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The accuracy of pharmacokinetic parameter measurement in DCE-MRI of the breast at 3 T
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2010 Phys. Med. Biol. 55 121
(http://iopscience.iop.org/0031-9155/55/1/008)
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IOP PUBLISHING PHYSICS IN MEDICINE AND BIOLOGY
Phys. Med. Biol. 55 (2010) 121–132 doi:10.1088/0031-9155/55/1/008
The accuracy of pharmacokinetic parameter
measurement in DCE-MRI of the breast at 3 T
P Di Giovanni1,4, C A Azlan1,2, T S Ahearn1, S I Semple3, F J Gilbert1
and T W Redpath1
1Aberdeen Biomedical Imaging Centre, University of Aberdeen, Aberdeen, UK
2Department of Biomedical Imaging, Faculty of Medicine, University of Malaya, Kuala Lumpur,
Malaysia
3Medical Physics, University of Edinburgh, Edinburgh, UK
E-mail: pierluigi.di.giovanni@abdn.ac.uk
Received 8 July 2009, in final form 23 October 2009
Published 10 December 2009
Online at stacks.iop.org/PMB/55/121
Abstract
The purpose of this work is to quantify the accuracy of pharmacokinetic
parameter measurement in DCE-MRI of breast cancer at 3 T in relation to
three sources of error. Individually, T1 measurement error, temporal resolution
and transmitted RF field inhomogeneity are considered. Dynamic contrast
enhancement curves were simulated using standard acquisition parameters of
a DCE-MRI protocol. Errors on pre-contrast T1 due to incorrect RF spoiling
were considered. Flip angle errors were measured and introduced into the
fitting routine, and temporal resolution was also varied. The error in fitted
pharmacokinetic parameters, Ktrans and ve, was calculated. Flip angles were
found to be reduced by up to 55% of the expected value. The resultant errors
in our range of Ktrans and vewere found to be up to 66% and 74%, respectively.
Incorrect T1 estimation results in Ktrans and veerrors up to 531% and 233%,
respectively. When the temporal resolution is reduced from 10 to 70 s Ktrans
drops by up to 48%, while veshows negligible variation. In combination,
uncertainties in tissue T1 map and applied flip angle were shown to contribute
to errors of up to 88% in Ktrans and 73% in ve. These results demonstrate the
importance of high temporal resolution, accurate T1 measurement and good
B1 homogeneity.
1. Introduction
Malignant breast cancer is the most common type of cancer amongst young women in the
world (Kamangar et al 2006); DCE-MRI has been shown to be a sensitive and predictive
technique for the characterization of breast lesions (Kuhl et al 2000, Hayes et al 2002,Yuet al
4Author to whom any correspondence should be addressed.
0031-9155/10/010121+12$30.00 © 2010 Institute of Physics and Engineering in Medicine Printed in the UK 121
122 P Di Giovanni et al
2007). During a DCE-MRI study, a small molecular weight contrast agent (e.g. Gd-DTPA) is
administered by intravenous injection and spreads into the interstitial space of the body where
it stays until renal filtration occurs. The uptake of contrast agent in tumours is greater than in
the surrounding tissues due to the leaky nature of tumour neovasculature. The contrast agent
concentration within the lesion area is tracked over a period of few minutes with a fast T1-
weighted sequence. A dynamic signal enhancement curve is the result of the contrast medium
exchange between different biological compartments of the tumour microenvironment. A
pharmacokinetic model can be used to mathematically describe the passage of the contrast
agent. This is done by defining discrete compartments within the body which the contrast
agent can move between. A commonly used pharmacokinetic model has been described by
Tofts and Kermode (1991). Compartments representing the intravascular space, extravascular
space and tumour are defined. Contrast agent is cleared from the system by the glomerular
filtration in the kidneys. The exchange of contrast medium between the intravascular and
extravascular compartments of the model is regulated by two quantities, the transfer constant
Ktrans and the fractional volume of interstitial space ve. Estimating these parameters allows us
to classify different types of tumours and to monitor their response to a given therapy (Hayes
et al 2002, George et al 2001, Semple et al 2006, Pickles et al 2005).
The practical implementation of a DCE-MRI scan depends on parameters such as the T1-
weighted imaging sequence used, the accuracy of the measurement of the tissue’s native T1
and the temporal resolution of the dynamic scan. The choice of these parameters is constrained
by the need for maintaining a good balance between spatial and temporal resolution (Li et al
2007).
To account for the dependence of signal enhancement on both concentration of contrast
agent and native T1, a baseline T1 map over the tumour area is commonly acquired prior to
contrast injection. When imaging patients, this must be performed in a short time. This implies
that an inversion recovery (IR) approach is not suitable and so a multiple flip angle T1-weighted
FLASH approach (Brookes et al 1999)oraT
1-weighted saturation-recovery-TurboFLASH
method (Brix et al 2004) is commonly used. The multiple flip angle T1-weighted FLASH
method consists of acquiring a set of images with different flip angles (e.g. 6◦,10
◦and 35◦),
with the native tissue T1 estimated by fitting the obtained MR signals to the FLASH equation.
The saturation-recovery-TurboFLASH approach extracts a measure of tissue T1 by combining
an effective saturation scheme with the MR signal obtained at different recovery times.
If a multiple flip angle approach is chosen, good B1 homogeneity is needed, and it is
crucial to verify by phantom experiments that the MR signal can be modelled according to
the theoretical FLASH equation. As demonstrated by Preibisch and Deichmann (2009), when
alowTR(e.g.<10 ms) is used, RF spoiling can be far from effective and this will affect
the T1-weighted signal producing a biased estimate of T1 and thus of the pharmacokinetic
parameters. This latter case will be analysed in this work using software simulations. An error
in the estimated tissue T1 has a significant and known impact on the parameters Ktrans and ve
(Tofts et al 1995).
T1-weighted FLASH is the sequence of choice in most clinical dynamic DCE-MRI
protocols (Heywang-K¨
obrunner et al 2001, Vomweg et al 2004). In order to cover both
breasts in a short time, a short TR/TE (∼10 ms/5 ms) is chosen. The flip angle has to be high
enough to obtain a reasonable SNR from the imaged tissues but cannot be too high for the
given TR in order to avoid signal saturation. The transmitted RF field inhomogeneity (referred
to as ‘B1 inhomogeneity’ throughout the paper) is assumed to be small up to 1.5 T, but it has
been shown to become an issue at 3 T (Dietrich et al 2008). When both breasts are scanned
in the axial plane, a large amount of signal change can be observed across the field of view
(Kuhl et al 2007).
Breast pharmacokinetic parameter estimation 123
Henderson et al (1998) showed by software simulations how the width of an arterial
input function together with dynamic temporal resolution influences the accuracy of the
fitted physiological parameters. Another simulation study conducted by Lopata et al (2007)
demonstrated that the higher the value of Ktrans is, the higher the sampling frequency of
the dynamic scan must be in order to obtain an accurate estimate of the pharmacokinetic
parameters.
The aim of this work is to evaluate, by simulation, the effect of temporal resolution on the
estimation of the physiological parameters Ktrans and ve. Also, the B1 inhomogeneity in axial
breast imaging at 3 T is measured using normal volunteers. Simulations are then performed
that demonstrate how this B1 error propagates into the pharmacokinetic parameters Ktrans
and ve. Lastly, potential errors in pre-contrast T1 calculation and their effect on parameter
measurement are assessed.
2. Materials and methods
2.1. Dynamic protocol parameters and data fitting
With a spoiled T1-weighted FLASH sequence, a TR/flip angle combination of 10 ms/15◦
was chosen. We assumed a very short echo time and thus negligible effect of T2∗relaxation
time on the MR signal. The T1 of glandular tissue at 3 T was set to 1.3 s (Rakow-Penner
et al 2006), while the T1 relaxivity of Gd-DTPA at 3 T was given by 4.5 mM−1s−1(Sasaki
et al 2005). We considered the Tofts model assuming a bolus injection of 0.1 mmol kg−1and a
dynamic scan lasting for more than 6 min. We also assumed that full coverage of both breasts
in a 3D imaging mode could be achieved in 10 s with a reasonable spatial resolution. In each
simulation the contrast concentration was derived from the signal dynamic curve by fitting the
data to the following equation:
SCM(t )
S0(t)
=1−exp −TR1
T10 +r1C(t)1−cos αexp −TR
T10
1−cos αexp −TR1
T10 +r1C(t)1−exp −TR
T10 (1)
where C(t) is the tissue contrast concentration, SCM(t ) is the MR signal with contrast and
S0(t) is the baseline MR signal before contrast injection, T10 is the pre-contrast T1 value of the
tissue, αis the flip angle and r1is the relaxivity value of Gd-DTPA at 3 T. The simulated MR
signal has been generated without any added random noise in order to evaluate the systematic
errors affecting the pharmacokinetic parameters due to three specific confounding factors,
these will be analysed in the next sections. In accordance with the Tofts model, the tissue
contrast concentration is given by the following equation (Tofts 1997):
C(t) =DKtrans
2
i=1
aT
ie−t(Ktrans
ve)−e−tmi
mi−Ktrans
ve(2)
where Dis the injected dose (0.1 mmol kg−1), aT
iand miare parameters describing the
plasma biexponential decay as measured by Weinmann et al (1984). Ktrans and veare the free
parameters fitted in our simulations. We have assumed that the plasma volume fraction is
negligible (vp=0) even though this might not be the case in breast tissue; however, we do
not believe that this would affect the results of the simulations.
The fitting procedure was based on the Levenberg–Marquardt algorithm and implemented
in IDL (Research Systems, Inc., Boulder, Colorado). Using multiple starting points for
Ktrans and vein the Levenberg–Marquardt algorithm has been shown to significantly improve
robustness of the technique, as this prevents the fitting routine falling in local minima (Ahearn
124 P Di Giovanni et al
et al 2004). A simulated signal enhancement curve was generated with 10 s temporal resolution
for a range of Ktrans and vefrom 0.5 min−1/0.3 to 2.0 min−1/0.7, respectively, these values of
Ktrans were observed in a clinical study (Hayes et al 2002), they represent a strong enhancing
and weakly enhancing lesion, while the chosen verange covers the spectrum of possible values
going from 0.0 to 1.0 (Planey et al 2009). In fitting the pharmacokinetic parameter Ktrans,an
upper threshold of 4.0 min−1was chosen.
2.2. Temporal resolution
We assumed that the central k-space lines of the dynamic images were acquired every 10 s in
a linear k-space sampling mode; we simulated lower temporal resolutions by averaging the
signals of two or more adjacent signal samples. This means that for a temporal resolution
of 20 s, two consecutive samples were averaged (Planey et al 2009) and the average value
was placed in a time location between the two original samples, and for lower resolutions, an
increasing number of points was averaged. This process was repeated at steps of 10 s up to
70 s temporal resolution. On the new time points, a dynamic curve was fitted according to the
Tofts model and the obtained Ktrans and vevalues were then compared with their true values.
Throughout the work error will be expressed as relative difference to the true value of
Ktrans and vein per cent according to the following equation:
Error (%)=100 ×(Fitted value −True value)/(True value). (3)
2.3. T1 error due to a multiple flip angle spoiled FLASH approach
An estimation of the T1 measurement error resulting from adopting a multiple flip angle
FLASH approach is described in the work of Preibisch and Deichmann (2009). Errors up
to +14% and −65% of the original T1 value were observed when incorrect RF spoiling in
combination with two flip angles (4◦and 18◦) and a short TR (7.6 ms) was applied. In order
to estimate how these errors affect Ktrans and ve, the Tofts model was fitted to simulated MR
signal samples generated with a native T1 equivalent to that of ductal tissue at 3 T. The range
of nominal Ktrans and vevalues described in section 2.1 was used introducing the T1 error in
the fit (equation (1)). The fitted pharmacokinetic values were then compared with their true
values.
2.4. B1 inhomogeneity effect
The amount of B1 inhomogeneity was assessed on a 3 T MRI scanner (Philips Achieva X
Series, Best, The Netherlands) by generating B1 maps of 25 healthy volunteers on the axial
plane using a seven-channel breast receiving coil and transmission through the conventional
whole body coil. The local Research Ethics Committee approved the study, and informed
consent was provided by the participants. B1 maps were produced with the technique ‘actual
flip-angle imaging’ (Yarnykh 2007), which is based on two acquisitions with different TRs
(TR1/TR2/TE/α =30 ms/150 ms/2.4 ms/60◦). The highest flip angle error across the field
of view was then used to quantify the equivalent error in the estimated Ktrans and ve.Asinthe
temporal resolution simulation, the starting values of Ktrans and veranged from 0.5 min−1/0.3
to 2.0 min−1/0.7. A range of dynamic signal curves was generated using flip angles higher
and lower than a nominal value of 15◦, and then each curve was fitted assuming that the flip
angle delivered by the RF system was equal to the nominal one. The new sets of Ktrans and ve
values were then compared with their true values.
Breast pharmacokinetic parameter estimation 125
2.5. Combined error effect on fitted Ktrans and ve
So far the bias affecting the pharmacokinetic parameters has been treated independently
for the three sources of error; we now sum together these components and derive the total
Ktrans and veerror for two values of ve(0.3 and 0.7) and two values of Ktrans (0.5 min−1and
2.0 min−1associated with a lower and a higher perfusion state in tumours). This was performed
by incorporating the native T1 error and the B1 inhomogeneity effect into a simulation where
the temporal resolution was chosen 10 s, the errors in the fitted parameters were then calculated.
3. Results
3.1. Temporal resolution
Figure 1shows plots of error in fitted parameters Ktrans and vevarying temporal resolution for
nine combinations of their true values. At 10 s temporal resolution, both Ktrans and veerrors
are zero (figure 1(c)) because that is the resolution at which the original simulated dynamic
curve was generated, so that the points lie precisely on the ‘true’ curve. For a veof 0.3, and
a temporal resolution of 70 s, Ktrans drops by 17% when its starting value is 0.5 min−1,by
36% when its starting value is 1.2 min−1and by 48% when its starting value is 2.0 min−1
(figure 1(a)). Figure 1(c) shows how Ktrans is underestimated when the temporal resolution
is decreased. This effect increases with Ktrans and is more pronounced at lower temporal
resolutions. Figures 1(b) and (c) show little variation of the vefit with respect to the temporal
resolution. The highest variation of veis an increase by 2% of its true value (0.7) at 70 s when
the true Ktrans is 0.5 min−1. However, figure 1(a) shows a clear ve-dependent underestimation
of Ktrans which is more severe for low vevalues.
3.2. T1 measurement error due to a multiple flip angle spoiled FLASH approach
Figure 2shows the calculated Ktrans and veerrors when the observed T1 error is applied to
equation (1) and different true values of Ktrans and veare used. Figures 2(a) and (b) show the
plots of Ktrans error for different true values of veand Ktrans associated with the two highest
values of native tissue T1 error, +14% error in figure 2(a) and −65% error in figure 2(b). Ktrans
is overestimated if the tissue’s T1 is underestimated while is less severely underestimated if
the tissue’s T1 is overestimated (figure 2(e)). The highest error (531%) in estimating Ktrans
is given when the ductal native tissue T1 is underestimated by 65% of its nominal value and
is associated with a true Ktrans of 0.5 min−1andatrueveof 0.5 (figure 2(b)). In this case,
the fitting routine is forced to set veto its upper threshold (ve=1.0); similarly for a nominal
Ktrans of 1.2 min−1and 2.0 min−1, the estimated Ktrans equals its upper threshold (4.0 min−1).
Figures 2(c) and (d) show the plots of the associated veerrors. veestimation hits its upper
threshold when T1 is 65% lower than its nominal value for any combination of true Ktrans and
vevalues. veincreases by 233% if its true value is 0.3, by 100% if its true value is 0.5 and
by 43% if its true value is 0.7. Figure 2(e) represents a typical error pattern when the native
tissue T1 error oscillates between +14% and −65%; in this case, the nominal Ktrans is equal to
1.2 min−1and the nominal veis 0.3.
3.3. B1 inhomogeneity effect
From the volunteer B1 maps, errors of up to around 55% of the nominal flip angle were
observed (Azlan et al 2009). This is in agreement with the work of Kuhl et al (2007). There
was a marked left–right difference, and the flip angle was significantly lower than the nominal
126 P Di Giovanni et al
0
10
20
30
40
50
0.00.51.0 1.52.02.53.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Percent error - underestimation (%)
Ktrans (min -1)
ve
Ktrans error with a temporal resolution of 70 s
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
0.5 1.0 1.5 2.0 2.5 3.0 0.10.2 0.30.40.5 0.60.70.8 0.9
Percent error (%)
Ktrans (min -1)
ve
ve error with a temporal resolution of 70 s
Temporal resolution (s)
0 1020304050607080
Percent error (%)
-40
-30
-20
-10
0
10
Ktrans error
ve error
(a) (b)
(c)
Figure 1. Software simulation results at 70 s temporal resolution, Ktrans error (a) and veerror
(b) at different combinations of true Ktrans values (0.5, 1.2 and 2.0 min−1)andvevalues (0.3, 0.5
and 0.7). (c) Example of error changes induced by dropping the temporal resolution when the true
Ktrans and veare equal to 1.2 min−1and 0.3, respectively.
flip angle on one side depending on whether the patient is positioned in the magnet bore head
or feet first. Figure 3shows plots of Ktrans and veerror versus flip angle error obtained with
the software simulations (with respect to the nominal 15◦flip angle). Figures 3(a), (b) and
(e) show how Ktrans error increases with flip angle error. Also, when the applied flip angle is
lower than the nominal angle, Ktrans is underestimated, while it is less severely overestimated
when the flip angle is higher than the nominal (figure 3(e)).
Measured Ktrans can drop by up to 66% from its true value of 2.0 min−1(ve=0.7) when
there is an underestimation of the flip angle by 55% of its nominal value (figure 3(a)). On
the other hand, if the flip angle is overestimated by 55% of its nominal value, Ktrans becomes
overestimated by up to 61% of its expected value (figure 3(b)). As veincreases, the sensitivity
of Ktrans to B1 inhomogeneity increases. Figures 3(c)–(e) show that veis underestimated when
the applied flip angle is lower than the nominal flip angle and is less severely overestimated
when the applied flip angle is higher than the nominal one. Different from Ktrans,veshows
Breast pharmacokinetic parameter estimation 127
0
5
10
15
20
0.0 0.5 1.0 1.5
2.0 2.5
3.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Percent error - underestimation (%)
Ktrans (min
-1
)
ve
Ktrans error with tissue T1 value 14% higher than nominal one
0
100
200
300
400
500
600
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Percent error (%)
Ktrans (min
-1
)
ve
Ktrans error with tissue T1 value 65% lower than nominal one
0
5
10
15
20
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Percent error - underestimation (%)
Ktrans (min
-1
)
ve
ve error with tissue T1 value 14% higher than nominal one
0
50
100
150
200
250
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Percent error (%)
Ktrans (min
-1
)
ve
ve error with tissue T1 value 65% lower than nominal one
Native tissue T1 error (%)
-80 -60 -40 -20 0 20
Percent error (%)
-50
0
50
100
150
200
250
Ktrans error
ve error
(a) (b)
(c) (d)
(e)
Figure 2. Software simulation results at native tissue T1 error of +14% and −65%,Ktrans error
(a), (b) and veerror (c), (d) at different combinations of true Ktrans values (0.5, 1.2 and 2.0 min−1)
and vevalues (0.3, 0.5 and 0.7). (e) Example of error changes induced by a bias in tissue native
T1 when the true Ktrans and veare 1.2 min−1and 0.3, respectively.
128 P Di Giovanni et al
0
20
40
60
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Percent error - underestimation (%)
Ktrans (min
-1
)
ve
Ktrans error with flip angle 55% lower than nominal one
0
10
20
30
40
50
60
70
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Percent error (%)
Ktrans (min -1)
ve
Ktrans error with flip angle 55% higher than nominal one
0
20
40
60
80
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Percent error - underestimation (%)
Ktrans (min
-1
)
ve
ve error with flip angle 55% lower than nominal one
0
10
20
30
40
50
60
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Percent error (%)
Ktrans (min
-1
)
ve
ve error with flip angle 55% higher than nominal one
Flip an
g
le error (%)
-80 -60 -40 -20 0 20 40 60 80
Percent error (%)
-80
-60
-40
-20
0
20
40
Ktrans error
ve error
(a) (b)
(c)
(e)
(d)
Figure 3. Software simulation results at flip angle errors of ±55%,Ktrans error (a), (b) and veerror
(c), (d) at different combinations of true Ktrans values (0.5, 1.2 and 2.0 min−1)andvevalues (0.3,
0.5 and 0.7). (e) Example of error changes induced by a bias in the applied flip angle when the
true Ktrans and veare 1.2 min−1and 0.3, respectively.
the same increase/decrease independent of the associated Ktrans (no more than 1% variation).
When veis as high as 0.7 and the flip angle is underestimated by 55% of its original value,
Breast pharmacokinetic parameter estimation 129
0
20
40
60
80
100
0.5
2.0
0.3
0.7
Percent error (%)
Ktrans (min
-1
)
ve
Ktrans error for -65% T1 error and -55% flip angle error
0
20
40
60
80
0.5
2.0
0.3
0.7
Percent error (%)
Ktrans (min
-1
)
ve
ve error for -65% T1 error and -55% flip angle error
(a) (b)
Figure 4. Software simulation results given by the native tissue T1 error of −65% combined with
a flip angle error equal to −55%. The error affecting Ktrans (a) and ve(b) has been plotted for a
different range of nominal Ktrans and vevalues.
veis underestimated by 74% of its expected value (figure 3(c)). When the flip angle is 55%
over its nominal value, veis overestimated by 43% of its expected value (figure 3(d)). In this
latter case (ve=0.7) fitted vereaches its upper threshold. Figure 3(e) represents a typical
error pattern when the applied flip angle is between −55% and +55% of its nominal value; in
this case, the nominal Ktrans is equal to 1.2 min−1and the nominal veis 0.3.
3.4. Combined error effect on fitted Ktrans and ve
The effects of the two major error sources, a T1 error of −65% and a flip angle error of −55%,
have been combined in a single simulation, and the results are reported in figure 4. The highest
error affecting Ktrans is 88%, and it was recorded for the combination of true Ktrans and vevalues
0.5 min−1and 0.3, respectively (figure 4(a)). For the same combination of nominal Ktrans and
vevalues, the highest error was recorded for estimated ve, which is equal to 73% (figure 4(b)).
4. Discussion
This work was focused on three sources of error for the quantitative tumour characterization
analysis by DCE-MRI using the Tofts model. These are temporal resolution, B1 inhomogeneity
(flip angle error) and native T1 calculation error. Our model has shown how each error source
affects the estimated Ktrans and vepharmacokinetic parameters when treated independently
and when the two major sources of error (native tissue T1 error and B1 inhomogeneity) are
combined together. Our simulations have been run with a range of nominal Ktrans and vevalues
in a way to build an error look-up table applicable in different clinical situations. Our model
assumes instantaneous mixing of contrast into the plasma space after injection; however, there
are other approaches which use the idea of arterial input function (Van Osch et al 2003,
Johnson et al 2004). The latter method can improve the pharmacokinetic model, but it is also
highly dependent on factors such as the choice of the region of interest on a well-defined artery
and the in-flow effect due to the moving spins.
130 P Di Giovanni et al
The simulated signal enhancement curves have been generated without any added random
noise; this was done in order to extract the systematic error on Ktrans and vedue to three error
sources only. However, in a clinical environment, the pharmacokinetic parameters estimate
will also be affected by unpredictable signal variations due to factors such as electrical noise
and patient movement.
In the previous work of Tofts et al (1995), the tissue native T1 effect on the pharmacokinetic
parameters has been expressed as the error propagation ratio (fractional change in fitted
parameter/fractional change in confounding parameter). While this was equal to 1 in their
work (e.g. a 50% change in T1 causes a 50% error in the fitted Ktrans or ve), it takes higher
values in our simulations because of the different flip angle used (15◦instead of 60◦).
In several simulations, we found that the error magnitude introduced by the confounding
factors indicated that the model was forced to choose the upper threshold of the
pharmacokinetic parameters (i.e. 1.0 for veand 4.0 min−1for Ktrans) in order to give the
best fit. This has to be carefully considered when analysing the errors derived from
the simulations, for instance an error in veequal to 100% of its true value when this is
0.5 and an error equal to 25% of its true value when this is 0.8 suggests that in both cases the
upper threshold of vewas reached (ve=1.0).
From the combined error analysis emerged how, if one confounding factor leads to the
overestimation of a pharmacokinetic parameter (e.g. see the effect of low native tissue T1 on
Ktrans), while a second error source leads to its underestimation (e.g. see the effect of the low
flip angle on Ktrans), these tend to cancel each other out (compare figures 2(b) and 3(a) with
figure 4(a)).
A measure of reduced chi-square was derived from each fit, and the highest values were
found in the native tissue T1 error simulations as expected given the high errors introduced
into the pharmacokinetic results. A better fit was given by the B1 inhomogeneity simulation,
while the lowest error introduced into Ktrans and veby dropping the temporal resolution was
reflected in the associated low reduced chi-square values. In figure 5an example of goodness
of fit versus each of the three confounding factors analysed so far is given.
Assuming that an error higher than 10% in either Ktrans or vecannot be tolerated in clinical
practice, for a tumour with a clinically high Ktrans of 2.0 min−1(Hayes et al 2002) and a veof
0.3 it can be seen that lowering the temporal resolution of the dynamic series has a negligible
effect on ve(figure 1(b)). However, to ensure that Ktrans is within 10% of its true value, a
temporal resolution higher than 20 s (14% error) must be used. Ktrans and veare estimated
with an error lower than 10% when T1 is overestimated by less than 7% of its nominal value
(91 ms) or when T1 is underestimated by less than 6% of its nominal value (78 ms). Concerning
the B1 inhomogeneity effect, in order to have a maximum of 10% error in Ktrans and ve,the
actual flip angle cannot be lower than 11% of its nominal value (13.3◦for a 15◦flip angle) or
higher than 13% of its nominal value (16.9◦for a 15◦flip angle).
The error analysis presented in this work can be brought forward in several ways, for
instance the effect of B1 inhomogeneity has been considered on the dynamic MR signal but
not on the T1 mapping procedure where we have explored the effect of incorrect RF spoiling
only. If both effects are considered simultaneously for T1 mapping, different errors will be
generated in fitted Ktrans and ve.
The T1 map errors due to a multiple flip angle FLASH method can be corrected in three
ways. In theory, the TR could be extended (e.g. >100 ms) but since this affects the total length
of the image acquisition process it is clinically impractical. Alternatively the RF spoiling
scheme can be improved (Preibisch and Deichmann 2009) or a calibration technique could be
used.
Breast pharmacokinetic parameter estimation 131
Temporal resolution (s)
0 1020304050607080
Goodness of fit - reduced Chi-Square
0
5e-5
1e-4
1e-4
2e-4
K
trans
= 0.5 min
-1
- v
e
= 0.3
K
trans
= 1.2 min
-1
- v
e
= 0.5
K
trans
= 2.0 min
-1
- v
e
= 0.7
Native tissue T1 error (%)
-80 -60 -40 -20 0 20
Goodness of fit - reduced Chi-Square
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Ktrans
= 0.5 min-1
- v
e = 0.3
Ktrans
= 1.2 min-1
- v
e = 0.5
Ktrans
= 2.0 min-1
- v
e = 0.7
Flip angle error (%)
-80 -60 -40 -20 0 20 40 60 80
Goodness of fit - reduced Chi-Square
0.00
0.01
0.02
0.03
0.04
0.05
K
trans
= 0.5 min
-1
- v
e
= 0.3
K
trans
= 1.2 min
-1
- v
e
= 0.5
K
trans
= 2.0 min
-1
- v
e
= 0.7
(a) (b)
(c)
Figure 5. Example of goodness of fit as a measure of reduced chi-square in respect to the temporal
resolution (a), T1 error (b) and B1 inhomogeneity (c).
In conclusion accurate and robust pharmacokinetic parameter estimation is important if
the DCE-MRI is to be used in routine clinical practice. We have identified three sources of
error in parameter estimation and quantified their magnitude. Steps to improve native T1
calculation, B1 inhomogeneity and ensuring that the dynamic temporal resolution is not lower
than 20 s will minimize these errors.
Acknowledgment
The first author would like to thank Philips Healthcare for funding his PhD studentship.
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