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j.mri.2017.10.004

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 Semi-Parametric Arterial Input Functions for Quantitative Dynamic Contrast
Enhanced Magnetic Resonance Imaging in Mice
Torfinn Taxt, Rolf K. Reed, Tina Pavlin, Cecilie Brekke Rygh, Erling
Andersen, Radovan Jiřı́k
PII:
DOI:
Reference:
S0730-725X(17)30227-8
doi:10.1016/j.mri.2017.10.004
MRI 8848
To appear in:
Magnetic Resonance Imaging
Received date:
Revised date:
Accepted date:
30 January 2017
15 September 2017
17 October 2017
Please cite this article as: Taxt Torfinn, Reed Rolf K., Pavlin Tina, Rygh Cecilie Brekke,
Andersen Erling, Jiřı́k Radovan, Semi-Parametric Arterial Input Functions for Quantitative Dynamic Contrast Enhanced Magnetic Resonance Imaging in Mice, Magnetic
Resonance Imaging (2017), doi:10.1016/j.mri.2017.10.004
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Semi-Parametric Arterial Input Functions for
Quantitative Dynamic Contrast Enhanced Magnetic
Resonance Imaging in Mice
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Torfinn Taxta,b , Rolf K. Reeda,c , Tina Pavlina,b , Cecilie Brekke Rygha , Erling
Andersend , Radovan Jiřı́ke
a Dept.
of Biomedicine, University of Bergen, Jonas Lies vei 91, N-5020 Bergen, Norway
of Radiology, Haukeland University Hospital, Jonas Lies vei 83, N-5020 Bergen,
Norway
c Centre for Cancer Biomarkers (CCBIO), University of Bergen, Jonas Lies vei 87, N-5021
Bergen, Norway
d Dept. of Clinical Engineering, Haukeland University Hospital, Jonas Lies vei 83, N-5020
Bergen, Norway
e Czech Academy of Sciences, Inst. of Scientific Instruments, Královopolská 147, 61264
Brno, Czech Rep.
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b Dept.
Abstract
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Objective: An extension of single- and multi-channel blind deconvolution
is presented to improve the estimation of the arterial input function (AIF) in
quantitative dynamic contrast enhanced magnetic resonance imaging (DCE-
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MRI).
Methods: The Lucy-Richardson expectation-maximization algorithm is used
to obtain estimates of the AIF and the tissue residue function (TRF). In the
first part of the algorithm, nonparametric estimates of the AIF and TRF are
obtained. In the second part, the decaying part of the AIF is approximated by
three decaying exponential functions with the same delay, giving an almost noise
free semi-parametric AIF. Simultaneously, the TRF is approximated using the
adiabatic approximation of the Johnson-Wilson (aaJW) pharmacokinetic model.
Results: In simulations and tests on real data, use of this AIF gave perfusion values close to those obtained with the corresponding previously published
nonparametric AIF, and are more noise robust.
Conclusion: When used subsequently in voxelwise perfusion analysis, these
semi-parametric AIFs should give more correct perfusion analysis maps less
Preprint submitted to Magnetic Resonance Imaging
October 20, 2017
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affected by recording noise than the corresponding nonparametric AIFs, and
AIFs obtained from arteries.
Significance: This paper presents a method to increase the noise robustness
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in the estimation of the perfusion parameter values in DCE-MRI.
Keywords: DCE-MRI, blind deconvolution, arterial input function
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1. Introduction
Quantitative DCE-MRI is an important tool in clinical disciplines such as
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oncology [1, 2, 3, 4], cardiology [5, 6, 7], and in biological research [8, 9]. DCEMRI is used in rodents to study perfusion changes in malignant tumors following
treatment with new vasoactive drugs [10] and changes in perfusion of animal
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strains with knock out deletions of genes.
To do quantitative DCE-MRI, the observed contrast tissue signals have to be
converted to contrast concentration time sequences (perfusion sequences) [11].
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Each perfusion sequence is a convolution of an AIF and a TRF multiplied by
plasma flow [12]. Thus, the perfusion sequence has to be deconvolved with the
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AIF to find the TRF with its associated perfusion parameters. There are several
approaches to estimate the AIF (a comprehensive review in [13]).
The AIF can be derived from one or more artery signals close to the re-
gion of interest (ROI). However, AIFs from arteries are degraded by saturation
effects. For high peak concentration of the contrast agent in the arteries, the
R1 relaxivity rate is no longer proportional to the contrast agent concentration
[14]. High contrast agent concentration in arteries leads also to non-negligible
T2* effects decreasing further the level of the measured arterial signal [15]. A
measured AIF can also be degraded by flow artifacts and partial volume effects
[16]. Furthermore, representative arteries are often difficult to obtain in the acquired image sequence, especially in preclinical applications. Finally, dispersion
(change of the AIF shape due to the passage of the contrast-agent bolus from
the site of the measured arterial signal to the tissue of interest) is ignored.
A population based AIF can also be used [17]. This ignores the differences in
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the vascular tree between different subjects and depends on the AIF acquisition
methods used for creation of these population-based ”standards”.
Another approach of AIF estimation is based on a reference tissue (e.g. mus-
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cle) [18]. The AIF is estimated from the tissue curve in the reference tissue and
the presumably known perfusion parameters. This approach has been shown for
the Tofts model. However, for more complex pharmacokinetic models described
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by more perfusion parameters, the complete set of perfusion parameters would
have to be known, which is not very realistic.
The AIF estimation approach used here is based on single- and multi-channel
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blind deconvolution [19, 20, 21]. The AIF is estimated simultaneously with
the TRF from the measured tissue contrast agent concentration curves. This
provides examination specific AIF estimates. These techniques avoid many of
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the problems described above. Because the contrast agent concentration in the
tissue is much lower than in arteries, the saturation effects mentioned above for
measured AIFs are avoided.
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The algorithm can be implemented as single- [22, 23] or multi-channel [24, 25]
deconvolution, where the number of channels is the number of tissue region sig-
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nals processed simultaneously. Multi-channel algorithms rely on the assumption
that the AIF is the same for all processed ROIs. This might be regarded as an
additional prior information if the assumption is true, or as a source of error if
the tissue signals differ in the dispersion term of the local tissue region specific
AIFs. This also means that single-channel blind deconvolution provides the local AIF (accounting for dispersion) while the multi-channel technique leads to
an AIF estimate specific for the whole region containing all channels. However,
for channels selected from a small region, multi-channel blind deconvolution can
also provide an estimate of a local AIF [26].
Blind deconvolution AIF estimation has been introduced in [24, 21, 25] for
clinical DCE-MRI and extended later to preclinical DCE-MRI [10, 22, 27, 28,
23]. To obtain an AIF using blind deconvolution, a realistic pharmacokinetic
model of the TRF for the tissue in the ROI has to be selected. Different pharmacokinetic models have a clear impact on the time course of the estimated
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AIF [23]. The pharmacokinetic models of the TRF used in blind deconvolution are largely based on Tofts [24] and extended Tofts [21, 25] models. More
advanced pharmacokinetic models used for blind deconvolution include the two
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compartment exchange model (2CXM [1]) [23], the adiabatic approximation to
the tissue homogeneity model (aaJW [29]) [22, 23], the distributed capillary adiabatic tissue homogeneity model (DCATH [30]) [31] and the Gamma Capillary
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Transit Time (GCTT [32]) [28] models.
Blind deconvolution has been applied with a nonparametric AIF [24, 10, 22,
23] or with a parametric AIF [21, 25, 28, 31]. Clinical applications of blind
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deconvolution have been based on Parker’s AIF model (sum of two Gaussian
functions and a sigmoid multiplied by an exponential function) [31], Fluckiger’s
AIF model (Parker’s model with the Gaussian functions replaced by gamma
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variate functions) [21], and its modifications where three gamma variate functions were used instead of two, while forcing some parameters to the same values
[25]. The gamma variate AIF model of [25] has been used also for preclinical
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DCE-MRI [28].
A realistic AIF model is an important prior information that can improve
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the reliability of blind deconvolution, compared to use of a nonparametric AIF.
On the other hand, the use of a less realistic AIF model might degrade the
accuracy of blind deconvolution. This paper is focused on small animal DCEMRI using an advanced TRF model (namely the aaJW model). For these
applications, estimation of intravascular perfusion parameters, such as plasma
flow and volume is very sensitive to the initial part of the AIF and of the
measured tissue concentration sequence. Hence, it is important to estimate the
initial part of the AIF accurately.
In this paper, a semi-parametric AIF formulation is proposed to keep a suffi-
cient degree of freedom for the initial AIF part by its nonparametric formulation
and to regularize estimation of the slowly decaying AIF part by its parametrization. This approach is proposed as an alternative to the existing blind deconvolution AIF estimation methods which are based either on a parametric or a
nonparametric AIF.
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Estimation of the semi-parametric AIF is a straightforward extension of the
previously published method of blind deconvolution for estimation of a nonparametric AIF [22]. As in [22], the algorithm is based on the Lucy-Richardson
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expectation-maximization algorithm and provides estimates of the AIF and the
TRF.
Simulated perfusion sequences were studied first. The AIFs and TRFs were
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derived from the inner part of the masseter muscle and the temporalis muscles of
normal mice and mice treated with the anaphylactic agent C48/80. This agent
gives a huge increase in blood flow and the permeability surface area product
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of muscle, thus leading to a large increase in the signal to noise ratio (SNR) of
the muscle DCE-MRI signals [33, 34, 35, 36, 23].
In the second part, real perfusion sequences from the masseter and tem-
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poralis muscle of normal mice and mice treated with the anaphylactic agent
C48/80 were studied. Perfusion parameter estimates derived using the aaJW
compared.
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model for the TRF and either the nonparametric or semi-parametric AIF, were
In the simulations and using real data, the semi-parametric AIF formulation
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was more noise robust than the nonparametric AIF. On the other hand, the
nonparametric AIF formulation was more accurate in high SNR cases.
2. Theory and Methods
2.1. Perfusion Analysis in DCE-MRI
The observed perfusion sequence, Ct [n] (n is the time index), is a convolution
of the AIF and TRF,
Ct [n] = Cp [n] ∗ H[n].
(1)
Here, Cp [n] is the AIF and H[n] is the TRF multiplied by plasma flow, Fp .
To find the relevant perfusion parameters in DCE-MRI, H[n] has to be modeled and isolated from Cp [n] through deconvolution and parameter estimation.
To estimate Cp [n] and H[n] simultaneously, blind deconvolution is used.
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Here, an extension of a previously published single- and multi-channel blind
deconvolution method [22] is used to solve this problem. The single-channel
algorithm is an iterative algorithm (Fig. 1) consisting of two parts. First, the
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TRF is updated based on the current AIF and TRF estimates, and then the
AIF is updated based on the current AIF and TRF estimates. In the first 200
iterations, only nonparametric AIF and TRF are used. In the following 90
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iterations, updating of the nonparametric TRF is followed by its approximation
with a parametric aaJW model and update of the nonparametric AIF is followed
by its approximation with the semi-parametric model. 90 iterations were found
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to be sufficient (200 iterations were used in [22]). More iterations did not lead to
any visible change in the AIF and TRF estimates, indicating that convergence
has been reached.
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The standard Lucy-Richardson deconvolution algorithm is formulated for
non-negative signals that start with zero values and end with zero values. This
leads to the need for extrapolation of the observed trunctated perfusion se-
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quences [22] to the length of approximately 3 hours (wash out time expected
for a small molecular weight gadolinium contrast agent). This is done as in [22]
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by approximation of the upper time tail of the tissue function with a gamma
function. A least-mean-square error procedure is used.
The multi-channel version of the blind deconvolution algorithm is a direct
extension of the single-channel approach [22]. The first 200 iterations (nonparametric deconvolution part) are performed as single-channel, separately for
each channel. The channel-specific AIF estimates are merged in the step ”SemiParametric approximation of nonparametric AIF” (Fig. 1) as follows. All AIFs
are first delay corrected (see [22] for more details) and averaged prior to the
semi-parametric approximation.
2.2. Nonparametric Updates of the TRF and AIF
A modification of the basic algorithm [22] is introduced to give improved
estimates of the nonparametric AIFs and TRFs. Extensive preliminary experiments showed less distortion of the decaying part of the AIF due to noise when
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using the regularization term for the Lucy-Richardson algorithm of [37], page
1062, instead of the total variation regularization term used in the basic algorithm. For simplicity, a low-pass filtered version of the residual defined in [37]
to the following update equations.
Cps+1 [n] = {
Ct [n]+R̄1s [n]
Cps [n]
∗ Cps [−n]}H s [n],
(2)
∗ H s+1 [−n]}Cps [n].
(3)
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H s+1 [n] = {
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is used as a special case of the wavelet transform approach in [37]. This leads
Ct [n]+R̄2s [n]
H s+1 [n]
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Here, s is the iteration index, H s [n] is the estimation of H[n] in the s-th iteration, Cps [n] is the estimation of the AIF in the s-th iteration and R̄is [n] is
the low-pass filtered residual (filter implemented as a shifted sigmoid in the
π
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rad/sample). The residual is R1s [n] =
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frequency domain, cutoff frequency
Ct [n] − Cps [n] ∗ H s [n] for Eq. (2) and R2s [n] = Ct [n] − Cps [n] ∗ H s+1 [n] for Eq. (3).
The AIF is initialized to the tissue perfusion sequence, Ct [n], after addi-
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tional median filtering (length 50 samples here, found experimentally). The
TRF is initialized as a constant for the whole time interval of the extrapolated
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tissue function (including the precontrast part). More details on the choice of
initialization can be found in [22].
For some of the noisy perfusion sequences of the controls, the parametric
TRF did not overlap well with the corresponding nonparametric TRF. Also,
the fit of the estimated perfusion sequence to the observed perfusion sequence
was suboptimal. Obviously, the convergence point of the algorithm was local
and wrong. Additional systematic variation of the optimization parameters always resulted in a global optimum point with the best fit of the observed and
estimated time sequences and a good overlap of the final nonparametric and
parametric TRFs. The systematically changed parameters were the intravascular transit time, Tc (given in samples: 6 to 17) and the number of samples,
N o, (100,200,300 and 400), of the tail of the observed signal sequence used to
estimate the extrapolation of this sequence in the preprocessing step. For all
pairs of Tc and N o values from the above given ranges, the whole perfusion
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estimation algorithm was run. The pair giving the best fit of the observed and
estimated time sequences was then selected.
2.3. Parametric Approximation of the TRF
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The nonparametric TRF is approximated with the aaJW model (described
below). In the first iteration with approximation (s=201, Fig. 1), the delay of
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the parametric TRF is estimated. The vertical rising edge of the parametric
TRF is aligned with the point of the steepest rise of the nonparametric TRF,
estimated as the maximum of the difference-operator processed sequence (see
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[22] for more details). The delay corresponding to this alignment is then applied
in all following iterations.
The aaJW model of the TRF is selected here as one of the simplest models
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providing a complete description of the intravascular and extravascular distribution of the contrast agent as a function of time. Other alternatives include the
2CXM [1], the tissue homogeneity model (TH) [38], the distributed parameter
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model (DP) [39], the distributed capillary adiabatic tissue homogeneity model
(DCATH) [30], and the Gamma Capillary Transit Time (GCTT) model [32].
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The TH and DP models are computationally demanding options because of
no time domain analytic solution of the TH model and the need for numeric
integration for the DP model. The DCATH and GCTT models include an additional perfusion parameter describing the distribution of the mean transit times.
This may lead to worse conditioning of the deconvolution problem. A recent
comparison of the aaJW and the 2CXM models indicated that the aaJW model
is more realistic for cross-striated muscle (used here for evaluation) [23]. For
cross-striated muscles, the plug flow assumption induced to the intravascular
space by the aaJW model is more realistic than the compartment assumption
of the 2CXM.
For the aaJW model, H[n] is modeled as [29]:
H[n]
=
Fp ,
0 ≤ nTs < Tc ,
H[n]
=
EFp · e−kep ·nT s , nT s ≥ Tc ,
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(4)
EFp
ve
is the rate of
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where Fp is plasma flow, Ts is the sampling interval, kep =
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contrast agent transport from the extravascular extracellular space to plasma,
Tc is the mean intravascular transit time, ve is the extravascular extracellular
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volume per unit volume of tissue (leakage space) and E ≤ 1 is the extraction
fraction. The first row of (4) is the vascular phase and the second row is the
parenchymal phase of H[k].
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The extraction fraction, E, is the fraction of the contrast agent that diffuses unidirectionally from plasma into the surrounding tissue during a single
contrast-agent transit through the capillary bed. It is related to Fp and the per-
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meability surface area product, P S, through [40] E = 1 − e−P S/Fp . The aaJW
model is completely described by the four independent physiological parameters
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Fp , Tc , E and kep .
2.4. Semi-Parametric Approximation of the AIF
The semi-parametric AIF formulation consists of a nonparametric initial
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part, as returned by the nonparametric update, followed by a decaying parametric part. This part is a sum of three decaying exponential functions with
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the same delay, i.e. the time from which the approximation by the model starts.
Hence, the parametric part of the AIF is described by six parameters (scaling
factor and time constant for each exponential) and the common delay.
In the first iteration where the semi-parametric approximation of the AIF is
done (s=201, Fig. 1), the common delay of the three exponential decaying functions is determined as the point of the steepest rise of the nonparametric AIF
estimated as the maximum value of the difference operator processed nonparametric AIF. In all following iterations, this delay is calculated as the location
of the maximum point of the nonparametric AIF. This approach gave the best
results, based on preliminary experiments.
In extensive preliminary experiments, only two decaying exponential func-
tions with the same delay were used to approximate the decaying phase of the
AIF [41, 42]. This approach always led to lower values of the extraction fraction
than the corresponding nonparametric AIF, while the other perfusion parame9
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ters remained fairly unchanged. Thus, two decaying exponential functions were
considered inadequate for a valid approximation of the decaying part of the AIF.
The six parameters of the semi-parametric AIF are initialized in the first
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semi-parametric AIF approximation step (s=201, Fig. 1). It is done using the
successive subtraction method [43]. Then, these initial parameter estimates are
used as starting values for the nonlinear least mean square error method of
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Levenberg-Marquardt (the algorithm mrqmin in Numerical Recipes in C [44]).
In the following iterations, the parameter values of the previous iteration are
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used as starting values for the Levenberg-Marquardt method.
2.5. Scaling
Blind deconvolution estimates the AIF and TRF except for a scaling factor.
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The scaling factor is estimated at the end of the blind deconvolution algorithm
(Fig. 1) from a reference tissue, the masseter muscle (see Discussion for other
approaches to AIF scaling). Based on literature values of ve [45, 46, 47] and
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the assumption that blood plasma volume, vp , is approximately 5% of ve , the
reference value ve + vp = 0.13 ml/ml is used. According to the formulation of
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the TRF model (Eq. (4)), ve + vp equals the area under the TRF curve.
From the definition of the convolution of two non-negative signals, it follows
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that the area under the curve of a result of a convolution (here n=1 Ct [n],
where N is the number of samples of the extrapolated perfusion sequence)
is the product of the areas under the curves of the convolution factors (here
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n=1 Cp [n] and
n=1 H[n]). Hence, the sought scaling factor of the blind de-
convolution result, i.e. the area under the curve of the AIF, is the area under
the curve of the contrast agent concentration sequence measured in a masseter
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muscle region,
Ct ref [n], divided by 0.13 ml/ml.
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To account for the variability in the real recordings, the median of the values
Ct ref [n] obtained from all real recordings was used. This relies on the fact
that the same contrast-agent dose and concentration were used for all mice and
that approximately the same kidney function can be assumed for both normal
and treated mice.
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2.6. Simulations
Time sequences were made to simulate the perfusion pattern of the masseter and temporalis muscles of control mice and mice treated with C48/80 (see
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below), with and without added noise. Thus, ignoring noise, four different perfusion sequences were generated, mimicking the perfusion sequences obtained
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from the real data recorded on one control and one treated mouse (Section 2.7).
The Four Basic Perfusion Sequences. Typical aaJW TRFs of the masseter and
temporalis muscles of the control and treated mice were generated using the
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reference perfusion parameters (Table 1). These perfusion parameters were obtained from in vivo mice perfusion sequences (see below) processed with the
single-channel blind deconvolution algorithm using the nonparametric AIF for-
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mulation. The length of the simulated TRFs was 16384 samples, i.e. the length
of real perfusion sequences after extrapolation and additional zeropadding for
use of the fast Fourier transform, see Section 2.7.
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The AIFs used for simulation were obtained from in vivo data after preprocessing (conversion to R1, median filtering and extrapolation, see below).
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Single arterial voxels selected manually in the lower part of both hemispheres
were used. The AIFs of the control mice were all much degraded by noise. To
obtain a usable AIF of these mice, AIFs from four different control mice were
selected and averaged. The AIFs of the treated mice were clearly less degraded
by noise and artifacts. Hence, only a single, typical AIF was selected for the
treated mice.
The two TRFs of the control mice were convolved with the AIF of the control
mice. In the same way, the two TRFs of the treated mice were convolved with
the AIF of the treated mice. No dispersion was modeled.
The resulting perfusion sequences were almost noise free (the remaining noise
was due to the noise in the AIFs) and consisted of 16384 samples (the length of
real perfusion sequences after extrapolation and additional zeropadding for use
of the fast Fourier transform) with sampling interval 1.079 s. These perfusion
sequences were denoted ”inf-full” on the SNR axis (Figs. 3, 5). No extrapolation
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was applied in processing of these sequences.
To evaluate also the effect of extrapolation on the blind deconvolution performance, the simulated sequences were truncated to 896 samples (the length of
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the real recordings, see below) and processed as the real data. These perfusion
sequences were denoted ”inf” on the SNR axis (Figs. 3, 5).
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Perfusion Sequences with Noise. The in vivo perfusion sequences had about
130 samples before the contrast arrived. By merging several of these precontrast
signal parts obtained from several recordings (converted to Ct [n]) and truncating
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the result, a signal of 896 samples with no contrast agent contribution and the
noise level of real perfusion sequences was generated.
Two such noise signals for the simulated perfusion sequences were gener-
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ated, one for the control mice and one for the treated mice. The noise signals
were added to the corresponding almost noise free signals, to simulate the real
perfusion sequences as closely as possible.
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To generate perfusion sequences with lower SNRs, the noise signals were
multiplied with a constant factor higher than 1.0 before they were added to the
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almost noise free signals. The noise corrupted simulated signals were median
filtered as the real signals (see below) before the blind deconvolution.
The SNR for a single perfusion sequence was defined as the mean value of
the perfusion sequence divided by the standard deviation of the noise [22]. Only
the signal corresponding to the recorded part of the signal was used to estimate
the SNR. The SNR for the multi-channel algorithms was defined as the mean
of the SNR over all channels.
2.7. Animals, Imaging and Relaxation Rates
All animal procedures were approved by the National Animal Research Authority (Oslo, Norway). All experiments were done in accordance with these
protocols. Female C57Bl6 mice were anesthetized with Isoflurane in oxygen,
and the body temperature was kept at 37o C throughout the experiment. One
group of mice (n=8) received an injection of C48/80 (50 µg in 0.1 ml saline)
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and one group of mice (n=8) received saline only. The data from these 16 mice
have been reported and used for analysis also in [22, 23]. C48/80 acts as a mast
cell degranulating agent and induces an inflammation due to the release of his-
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tamine [36]. Imaging was done using the FLASH sequence of a 7T Pharmascan
(Bruker Biospin, Germany) and a mouse head volume coil.
Precontrast T1(0) maps were computed based on precontrast T1-weighted
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recordings with fixed TR/TE = 15.0/2.5 ms and flip angles α = 5◦ , 10◦ , 15◦
and 20◦ , and 10 repetitions per flip angle. The linear form of the FLASHequation [48] was used. This approach allows computation of T1(0) values
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without neglecting the T2∗ effect.
In the following steps, the T2∗ effect was neglected, which is the standard
approach, valid for short TE, as used here. The k · ρ map (k is a spatially
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invariant constant dependent on the scanner and its gain setting, ρ is spin
density) was computed using the FLASH-equation, S = k·ρ·sin(α)(1−E1 )/(1−
cos(α)E1 ). Here, S is the voxel intensity in the precontrast recordings with
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α = 5◦ , E1 = e−T R/T 1(0) and T 1(0) is the map calculated in the previous step.
DCE-MRI (dynamic T1-weighted 2D FLASH sequence, TR/TE = 15.0/2.5
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ms, flip angle 25◦ , number of frames K = 1000, temporal resolution 1.079
s, slice thickness 1 mm, one axial slice through the left masseter muscle, one
oblique sagittal slice along the thoracic aorta and at least one arteria carotis
communis) was performed immediately after administration of the drug/saline.
The conversion from signal intensity to R1 time sequences was done using the
FLASH-equation, where the term k · ρ was known from the analysis of the
multiple flip angle precontrast images.
The contrast agent (gadodiamide, 0.1 mmol (kg BW)−1 in saline) was injected manually through the tail vein after acquisition of 130 frames. Care was
taken to use as similar injection speed as possible for all mice.
To increase the SNR to acceptable levels, average signals in manually drawn
regions of interest were used. The average masseter muscle signal was derived
from 40-80 voxels in the left and right masseter muscles. The average temporalis
muscle signal was derived from 13-26 voxels in the left and right temporalis
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muscles.
The resulting two time sequences were median filtered (window length 11
samples). This filtering was selected based on previous experiments [22] as
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a compromise allowing sufficient noise reduction while not degrading the highfrequency content of the underlying signal. The fairly long window of the median
filter is acceptable for tissue perfusion sequences of muscles due to their slow
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changes in shape. For arterial signals, the smoothing effect of the median filter
affected mainly the shape of the peak (Fig. 4). However, this slight distortion
is acceptable since the arterial signals are used only for generation of synthetic
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perfusion sequences.
The real perfusion sequences were extrapolated to 13800 samples (3 hours)
and zeropadded to 16384 samples (4 hours 33 min). This size was chosen as
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the lowest power of two (to allow use of fast Fourier transform) giving a signal
which is longer than the assumed minimum washout of 3 hours (see section 2.1).
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2.8. Evaluation Methods
The simulated and real sequences were processed by single- and multi-
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channel blind deconvolution using the nonparametric and semi-parametric AIF.
Multi-channel blind deconvolution was applied to the masseter and temporalis
muscle perfusion sequences, i.e. two channels were used. For simulated data,
AIF scaling was done with respect to the true reference AIF, i.e. an ideal case
of accurate AIF scaling was simulated. For real data, AIF scaling was done
according to Section 2.5.
Simulations. To quantify the fit between a simulated perfusion sequence, Ct [n],
and the corresponding estimated perfusion sequence, Ĉt [n], the approximation
difference between these two sequences was computed. First, both Ct [n] and
Ĉt [n] were normalized to have area under the curve 1.0. Then, the sum of
absolute values of all sample based differences between the two sequences was
computed, and taken as the approximation difference.
To quantify the fit between a simulated true AIF and an estimated AIF,
the overlap of the two sequences was computed. First, both the simulated and
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estimated AIFs were normalized to have area under the curve 1.0. Then the
absolute value difference, absdiff, between the two AIFs was computed as the
mean of absolute values of all sample based differences. The percent overlap
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was defined as (1−absdiff/2) · 100%.
To evaluate the accuracy of the perfusion-parameter estimation, the measure
e was introduced as a mean relative error. It is defined as [22]
1 |Fb − Fˆb | |Tc − Tˆc | |E − Ê| |kep − k̂ep |
(
+
+
+
) · 100%,
4
Fb
Tc
E
kep
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e=
(5)
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where the symbols with accent are the estimated values and the values without
accent are the known reference values. The parameters Tc , E and kep are
independent of AIF scaling. The parameter Fp depends on both the shape and
the scaling of the estimated AIF. As an ideal (accurate) scaling is assumed in the
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processing of the simulated data, only the effect of the estimated AIF shape on
Tc , E, kep and Fp is evaluated by Eq. (5). As a simplification, all four perfusion
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parameters are weighted equally in Eq. (5), supposing equal diagnostic value
and impact on the precision and accuracy of the derived perfusion parameters
(e.g. P S, ve ).
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The paired replicate data were compared using Wilcoxon’s two-sided distri-
bution free signed rank test [49]. The significance level was 0.05.
In-Vivo Data. To quantify the fit between an observed perfusion sequence,
Ct [n], and the corresponding estimated perfusion sequence, Ĉt [n], the approximation difference between the two sequences was computed in the same way as
described for the simulated perfusion sequences.
To quantify the accuracy of the estimated AIFs, an indirect method was
used because no ground truth AIF was available. The measured AIFs were
not considered as a ground-truth because of acquisition artifacts and because
reasonable AIF measurements were possible only for a limited number of animals (due to acquisition artifacts). Hence, the percent overlap, as defined for
the simulated data, was computed between the AIF estimated from the masseter and temporalis muscles for each animal. This provided a comparison of
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the nonparametric and semi-parametric AIF formulations for the single-channel
algorithm. For multi-channel deconvolution this comparison was not available
because it estimated one AIF common for both muscles/channels. To compare
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the single- and multi-channel algorithms, the overlaps between the nonparametric and semi-parametric AIF estimates was compared.
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3. Results
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3.1. Simulations
Perfusion Sequences. The approximation difference (see methods for definition)
was in general clearly lower for the nonparametric AIF than for the semiparametric AIF. This was observed for both the single- and multi-channel al-
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gorithms (Fig. 3). This shows that the nonparametric AIF has clearly more
degrees of freedom to adapt to the signal noise. It also reflects the noise in
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the true AIFs (Fig. 4), and that the decaying phase of even noiseless true AIFs
is not exactly modeled by three exponential functions. Examples of simulated
perfusion sequences and their approximations are shown in Fig. 2 (the ripples
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in the simulated curves are due to the added noise and the subsequent median
filtering of the simulated signals).
The approximation difference was higher for the multi-channel algorithms
than the single-channel algorithms (Fig. 3, note the different scale on the vertical axes). The reason for this difference was that compared to single-channel
deconvolution, multi-channel deconvolution had less degrees of freedom due to
the assumption of the same AIF for all channels. Hence, the single-channel
estimates of the perfusion sequences could adapt to the given noise realization
more closely.
AIFs. For both, single- and multi-channel deconvolution, with no noise, the
nonparametric AIF gave higher overlap with the true AIF than the semi-parametric
AIF (Figs. 4, 5). However, when SNR decreased, the semi-parametric AIF had
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consistently higher overlap. This clearly shows that semi-parametric AIF estimation is more robust with respect to noise (Figs. 4, 5).
Comparing the multi- and single-channel algorithms, with no noise, the over-
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lap of the estimated AIFs with the true AIF was about the same. When SNR
was about 10 or lower, the multi-channel algorithm gave better overlap of the
estimated and the true AIF than the single-channel algorithm (Fig. 5). This was
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the case for both the nonparametric and semi-parametric AIFs in controls and
treated mice. This could be explained by the stabilizing function of the prior
assumption in multi-channel blind deconvolution that the AIFs of all channels
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were the same. This assumption took effect mostly in low SNR conditions.
TRFs. The percent error of the perfusion parameters (Eq. (5)) corresponded
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to the observations obtained from overlaps of the estimated and reference AIFs
described above. However, for the percent error of the perfusion parameters, the
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effect of randomness was higher than for the AIF-overlap measure. Hence, the
percent error of the perfusion parameters was treated as a supportive measure
and the corresponding tables are not shown.
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Comparing the nonparametric and semi-parametric AIF formulations, the
percent errors of the perfusion parameters were comparable (mostly under 10%
for SNR>15 and in the range between 4% and 57% for SNR<15). For high SNR
(SN R >15), the nonparametric AIF led to clearly better perfusion-parameter
estimates. For low SNR (SN R <15), the semi-parametric AIF approach was
slightly (but non-significantly) better.
Comparing single- and multi-channel deconvolution, the obtained percent
errors of the perfusion parameters were comparable. The multi-channel algorithm gave clearly less accurate perfusion-parameter estimates for SNR>5 and
slightly better (but not significantly) for low SNR (SN R < 5).
3.2. Real Perfusion Time Sequences
SNR and Perfusion Sequences. As expected from the known effects of C48/80
on muscle perfusion, the SNR of the perfusion sequences of the treated muscles
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was clearly higher than that of the corresponding control muscles (Table 2).
Example perfusion sequences are shown if Fig. 6 (noise is manifested as ripples
remaining after median filtering). The SNR of the perfusion sequences of the
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control masseter and temporalis muscles did not differ significantly, while the
SNR in the treated mice was significantly higher for the masseter muscle than
for the temporalis muscle (Table 2).
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The approximation differences of the observed and estimated perfusion sequences for the single-channel algorithm with a nonparametric and semi-parametric
AIF (Table 2) were comparable to the corresponding approximation differences
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of the simulations with a similar SNR (Fig. 3). These approximation differences
were smaller for the treated mice than for the control mice (Table 2). For the
multi-channel algorithm, the approximation differences were slightly higher for
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the real data (Table 2) than for the corresponding simulated data (Fig. 3).
The approximation differences of the observed and estimated perfusion sequences for the single-channel algorithm with a semi-parametric AIF were larger
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than the corresponding differences of the single-channel algorithm with a nonparametric AIF (Table 2, p = 0.008 for both muscles). This was the same
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observation as for the simulated data. It showed that the use of the nonparametric AIF allowed more degrees of freedom of the model for the perfusion
sequences than the use of the semi-parametric AIF.
Compared to single-channel deconvolution, the multi-channel algorithm gave
much larger approximation differences of the observed and estimated perfusion
sequences (Table 2). This indicated that the AIF of the two muscles differed
due to dispersion. Possible different delays were corrected for automatically in
the algorithm.
The multi-channel algorithm with a semi-parametric AIF gave approximation differences of the observed and estimated perfusion sequences that were
similar to those of the multi-channel algorithm with a nonparametric AIF (Table 2). Again, the most likely explanation for this observation was that the
AIFs of the two muscles differed due to dispersion. Noise was only a minor
contributing factor.
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AIFs. The overlap of the AIFs estimated from the masseter and temporalis
muscles of the same mouse (Table 3, columns 2, 3) was clearly higher for the
semi-parametric AIFs than for the nonparametric AIFs. For in vivo control mice
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recordings, this corresponded well to the results obtained on simulated data for
low SNRs (including the SNRs of in vivo control mice, Fig. 5, left). However,
the in vivo recordings of the treated animals had SNRs in the high SNR range
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of the simulated data (Fig. 5, left) where the semi-parametric AIF gave worse
results than the nonparametric AIF (see the Discussion section below).
The overlap of the nonparametric and semi-parametric AIF estimates for
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the same animal (Table 3, columns 4–6) showed a slightly better consistency of
the multi-channel algorithm than the single-channel algorithm. Compared to
the simulation results (Fig. 5), this corresponds to the low SNR range where
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the multi-channel algorithm is more reliable. This was true even for the treated
group where the SNR was in the range where the accuracy of the single- and
multi-channel algorithms was comparable in the simulations (see the Discussion
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section below).
For the semi-parametric algorithms, the parameters of the three exponential
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functions used in the AIF approximation varied to a large extent. This was the
case for each group of mice for both semi-parametric algorithms and between
the masseter and temporalis muscles of the same mouse for the single-channel
semi-parametric algorithm.
TRFs. Perfusion parameters were compared for all real perfusion sequences
for the single- and multi-channel algorithms with the nonparametric and semiparametric AIF formulations. Except for two cases (out of 32 cases: 4 perfusion
parameters x 2 muscle tissues x 2 AIF formulations x 2 animal groups), the
perfusion parameters (Fb , Tc , E and kep ) did not differ significantly.
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4. Discussion
For both the nonparametric and semi-parametric AIFs, the estimated AIFs
of the single- and multi-channel algorithms had a good overlap with the true
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AIFs when the SNR of the perfusion sequences was reasonable. The nonparametric and semi-parametric AIFs led to comparable perfusion parameter esti-
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mates. Thus, modeling the decay phase of real AIFs of mice with three exponential decaying functions with the same delay was adequate for finding an
almost noise free semi-parametric AIF specific for each mouse.
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In contrast, as done previously [41, 42], using only two exponential decaying
functions with the same delay was not sufficient in the present setup. The AIF
model with two exponentials might be sufficient when a simpler TRF model is
used, as in [41, 42] (extended Tofts model). However, using a more complete
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TRF model (such as the aaJW model used here) imposes higher demands on
the accuracy of the AIF model.
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As expected, a semi-parametric AIF led to a better noise robustness than the
nonparametric AIF. This conclusion follows from the overlaps of the estimated
and true AIFs for low SNR in the simulation experiments (Fig. 5) and from the
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consistency of the AIFs estimated from real recordings (Table 3, rows 2–3). In
low SNR conditions, the semi-parametric AIF formulation provides additional
prior information which stabilizes the estimation process.
On the other hand, the nonparametric AIF estimates adjusted to the noise
and diverged from the true AIF for the low SNR case. This is illustrated
by the fact that the approximation differences of the estimated and simulated/measured perfusion sequences showed a better fit for the nonparametric
AIF also for low SNR (Fig. 3, Table 2).
For real data, the superiority of the semi-parametric AIF formulation was
shown by the consistency of the AIFs estimated independently from two muscles.
The semi-parametric AIF formulation gave better results than the nonparametric AIF (Table 3) even for high SNR conditions (treated mice, see Table 2)
which was not in line with the simulations (Fig. 5). This observation suggests
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the effect of an additional source of ”’noise” present in the real data and not
accounted for in the simulations. This could be the deviation of the theoretical
pharmacokinetic model and the real pharmacokinetics of the muscles.
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Another source of inaccuracy could be conversion from signal intensity to
contrast agent concentration, as described in [50]. A solution would be to adjust
the acquisition parameters according to the recommendations in [50]. Also, the
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nonlinear relation of the contrast agent concentration to the R1 relaxation rate
due to the finite water exchange rate between blood plasma and tissue interstitial
space was not considered here. Taking this effect into account would require
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extension of the pharmacokinetic model at the expense of two additional free
parameters to be estimated [51]. It has not been shown, so far, whether such a
model extension can lead to more accurate and precise blind deconvolution.
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The approximation of a function with three decaying exponential functions
is an ill-conditioned problem, known for a long time [43]. The ill-conditioning
is thought to be related to the fact that decaying exponential functions do not
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form an orthogonal basis. The observed large variation of the estimated AIF
parameters stated above is likely to have the same origin.
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Comparing the single- and multi-channel algorithms for low SNR, multi-
channel deconvolution lead to better accuracy of the AIF estimation for simulated data (Fig. 5). The reason was most probably that the assumption of the
same AIF for all channels provided additional prior information.
The comparison in terms of approximation differences of the estimated and
simulated / measured perfusion sequences (Fig. 3, Table 2) showed the best fit
for single-channel deconvolution for all SNR cases. This is reasonable since the
single-channel deconvolution algorithm has more degrees of freedom to adjust
to the actual noise realization than the multi-channel approach. For real data,
an additional drawback of the multi-channel approach has to be taken into
account, namely the dispersion differences of the AIFs in the different channels
(not modeled in the simulations).
The choice of the aaJW pharmacokinetic model for the TRF was based on
previous work on perfusion in mouse masseter muscle [22, 23]. The choice of
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a realistic TRF model for a given tissue is critical in obtaining a realistic AIF
estimate, since the TRF model will have a strong influence on the time course
of the estimated AIF [23].
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Muscle tissue is present in most perfusion images of the body. Here, the
aaJW model seems adequate for the TRF. Hence, a realistic blind estimate for
the AIF in the region of interest can be obtained, and subsequently used in
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non-blind voxel-by-voxel perfusion analysis of the whole image. However, the
challenge of choosing a realistic TRF model for tissues other than muscle in the
image remains.
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The presented AIF scaling method was based on a known value of ve + vp in
a reference tissue and on the assumption that the area under the AIF curve is
the same for all animals in the study, due to the same contrast agent dose and
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the same kidney function. These are certainly simplifications.
In [31], the present AIF scaling was compared to other two approaches. The
first method was scaling to the area under the curve of a measured AIF (an
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arterial perfusion sequence derived from the DCE-MRI recording). The second
method was scaling to the area under the curve of the tail of a measured AIF
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(the part of the curve after the peak of signal intensity where the saturation
artifacts are reduced, used previously in [21, 52]). The AIF scaling method
presented here gave clearly the most consistent results [31].
In [53, 52], the use of one venous sample at a late time (when arterial and
venous concentrations are the same) was mentioned. Another option, mentioned
in [21, 52], is to apply an additional scan after the DCE-MRI acquisition which
would provide an accurate measurement of contrast agent concentration in a
nearby artery or vein. These techniques need to be implemented and compared
to decide on the best AIF scaling method for a given application.
This work was limited to DCE-MRI in small animals, where the decay phase
is monotonously decaying [54, 15, 42]. In humans, the decay phase of the AIF
usually has a bimodal or three-modal time course. Several fully parametric
models for these AIFs exist [21, 17, 55]. A future challenge is to incorporate
such an AIF model in the presented framework to allow estimation of almost
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noise free AIFs for single voxel perfusion analysis in humans.
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The present blind deconvolution method was computationally demanding.
For one perfusion sequence, the current implementation (Linux C-shell scripts
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and C files, not optimized for speed) gave computation time of approx. 10 min
for single-channel and 12 min for a two-channel deconvolution, on a PC with
Intel Core 2 Duo P8400 processor (2.53 GHz, 725 3 MB L2 cache), 3.5 GB
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RAM, Xubuntu Linux Xfce 4.4. By a speed optimized reimplementation of the
algorithm, a very large reduction in compuation time is expected.
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5. Conclusion
For quantitative preclinical DCE-MRI perfusion analysis on simulated and
real data, a semi-parametric model of the AIF was introduced in the estimation
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of the AIF. The performance of this model combined with the aaJW pharmacokinetic model for the TRF was studied using single- and multi-channel
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Lucy-Richardson blind deconvolution expectation-maximization algorithms.
The algorithm consists of two steps. In the first part, nonparametric estimates of the AIF and TRF are obtained. In the second part, the TRF is
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approximated using the aaJW pharmacokinetic model and the decaying part of
the AIF is approximated by three decaying exponential functions. This leads to
almost noise free semi-parametric AIFs.
The tests on simulated and real data showed that use of semi-parametric
AIFs led to better noise robustness of the AIF estimation for low SNR, when
compared to the corresponding nonparametric AIF approach. The use of such
blind deconvolution semi-parametric AIFs in the subsequent nonblind voxelwise
deconvolution should lead to more accurate perfusion parameter maps than use
of the corresponding nonparametric AIFs.
Acknowledgment
This study was supported by grants from the the Czech Science Foundation
(grant no. 16-13830S) and by the Ministry of Education, Youth, and Sports of
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the Czech Republic (project No. LO1212). The MR scanning was performed
at the Molecular Imaging Center (MIC) at the Department of Biomedicine,
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University of Bergen.
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Tables
Table 1: Parameters of the simulated TRFs..
tissue
Fb
Tc
E
[ml/min/100ml]
[s]
[-]
controls, masseter m.
6.9
11.9
66.1
0.567
treated, masseter m.
29.4
6.5
51.3
0.661
controls, temporalis m.
4.8
7.6
59.1
0.472
treated, temporalis m.
17.4
[1/min]
SC
NU
MA
ED
PT
AC
CE
31
kep
7.6
56.5
0.577
Approximation differences of the perfusion sequences (see Sec. 2.8 for definition).
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Table 2:
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Real data. Each difference multiplied by 100. Single-channel algorithm: column 2-7. Multichannel algorithm: column 8-10. Nonp: Nonparametric. Semi: Semi-Parametric.
mouse
SNR
nonp
temporalis m.
semi
SNR
nonp
controls
14.3
0.22
0.60
16.3
22
14.6
0.28
0.72
14.6
23
20.9
0.21
0.59
15.2
24
13.6
0.28
0.82
20.4
25
21.3
0.32
3.79
21.5
26
30.0
0.33
1.58
28
13.6
0.35
mean
18.3
0.28
SD
6.1
0.05
0.26
0.66
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19
semi
mean of both muscles
SNR
nonp
semi
15.3
2.67
2.54
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masseter m.
0.71
14.6
6.78
6.38
0.19
0.38
18.1
6.50
6.23
0.31
1.33
17.0
5.92
5.99
0.18
0.72
21.4
14.80
14.65
24.2
0.36
1.33
27.1
2.13
2.86
0.73
13.5
0.41
1.12
13.6
6.24
6.17
1.26
18.0
0.25
0.89
18.2
6.43
6.40
4.1
0.12
0.37
4.7
4.15
3.99
ED
MA
0.26
1.17
treated
35.1
0.18
1.14
30.6
0.21
0.75
32.9
5.75
5.77
6
86.2
0.07
0.48
34.1
0.11
0.58
60.2
1.92
1.99
7
83.8
0.07
0.27
42.7
0.08
0.47
63.3
8.04
8.18
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PT
5
8
49.7
0.17
1.12
53.5
0.09
0.54
51.6
2.02
1.99
10
62.8
0.10
0.55
54.2
0.12
0.58
58.5
0.43
0.67
12
49.3
0.10
0.66
44.4
0.12
0.65
46.9
2.42
2.46
13
53.0
0.09
0.56
34.1
0.15
0.85
43.6
6.52
3.83
mean
60.0
0.11
0.68
41.9
0.13
0.63
51.0
3.87
3.55
SD
18.3
0.05
0.33
9.5
0.04
0.13
10.7
2.86
2.61
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Overlap of AIFs in percent. Real data. Single-channel algorithm (single), overlap
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Table 3:
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of AIFs estimated from the masseter and temporalis muscles using nonparametric (nonp,
column 2) and semi-parametric (semi, column 3) AIFs. Overlap of AIFs estimated using the
nonparametric and semi-parametric AIFs, single-channel algorithm applied to the masseter
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(column 4) and temporalis (column 5) muscles and multi-channel algorithm (column 6).
mass versus
nonp versus semi
multi
19
95.1
NU
temp
mouse
single
single
single
single
nonp
semi
mass
temp
94.8
92.0
94.7
94.4
22
94.1
95.6
96.4
94.4
96.5
23
93.1
96.1
96.6
94.8
96.0
24
91.9
96.8
95.7
95.2
96.2
ED
MA
controls
90.8
95.9
93.4
94.4
96.0
26
92.3
95.9
94.3
93.6
95.5
28
91.4
96.9
92.9
92.9
95.0
mean
92.7
96.0
94.5
94.3
95.7
SD
1.5
0.7
1.8
0.8
0.7
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CE
PT
25
treated
5
94.6
95.7
94.5
96.3
96.3
6
96.0
98.6
97.4
96.5
97.4
7
90.4
96.2
95.7
93.4
97.5
8
95.6
96.1
94.9
96.1
96.8
10
97.5
97.3
97.5
97.3
98.0
12
95.7
97.4
96.6
96.1
96.9
13
96.9
98.8
97.5
97.3
98.0
mean
95.3
97.2
96.3
96.1
97.3
SD
2.3
1.2
1.3
1.3
0.6
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Figure Captions
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Figure 1. Flowchart of the single-channel blind deconvolution algorithm.
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Figure 2. Perfusion sequences. Two simulated sequences of the temporalis
muscle and the corresponding estimated sequences with a semi-parametric AIF.
Control-mouse conditions (SNR = 8.0) (a) and treated-mouse conditions (SNR
MA
= 23.3) (b).
Figure 3. Approximation differences of the simulated perfusion sequences (see
ED
Sec. 2.8 for definition). Each difference multiplied by 100. Nonp: nonparametric
estimate of the AIF. Semi: semi-parametric estimate of the AIF. Horizontal axis:
PT
”inf” – noiseless sequences, ”inf-full” – noiseless sequences with no truncation
Figure 4. Reference and estimated AIFs. Single-channel algorithm. The two
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reference AIFs used in the simulations (solid lines) and the corresponding semiparametric AIFs estimated by blind deconvolution of the simulated time sequences (dotted and dashed lines). (a) – The reference AIF for control mice
and the AIFs estimated from masseter (Mass. AIF, SNR=22.7) and temporalis
(Temp. AIF, SNR=16.0) muscles. (b) – The reference AIF for treated mice
and the AIFs estimated from masseter (Mass. AIF, SNR=63.5) and temporalis
(Temp. AIF, SNR=46.6) muscles. SNR values correspond to those of the real
recordings. Insets show magnified AIF segments.
Figure 5.
Percent overlap between true and estimated AIFs (see Sec. 2.8
for definition). Nonp: nonparametric estimate of AIF. Semi: semi-parametric
estimate of AIF. Horizontal axis: ’inf’ – noiseless sequences, ’inf-full’ – noiseless
sequences with no truncation.
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Figure 6. Observed and approximated perfusion sequences, real data, singlechannel algorithm, semi-parametric AIF. Temporalis muscle of control mouse
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24 (a) (SNR = 20.4) and treated mouse 6 (b) (SNR = 34.1).
Figure 7. Estimated AIFs of control mouse 24. Single-channel algorithm. Nonparametric and semi-parametric AIFs. Masseter muscle with SNR = 13.6 (a).
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Temporalis muscle with SNR = 20.4 (b).
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Figure 8. Estimated AIFs of treated mouse 6. Single-channel algorithm. Nonparametric and semi-parametric AIFs. Masseter muscle with SNR = 86.2 (a).
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PT
ED
Temporalis muscle with SNR = 34.1 (b).
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Preprocessing
Initialization of AIF, TRF, iteration index s=1
S>200
-
Parametric approximation
of nonparametric TRF
+
Nonparametric updating of AIF
Semi-Parametric approximation
of nonparametric AIF
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s=s+1
+
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S>200
-
SC
Nonparametric updating of TRF
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Figures
S>290
Scaling of AIF and TRF
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Figure 1: Flowchart of the single-channel blind deconvolution algorithm.
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(a)
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1
Simulated
Estimated
0.8
SC
0.6
0.4
0
0
5
NU
0.2
10
time [min]
15
(b)
MA
1
0.8
0.6
Simulated
Estimated
ED
0.4
0.2
0
PT
0
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Figure 2: Perfusion sequences.
5
10
time [min]
15
Two simulated sequences of the temporalis muscle and the
corresponding estimated sequences with a semi-parametric AIF. Control-mouse conditions
(SNR = 8.0) (a) and treated-mouse conditions (SNR = 23.3) (b).
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12
10
8
SC
Approximation difference [a.u.]
3
1
20
40
Inf Inf−full
ED
Masseter, nonp
Masseter, semi
Temporalis, nonp
Temporalis, semi
PT
1
0.5
0
Figure 3:
20
40
60 80
SNR [−]
Inf Inf−full
Approximation difference [a.u.]
Treated, single−channel dec.
1.5
0
60 80
SNR [−]
MA
0
NU
2
0
Controls, multi−channel dec.
Masseter, nonp
Masseter, semi
Temporalis, nonp
Temporalis, semi
4
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Approximation difference [a.u.]
Approximation difference [a.u.]
Controls, single−channel dec.
Masseter, nonp
Masseter, semi
Temporalis, nonp
Temporalis, semi
6
4
2
0
0
3.5
3
2.5
2
1.5
1
0.5
0
20
40
60 80
SNR [−]
Treated, multi−channel dec.
Masseter, nonp
Masseter, semi
Temporalis, nonp
Temporalis, semi
0
20
40
60 80
SNR [−]
Approximation differences of the simulated perfusion sequences (see Sec. 2.8 for
definition). Each difference multiplied by 100. Nonp: nonparametric estimate of the AIF.
Semi: semi-parametric estimate of the AIF. Horizontal axis: ”inf” – noiseless sequences,
”inf-full” – noiseless sequences with no truncation
38
Inf Inf−full
Inf Inf−full
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(a)
0.02
High SNR
0.01
0
5
0.5
1
1.5
10
time [min]
2
NU
0
SC
0.02
0.01
0
Low SNR
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Reference
15
Low SNR
High SNR
MA
(b)
Reference
0.04
0.04
0.03
0.01
0
5
0
0.5
10
time [min]
AC
CE
PT
0
0.02
ED
0.02
1
1.5
15
Figure 4: Reference and estimated AIFs. Single-channel algorithm. The two reference AIFs
used in the simulations (solid lines) and the corresponding semi-parametric AIFs estimated
by blind deconvolution of the simulated time sequences (dotted and dashed lines). (a) – The
reference AIF for control mice and the AIFs estimated from masseter (Mass. AIF, SNR=22.7)
and temporalis (Temp. AIF, SNR=16.0) muscles. (b) – The reference AIF for treated mice
and the AIFs estimated from masseter (Mass. AIF, SNR=63.5) and temporalis (Temp. AIF,
SNR=46.6) muscles. SNR values correspond to those of the real recordings. Insets show
magnified AIF segments.
39
Controls, single−channel dec.
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95
90
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Masseter, nonp
Masseter, semi
Temporalis, nonp
Temporalis, semi
85
0
20
AC
CE
90
0
20
40
Masseter, nonp
Masseter, semi
Temporalis, nonp
Temporalis, semi
60 80
SNR [−]
Inf Inf−full
Controls, multi−channel dec.
95
90
85
80
Inf Inf−full
ED
PT
95
Figure 5:
60 80
SNR [−]
Treated, single−channel dec.
100
85
40
Nonp
Semi
0
20
40
Inf Inf−full
95
90
85
Nonp
Semi
0
20
40
Percent overlap between true and estimated AIFs (see Sec. 2.8 for definition).
Nonp: nonparametric estimate of AIF. Semi: semi-parametric estimate of AIF. Horizontal
axis: ’inf’ – noiseless sequences, ’inf-full’ – noiseless sequences with no truncation.
40
60 80
SNR [−]
Treated, multi−channel dec.
100
Aif overlap [%]
80
Aif overlap [%]
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MA
Aif overlap [%]
100
Aif overlap [%]
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60 80
SNR [−]
Inf Inf−full
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(a)
−3
1.5
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x 10
SC
1
0
5
10
time [min]
Measured
Estimated
15
MA
0
NU
0.5
(b)
ED
x 10−3
1.5
PT
1
0.5
AC
CE
0
0
Measured
Estimated
5
10
time [min]
15
Figure 6: Observed and approximated perfusion sequences, real data, single-channel algorithm, semi-parametric AIF. Temporalis muscle of control mouse 24 (a) (SNR = 20.4) and
treated mouse 6 (b) (SNR = 34.1).
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(a)
−3
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6
x 10
Nonparametric
Semi−parametric
2
0
5
10
time [min]
15
MA
0
NU
SC
4
(b)
Nonparametric
Semi−parametric
ED
x 10−3
6
PT
4
AC
CE
2
0
0
5
10
time [min]
15
Figure 7: Estimated AIFs of control mouse 24. Single-channel algorithm.
Nonparametric
and semi-parametric AIFs. Masseter muscle with SNR = 13.6 (a). Temporalis muscle with
SNR = 20.4 (b).
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(a)
−3
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x 10
Nonparametric
Semi−parametric
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6
4
5
0
10
time [min]
MA
0
NU
2
15
0.01
AC
CE
PT
0.005
ED
(b)
0
0
5
Nonparametric
Semi−parametric
10
time [min]
15
Figure 8: Estimated AIFs of treated mouse 6. Single-channel algorithm. Nonparametric and
semi-parametric AIFs. Masseter muscle with SNR = 86.2 (a). Temporalis muscle with SNR
= 34.1 (b).
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