Julie C. DiCarlo^{1,2} and Thomas E. Yankeelov^{1,2,3,4}

^{1}Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, AUSTIN, TX, United States, ^{2}Livestrong Cancer Institutes, The University of Texas at Austin, AUSTIN, TX, United States, ^{3}Department of Biomedical Engineering, The University of Texas at Austin, AUSTIN, TX, United States, ^{4}Department of Diagnostic Medicine, The University of Texas at Austin, AUSTIN, TX, United States

### Synopsis

A new
method based on analysis of simplicial complexes (ASC) is presented to select
three time points at which to sample dynamic contrast-enhanced MRI uptake
curves. The technique maps expected
enhancement curve amplitudes to simplicial complex vertices and searches for the
best-discriminating set of time samples.
Simulation results indicate it should be possible to estimate Kety-Tofts
kinetic parameters from images whose acquisition times are increased above 16
seconds per volume, in a 2-minute shorter imaging time than needed for signal
enhancement ratio (SER) measurement.

### Introduction

Radiological assessment for breast cancer screening and diagnosis
requires a series of dynamic contrast-enhanced images (DCE-MRI) to visualize
areas of signal enhancement with sufficiently high spatial resolution to
describe tumor boundary shape and structure. Acquiring these images with high
enough signal to noise ratio (SNR) comes at the cost of requiring longer
volumetric acquisition times, between 60-90 seconds per volume. More quantitative assessment of signal
enhancement has been shown to improve (for example) specificity in
distinguishing benign from malignant lesions^{1}. Estimating
pharmacokinetic (PK) parameters with less than 10% error requires acquisition
times to be at or below 16 seconds per volume of coverage^{2}. The signal
enhancement ratio (SER) is a semi-quantitative approach for characterizing
uptake that improves specificity^{3} and is correlated to PK parameters
under certain acquisition conditions^{4}. Importantly, the SER requires
only three acquisition time points, *S*_{0}, S_{1}, and *S*_{2},
where *S*_{0} is pre-contrast, *S*_{1} is nearly the time
of peak flow, and *S*_{2} is chosen far enough to discriminate
between flow that is persistently increasing, flattens in a plateau, or drops
with washout. The SER is then defined as
the ratio of (*S*_{1}-S_{0}) to (*S*_{2}-S_{0}).
SER post-contrast time points vary, with *S*_{1} often falling
around 1.5 minutes post-injection of the contrast agent, and *S*_{2} at
the last-acquired time point, often 6-7 minutes later. It has been shown that decreasing *S*_{2}
from 7.5 to 4.5 minutes reduces the imaging time needed for a SER measurement
without diminishing the ability to discriminate between benign and malignant
breast lesions^{5}. In this
work, we seek to identify a set of three time points that allow for direct
estimation of PK parameters by presenting a new method of searching
mathematical mappings between PK parameters and uptake level at various
possible sampling times.### Methods

An
analysis of simplicial complexes (ASC) was performed to select three time points
at which to sample contrast agent concentration curves to maximize curve
separation. The method can be used to
select any number of time points, but is most easily visualized for 2-point
selection as illustrated in Figure 1. For
the standard Kety-Tofts model, parameter permutations of *K*^{trans}
and *v*_{e} values are used to compute the set of associated
tissue contrast-agent concentration curves using an available population
arterial input function (AIF). Next, the set of all possible combinations of
sample times is computed. For each sample time combination, a simplicial
complex is generated whose vertices map to uptake curve values at the sample
times (see Figure, 1 right panel.) The
simplicial complexes are searched to select the one with maximum area, which
indicates the best ability to discriminate between the curves. For the case of selecting three time points,
simplicial complexes are curved surfaces in 3-dimensional space.

ASC
was used to select three time points at 36, 65, and 153 seconds from injection
using 21×21 (*K*^{trans}, *v*_{e})
permutations with values between 0 and 1.
A second set of 961 (31×31) permutations of (*K*^{trans}, v_{e})
parameter values were used to generate simulated uptake curves by adding 10% Gaussian
noise to computed curves. The addition of random noise was repeated 500 times to
generate a test set of 480,500 noisy sample curves. The noisy sample curves
were each sampled at the ASC time points of 36, 65, and 153 s and at another set
of three time points that define the SER samples: 0, 90, and 270 s. For both
sets of three-point samples, nonlinear least-squares fits to the standard
Kety-Tofts model were used to generate fitted uptake curves and (*K*^{trans},
v_{e}) estimates. Figure
2 shows three examples of repeated generation of noisy curve samples for the
entire time course, ASC and SER selected three-point samples, and the result of
applying fits to both 3-sample sets.### Results

The SER
three-time point fits had the largest range in average fit error across
parameter values: from 1-1000% for *K*^{trans} and 0.03 to 686%
for *v*_{e}. The ASC three-time point fits had the next largest range
in average fit error across parameter values, from <0.01 to 73% for *K*^{trans}
and <0.01 to 659% for *v*_{e}. The full-length noisy time course had the
smallest range in average fit error, from < 0.01 to 20% for *K*^{trans}
and <0.01 to 105% for *v*_{e}. In all sampling cases, largest
errors were at the smallest values of *K*^{trans} or *v*_{e}.
The contour maps in Figure 3 show how the error in fit as averaged over 500
repetitions is distributed across parameter ranges.### Discussion and Conclusion

Simulations
indicate that using analysis of simplicial complexes, three-timepoint sampling
can be used to estimate standard Kety-Tofts kinetic parameters. Since increasing acquisition time should
result in lower noise levels, error in estimated parameters should be even more
decreased than our simulations indicate.
Further work will be carried out to validate the method in our current
patient data set. ### Acknowledgements

We
gratefully acknowledge the support of NCI U01CA142565, U01CA174706, and CPRIT
RR160005.### References

[1] Sorace,
AG, Partridge, SC, Li, X, Virostko, J, Barnes, SL, Hippe, DS, Huang, W and
Yankeelov, TE. Distinguishing benign and malignant breast tumors: preliminary
comparison of kinetic modeling approaches using multi-institutional dynamic
contrast-enhanced MRI data from the International Breast MR Consortium 6883
trial. J Med Imaging, 2018; 5(1): 011019.

[2] Henderson,
E, Rutt, BK and Lee, TY. Temporal sampling requirements for the tracer
kinetics modeling of breast disease. Magn Reson Imaging. 1998; 16(9): 1057-1073.

[3] Arasu, VA,
Chen, RC, Newitt, DN, Chang, CB, Tso, H, Hylton, NM and Joe, BN. Can signal
enhancement ratio (SER) reduce the number of recommended biopsies without
affecting cancer yield in occult MRI-detected lesions? Acad
Radiol. 2011; 18(6): 716-721.

[4] Li, KL, Henry,
RG, Wilmes, LJ, Gibbs, J, Zhu, X, Lu, Y and Hylton, NM.
Kinetic Assessment of Breast
Tumors Using High Spatial Resolution Signal Enhancement Ratio (SER) Imaging.
Magn Reson Med. 2007; 58(3): 572–581.

[5] Partridge, SC, Stone, KM, Strigel,
RM, DeMartini, WB, Peacock, S. and Lehman, CD. Breast DCE-MRI: influence of postcontrast
timing on automated lesion kinetics assessments and discrimination of benign
and malignant lesions. Acad Radiol. 2014; 21(9): 1195-1203.