Fig 1.
(a) An example of an SW1222 tumour vascular network enhanced using Frangi filters and extracted from the tumour image stack generated by OPT. (b) The skeletonised vasculature is then segmented into a series of interconnected nodes and vessel segments with known diameters, di and lengths, li, for i = 1, 2, 3 shown here. (c) A schematic of sensitivity analysis performed on the source parameters: 1) updating intravascular pressure pi,b for k iterations where p0,b is the initial network pressure distribution approximated by the flow estimation algorithm; 2) the spacing δ between sources distribution across a branching vessel; and 3) the size of the source radius, r0. (d) A flow diagram of the computational framework. In vivo imaging is performed on vascularised tissue to obtain perfusion data (and literature values of vascular pressures when available) which are used to parameterise and validate the framework. Ex vivo imaging is performed on equivalent tissue samples to obtain data on the vascular architecture, including coordinates, vessel diameters and lengths, which are then used to parameterise the vascular flow model. Boundary conditions are assigned (see Fig 2a and 2b) and network intravascular blood pressure is solved. Sources of fluid flux are distributed across the vasculature and assigned a radius equal to their corresponding vessel radius. Interstitial flow parameters are assigned and the model is coupled to the vascular flow compartment via Starling’s Law. Solved source strengths are used to update Starling’s law (i ∈ Ns). This iterative scheme is terminated once predefined tolerances are reached.
Table 1.
Tumour vascular network statistics.
Table 2.
Fluid transport parameters.
Fig 2.
(a, b) The optimisation scheme used to assign boundary conditions to the tumour networks.
(a) The process to simulate physiological tissue perfusion. (b) The flowchart for the subroutine “Assign Pressure Conditions” given in (a). (c) Perfusion through a tumour is calculated by generating a convex hull across the surface of the tumour to accurately extract tumour volume. (d) Discretising the hull into a finer mesh and calculating IFP at coupled points across, and normal, to the tumour hull. (e) A sphere packing algorithm is then applied to the points on the tumour surface with inflow averaged across the great circles of each sphere, enabling an approximation of perfusion.
Fig 3.
Simulated vascular blood flow in (a) GL261 and (b) LS147T tumours.
Distributions are shown for vessel radii, blood pressure, flow and vessel wall shear stress, respectively.
Fig 4.
Simulated fluid transport through the interstitium in GL261 and LS147T tumours.
(a, b) (Left) Predicted interstitial fluid pressure (IFP) fields for {X,Y}-planes through the tumours, emulating the traditionally high pressure in the tumour core but with predicted spatial heterogeneities. (Middle) Simulated interstitial perfusion maps discretised into ∼ 140 μm2 pixels. Results replicate the traditionally elevated perfusion existing at the periphery of the tumours. (Right) Interstitial fluid velocity (IFV, overlaid onto greyscale image of interstitial perfusion) predictions depicting spatial interstitial flow heterogeneities across the entire tumours. Note, perfusion and interstitial fluid velocity maps are shown for the central slice in the interstitial fluid pressure graphics. (c, d) Fitted curves with error bars indicating standard deviation for (left) IFP and (right) IFV in (c) GL261 and (d) LS147T, plotted against normalised radius, corresponding to the simulations shown in (a, b).
Table 3.
Simulated fluid transport statistics.
Fig 5.
Sensitivity of the computational framework.
(a, e) Cross-sectional slice from the core of the (top) GL261 and (bottom) LS147T tumours, showing standard deviation of intravascular pressures (mmHg) across all 12 simulations. (b, f) the mean segment pressure error between consecutive iterations for the (top) GL261 (top) and (bottom) LS147T networks. (c, g) IFP distribution from the centre of the tumour to its periphery for a maximum spacing of (blue) 10 and 25 μm and (orange) 50 and 100 μm for GL261 and LS147T, respectively. (d, h) IFP distributions for the source radii set to the minimum vessel radius multiplied by a factor of (blue) 10−1, (red) 100 and (yellow) 101. Note, error bars correspond to standard deviation.
Fig 6.
Data-fitted curves of IFP (top) and IFV (bottom) for modulation of interstitial model parameters: (a, e) p∞ (mmHg), (b, f) σ, (c, g) Lp(cm / mmHg s) and (d, h) κ (cm2 / mmHg s).
Arrows indicate increasing parameter values, with exception of (d) in which a range of κ (cm2 / mmHg s) is indicated. IFP profile gradients across LS2 sensitivity analysis of (a) far-field pressure, p∞, (b) oncotic reflection coefficient, σ, (c) vascular hydraulic conductivity, Lp, and (d) interstitial hydraulic conductivity, κ (cm2 / mmHg s). Arrows indicate increasing values of the given parameter and colours in each column indicate the equivalent simulation.
Fig 7.
Predicting changes in vascular perfusion in response to vascular normalization therapy.
Simulated vascular perfusion (left) pre- and (right) post-normalization in (a) GL261 and (b) LS147T tumours. Perfusion maps are discretised into isotropic pixels of width ∼ 140 μm.
Fig 8.
Simulated normlization of Lp, σ, vessel diameters and κ in (a) GL261 and (b) LS147T tumours.
Plots show (left) IFP and (right) IFV for (blue) pre- and (red) post-normalization. Error bars represent standard deviation.
Fig 9.
Normalization of vascular hydraulic conductance, oncotic reflection coefficient and interstitial hydraulic conductivity, and stabilisation of vessel diameters to physiological values in (a, b) GL261 and (d, e) LS147T tumours.
(a, d) Planar contour plots of IFP and interstitial fluid speed (IFS) for the baseline (top) and normalised (bottom) predictions. (b, e) Predictions of normalised interstitial fluid spatial maps (left) and IFV (right—with greyscale perfusion underlaid) where the isotropic pixels are ∼ 140 μm wide. For comparison, equivalent {X,Y}-planes for baseline simulations can be viewed in Fig 4a and 4b, respectively. (c) A comparison of change in the source density, the ratio between sources and sinks (= [n − m]/[n + m] where n and m are the sum of sources and sinks, respectively, in the tumour region), across GL261 and LS147T (blue) pre- and (orange) post-normalization.