Alina Leandra Bendinger^{1,2}, Charlotte Debus^{3,4,5}, Christin Glowa^{4,5,6}, Christian Peter Karger^{4,6}, Jörg Peter^{1}, and Martin Storath^{7}

^{1}Department of Medical Physics in Radiology, German Cancer Research Center (DKFZ), Heidelberg, Germany, ^{2}Faculty of Biosciences, University of Heidelberg, Heidelberg, Germany, ^{3}Translational Radiation Oncology, National Center for Tumor Diseases (NCT), German Cancer Research Center (DKFZ), Heidelberg, Germany, ^{4}Heidelberg Institute for Radiation Oncology (HIRO) and National Center for Radiation Research in Oncology (NCRO), Heidelberg, Germany, ^{5}Department of Radiation Oncology and Radiotherapy, University Hospital Heidelberg, Heidelberg, Germany, ^{6}Department of Medical Physics in Radiation Oncology, German Cancer Research Center (DKFZ), Heidelberg, Germany, ^{7}Natural Sciences and Humanities, University of Applied Sciences Würzburg-Schweinfurt, Schweinfurt, Germany

### Synopsis

Accuracy in pharmacokinetic modelling of DCE-MRI data can be impaired due to
a delay between the contrast agent arrival in the tissue of interest and an
artery further upstream. To correct the delay, bolus arrival times (BATs) are
estimated from the concentration curves. However, the state-of-the-art method
for estimating BATs may give unsatisfactory results if the curves do not
exhibit a fast up-slope. We propose a spline-based method for BAT estimation for
concentration curves without fast up-slopes which are often observed in small
animal data. The proposed method gives accurate results on simulated and *in
vivo* acquired rat data.

### Objectives

Accuracy in pharmacokinetic modelling
of dynamic contrast-enhanced MRI (DCE-MRI) data can be impaired due to a delay between the arrival of contrast
agent (CA) bolus in the artery, selected to extract the arterial input function
(AIF), and the arrival of CA in the tissue of interest. When the concentration
curves exhibit a fast up-slope in CA concentration, this delay can be corrected
for by estimating the difference between both bolus arrival times (BATs) using the
state-of-the-art method proposed by Cheong *et
al.*^{1}. We introduce a method for continuous BAT estimation of concentration
curves that do not have fast up-slopes. Such data is particularly observed in
small animal data.### Proposed method

The MR-signal intensities *c*_{n} are assumed to be recorded
at *N *uniformly sampled time points *t*_{n}, and the CA to be injected as a bolus
after starting the DCE-MRI sequence. The proposed model approximates the data
points *c*_{n} by a continuous
piecewise defined function *$$$u^*$$$* which is
a constant function before the BAT *t*_{BAT}
and a smoothing spline afterwards. Hence, this approach is able to adapt to the
shape of the curve (Fig. 1, equ. 1). The BAT *t*_{BAT} is estimated along with finding the most accurate fit
for the sample points *c*_{n}. The
model is described by:

$$\tilde{u}^* = \text{argmin}_{\tilde{u}}\,\sum_{n=1}^{N'} (\tilde u(t_{\text{BAT}}) -c_n)^2 + \sum_{n=N'+1}^{N} (\tilde u(t_n) -c_n)^2 + \alpha\int_{t_{\text{BAT}}}^{t_N} (\tilde u^{(k)}(\tau))^2 \,d\tau \qquad\qquad(1) $$

where $$$\tilde u^{(k)}$$$ denotes the *k*-th derivative of the function $$$\tilde{u}$$$, α > 0 adjusts the
relative weight of data fidelity and smoothness, and *N’* is the index of the last sample point before *t*_{BAT}. The principal interest
is to estimate *t*_{BAT} from
the concentration curve. Alongside, the spline parameters *α* and *k* are determined,
too, as they adapt the model to the measured data points *c*_{n}. *t*_{BAT}
is estimated on a continuous time scale and therefore is not restricted to the
discrete sample points *t*_{n}.
All three parameters are automatically determined by generalized cross
validation.

### Model validation

The proposed model was validated
using simulated and *in vivo *acquired
rat data. Three tissue concentration curves (TCCs) were simulated using an
in-house developed software ^{2} by forward convolution of a
model-based rat AIF ^{3} with the tissue response
function of the extended Tofts model ^{4} (*K*_{trans} = 15 ml/min/100ml, *v*_{p} = 0.05, *v*_{e}
= 0.3, 0.6, or 0.1, Fig. 2 A). Data were adapted for three temporal resolutions (2 s, 5 s, and
7 s) and four noise levels (SNR = 100, 50, 25, 10). BATs were estimated by the
proposed method for 500 noise realizations for each configuration. For *in vivo* validation, DCE-MRI data of five
male Copenhagen rats bearing anaplastic Dunning R3327-AT1-tumors transplanted
onto both thighs were acquired. Animals were imaged when tumors reached a size
of 10×10 mm.
DCE-MRI (TURBO-FLASH sequence, temporal resolution: 0.75 s, 380 s total
acquisition time) was acquired per animal for single slices through the largest
diameter of the tumor and through the heart to acquire the AIF. Image signals
were converted to concentrations using absolute signal enhancement.### Results

Fig. 2 B displays representative fits of the proposed
model to the simulated TCC type 3 and the corresponding estimated BATs of all
configurations. The proposed method could accurately estimate the BAT of the
simulated TCCs even at low SNR and low temporal resolution (Fig. 3). Accuracy
was only compromised for TCC type 1 at low SNR. The BAT of the AIF was
accurately estimated for all configurations. BAT estimation by the
state-of-the-art gave unsatisfactory results for most configurations (Fig. 3). Qualitative analysis of*
in vivo* acquired data reveals that
BATs were estimated at reasonable time points (exemplary fitting results and
BAT estimations for animal 4 are displayed in Fig. 4 A-F) and within small
margins per animal (Fig. 4 G).### Discussion and conclusion

A
new method for BAT estimation on a continuous time scale of DCE-MRI is proposed
for dynamic data lacking a fast up-slope, as often found in small animal data. A
simulation study with known ground truth showed that the accuracy of BAT estimation
was superior to other methods even at low SNR and low temporal resolution. Results
of *in vivo* acquired data were found
to be at reasonable time points.### Acknowledgements

This work was supported by the German Research Foundation (DFG STO1126/2-1, GL893/1-1, KA2679/3-1, and KFO 214).

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