DS
SC-MRI first-p
pass curve fittting and modeelling is improoved with a noovel cosine-based function
1
Matthew R Orrton1, James A d'A
Arcy1, Keiko Miyazzaki1, Nina Tunariiu1,2, David J Colliins1,2, and Martin O Leach1
CR
R-UK and EPSRC Cancer Imaging Centre,
C
Institute off Cancer Research,, Sutton, Surrey, U
United Kingdom, 2C
Clinical MRI Unitt, Royal Marsden H
Hospital, Sutton,
Surreyy, United Kingdom
m
Introd
duction
First-p
pass contrast-time curves from DSC
C-MRI data can bee characterized by fitting a suitable function to the daata, from which suummary parameterrs
(bolus arrival time, time--to-peak, maximum
m peak value, etc.)) can be derived [1
1]. In this applicattion the fitting proocedure is essentiaally a de-noising opperation, in that th
he
d in principle be (n
noisily) estimated from the data themselves. A less w
widely used appliccation is to fit a moodel to the data thhat includes specifiic
summaary measures could
parameeters describing th
he tissue propertiess (mean transit tim
me, blood volume and blood flow), in a similar mannner to the methodoology routinely useed to fit DCE-MR
RI
data [22]. In both applicaations the success of the technique depends on how well
w the model funnction describes ssuch data. In this abstract we propoose a new model to
t
describbe DSC-MRI first--pass data and dem
monstrate that it giv
ves improved fits to
t clinical DSC-M
MRI data comparedd with two establisshed models.
α
A gam
mma-variate functio
on of the form Atα–1
exp(–μt)/Γ(α) iss widely used for fitting first-pass ccontrast changes inn DSC-MRI data [3]. Less often ussed is a log-norma
al
functioon of the form At–11exp(– (ln(t) –μ)2/((2σ2)) [4], which is
i empirically motivated and has no physical interprettation. First-pass ccurves arise after a bolus injection of
o
contrasst into a peripheraal vein, after which the contrast passses through the riight-heart, the lunngs and the left-heeart before it arrives at the brain or other tissues. Th
he
transit times of these reg
gions will not be id
dentically distributted or have the sam
me mean value, wh
which means that thhe commonly citedd interpretation off the gamma-variatte
model as a series of α mixing chambers wiith equal exponenttially distributed trransit times [5] is uunsatisfactory.
We proopose a new modeel to describe DSC
C-MRI data that is based on the conv
volution between a raised-cosine funnction (of the form
m A(1 – cos(κt)) foor 0< t < 2π/κ), an
nd
a gamm
ma-variate. The id
dea behind this is that
t the gamma-vaariate is appropriatte for describing trransit times througgh regions that opeerate as simple mixxing chambers (e.g
g.
the heaart), while a moree symmetric functtion (the raised-co
osine function) is potentially better suited to modelinng transit times thhrough tissues conntaining a vascula
ar
networrk (e.g. the lungs). Other symmetricc functions (e.g. a Gaussian) are feassible and perhaps m
more theoreticallyy plausible, but thee raised-cosine hass the advantage tha
at
it is sym
mmetric, has a fin
nite duration in tim
me and the convolution can be analytiically solved givinng a function that ccan be easily and qquickly evaluated.
Methood The gamma-v
variate function, th
he log-normal fun
nction and three co
osine-based modeels with α = 1, 2 aand 3 were evaluaated by comparingg how well they fit
f
DSC-M
MRI data. Since all
a these models haave the same num
mber of unknown parameters
p
it is suffficient to comparre the residual sum
m of squared errorss (RSS) from leasttsquares fitting. We also compare the execu
ution time, numbeer of iterations for the fitting to convverge, and the execcution time per evaaluation of the model function.
Data A
Acquisition and Processing
P
Fourrteen patients with
h advanced glioblaastoma multiformee (GBM) were im
maged at DSC-MR
RI with a Philips A
Achieva 3T and th
he
following parameters: multi-slice
m
FE-EPI with
w 45 echoes and
d SENSE factor 2, TR/TE = 1554/400 ms, flip-angle = 75o, 25×4mm axiial slices, 962 acquuisition matrix, 224
42
OV, 40 dynamic points
p
at 1.5 sec/v
volume. Magneviist contrast agent was used at a dosse of 0.2ml/kg, deelivered at 3ml/seec using a power iinjector, and signa
al
mm FO
changees were converted
d to concentration changes assumin
ng exponential sign
nal dependence an
and a relaxivity off 4.4 mM/ms. Froom each data set a slice through th
he
cerebraal ventricles was selected
s
and an ROI drawn to inclu
ude the whole braiin, excluding the vventricles and anyy pathology. A cuut-off time was m
manually selected to
t
removee recirculation datta leaving only thee first pass curve, which
w
was fitted pixel-wise
p
using alll five models. A delay parameter w
was included in thhe model to accoun
nt
for varriations in the arriv
val time of the con
ntrast. Fitting was implemented in IDL (Research Syystems Inc, Bouldder, Colorado) runnning in Windows XP under VMwarre
Fusionn 3.1.3 on a Mac Prro 2.26 GHz Quad
d-Core Intel Xeon.
Formuula
RSS
Mean eexecution time/pix
xel (ms)
Mean iiterations per pixell
Mean eex. time per iteratiion (ms)
Gamma-variatte
Log-Norm
mal
Cos-Gamma
C
α=2
mma α = 3
Cos-Gaamma α = 1
Cos-Gam
Aexp(–μ
μt)⊗(1 – cos(κt)) Atexp(–μt)⊗(1 – cos(κt))
At2exxp(–μt)⊗(1 – cos((κt)) Atα–1exp(–μ
μt)/Γ(α) At–1expp(–(ln(t)–μ)2/(2σ2))
)
0.6596
0.6434
0
0.6644
1.1834
0.7611
130
54
5
360
74
58
436
143
1
705
220
209
0.32
0.38
0
0.51
0.34
0.27
Resultts and Discussion
n The mean overr pixels of the RSS
S was reported fo
or each patient andd the mean over thhe 14 patients is ggiven in the abovee table. The figurre
shows an example fit to the mean curve from
f
one patient. The cosine and non-cosine
n
models have been
T residuals plot indicates
i
the improovement in
plottedd on separate axes to better show theeir fit accuracy. The
the fit of the cosine-baseed models, and theeir similarity with each
e
other. Thesee figures are repressentative of
w
data set.
the pattterns seen in the whole
Overalll the three cosine--based models hav
ve very similar errrors: α = 2 is sligh
htly better than α = 1 and 3,
althouggh a paired t-test iss not significant fo
or all comparisons between these mo
odels. This impliees that there
is littlee to be gained by including
i
α as a fit
f parameter with these models, and
d that restricting α to integer
values has little impact. The execution tim
me per iteration off the cosine-based models
m
is longer ffor larger α
t
in the form
mula with the conv
volution written out), although the number of
(due too the number of terms
iteratioons per pixel mean
ns that this pattern
n is not seen in th
he overall executio
on time. Overall, the model
with α = 2 is preferred am
mong the cosine-b
based models due to
t its fitting accuraacy and more impoortantly, its
executiion time.
The coosine-based modeels have lower errrors than the gam
mma-variate and log-normal modeels, and all
compaarisons between th
he cosine and non--cosine models aree significant (p<0.05). In particulaar the mean
errors for the Gamma-vaariate model are 84%
8
larger than th
he Cos-Gamma α = 2 mode1, (p = 0.016) and
del they are 18% laarger (p = 0.0098)..
for the Log-Normal mod
Conclu
usions In this stu
udy we have presen
nted a novel modeel for describing DSC-MRI
D
first-passs data that gives im
mproved fitting acccuracy and speed (for α = 2) relativ
ve
to twoo established models. Further worrk is needed to esstablish if the com
mponents of the m
model correspondd to the transit off contrast throughh different vascula
ar
compoonents (mixing chaambers/capillary beeds) and therefore if the model param
meters derived havve a direct physicaal interpretation.
Acknoowledgements We
W acknowledge th
he support received
d for the CRUK an
nd EPSRC Cancerr Imaging Centre iin association withh the MRC and Deepartment of Healtth
(Englaand) grant C1060/A
A10334 and also NHS
N
funding to thee NIHR Biomedicaal Research Centree and the Wolfsonn Foundation.
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