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Perfusion MRI Deconvolution with Delay Estimation
and Non-Negativity Constraints
Marco Pizzolato, Aurobrata Ghosh, Timoth´e Boutelier, Rachid Deriche
To cite this version:
Marco Pizzolato, Aurobrata Ghosh, Timoth´e Boutelier, Rachid Deriche. Perfusion MRI
Deconvolution with Delay Estimation and Non-Negativity Constraints. International Sym-
posium on Biomedical Imaging, Apr 2015, Brooklyn, New York, United States. 2015,
<http://biomedicalimaging.org/2015/>.<hal-01143213>
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Perfusion MRI Deconvolution with Delay Estimation and Non-Negativity Constraints
Marco Pizzolato?Aurobrata Ghosh?Timoth´
e Boutelier†Rachid Deriche?
?Athena Project-Team, Inria Sophia Antipolis - M´
editerran´
ee, France
†Olea Medical, La Ciotat, France
ABSTRACT
Perfusion MRI deconvolution aims to recover the time-
dependent residual amount of indicator (residue function)
from the measured arterial and tissue concentration time-
curves. The deconvolution is complicated by the presence of
a time lag between the measured concentrations. Moreover
the residue function must be non-negative and its shape may
become non-monotonic due to dispersion phenomena. We
introduce Modified Exponential Bases (MEB) to perform de-
convolution. The MEB generalize the previously proposed
exponential approximation (EA) by taking into account the
time lag and introducing non-negativity constraints for the
recovered residue function also in the case of non-monotonic
dispersed shapes, thus overcoming the limitation due to the
non-increasing assumtion of the EA. The deconvolution prob-
lem is solved linearly. Quantitative comparisons with the
widespread block-circulant Singular Value Decomposition
show favorable results in recovering the residue function.
Index Terms—Perfusion, Deconvolution, Exponential
Bases, Delay, Dispersion, Non-Negative, DSC-MRI
1. INTRODUCTION
Perfusion imaging aims to recover parameters related to the
passage of blood in the parenchyma (i.e. the functional part)
of a tissue. The amount of perfusion is related to the function-
ality of the parenchyma and its grade of activity.
The perfusion can be characterized in vivo using Dynamic
Susceptibility Contrast MRI (DSC-MRI) where a bolus of
paramagnetic contrast agent (PA) is injected in the subject’s
vascular system. In each voxel, a signal related to the PA con-
centration is acquired on a time sampling grid to derive the
correspondent tissue concentration curve Cts(t). According
to the indicator-dilution theory [1], the measured Cts(t)in a
voxel is expressed as the convolution between the arterial in-
put concentration to the voxel Ca(t), and a curve R(t)which
expresses at each time the residual amount of PA in the voxel
Cts(t) = Zt
0
Ca(θ)R(t−θ)dθ (1)
The authors express their thanks to Olea Medical and to the Provence-
Alpes-Cˆ
ote d’Azur Regional Council for providing grant and support.
where R(t)is unknown and must be recovered from the ob-
served Ca(t)and Cts(t)by means of deconvolution. The
quality of the reconstructed R(t)directly affects the estima-
tion of the perfusion parameters such as the blood flow (BF),
the blood volume (BV) and the mean transit time (MTT).
Since deconvolution is ill-posed [2] several regularization
methods have been proposed to suppress noise, such as the
truncated Singular Value Decomposition (tSVD) [3]. More
recently a deconvolution method [2] based on exponential
bases approximation (EA) showed to outperform tSVD by
modeling R(t)as a weighted sum of exponentially decaying
functions.
However those techniques show some limitations related
to a technical issue inherent the perfusion data acquisition and
the physiological nature of the residue function. In real prac-
tice, indeed, it is impossible to measure the actual arterial con-
centration in input to a specific voxel and normally Ca(t)is
selected as the tissue concentration of a voxel within a region
containing only arterial blood. Therefore the concentrations
Cts(t)of all the other voxels are physiologically delayed with
respect to the selected arterial input Ca(t), or can anticipate
it, causing the failure of the deconvolution due to the lack
of causality assumptions. In addition, the bolus of PA may
undergo dispersion which causes the shape of Ca(t)to mod-
ify, thus changing the natural decreasing shape of R(t)into
a non-monotonic one [4]. Such a shape is sometimes con-
sidered in myocardial perfusion applications [5]. Moreover,
since R(t)expresses the time-dependent remaining quantity
of PA in the voxel, negative values are non-physiological and
need to be avoided. The EA does not take into account the
time lag (delay) between Ca(t)and Cts(t). In addition it
does not consider dispersion effects since it cannot guaran-
tee non-negativity unless the residue function is constrained
to be monotonically decreasing [2].
We propose Modified Exponential Bases (MEB) allowing
time delay estimation and a straight forward implementation
of non-negativity constraints also in case of non-monotonic
dispersed residue functions. We discuss an iterative linear
implementation of the MEB and compare the results with the
standard block-circulant formulation of tSVD [6], henceforth
addressed as oSVD, which is insensitive to the delay and thus
comparable with the proposed technique.
2. THE DECONVOLUTION PROBLEM
Assuming the tissue concentration Cts(t)and the arterial
one Ca(t)both measured on an equally spaced time grid
t1, t2, . . . , tMof size M, with ∆t=ti+1 −ti, then Eq. 1 is
discretized as
Cts(tj) = ∆t
j
X
i=0
Ca(ti)R(tj−ti)(2)
which can be formulated in matrix form as cts =Ar, where
Ais the M×Mconvolution matrix containing the samples of
the arterial input concentration, cts contains the Msamples of
the tissue concentration and rcontains the Munknown sam-
ples of the residue function. If a model R(t) = G(t, p)for
the residue function is defined, then the deconvolution prob-
lem is finding the set of parameters p. In case of linearity in
the parameters the residue function can be written as r=Gp,
where Gis the M×Ndesign matrix and pthe N×1vector
of coefficients with N≤M. Therefore the convolution is
formulated as cts =AGp, where pis unknown.
In [2] it is proposed the exponential approximation (EA)
to perform deconvolution, which searches a solution for R(t)
in the span of decaying exponential functions, thus modeling
the residue function as a sum truncated to an order N
R(t) =
N
X
n=1
kne−λntu(t)(3)
where u(t)is the Heaviside step function and the time
rates λnare chosen according to the harmonic distribution
λn=n/T , with Tthe observation time interval. The un-
known parameters p= [k1, ..., kN]are found with R(t)
constrained to be at the same time non-negative R(t)≥0and
non-increasing R0(t)≤0for t > 0, where the non-negativity
holds due to the non-increasing assumption [2]. Therefore
EA, in order to guarantee non-negativity, neglects dispersion
effects assuming a non-increasing shape for R(t). In addition
it does not take into account the time delay between Ca(t)and
Cts(t). In the following section we propose a more general
solution solving these limitations.
3. MODIFIED EXPONENTIAL BASIS
In order to estimate the time delay τand guarantee the
non-negativity for both monotonically decreasing and non-
monotonic residue functions, we propose Modified Exponen-
tial Bases (MEB). Each basis is constituted as the sum of an
exponential decay and the opposite of its derivative with re-
spect to the time rate, respectively weighted by two different
constants, anand bn. Each basis shape can thus vary between
a pure exponential decay (bn= 0) and its derivative term
(an= 0). Therefore the MEB include the EA and generalize
it by integrating the exponential derivative terms. Moreover
the time delay τis explicitly considered with a Heaviside step
function and is included in the exponentials, giving
R(t) = u(t−τ)
N
X
n=1
(an+bn(t−τ))e−αn(t−τ).(4)
Hence, we can guarantee non-negativity just by constraining
all the anand bnto be non-negative, and at the same time
recover both monotonically decreasing and non-monotonic
convolution kernels.
The deconvolution problem incorporating Eq. 4 has a non-
linear solution since the unknown αnand τappear in the ex-
ponential. However the time rates αncan be preset without
loss of generality [2], thus allowing to find an iterative linear
procedure to estimate the unknowns.
In order to fix the rates αnwe found that the harmonic
rule proposed in [2] is too sensitive to the chosen maximum
basis order N, as also reported in [7]. Despite such a rule
can be used, we decide to calculate the rates based on the
signal rather than the observation time interval. We define
M T Tmax as the maximum M T T expected. This can be
fixed empirically, or set according to a conservative value
calculated from the MTT found with mono-exponential ap-
proximation or via oSVD. Then the time rates αnof Eq. 4
are calculated as the reciprocal of the M T T values harmon-
ically distributed in the range [M T Tmin, M T Tmax]where
M T Tmin =M T Tmax/N.
In order to consider the delay τin the estimation we no-
tice that, due to the linearity of the convolution, it can be in-
corporated in the residue function estimation. To account for
this, the problem formulated in Sec. 2 must consider the circu-
lar formulation of the convolution [6]. More precisely Cts(t)
and Ca(t)of length Mare extended by zero-padding up to a
length L≥2Mto avoid aliasing. Then the new L×Lsquared
circular convolution matrix Achas entries Ac
i,j =Ca(ti−j+1)
if j≤iand Ac
i,j =Ca(tL+i−j+1)if j > i. In the rest we
drop the superscript for clarity, Ac
i,j =Ai,j . The convolution
problem is then formulated as cts =AG(τ)p, where G(τ)is
the L×2N τ-dependent design matrix extended to the circu-
lar time sampling grid, and pis the vector containing the 2N
coefficients aiand bito be estimated. We perform the estima-
tion of τvia grid search over a range [τmin, τmax]seconds,
with a time step τs≤∆t. The set of estimated parameters ˆ
p
is obtained as the Least Squares solution subject to ai, bi≥0,
∀i= 1, ..., N when the estimated delay ˆτis that minimizing
||cts −AG(τ)ˆ
p||2
2among all τ∈[τmin, τmax ].
4. EXPERIMENTAL DESIGN
We test the performance of the compared methods with
two synthetic datasets, one generated with a decreasing
residue function and the other with a non-monotonic (dis-
persed) one. In the first case we adopt the bi-exponential
model Rbi−exp(t) = A·e−τ1t+ (1 −A)·e−τ2t, where
(a) SN R = 80.
(b) Bi-exponential.
(c) Pharmacokinetic.
Fig. 1. Absolute relative error of estimates (mean and standard deviation).
τ1= 0.68,τ2= 0.05 and A= 0.95, according to the nor-
mal tissue values given in [8]. In the second case, we use
the Pharmacokinetic model as formulated in [7], Rpk (t) =
(e−λ1t−e−λ2t)/(λ2−λ1)−(e−λ1t−e−λ3t)/(λ3−λ1)
where we fix λ2= 0.21,λ3= 0.36 according to [5] and
λ1= (λ3−λ2)/(M T T ·λ2·λ3)with M T T ≈2.2s, as
obtained from the bi-exponential model. We generate the
actual residue function via multiplication of Rbi−exp(t)or
Rpk(t)with BF = 30 ml/100g/min, giving peak values of
BF for the bi-exponential model and ≈5.7ml/100g/min
for the pharmacokinetic one. In the latter case, the peak
value will be considered as BF for the experiments. With
the chosen parametrization, BV ≈1.1ml/100gin all cases.
The arterial concentration is generated according to [4] as
Ca(t) = γ0(t−t0)ν·e−(t−t0)/β with ν= 3,β= 1.5,γ0= 1,
and t0= 30 s. The tissue concentration Cts (t)is then ob-
tained via Eq. 2. The concentration curves are then converted
to signal intensities according to S(t) = S0·e−κ·C(t)·T E ,
where T E is the echo time and κis the susceptibility con-
stant fixed according to [3]. We choose S0and T E separately
for the tissue concentration St
0= 200, T Et= 55 ms and
for the arterial one Sa
0= 600, T Ea= 13 ms, according to
[9]. The signals, obtained with repetition time ∆t= 1 s,
are then corrupted with Gaussian noise with zero mean and
standard deviation σ=S0/SN R, and finally reconverted in
concentration curves.
We performed the experiments separately for Cts(t)ob-
tained after convolution of Ca(t)with both the bi-exponential
and the pharmacokinetic kernel. In each case we delayed
Ca(t)and Cts(t)with integer values in range [−5,5] sec-
onds. For each delay we calculated the residue function via
oSVD (20%) and MEB (N= 30, harmonic αn,M T Tmax =
4·M T ToSV D,τmin =−10 s,τmax = 15 s,τs= 0.25 s)
deconvolution, and obtained the estimates of BF ,BV and
M T T . We tested four SN R values [9], [40,60,80,100], gen-
erating 100 noisy repetitions for each possible combination of
kernel, delay and SN R.
(a) Pharmacokinetic kernel (b) fitting, SN R = 80
(c) Bi-exponential kernel (d) fitting, SN R = 80
Fig. 2. Residual error (a,c) and residue function fitting (b,d).
5. RESULTS AND DISCUSSION
Figure 1(a) illustrates the mean absolute relative error in the
estimation of each parameter for the representative case of
SN R = 80. Figures 1(b) and 1(c) show the tables giving
the estimates mean errors and standard deviations for all the
SN R, in the case of bi-exponential and pharmacokinetic
kernel respectively. For both oSVD and MEB, we computed
the mean residual error of the recovered residue function
with respect to the ground truth, as shown in Fig. 2(a) and
2(c), for increasing noise levels, in case of dispersed and the
bi-exponential kernel respectively; the std areas are shown.
Fig. 2(b) and 2(d) show examples of the correspondent MEB
and oSVD fittings against the ground truth.
We notice that, for both MEB and oSVD, the quality of
the estimated parameters depends on the chosen convolution
kernel (Fig. 1). In the case of a smooth kernel, such as the
pharmacokinetic one, the MEB show less effective results
than oSVD in estimating the parameters (table. 1(c)). Indeed
oSVD cuts the high frequency oscillations of Ca(t)which, in
such a case, mainly correspond to noise. A possible explana-
tion of the MEB performance is that the delay estimation is
harder on the dispersed kernel due to its smoothness. Indeed,
we found that the delay estimation error for the pharmacoki-
netic kernel doubles that obtained for the bi-exponential one
(≈42% against ≈19%,SN R = 80).
On the other hand oSVD show major limitations in the
case of bi-exponential residue function. For instance the es-
timation error of M T T almost doubles the magnitude of the
correspondent ground truth value. This performance drop of
oSVD is highly related to the marked underestimation of BF ,
with an error around the 57% of the actual blood flow as
shown in Fig. 1(a) and in table 1(b).
The MEB outperform oSVD for the bi-exponential residue
function as shown in Fig. 1(a) and in table 1(b). Indeed the
MEB manage to reduce the underestimation of BF giving
an average relative error in a range [15%,21%] of the ground
truth value. The MEB also allow to obtain an error for M T T
always at least three times smaller than that of oSVD. The
calculation of BV as the analytical integral of R(t)obtained
via MEB is comparable to that obtained via oSVD. However
it is normally preferable to estimate BV directly from the
concentration curves Ca(t)and Cts(t)to avoid introducing
errors related to the deconvolution procedure.
The MEB show less residual error then oSVD for both of
the tested kernels and at any SN R, as shown in Fig. 2(a) for
the dispersed residue function and 2(c) for the bi-exponential
one. We further notice that the MEB fitting is more robust
to noise. Indeed Fig. 2(a) shows that with dispersed residue
function, the difference between the oSVD residual error at
SN R = 40 and SN R = 100 is almost four times higher
than the correspondent difference for the MEB. The fitting
capability of the MEB is more marked for the bi-exponential
kernel, Fig. 2(c), where the amount of residual error stands
between the 35% and the 60% of the oSVD error for any
SN R. In general, the residue function recovered via MEB
looks smoother and without ripples, which are a contributing
cause to the oSVD residual error.
6. CONCLUSIONS
We introduced Modified Exponential Bases to perform per-
fusion deconvolution. The proposed MEB generalize the
exponential approximation by including both exponentially
decaying terms and their derivatives. Indeed the MEB allow
to take into account residue functions which are either purely
decreasing or dispersed, providing at the same time a straight
forward implementation of non-negativity constraints. The
MEB also explicitly take into account the time delay be-
tween the arterial and the tissue concentrations. The oSVD
showed a lower error in the perfusion parameters estimation
for the dispersed residue function, in agreement with the lat-
ter’s smoothness. However, in the case of the bi-exponential
kernel the MEB outperform oSVD. In addition, for any type
of kernel, the fitting results of the MEB compare favorably
with respect to those obtained via oSVD, and are more ro-
bust to noise. We believe that time delay estimation and
non-negativity constraints help in recovering a physiologi-
cally meaningful residue function. Moreover we find that
performing deconvolution with continuous basis functions
improves the robustness to noise and we think that proposed
bases should take into account the variability in the possible
shapes of the residue function.
7. REFERENCES
[1] Meier et al., “On the theory of the indicator-dilution
method for measurement of blood flow and volume,” J
Appl Physiol, vol. 6(12), pp. 731–744, 1954.
[2] Keeling et al., “Deconvolution for dce-mri using an ex-
ponential approximation basis,” Med Image Anal, vol.
13(1), pp. 80–90, 2009.
[3] Østergaard et al., “High resolution measurement of cere-
bral blood flow using intravascular tracer bolus passages.
part i: Mathematical approach and statistical analysis,”
Magn Reson Med, vol. 36(5), pp. 715–725, 1996.
[4] Calamante et al., “Delay and dispersion effects in dy-
namic susceptibility contrast mri: simulations using sin-
gular value decomposition,” Magn Reson Med, vol.
44(3), pp. 466–473, 2000.
[5] Zarinabad et al., “Voxelwise quantification of myocardial
perfusion by cardiac magnetic resonance,” Magn Reson
Med, pp. 1994–2004, 2012.
[6] Wu et al., “Tracer arrival timinginsensitive technique for
estimating flow in mr perfusionweighted imaging using
singular value decomposition with a blockcirculant de-
convolution matrix,” Magn Reson Med, vol. 50(1), pp.
164–174, 2003.
[7] Batchelor et al., “Arma regularization of cardiac perfu-
sion modeling,” ICASSP, pp. 642–645, March 2010.
[8] Mehndiratta et al., “Modeling the residue function in
dscmri simulations: Analytical approximation to in vivo
data,” Magn Reson Med, 2013.
[9] Wirestam et al., “Wavelet-based noise reduction for
improved deconvolution of time-series data in dynamic
susceptibility-contrast mri,” Magn Reson Mater Phy, vol.
18(3), pp. 113–118, 2005.
