Multiscale biphasic modelling of peritumoural collagen microstructure: The effect of tumour growth on permeability and fluid flow

<div><p>We present an in-silico model of avascular poroelastic tumour growth coupled with a multiscale biphasic description of the tumour–host environment. The model is specified to in-vitro data, facilitating biophysically realistic simulations of tumour spheroid growth into a dense collagen hydrogel. We use the model to first confirm that passive mechanical remodelling of collagen fibres at the tumour boundary is driven by solid stress, and not fluid pressure. The model is then used to demonstrate the influence of collagen microstructure on peritumoural permeability and interstitial fluid flow. Our model suggests that at the tumour periphery, remodelling causes the peritumoural stroma to become more permeable in the circumferential than radial direction, and the interstitial fluid velocity is found to be dependent on initial collagen alignment. Finally we show that solid stresses are negatively correlated with peritumoural permeability, and positively correlated with interstitial fluid velocity. These results point to a heterogeneous, microstructure-dependent force environment at the tumour–peritumoural stroma interface.</p></div

Initial state meshes at both scales.

<p>The tumour spheroid octant is shown as a red wire frame and has radius 0.1 mm; the peritumoural stroma is shown as an opaque blue volume has thickness 0.5 mm; and each RVE has a side length of approximately 20 microns (axes are not to scale).</p

Final state variables in space and time.

<p>A: Final spherical solid stress components and IFP over radial distance. B: Final spherical solid stress components and IFP over time, integrated over the PTS volume. C: Final spherical permeability components over radial distance; inset shows a zoom into the tumour-PTS boundary. D: Final spherical permeability components over time, integrated over the PTS volume; inset shows the same for the region 0.1mm <<i>r</i> < 0.2mm. E: Final spherical IFV components over radial distance. F: Final spherical IFV components over time, integrated over the PTS volume. All time-dependent plots are normalised with respect to the initial volume of the PTS.</p

Tumour growth causes collagen remodelling.

<p>From top to bottom: final state peritumoural stroma mesh (both scales); final state RVEs from the (A) 5th, (B) 3rd and (C) 1st layers of the PTS; (D) histograms of the scalar product between the principal network permeability, <i>K</i>_<i>pvec</i>, and principal network orientation vectors, Ω_<i>pvec</i>; and (E) the scalar product between the principal network permeability and interstitial fluid velocity, <i>V</i>_<i>pvec</i>, for each RVE in the 1st layer of elements in the PTS. The arrows centred on each RVE show the principal eigenvectors of the network orientation (grey) and permeability (red), and the interstitial fluid velocity (blue).</p

Initial collagen microarchitecture influences final collagen alignment and fluid flow.

<p>Row-wise from top: Final state network principal orientation eigenvectors in the first layer of the PTS, for an initially (A) random and (B) aligned PTS. The colour bar indicates alignment with respect to the tumour boundary: 0 for circumferential, 1 for perpendicular. Final state network alignments for an initially (C) random and (D) aligned PTS. Final spherical IFV components near the tumour boundary, for an initially (E) random and (F) aligned PTS.</p

Capillary-tip extension velocity and vessel wall mechanical model.

<p>A: Extension rate of vascular-tip endothelial cells versus the capillary radius expressed by an exponential decay function, fitted to reported in-vitro angiogenesis experiments [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005259#pcbi.1005259.ref044" target="_blank">44</a>]. B: Stress–strain plot of the constitutive equation used to describe the biomechanics of the blood vessels, including the pressure which induces vessel collapse, <i>p</i>_c.</p

A Validated Multiscale In-Silico Model for Mechano-sensitive Tumour Angiogenesis and Growth

<div><p>Vascularisation is a key feature of cancer growth, invasion and metastasis. To better understand the governing biophysical processes and their relative importance, it is instructive to develop physiologically representative mathematical models with which to compare to experimental data. Previous studies have successfully applied this approach to test the effect of various biochemical factors on tumour growth and angiogenesis. However, these models do not account for the experimentally observed dependency of angiogenic network evolution on growth-induced solid stresses. This work introduces two novel features: the effects of hapto- and mechanotaxis on vessel sprouting, and mechano-sensitive dynamic vascular remodelling. The proposed three-dimensional, multiscale, in-silico model of dynamically coupled angiogenic tumour growth is specified to in-vivo and in-vitro data, chosen, where possible, to provide a physiologically consistent description. The model is then validated against in-vivo data from murine mammary carcinomas, with particular focus placed on identifying the influence of mechanical factors. Crucially, we find that it is necessary to include hapto- and mechanotaxis to recapitulate observed time-varying spatial distributions of angiogenic vasculature.</p></div

Normalised vascular density when mechano- and haptotaxis is included or discarded in the angiogenesis model.

<p>Comparison of the numerically predicted normalised vascular density when blood vessel sprouting is modulated by chemo-, mechano- and haptotaxis against the simplified chemotaxis case. Vascular density is defined as the ratio of the surface area of the blood vessels to the tissue volume, and is normalised against the corresponding initial value (day-0).</p

History plots of the parameters characterising the morphology of the microvascular tree.

<p>Numerically predicted parameters λ_v and <i>δ</i>_v-max with respect to time, compared to in-vivo measurements in murine mammary carcinoma (MCaIV-type) [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005259#pcbi.1005259.ref022" target="_blank">22</a>]. A: The geometrical exponent, λ_v, is obtained after linear regression on the pair of data: frequency of voxels versus the distance to nearest vessel (<i>δ</i>_v) [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005259#pcbi.1005259.ref065" target="_blank">65</a>] while the vertical bars denote standard deviation of the mean.</p

Comparison snapshots showing the effect of vascular wall stiffness to the tumour angiogenesis model predictions.

<p>Microvascular wall stiffness (from left to right) takes values: <i>E</i>_w-max = 1.3–5.22, <i>E</i>_w-max = 13.—52.2 and <i>E</i>_w-max = 130.—522. respectively, while each row corresponds (from top to bottom) to day-15, day-20, day-25 and day-30. Blood vessels are coloured green if they reside at the peri-tumoural area, red if they are located in the rest of the healthy tissue domain, and blue if the are inside the tumour. Note that for lower values of stiffness, <i>E</i>_w-max, nascent vessels are non-existent inside the tumour as opposed to higher values. The green-colour cloud denotes the TAF concentration level in the extracellular matrix, where both the host-tissue and tumour domain is invisible for illustration purposes.</p