Ben Hipwell^{1}, Tom Roberts^{1}, Paul Sweeney^{2}, Morium Ali^{2}, Angela D'Esposito^{3}, Eleftheria Panagiotaki^{2}, Mark Lythgoe^{2}, Daniel Alexander^{2}, Rebecca Shipley^{2}, and Simon Walker-Samuel^{2}

Compartmental models are increasingly being used to quantify diffusion MRI signals from tumours. We have developed a complex, multiscale mathematical modelling platform for simulating tumour pathophysiology, using high-resolution optical imaging data from complete tumour samples. Diffusion MRI signals from these tumours were simulated, including vascular flow and intra- and extracellular diffusion. These data were fitted to the VERDICT compartmental model, and the resulting parameters compared against ground truth simulation values. Cell radius and intra/extracellular fractional volume parameters and respective ground truth values were strongly correlated. A more complex relationship was found in vascular volume fractions.

**Substrate Generation:** The vascular compartment of the
simulation substrates (figure 1) was generated from optical projection
tomography data (5-10μm resolution) acquired from optically-cleared LS174T tumours grown
subcutaneously on balb/c nu/nu mice, with vessels labelled with fluorescently
conjugated lectin^{4,5}. Optical clearing was performed
with benzyl alcohol benzyl benzoate (BABB). The vessel structures were
segmented by applying a Frangi vesselness filter^{6}, binary threshold and
skeletonisation algorithm. Volumetric blood flow values were estimated
throughout the networks using a discrete-network model simulating Poiseuille
flow with conservation of flux at network bifurcations^{7}. Four voxel-sized sections
were randomly isolated for separate Monte Carlo simulations. The intracellular
and extracellular compartments of the simulation were created by randomly generating
spherical cells with a sphere-packing algorithm, which packed non-overlapping
cells of a specified radii around the vessel network.

**Synthetic MRI Data Generation:** Walkers, representing water
molecules, were initialised in one of the three compartments within the
substrate, with the total in each compartment weighted by pre-specified volume
fractions. In the intra/extracellular space the walker position was updated at
each time-step using a random-walk protocol^{8} (figure 2(c)). In the
vascular compartment, the walkers followed the flow velocity of their current
vessel segment. Once all the walkers had been propagated, synthetic VERDICT
data (46 b-value, 3 direction PGSE) was generated from their recorded
trajectories using the Camino Diffusion MRI Toolkit^{9}.

**VERDICT Parameter Validation:** The VERDICT models
‘BallSphereStick’ (Anisotropic vascular compartment) and ‘BallSphereAstrosticks’
(Isotropic vascular compartment) were each fitted to the synthetic data. The
model parameters produced from these fits were then compared against the ground
truth values specified in the simulation. This process was repeated for a range
of simulation parameters to ascertain the correspondence between the VERDICT parameter
estimates and the ground truth, over a wide range of values. The synthetic data
produced by the simulation was also compared against subject-matched in-vivo data acquired previously.

1. Panagiotaki E, Walker-Samuel S, Siow B, Johnson SP, Rajkumar V, Pedley RB, Lythgoe MF, Alexander DC. Noninvasive quantification of solid tumor microstructure using VERDICT MRI. Cancer research. 2014 [accessed 2015 May 26];74(7):1902–12. http://cancerres.aacrjournals.org/content/74/7/1902.full

2. Zhang H, Schneider T, Wheeler-Kingshott CA, Alexander DC. NODDI: practical in vivo neurite orientation dispersion and density imaging of the human brain. NeuroImage. 2012 [accessed 2015 Mar 3];61(4):1000–16. http://www.sciencedirect.com/science/article/pii/S1053811912003539

3. Novikov DS, Fieremans E, Jensen JH, Helpern JA. Random walk with barriers: Diffusion restricted by permeable membranes. 2010 Apr 15 [accessed 2017 Jul 7]. http://arxiv.org/abs/1004.2701

4. Oldham M, Sakhalkar H, Oliver T, Wang YM, Kirpatrick J, Cao Y, Badea C, Johnson GA, Dewhirst M. Three-dimensional imaging of xenograft tumors using optical computed and emission tomography. Medical Physics. 2006 [accessed 2017 Jan 11];33(9):3193. http://www.ncbi.nlm.nih.gov/pubmed/17022212

5. d’Esposito A, Nikitichev D, Desjardins A, Walker-Samuel S, Lythgoe MF. Quantification of light attenuation in optically cleared mouse brains. Journal of biomedical optics. 2015 [accessed 2017 Jan 11];20(8):80503. http://www.ncbi.nlm.nih.gov/pubmed/26277988

6. Frangi AF, Niessen WJ, Vincken KL, Viergever MA. Multiscale vessel enhancement filtering. Springer, Berlin, Heidelberg; 1998. p. 130–137. http://link.springer.com/10.1007/BFb0056195

7. FRY BC, LEE J, SMITH NP, SECOMB TW. Estimation of Blood Flow Rates in Large Microvascular Networks. Microcirculation. 2012 [accessed 2017 Jun 30];19(6):530–538. http://www.ncbi.nlm.nih.gov/pubmed/22506980

8. Hall MG, Alexander DC. Convergence and Parameter Choice for Monte-Carlo Simulations of Diffusion MRI. IEEE transactions on medical imaging. 2009;28(9):1354–1364.

9. Cook, Bai Y, Gilani N, Seunarine K, Hall M, Parker G, Alexander D. Camino: Open-Source Diffusion-MRI Reconstruction and Processing. 14th Scientific Meeting of the International Society for Magnetic Resonance in Medicine. 2006 [accessed 2017 Jul 4]. http://www.citeulike.org/user/birdyyo/article/3805628

Figure 1: (a) An
image showing a 2D projection of the segmented and graph-converted vessel
structure from an LS174T tumour overlaid on the structural MR image of the
tumour. (b) 3D representations of the sub-voxels that were selected from the
vessel structure, colour-coded according to the blood-flow velocity estimated
by the finite network model. The thickness of the graph segments corresponds to
the radius of the blood vessel calculated from the OPT data.

Figure 2: (a) Graph
showing the normalised signal curves from subject-matched in-vivo data and
simulated data (The vessel structure used for the substrate in the simulations
was taken from the same tumour). The solid lines represent fitting the 'BallSphereAstrosticks' model to both datasets. (b) b0 image of a subcutaneous LS174T tumour,
with a representation of a ROI overlaid (imaged using parameters of Panagiotaki
et al^{1}). (c) 3D plot of the
trajectories from a typical Monte Carlo simulation.

Figure 3: Plots
showing the correlation between the measured and simulation ground-truth values
of the cell radius and intracellular volume fraction parameters produced using
two different VERDICT models. The different colours represent different
simulation voxels with different vessel structures. The black lines represent identity (a 1-1 correspondence). (a) Correlation between
ground truth and measured cell radius using the ‘BallSphereStick’ model. (b)
Correlation between ground truth and measured cell radius using the
‘BallSphereAstrosticks’ model. (c) Intracellular volume fraction correlation
using ‘BallSphereStick’ model. (d) Intracellular volume fraction correlation
using ‘BallSphereAstrosticks’ model.

Figure 4: (a) Shows
the correlation between the ground truth and measured values of vascular volume
fraction produced by the ‘BallSphereAstrosticks’ model. The different colours
represent different simulation voxels with different vascular structures and
blood-flow velocity distributions. (b) Shows the vascular volume fraction
correlations produced from a single simulation voxel while varying the
blood-flow velocity by a scalar factor between 0 and 2. The colour of the
points represent the flow-adjustment factor.