

determined by the concentration of the agent. By considering

0730-725X/$ – see front matter D 2005 Elsevier Inc. All rights reserved.

doi:10.1016/j.mri.2005.02.013

* Corresponding author. Institute of Imaging Science, Vanderbilt

University, Nashville, Tennessee 37232-2675, USA.

E-mail address: [email protected] (T.E. Yankeelov).

a set of images acquired before, during and after a CA

infusion, a region of interest (ROI) or individual voxel will

display a characteristic signal intensity time course that can

be related to CA concentration. This time course can be

analyzed with an appropriate mathematical pharmacokinetic

model. By fitting the DCE-MRI data to such model,

physiological parameters can be extracted that relate to, for

example, tissue perfusion, microvascular vessel wall perme-

ability and extracellular volume fraction [1]. It has been

shown that both healthy and pathologic tissues exhibit

characteristic signal intensity time courses as well as

pharmacokinetic parameter values (see, e.g., Ref. [2–4]).

Furthermore, since these parameter values are probes of

aging 23 (2005) 519–529

T.E. Yankeelov et al. / Magnetic Resonance Imaging 23 (2005) 519–529520

tissue status, theymay be used to differentiate malignant from

benign tumors [5], aid in tumor staging [6,7] and monitor

treatment response [6,8–10]. Thus, there is considerable and

continuing interest in developing new and improved methods

to obtain these parameter values accurately and precisely.

Analysis of DCE-MRI data is usually based on indicator

dilution theory [11] and requires knowledge of the concen-

tration of the CA in the blood plasma, Cp, the so-called

arterial input function (AIF). This is a notoriously difficult

problem and three main approaches have been developed to

estimate the AIF in DCE-MRI studies. One approach

involves introducing an arterial catheter into the subject and

sampling blood during the imaging process for later analysis

[12,13]. An advantage of this approach is that the Cp in each

sample can be determined accurately through standard

spectroscopic methods (e.g., inductive coupled plasma

emission spectroscopy), thereby allowing for characteriza-

tion of Cp as a function of time. However, the disadvantages

include its invasive nature, poor temporal resolution and

relative ambiguity concerning the actual time at which the

sample was drawn. In laboratory mice, which are used in

many DCE-MRI experiments, the total blood volume is very

small (~2 ml) so that very few samples (2–5; assuming a

volume of 50–100 Al per sample, the minimum amount

needed for standard spectroscopic measurements) can be

taken in total, and fewer still can be used to characterize the

uptake portion of the Cp curve.

A second method assumes that the AIF is similar for all

subjects. The AIF is first measured via blood samples in a

small cohort of subjects [14], and the resulting average AIF is

then assumed to be valid for subsequent studies [15,16]. A

major advantage of this approach is its simplicity in both data

acquisition and data analysis; no AIF measurement is

required for the experimental subjects, and the subsequent

curve fitting uses a common AIF on all data sets. The

disadvantages include the influences of both inter- and intra-

subject variations in AIF, which can introduce large errors in

both AIF characterization and subsequent pharmacokinetic

analysis [16]. Also, by measuring the AIF in one cohort of

subjects and applying it to another, changes in the AIF that

may be introduced by the pathology under investigation are

ignored, reducing the validity of the assumption in important

practical situations.

A third method obtains the AIF from the DCE-MRI data

sets themselves [11,17,18]. Methods have been developed

that simultaneously measure signal intensity changes (due to

CA passage) in both the blood and tissue. A calibration is then

employed to convert the blood signal intensity to the

intravascular concentration of CA. Such a method has the

potential advantage of measuring the AIF accurately on an

individual basis, and since it does not require any further

measurements, being completely noninvasive. However, it

requires the presence of a large vessel within the field of view

(FOV) [19]. Additionally, the images must be acquired such

that the lumen signal is devoid of partial volume or flow

effects. Specialized pulse sequences can be employed that

selectively saturate spins to avoid inflow effects, and this

allows acquisition of a set of slices containing a feeding

vessel without inflow effects. A recent elegant example of

such an approach was presented by McIntyre et al. [20].

However, this method still requires high temporal resolution

and restricts the choices of both which regions can be

characterized and the imaging slice orientation. In general,

accurate AIF measurements require (significantly) higher

temporal resolution (less than 10 s) than tissue measurements

(30–60 s), so the temporal resolution is dictated by the AIF

measurements process, and the spatial resolution and signal-

to-noise ratio are compromised. The former is potentially an

especially important drawback as a major use of DCE-MRI is

to assess heterogeneous tissues (such as tumors) that demand

high spatial resolution.

We present here a general method derived from the

positron emission tomography (PET) literature [21], which

allows for quantitative pharmacokinetic analysis of

DCE-MRI data without knowledge of the AIF. The PET

community refers to this formalism as the reference region

(RR) model because it relies on finding a well-characterized

RR from which to bcalibrateQ the signal intensity changes in aTOI. We have amended this model to allow for analysis of

T1-weighted DCE-MRI data. It should be noted that a similar

method has previously been proposed by Kovar et al. [22].

The two main differences between that approach and this

contribution are that Kovar et al employed the differential

form of the Kety equation [11] to estimate the AIF from an

RR, whereas our theory incorporates the integral form of the

Kety equation that allows for the development of an opera-

tional equation that is independent of the AIF and therefore

eliminates the requirement for AIF estimation entirely. This

allows for the application of a simple (one-step) curve-fitting

algorithm to obtain estimates on the pharmacokinetic

parameters. Additionally, employing the integral versions

of theKety theory offers the possibility for refined RRmodels

as presented in the Discussion.

In this report, we present the mathematical framework of

a model for DCE-MRI analysis that does not require

knowledge of an AIF. We then test the model’s accuracy,

precision and sensitivity to incorrect RR assumptions

through simulations and show the practical effectiveness

of the model. The potential ease of implementation of this

method promises to provide quantitative DCE-MRI analy-

ses in both experimental and clinical settings. Future efforts

will seek to validate this method.

2. Theory

In an effort to encourage the use of standardized nota-

tion, we use only those symbolic conventions described by

Tofts et al. [1].

Fig. 1 displays a simple two-compartment model in

which CA diffuses from the blood plasma into the

extravascular–extracellular spaces of the RR and the TOI.

Fig. 1. A cartoon depiction of the RR model. Contrast agent from the blood

plasma diffuses bidirectionally from the intravascular space to the

extravascular–extracellular space of the reference tissue and to the

extravascular–extracellular space of the tissue-of-interest. The equations

describing this simple two-compartment model are given as Eqs. (1) and (2).

T.E. Yankeelov et al. / Magnetic Resonance Imaging 23 (2005) 519–529 521

The linear, ordinary differential equations that describe this

system are given as Eqs. (1) and (2):

d

dtCRR tð Þ ¼ K trans;RR

bCp tð Þ � K trans;RR=ve;RR�bCRR tð Þ

�ð1Þ

d

dtCTOI tð Þ ¼ K trans;TOI

bCp tð Þ�ðK trans;TOI=ve;TOIÞbCTOI tð Þ ð2Þ

where CRR and CTOI are the tissue concentrations (expressed

in millimolar) of CA in the RR and the TOI, respectively;

K trans,RR and K trans,TOI are the CA extravasation rate

constants for the RR and TOI, respectively; and ve,RR and

ve,TOI are the extravascular–extracellular volume fractions

for the RR and TOI, respectively [1]. Eqs. (1) and (2) make

clear how the concentration of CA in the blood [Cp(t), i.e.,

the AIF] is incorporated into the analysis: if Cp and CTOI [as

in Eq. (2)] can be measured, then a curve-fitting routine can

return estimates on Ktrans and ve. We seek to remove the

dependence of Eqs. (1) and (2) on the AIF, and to do so, we

must eliminate all Cp terms from the formalism.

Solving Eq. (1) for Cp(t) yields Eq. (3):

Cp tð Þ ¼ ð1=K trans;RRÞ d

dtCRR tð Þ þ 1=ve;RR

�bCRR tð Þ:

�ð3Þ

Eq. (3) can then be substituted into Eq. (2) to yield Eq. (4):

d

dtCTOI tð Þ þ ðK trans;TOI=ve;TOIÞbCTOI tð Þ

¼ Rbd

dtCRR tð Þ þ K trans;TOI=ve;RR

�bCRR tð Þ;

�ð4Þ

where RuKtrans,TOI/Ktrans,RR. Note that Eq. (4) is indepen-

dent of Cp(t ). Next define the integrating factor

Iuexp[(Ktrans,TOI/ve,TOI)t] and note that (via the product

rule of calculus)

d

dtðCTOI tð ÞbIÞ ¼ Ib

� d

dtCTOI tð Þ

þ ðK trans;TOI=ve;TOIÞbCTOI tð Þ�; ð5Þ

which is identical to the left-hand side of Eq. (4) multiplied

by I. Then Eq. (4) may be rewritten as

d

dtðCTOI tð ÞbIÞ ¼ Rb

d

dtCRR tð ÞbI

þ ðK trans;TOI=ve;RRÞbCRR tð ÞbI ; ð6Þ

which is an exact equation and therefore can then be inte-

grated directly (by parts) to obtain a relationship between

CTOI and CRR that is independent of Cp (see, e.g., Ref. [23]):

CTOI Tð Þ ¼ RbCRR Tð Þ þ Rb½ðK trans;RR=ve;RRÞ

� ðK trans;TOI=ve;TOIÞ�bZT

0

CRR tð Þ

� expðð � K trans;TOI=ve;TOIÞb T � tð ÞÞdt: ð7ÞIt is important to note that CTOI and CRR are the bulk

concentrations in the tissue water and are related to the real

concentration of CA by the extravascular–extracellular

water volume fraction; that is, CTOI and CRR are only

proportional to the true concentrations (those in the volumes

in which the CA is actually dissolved) — and then only if

the compartment volumes do not change during the

experiment [24]. The transformations from tissue concen-

tration, Ct, to concentration in the extravascular–extracellu-

lar space is given by Eq. (8):

Ct ¼ vebCe; ð8Þ

where Ce denotes concentration of CA in the extravascular–

extracellular space [1]. Thus, the CTOI and CRR expressions

of Eq. (7) can be converted to the true concentrations via the

following transformations:

CTOI ¼ ve;TOIbCe;TOI ð9Þ

CRR ¼ ve;RRbCe;RR; ð10Þ

where Ce,TOI and Ce,RR denote the concentrations of CA in

the extravascular–extracellular (water) space of the TOI and

RR, respectively [24,28]. Furthermore, since concentration

of CA is not measured directly in a DCE-MRI experiment,

a calibration to the measured longitudinal relaxation rate

constant, R1 (u1/T1), is required [1]. In the fast exchange

limit approximation [1,24,25], the following relations

are assumed:

R1;TOI ¼ r1;TOIbve;TOIbCe;TOI þ R10;TOI ð11Þ

R1;RR ¼ r1;RRbve;RRbCe;RR þ R10;RR; ð12Þ

where r1,TOI and r1,RR are CA longitudinal TOI and RR

relaxivities (in mM�1 s�1), respectively, and R10,TOI and

2D Graph 4

time (min)0 10 20 30 40

R1(t

) (s

-1)

1.0

2.0

3.0

R1bR1,TOIR1,RR

C (

mM

)

0.0

0.4

0.8

1.2

1.6

2.0

CpCe,TOICe,RR

A

B

Fig. 2. Panel A depicts simulated Cp, Ce,TOI and Ce,RR time courses. Cp was

converted to Ce,TOI and Ce,RR via Eq. (15) with Ktrans,RR and ve,RR set to

0.1 min�1 and 0.1, respectively, and K trans,TOI and ve,TOI set to 0.1 min�1

and 0.1. Panel B represents the linear transformation for concentration of

CA time courses to R1 time courses as described by Eqs. (10) and (11).


R10,RR are the TOI and the RR R1’s before contrast

administration, respectively. Though a simplifying assump-

tion, it is common to assume that r1 is constant throughout the

FOV, and we do the same here [1,24,28]. Combining Eq. (9)

with Eq. (11) and Eq. (10) with Eq. (12) yields Eqs. (13) and

(14), respectively:

R1;TOI ¼ r1;TOIbCTOI þ R10;TOI ð13Þ

R1;RR ¼ r1;RRbCRR þ R10;RR: ð14Þ

Solving Eqs. (13) and (14) for CTOI and CRR gives

expressions that are readily substituted into Eq. (7), yielding

an operational equation that can be employed in a curve-

fitting routine to extract Ktrans,TOI and ve,TOI if R1,TOI,

R10,TOI, R1,RR and R10,RR can be measured:

ðR1;TOI Tð Þ � R10;TOIÞ ¼ RbðR1;RR Tð Þ � R10;RRÞ

þ R½ðK trans;RR=ve;RRÞ� ðK trans;TOI=ve;TOIÞ�

�ZT

0

ðR1;RR tð Þ � R10;RRÞbexp

�ðð�K trans;TOI=ve;TOIÞ� T� tð ÞÞdt;

ð15Þ

or, alternatively, as

R1;TOI Tð Þ ¼ RbðR1;RR Tð Þ � R10;RRÞ þ Rb½ðK trans;RR=ve;RRÞ

� ðK trans;TOI=ve;TOIÞ�bZT

0

ðR1;RR tð Þ�R10;RRÞ

� expðð � Ktrans;TOI=ve;TOIÞb T � tð ÞÞdt

þ R10;TOI: ð16Þ

3. Methods

3.1. Simulations

To test the method, we simulated AIF, RR and TOI CA

curves. The AIF curve (Cp time course) was generated using

Eq. (17):

Cp tð Þ ¼ Abtbexp � tbBð Þ

þ C 1� exp � tbDð Þ½ �bexp � tbEð Þ; ð17Þ

where A= 0.6 mM, B = 0.18 min�1, C = 0.45 mM,

D=0.5 min�1, E=0.013 min�1. This AIF is reasonable

and similar in form to that of Simpson et al. [16]. The

resulting AIF was discretized with 1-min temporal resolu-

tion and truncated after the first 40 min to produce the AIF

shown in Fig. 2A. This Cp AIF was then converted to the

Fig. 2A RR and TOI CA curves via the integral form of the

Kety-Schmidt equation [1,11], which is less sensitive to

noise than the differential form (Eqs. (1) and (2)):

Ct Tð Þ ¼ K trans

ZT

0

Cp tð Þexp � K trans=veÞ T � tð Þð Þdt;ð ð18Þ

with Ktrans=0.10 min�1 and ve=0.10 assigned for the Ce,RR

curve, whileKtrans=0.25min�1 and ve=0.4 were assigned for

the Ce,TOI curve. The RR values are reasonable for muscle

tissue [25–27], and the TOI values are reasonable for

enhancement kinetics seen in a variety of tumor types (e.g.,

Refs. [9,28]). Ct curves are then converted to R1 curves via

the fast exchange limit approximation [24,25], Eqs. (13) and

(14). The CA relaxivity was set at 3.6 mM�1 s�1

[appropriate for gadolinium diethylenetriamine pentaacetic

acid (Gd-DTPA) at 7.0 T], and R10 values of 0.7 s�1 and

0.55 s�1 were used for the RR and TOI, respectively. These

transformations from Ct(t) to R1(t) yield the curves depicted

in Fig. 2B. The Fig. 2B blood R1 time course, R1b,

was computed from Cp via R1b(t)=r1(1�h)Cp(t)+R1b0,

where h is the hematocrit (set to 0.5) and R1b0 is the pre-

CA blood R1.


We then input the RR and TOI R1(t) curves with the RR

model (Eq. (18)) into a curve-fitting routine written in IDL

(Research Systems, Boulder, CO). The Ktrans,RR and ve,RRvalues were fixed at their assigned values (0.1 min�1 and

0.1, respectively), while the Ktrans,TOI and ve,TOI values were

allowed to vary. This tests the accuracy of the RR model. To

test the precision of the model, the curve-fitting procedure

was then repeated with n% random Gaussian noise [n%

noiseu(maximum R1 value)(n/100)] added to the RR and

TOI R1 curves. First 5% and 2% noise was added to the TOI

and RR curves, respectively. Then the entire process was

repeated with 10% and 4% noise added to the TOI and RR

curves, respectively. Assigning a higher noise values to the

TOI over the RR is justified since the RR curve is the

average of many voxels (below, we use 20), whereas the

TOIs will ultimately be single voxels. Additionally, the RR

curve was smoothed with a boxcar average of length 5

before fitting. The process of adding random noise was

repeated a hundred times (runs) to yield arrays of each

parameter (one value per run) from which means and

standard deviations were computed.

One of the potential drawbacks of the RR method is that

it requires assigning the K trans,RR and ve,RR values.

Potentially, even for a well-characterized tissuelike muscle,

these values could vary significantly from subject to subject,

and therefore, knowledge of Ktrans,RR and ve,RR might be

limited. To address this issue, we have run a series of

simulations aimed at elucidating the errors inherent in

assuming an incorrect Ktrans,RR and/or ve,RR value. We

constructed data using the Ktrans,RR and ve,RR values listed

previously (0.1 min�1 and 0.1, respectively) and then ran

the fitting routines with Ktrans,RR incorrectly assigned at a

value of xKtrans,RR, where x was first assigned the value of

0.70. This procedure was repeated with x varied from

0.70 to 1.30 incremented in units of 0.05. An analogous

procedure was performed for ve,RR. In this way, errors of

F30% in both Ktrans,RR and ve,RR were investigated. All

permutations of these incorrect parameter assignments and

their subsequent errors in the returned parameters were

assessed. The values returned by the fitting routines were

stored as two-dimensional arrays and rendered as three-

dimensional error plots. These simulations were performed

without noise so that systematic errors (rather than random

errors) could be assessed.

3.2. Experimental

A male C57/BL mouse (22 g) received a hind limb

subcutaneous injection of 2�105 Lewis Lung Carcinoma

cells. The mouse was fed a standard diet in a controlled

environment with a 12/12-h light/dark cycle. Just prior to

imaging, anesthesia was induced via a 5%/95% isoflurane/

oxygen mixture; anesthesia was maintained via a 2%/98%

isoflurane/oxygen mixture. A 26 G catheter (Abbott Labo-

ratories, Switzerland) was inserted into the tail vein. The

temperature of the animal was maintained via a flow of warm

air through the magnet bore. Heart and respiratory rate were

monitored throughout the experiment. All procedures

adhered to Vanderbilt University’s IACUC guidelines.

The mouse was imaged in a Varian 7.0-T scanner

equipped with a 38-mm quadrature birdcage coil on 16,

20, 24 and 33 days postinjection (we present here the results

from the day 33 study). With TR=400 ms, a variable flip

angle spoiled gradient echo (with flip angles of 158, 308, 458,608 and 758) approach was employed to produce a 1282 R10

map over a 30-mm2 FOV. This method of T1 measurement is

attractive as the total measurement time required (for large

volumes of interest) is drastically reduced compared to spin

echo or inversion recovery methods, while still maintaining

high the SNR of those methods [29]. The slice thickness was

1.5 mm, TE=4.1 ms and NEX=4. The DCE-MRI protocol

employed a standard T1-weighted, GRE sequence to obtain

50 serial images for each of 12 axial planes in 60 min of

imaging. The DCE-MRI parameters were TR=100 ms, flip

angle=308, with other imaging parameters the same as

above. Three images were acquired before a bolus of

0.3 mmol/kg Gd-DTPA was delivered within 30 s via the tail

vein catheter. The tumor TOI was manually selected to allow

both whole-tumor and single-voxel analyses. Twenty voxels

within the perivertebral muscle were selected as the RR.

All images were first coregistered to the first precontrast

image using a mutual information-based rigid registration

algorithm [30] and then analyzed on both a TOI and voxel-

by-voxel basis. Tumor volume was calculated by manually

outlining the visible tumor and multiplying the number of

voxels within the outline by the voxel volume (0.08 mm3);

repeating this procedure twice (by different investigators)

provided a standard deviation estimate for the tumor

volume. An R10 map was constructed by fitting the multiflip

angle spoiled-GRE data to Eq. (19):

S ¼ S0 sina 1�exp �TRbR1ð Þð Þ= 1� exp �TRbR1Þbcosað Þð �;½ð19Þ

where a is the flip angle, S0 is a constant describing the

scanner gain and proton density, and we have assumed that

TEbT2*. Voxels for which Eq. (19) could not fit the data

were set to 0 and colored white on the parametric R10 map.

The R10 map was then used to estimate R1 time courses for

each voxel within the FOV from the raw signal intensity

time courses in the manner of Landis et al. [24]:

R1 tð Þ ¼ � 1=TRð Þblnf½S0bsinabexpð�TE=T2TÞ

�S tð Þbcosa�= S0bsinabexpð�TE=T2TÞ�S tð Þ½ �g; ð20Þ

where S0 represents fully relaxed magnetization for a given

pixel and is computed via Eq. (21):

S0 ¼ S� 1� expð � TR=T1Þcosa½ �=f 1� expð � TR=T1Þ½ �

� sinabexpð � TE=T2TÞg; ð21Þ

where S- is the steady-state pixel-averaged intensity before

CA was injected. In the computation of the R1 time course

from Eqs. (20) and (21), we again took TEbT2*.

R1 (

s-1)

0.5

1.0

1.5

2.0

2.5

time (min)0 10 20 30 40

R1

(s-1

)

0.5

1.0

1.5

2.0

2.5

R1 (

s-1)

0.5

1.0

1.5

2.0

2.5

3.0

RRTOIfit

RRTOIfit

RRTOIfit

no noise

5% noise

10% noise

A

B

C

Fig. 3. Results of fitting the RR model (Eq.(13)) to simulated TOI curves

with 0% (A), 5% (B) and 10% (C) noise added to the bdataQ sets. See

Table 1 for the returned parameter values.


We then input the RR and TOI R1(t) curves with the RR

model (Eq. (16)) into the curve-fitting routine. Again, the

Ktrans,RR and ve,RR values were fixed at their assigned

muscle values (0.1 min�1 and 0.1, respectively), while the

Ktrans,TOI and ve,TOI values were allowed to vary. For the

ROI analysis, a cluster of nine (3�3) contiguous voxels was

selected, averaged and input into the curve-fitting routine.

To compute parameter uncertainties for this analysis, we

Table 1

Summary results of the RR fits to simulated data with 0%, 5% and 10% noise a

Parameter 0% Noise 5% No

Actual Returned Accuracy Precision Return

Ktrans (min�1) 0.25 0.25 – – 0.23

ve 0.40 0.40 – – 0.40

first find the average absolute deviation of the data points

from the best-fitted curve returned by Eq. (16); that is, we

compute du 1n

Ptnti¼t0

je tiÞjð , where e(ti)ufit(ti)�data(ti).

Then each point in the best-fitted curve is summed with a

random value from �d to +d, yielding a new bdataQ set.

This new curve is then fit with Eq. (16) to yield a new set of

parameters. Repeating this process 100 times yields

100 values each for Ktrans,TOI and ve,TOI from which means

and standard deviations are computed.

Voxel-by-voxel analysis allows for the production of

pharmacokinetic parameter maps to probe tumor heteroge-

neity. For the Ktrans parametric map, each voxel was then

assigned a color based on the Ktrans value returned from the

fit routine. An identical procedure was used to construct the

ve map. Voxels for which the model either did not converge

or converged to unphysical values (i.e., Ktransb0.0 min�1,

KtransN2.5 min�1; veb0.0, veN1.0) were displayed as black.

Each slice in which the tumor was visible was mapped

(slices 4–8).

4. Results

4.1. Simulations

As stated previously, the R1,TOI(t) and R1,RR(t) curves of

Fig. 2B (the two data sets that would actually be measured

in a DCE-MRI experiment) were discretized with 1-min

temporal resolution and input into a curve-fitting routine for

extraction of Ktrans,TOI and ve,TOI. The results of those

simulations are presented in Fig. 3A and Table 1. The fit is

good and the parameters returned (Ktrans,TOI=0.25 min�1,

ve,TOI=0.4) are identical to the values used to construct the

simulated curve. Next, we tested the model’s sensitivity to

noise as described previously. The data and the subsequent

curve fits are seen in Fig. 3B (5% noise) and C (10% noise),

while the parameters output by themodel are given in Table 1.

Again, the fits are good and the model returns

Ktrans,TOI=0.25F0.02 min�1, ve,TOI=0.4F0.02, for the 5%

case and Ktrans,TOI=0.25F0.04 min�1, ve,TOI=0.42F0.05,

for the 10% case. The v2 for the slowly and rapidly enhancingregions were 6.32�10�4 and 6.05�10�4, respectively. These

results indicate that the RR model is robust enough to

accommodate reasonable experimental noise levels. We next

investigate the systematic errors that can enter if an incorrect

assignment of Ktrans,RR and/or ve,RR is made.

Fig. 4 displays three-dimensional plots of the errors in

the returned Ktrans,TOI and ve,TOI values if incorrect RR

values are assigned. A horizontal plane at 0% on the ver-

tical axis indicates zero error in the returned parameter.

dded to the TOI (see Fig. 3)

ise 10% Noise

ed Accuracy Precision Returned Accuracy Precision

0.92 F0.03 0.20 0.80 F0.03

1.00 F0.01 0.40 1.00 F0.01

-30

-20

-10

0

10

20

30

-30-20

-100

1020

30

-30-20

-100

1020

Ktr

ans,

TO

I err

or

Ktrans,RR error

ve,RR error

Ktrans,TOI error

-30

-20

-10

0

10

20

30

-30-20

-100

1020

30

-30-20

-100

1020

v e,T

OI e

rro

r

ve,RR error

ve,TOI error

0

30

Error scale

KKtrans,RR error

-10 0

20

-30-20

10

Fig. 4. Three-dimensional renderings of systematic errors resulting from incorrect assignment of the RR parameters K trans,RR and ve,RR. The vertical axis

represents the degree to which the incorrect RR parameter assignments affects the Ktrans,TOI (left panel) and ve,TOI (right panel) output parameters.


Considering the ve,TOI error plot, maximum ve,TOI errors

occur at the points where ve,RR is at its maximum incorrect

value. In particular, the ve,TOI errors scale linearly with error

in ve,RR; that is, a �30% error in ve,RR yields a �30% error

in ve,TOI, and a +30% error in ve,RR yields a +30% error in

ve,TOI. The ve,TOI error is almost completely independent of

the Ktrans,RR value; if we pick any ve,RR error threshold, the

error introduced into the ve,TOI value is essentially the same

as we move along the Ktrans,RR error axis. Thus, the ve,TOIparameter is extremely stable to incorrect Ktrans,RR assign-

ment. This is reasonable because Ktrans determines the initial

uptake portion of the enhancement curve [1,31] and has

very little effect on the washout slope, whereas vedetermines the washout slope (and to a lesser extent, the

peak height achieved by the enhancement curve). Thus, it is

reasonable that mischaracterizing the uptake portion of the

curve (through assigning an incorrect value to Ktrans,RR) will

have little effect on the ve value. Eq. (16) essentially

calibrates the TOI curve to the RR curve, so errors in

Fig. 5. Panel A displays the slice 4 t =0 MR image, while panel B displays the t =

the tumor periphery, and to a much lesser extent, the tumor core, from panel B. T

and slowly enhancing TOIs, respectively, depicted in Fig. 6. Panel C displays the T

T10 map.

Ktrans,RR will translate into the Ktrans,TOI parameter almost

exclusively with little effect on ve,TOI. Similarly, error in

ve,RR effects both ve,TOI and (to a lesser extent) Ktrans,TOI

since (as mentioned previously) ve determines the washout

slope and influences the ultimate height achieved by the

enhancement curve. Consequently, the plot of Ktrans,TOI

error demonstrates more structure. Maximum errors again

occur only at the points where ve,RR and Ktrans,RR are both

off by +30% of their true values, and when they are both off

by �30% of their true values. However, when ve,RR is off by

�30% and Ktrans,RR is off by +30%, the error in Ktrans,TOI

approximately vanishes. A similar pattern occurs when ve,RRand Ktrans,RR are off by +30% and �30%, respectively, with

K trans,TOI error dipping to ~17%. This increases the

robustness of the returned Ktrans,TOI parameters as there is

a smaller area within the Ktrans,TOI error plot for which

Ktrans,TOI is significantly affected by incorrect assignments

of the RR parameters. These results are encouraging as the

amount of systematic error inherent in the model is tractable

40 min image obtained from this study. Note the significant enhancement in

he red and blue circles in panels A and B represent the rapidly enhancing

10 map from this same slice. A grayscale is provided at the far right for the

Fig. 7. The results of RR parameter mappings for (representative) slices 4–6

(panels A, C and E, respectively) and the associated color scale on the far

right. The superior portion of the displays many red voxels on both the

K trans (the left panels) and ve map (right panels), indicating K trans values

z0.27 min�1 and 0.20bveb0.45. These voxels are most likely associated

with highly perfused and leaky vessels with increased extravascular–

extracellular space volume fractions over healthy tissue (vec0.10).


and directly related to the amount of error in the RR

parameters. Many literature values for muscle Ktrans and veare within the 0.1 min�1 F30% and 0.1F30% explored

here. Neither parameter returns values worse than F30% of

the true value even in those cases when both Ktrans,RR and

ve,RR are off by that amount. In an elegant series of

experiments, Simpson et al. [16] showed that incorrect

characterization of the AIF can lead to errors in perfusion

estimation by as much as 60%. Nevertheless, these errors

are still of concern and we therefore propose a variation on

Eq. (16) in the Discussion.

4.2. Experimental

Fig. 5 displays an axial slice through the tumor hind limb

at t=0 (panel A) and 40 min obtained from this study, as well

as the corresponding T10 map (panel C). The tumor volume

was calculated at 536.0F26.8 mm3. The signal-to-noise ratio

is approximately 52 and 35 for a 20-voxel TOI and an

arbitrary voxel, respectively — both above the range

explored in the above simulations. There is a significant

change in signal intensity from panels A to B in the tumor

periphery, and to a much lesser extent, in the tumor core. The

T10 map is reasonable (for 7.0 T) with most muscle voxels

between 1.45 and 1.85 s, while most tumor voxels are

between 1.8 and 2.2 s. The t=4-min frame indicates two

voxel groups used for TOI analysis (red circles denote a

rapidly enhancing region, while the blue circle indicate a

slowly enhancing region), as well as the RR (white circle)

that was used for both the ROI and voxel analysis. The Fig. 6

circles, triangles and squares indicate the RR, a slowly

enhancing TOI (blue circle in Fig. 5) and a rapidly enhancing

TOI (red circle in Fig. 5), respectively. The results of the

fitting routines are displayed as solid (slowly enhancing) and

dashed (rapidly enhancing). Again, the fits are good and the

parameters are well within the range of reported values for a

9 pixel TOIs22 pixel RR

time (min)

0 10 20 30 40

R1

(s-1

)

0.4

0.8

1.2

1.6

RR slowly enhancing TOIrapidly enhancing TOI

Fig. 6. The results of the RR model fits to data taken from the TOIs labeled

in Fig. 5. The model accommodates both the rapidly enhancing data (filled

squares) and the slowly enhancing data (filled triangles). The data curve

used as the RR for every slice in the study is depicted as the filled circles.

number of tumor types [3,4,9,32]: Ktrans=0.039F0.002

min�1 and ve=0.46F0.01 for the slowly enhancing region,

Ktrans=0.35F0.05 min�1 and ve=0.31F0.01 for the rapidly

enhancing region. We proceed to voxel-by-voxel analysis to

construct pharmacokinetic parameter maps of Ktrans and ve.

As noted previously, the tumors were manually outlined

and each voxel within the outline was fit with Eq. (16) and

characterized by a Ktrans and ve value. These parameter

values are then assigned a color. All voxels for which the

fitting routine did not converge or return physical values are

colored black. The results of these mappings for slices 4–6

(representative slices) and the associated color scale are

displayed in Fig. 7. (The bottom slice is that depicted in

Fig. 5). First, consider the Ktrans parametric map of panel A.

The superior portion of the tumor displays many red voxels

indicating Ktrans values z0.27 min�1. These voxels are

most likely associated with highly perfused and leaky

vessels where angiogenesis-mediated neovascularization is

occurring, as is commonly the case in the periphery of many

tumor types [33,34]. A similar pattern, though not as

pronounced, is also seen in the inferior portion of the tumor.

Both regions slowly fade into the central portion of the


tumor, which is characterized by reduced perfusion and/or

permeability (Ktransb0.15 min�1) and many black voxels.

Since nearly all of these voxels occur within the tumor core,

we assume that these are located in dense necrotic regions that

are poorly vascularized as well as containing high intra-

tumoral pressure to prevent both active and passive delivery,

respectively, of the CA to those voxels. As mentioned in the

Discussion, this is a presumption that needs to be verified by

comparison with histology (which we are actively pursuing).

Moving from panel A to C to E, the number of red pixels

greatly diminishes in the superior portion of the tumor, while

the inferior portion of the tumor maintains high perfusion

and/or vessel permeability. All slices seem to display a central

necrotic zone as manifested by many black pixels for which

there was little or no enhancement.

The ve parameter map of slice 4 (panel B) displays, in the

superior and inferior portions of the tumor, increased

extravascular–extracellular space volume fractions

(0.20bveb0.45) over healthy tissue (ve~0.10). Again, the

values reach a maximum in the tumor periphery and begin

to decrease toward the tumor core. This pattern is seen in

both panels D and F as well, though the central necrotic

zone appears to expand just as in the Ktrans maps.

5. Discussion

We have presented a method by which DCE-MRI data

can be quantitatively analyzed on a voxel-by-voxel basis for

the extravasation transfer constant, Ktrans, and extravascular–

extracellular space, ve, without direct measurement of the

AIF. The assumptions inherent in the method are those

common to all compartmental models (Eq. (1)); principally,

that the subject’s body may be represented by one or more

pools, or bcompartments,Q into and out of which the CA

dynamically flows, and that each compartment is assumed

to be bwell mixedQ in the sense that CA entering the

compartment is immediately distributed uniformly through-

out the entire compartment. The method is fast [the results

presented here indicate that the method can analyze an entire

1282 DCE-MRI data set in less than 5 min on a Pentium P4

(Intel, Santa Clara, CA) running at 2.4 GHz], easily applied

and straightforward in implementation, thereby making it

useful in, for example, experiments to measure tumor

kinetics before and after treatment. The results are reason-

able and consistent with other methods that attempt to

measure the AIF directly. We now discuss some of the

assumptions inherent in the RR model.

It should be noted that an additional assumption inherent

in Eqs. (11) and (12) (and, indeed, nearly all of the DCE-

MRI literature), though simplifying, may not be accurate. It

has recently been shown that there frequently exists signi-

ficant water exchange effects between separate compart-

ments, and this can lead to errors in the analysis of dynamic

MR data. Significant transendothelial water exchange

effects have been seen in arterial spin labeling [35,36] and

DCE-MRI data [37,38], while significant transcytolemmal

water exchange effects have been seen in diffusion-

weighted [39] and DCE-MRI [24,25,28] data. We acknowl-

edge that the theory presented here does not account for

those complicating factors, and we are currently working to

modify Eq. (16) to account for water exchange affects. We

also note that though Fig. 1 implies that the RR and TOI are

in close spatial proximity, they are actually separated by tens

of millimeters as indicated by Fig. 5. Thus, water exchange

between the RR and TOI compartments should not be

incorporated into the model.

The simulations reveal that systematic errors in Ktrans,TOI

and ve,TOI may be caused by incorrect assignments of

Ktrans,RR and ve,RR. That is, there could be both intra- and

interanimal variation in RR parameter values, and by

assigning literature reported values (particularly to

Ktrans,RR), systematic error will be introduced as manifested

by the above simulations. Thus, both the assignment of RR

parameters (interanimal variation) and the selection of the

RR itself (intraanimal variation) can confound the results

obtained by the RR model. An alternate approach would be

to formulate Eq. (16) in terms of ratios of Ktrans,TOI to

Ktrans,RR and ve,TOI to ve,RR, thereby allowing for the

production of relative Ktrans and ve maps. This could

potentially reduce the model’s systematic error due to

incorrect RR parameter assignments since a model reporting

relative parameter values requires no assignments on the

RR. If we make the following assignments:

R1uRuK trans;TOI=K trans;RR ð22Þ

R2uK trans;RR=ve;RR ð23Þ

R3uK trans;TOI=ve;TOI ð24Þ

then Eq. (18) may be expressed as Eq. (25):

R1;TOI Tð Þ ¼ R1bðR1;RRðTÞ � R10;RRÞ

þ R1½R2 � R3�bZT

0

ðR1;RRðtÞ � R10;RRÞ

� expð � R3 T � tð ÞÞdt þ R10TOI: ð25Þ

By noting that (R1/R3)R2=ve,TOI/ve,RR, Eq. (25) can pro-

vide a three-parameter fit to a DCE-MRI time course to

extract rKtrans(R1), rve((R1/R3)R2) and rkep(R

3) values (kepis the rate constant describing the flow of CA from the

extravascular–extracellular space to the blood plasma [1]),

where brQ denotes relative. Such relative measurements have

been shown to be of clinical value, perhaps most noticeably

in investigations of cerebral perfusion. Our preliminary

results have shown that Eq. (25) is very robust to exper-

imental noise, returns accurate values and slightly increases

standard deviations over the Eq. (16) model — as one would

expect for a model with an additional degree of freedom.

Furthermore, we have made some progress on applying an

iterative approach in which Eq. (25) is used to first fit an


enhancement curve (to acquire relative parameter measure-

ments), and then Eq. (16) is used to conduct a full four

parameter fit (Ktrans,TOI, Ktrans,RR, ve,TOI and ve,RR) subject to

the results of the relative parameter values. If realized, such an

approach would eliminate the requirement for assumptions

on the RR and effectively eliminate the problems associated

with inter- and intraanimal variation discussed previously.

This represents another advantage of working with the

integral form of the Kety equation rather than the differential

version as in Ref. [22]; in Kovar et al., RR values must be

assumed to acquire the AIF. In the iterative approach just

described, one need not do that and it is a direct consequence

of formulating the RR method as an integral equation.

In conclusion, we have proposed a simple compartmental

model that allows for estimation of the Ktrans and vepharmacokinetic parameters on a voxel-by-voxel basis

without needing to characterize the AIF. The method needs

to be verified by correlation with accepted methods such as,

for example, standard histological analysis. To improve the

accuracy of the analysis, Eq. (16) needs to be amended to

account for transcytolemmal water exchange. It also needs

to be verified that the RR method can detect subtle

longitudinal changes in tumor vasculature so that the

approach could be applied in, for example, studies

evaluating the efficacy of novel anticancer therapies.

Acknowledgments

We thank Dr. Calum Avison for reviewing an early

version of this manuscript. We thank Drs. Adam Anderson,

Bruce Damon, Natasha Deane, Robert Kessler and Kenneth

Niermann for many stimulating and informative discussions.

Mr. Richard Baheza and Mr. George Holburn gave excellent

assistance in managing technical imaging and animal care

issues. We thank the National Institutes of Health for funding

through NCI 1R25 CA92043 and 5RO1 EB00461.

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