Ryan Thomas Woodall^{1}, Stephanie L Barnes^{1}, Anna G Sorace^{2}, David A Hormuth II^{1}, C Chad Quarles^{3}, and Thomas E Yankeelov^{1}

Standard compartmental models for quantitative dynamic contrast enhanced MRI (DCE-MRI) typically assume active delivery of contrast agent that is instantaneously distributed within the extravascular extracellular space within each imaging voxel. The goal of this study is to determine the error accumulated in the estimated pharmacokinetic parameters when these assumptions are not satisfied. Using finite element methods to model contrast agent arrival and diffusion throughout realistic tissue domains (obtained from histological stains of tissue sections from a murine cancer model), it was rigorously determined that parameterization error is highest in regions of low vascularity, and lowest in well-perfused regions.

Nude athymic mice were subcutaneously implanted
with BT474 cancer cells which were then allowed to grow into ~300 mm^{3}
tumors for six weeks prior to performing DCE-MRI (6). Following imaging, the
tumors were extracted, sectioned, and stained for vascularity (CD31) and
cellularity (H&E). Stained central slice sections were digitized at high
resolution (0.5 mm),
and segmented and meshed in MATLAB (Natick, MA) (**Figure 1**). A finite element
model (FEM) with 2 mm
resolution (down-sampled from the 0.5 mm
resolution) was developed based on the 2D diffusion equation: $$[1], \frac{dC(x,y,t)}{dt}=\nabla\cdot D\nabla C(x,y,t),$$ where $$$C(x,y,t)$$$ is the concentration of contrast
agent, and $$$D$$$ is the diffusivity (set
to 2 E-4 mm^{2}/s). Impermeable boundaries were assigned at the tumor
periphery and at cell membranes. Flux of contrast agent across the boundaries at
blood vessels was defined according to Eq. [2]: $$[2], \nabla C\cdot\hat{n}=P(C_p(t)-C(t)),$$ where $$$P$$$ is equal to $$$K^{trans}\times\frac{V}{S}$$$, $$$S$$$ is the total vessel
surface area within a voxel, $$$K^{trans}$$$ is the volume transfer coefficient, $$$V$$$ is the volume of tissue perfused, $$$\hat{n}$$$ is the normal vector with respect to the vessel boundary, and $$$C_p(t)$$$ is a population arterial input function (AIF) (6). All vessels
are assumed to contain the AIF concentration for each time step, and flux from
a vessel does not affect the vascular concentration. Using the resulting contrast
agent distribution, a signal intensity is calculated for each MRI voxel at 438 $$$\mu$$$m in plane resolution (down-sampled from the 2 mm resolution) at each AIF time step. The
extended Tofts’ model (Eq. [3]) is then fit to the simulated signal intensity
of each MRI voxel to provide estimates of $$$K^{trans}$$$, $$$v_e$$$ (extravascular,
extracellular volume fraction), and $$$v_p$$$ (plasma volume fraction):$$[3], C_t(t)=K^{trans}\int_{0}^{t}C_p(u)exp(\frac{K^{trans}}{v_e}(t-u))du+v_pC_p(t),$$ Finally, the fit values for $$$K^{trans}$$$, $$$v_e$$$,
and $$$v_p$$$ are then compared
to the histological ($$$v_e$$$ and $$$v_p$$$) and assigned ($$$K^{trans}$$$) model values used in
the forward model, and a percent error is calculated for each simulated MRI
voxel.

(1) Tofts PS, Kermode AG, et al. Measurements of the blood-brain barrier permeability and leakage space using dynamic MR imaging 1. Fundamental Concepts. Magn Reson Med. 1991:17(2):357-67

(2) Barnes SL, Quarles CC, et al. Modeling the Effect of Intra-Voxel Diffusion of Contrast Agent on the Quantitative Analysis of Dynamic Contrast Enhanced Magnetic Resonance Imaging. PlosOne.2014:9(9)

(3) Fluckiger JU, Loveless ME, et al. A diffusion-compensated model for the analysis of DCE-MRI data: theory, simulations and experimental results. Physics in Medicine and Biology.2013:58:1983-1998

(4) Koh TS, Hartono S, et al. In vivo measurement of gadolinium diffusivity by dynamic contrast-enhanced MRI: A preclinical study of human xenografts. Magnetic Resonance in Medicine.2013:69:269-276

(5) Sorace AG, Quarles CC, et al. Trastuzumab improves tumor perfusion and vascular delivery of cytotoxic therapy in a murine model of HER2+ breast cancer: preliminary results. Breast Cancer Res Treat.2016:155(5):273-83

(6) Loveless ME, Halliday J, et al. A quantitative comparison of the influence versus population-derived vascular input functions on dynamic contrast enhanced-MRI in small animals. Magnetic Resonance in Medicine.2012:67:226-236