Impact of fitting strategy on DCE parameter estimates and performance : a simulation study in image space

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

Time-resolved
concentration images with gaussian noise and contrast-to-noise ratio
CNR=300 (expressed as ratio between the maximum of the AIF and the standard deviation of the noise) were
simulated, using a measured arterial input function resulting from a
double-bolus injection (temporal resolution 2.11s, 169 timepoints, Ref. 7).
All curves were
fitted with an in-house written software module developed within the
Medical Imaging and Interaction Toolkit MITK^{8},
which uses the levenberg-marquard optimizer implementation of the
visual numeric library (VNL). The differential equations were solved
using the boost implementation of the Runge-Kutta method. For the
convolution approach, the arterial input function was interpolated
linearly. 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, 0<PS< 100 ml/min/100ml. The optimizer
configuration and start parameters were kept constant for all fits.

Accuracy and robustness were evaluated on concentration-curves using 5 different parameter combinations (Figure 1a). For each of these combinations, a homogeneous 30x30 pixel concentration image of 169 time points with random noise was simulated. The resulting fitted parameters were compared to the input value by calculating the bias, e.g. the difference between input value and the mean of the fitting result, and the variance of the mean.

For illustration a
50x50 pixel concentration image of 169 timepoints with different
lettering for F_{p}
and v_{p}
and constant PS, v_{e}
was fitted with both approaches.

The convolution and
the differential equation approach yielded similar results for a
given parameter combination (Figure 2). The mean estimates of F_{p}
compared to the true value showed no considerable deviation except for
low F_{p}.
(Figure 3). For the reference situation as well as low PS and low v_{e}
the parameter estimates of PS and v_{p}
were close to the original values. The fitting result for v_{e}
yielded a poor estimate with many outliers. The mean and median
parameter estimates differed strongly. For
low values of F_{p},
both fitting strategies yielded poor fit stability, large
uncertainties and many outliers.
The convolution approach
excelled in terms of computational time (Figure 4).

In the visual
approach (Figure 5) the text „ISMRM“ and „Singapore“ from the
input parameter maps could be reproduced by both fitting routines.
However, the value of v_{p}
showed interference with parameter estimate of PS and v_{e}
as fragments of the textline „Singapore“ could be observed in the
corresponding maps. PS showed good fit stability in the rest of the
image, whereas v_{e}
again presented poor fitting results.

[1] Sourbron, S. P., and D. L. Buckley. "Tracer kinetic modelling in MRI: estimating perfusion and capillary permeability." Physics in medicine and biology 57.2 (2012): R1.

[2] Brix, Gunnar, et al. "Tracer kinetic modelling of tumour angiogenesis based on dynamic contrast-enhanced CT and MRI measurements." European journal of nuclear medicine and molecular imaging 37.1 (2010): 30-51

[3] Sourbron, Steven, et al. "Quantification of cerebral blood flow, cerebral blood volume, and blood–brain-barrier leakage with DCE-MRI." Magnetic Resonance in Medicine 62.1 (2009): 205-217.

[4] Luypaert, Robert, et al. "The Akaike information criterion in DCE-MRI: Does it improve the haemodynamic parameter estimates?." Physics in medicine and biology 57.11 (2012): 3609.

[5] Brix, Gunnar, et al. "Pharmacokinetic analysis of tissue microcirculation using nested models: multimodel inference and parameter identifiability." Medical physics 36.7 (2009): 2923-2933.

[6] Luypaert, R., et al. "Error estimation for perfusion parameters obtained using the two-compartment exchange model in dynamic contrast-enhanced MRI: a simulation study." Physics in medicine and biology 55.21 (2010): 6431.

[7] 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.

[8] 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, 2004.

Parameter combinations with
representative corresponding simulated concentration-time curves for
homogeneous parameter images for reference values (b), low F_{p}(c),
low PS(d), low v_{p}(e) and low v_{e}(f). For
illustration, b) shows the simulated curve without noise (black line)
in comparison to the used curve with gaussian raondom noise (red dots).

Parameter estimate for F_{p},
PS, v_{p} and v_{e} from fitting homogeneous 30x30
pixel images of 5 different parameter combinations with
convolution(green) and differential
equation approach(blue). The whiskers are defined as
first/third quartile -/+1.5*IQR*. The parameter median is represented
by the black line, the black dot shows the mean value.

Accuracy and precision of parameter estimates F_{p},PS,v_{p}
and v_{e}
with the convolution(blue) and differential equation(green) approach in terms
of deviation from the true value and variance. Accuracy is calculated as difference between mean parameter
estimate of all 900 fitted curves and input value, precision as standard deviation of the
mean.

Optimization time per pixel for fitting
homogeneous 30x30 pixel images of 5 different parameter combinations,
respectively. The whiskers are defined as first/third quartile -/+1.5*IQR*. The parameter median is represented by the black line, the
black dot shows the mean value.

Parameter result images from fitting
the 50x50 pixel concentration image with the convolution and
differential equation model function compared to the input parameter
map(vertical) for model parameters F_{p}, PS, v_{p}
and v_{e} (horizontal).

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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