Charlotte Debus^{1,2,3,4}, Ralf Floca^{5}, Amir Abdollahi^{1,2,3,4}, and Michael Ingrisch^{6}

In pharmacokinetic analysis of DCE-MRI data, the choice of initial parameter values for fitting has been reported to have a significant impact on the outcome of the optimization and hence, on parameter estimates. In this study, we investigated the influence of initial values by fitting simulated concentration time curves with varying combinations of initial parameters, using the two compartment exchange model. The resulting parameter estimates were visualized and compared to the true values, used for simulation, by means of relative errors. Results showed that the choice of initial values has little influence on the precision of the pharmacokinetic analysis.

Five types of tissue curves, defined by
specific combinations of perfusion parameters (figure 1, table A),
were used to simulate time-resolved concentration images of 15x60 px
by convolution of a measured arterial input function (temporal
resolution 2.11s, 169 timepoints^{5})
with 2CXM tissue response functions. To account for acquisition
noise, Gaussian random numbers were added to each data point.
Standard deviation of the noise was chosen to achieve a contrast-to-noise ratio (CNR) of 300. CNR is defined as the ratio between AIF
peak and standard deviation of the noise.

All curves were fitted with the 2CXM, using an
in-house written software module^{6}, implemented in the Medical Imaging Interaction Toolkit^{7}.
This tool allows definition of initial optimization values for each
parameter on a pixelwise basis, which opens the possibility to use
various combinations of initial parameters and graphically visualize
the resulting parameter estimates.
Parameter constraints were applied to assure
model consistency : $$$ 0<v_{p},v_{e}<
1,\: v_{p}+v_{e}<1,\:
0<F_{p}<100\: [ml/min/100ml], \:-1<PS< 100\: [ml/min/100ml] $$$. Initial parameter values were
defined in 15x60 px images, with one parameter varying along the
x-axis and the other three kept fixed (figure 1, table B). Parameter
estimates were averaged over all 60 px with the same set of initial
values.
Accuracy
of the average parameter estimate of all four model parameters $$$\bar{P}_{fit}$$$
was evalutated by means of the relative error with respect to the simulation input
values $$$P_{input}$$$:
$$E_{rel}= \frac{\bar{P}_{fit} - P_{input}}{P_{input}}$$

^{1}
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^{2} Brix,
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based on dynamic contrast-enhanced CT and MRI measurements."
European journal of nuclear medicine and molecular imaging
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^{3} Brix, Gunnar, et al. "Pharmacokinetic analysis of tissue
microcirculation using nested models: multimodel inference and
parameter identifiability.", Medical Physics 36.7 (2009),
2923–2933

^{4}
Buckley, David L. "Uncertainty in the analysis of tracer
kinetics using dynamic contrast-enhanced T1-weighted MRI."
Magnetic Resonance in Medicine 47.3 (2002): 601-606.

^{5}
Ingrisch, Michael, et al. "Quantification of perfusion and
permeability in multiple sclerosis: dynamic contrast-enhanced MRI in
3D at 3T." Investigative radiology 47.4 (2012): 252-258.

^{6}
Debus, Charlotte et al „Impact
of fitting strategy on DCE parameter estimates and performance : a
simulation study in image space“, 24rd
Annual meeting of the International Society of Magnetic Resonance in
Medicine (2016),
Vol. 24, International Society of Magnetic Resonance in Medicine
(ISMRM)

^{7} Wolf, Ivo, et al. "The
medical imaging interaction toolkit (MITK): a toolkit facilitating
the creation of interactive software by extending VTK and ITK."
Medical Imaging (2004). International Society for Optics and
Photonics.

A: Parameter combinations
used to simulate concentration-time curves. Together with a measured
arterial input function, each of the parameter combinations was used
to generate a homogeneous 15 × 60 px dynamic image (169 time points,
temporal resolution 2.11s). These images were fitted with different
combinations of initial values for the four model parameters. Each
parameters' initial value was either constant over the whole image or
varied between 15 different values, as listed in table B.

The arterial input
curve used for simulations and representative simulated
concentration- time curves of the homogeneous parameter images for
different curve types „Reference“, „Low F_{p}“,
„Low PS“, „Low v_{p}“
and „Low v_{e}“. For illustration, the reference curve shows the
simulated curve without noise (black line) in comparison to the used
curve with Gaussian random noise (red dots). The AIF was taken from
measurements of a double bolus injection in the middle cerebral
artery (temporal resolution of 2.11s)

Visualization
of mean relative errors on parameter estimates of F_{p}
(A: left column) and PS (B: right column) for different curve types
„Reference“, „Low F_{p}“
,
„Low PS“ , „Low v_{p}“
and „Low v_{e}“
(indicated on the x-axis of each plot). Each row of plots shows
results from fits with varying initial values of either F_{p}
(top row), PS (2^{nd} row), v_{p}
(3^{rd}
row) or v_{e}
(bottom row), while the other three parameter initial values were
kept at a fixed value. The respective initial parameter values are
labeled on the y-axis of the plots.

Visualization
of mean relative errors on parameter estimates of v_{p}
(C: left column) and v_{e}
(D: right column) for different curve types „Reference“, „Low
F_{p}“
,
„Low PS“ , „Low v_{p}“
and „Low v_{e}“
(indicated on the x-axis of each plot). Each row of plots shows
results from fits with varying initial values of either F_{p}
(top row), PS (2^{nd} row), v_{p}
(3^{rd}
row) or v_{e}
(bottom row), while the other three parameter initial values were
kept at a fixed value. The respective initial parameter values are
labeled on the y-axis of the plots.

Relative
errors on parameter estimates of F_{p}
over varying initial values of v_{p}
(A) and v_{e}
(B) and on parameter estimates of v_{p}
(C) over varying initial values of v_{p}
(A) for concentration curves with low F_{p}.
The starting value of a single parameter took 15 different values
along the x-axis, while the other starting values were kept at a
fixed value. Parameter estimates were averaged over 60 curves and the
relative error was calculated with respect to the original parameter
value, used for data simulation. The error bars represent the
standard deviation of the mean.