Fig 1.
Motivation for sustained–delivery implants for treatment of PDAC.
Sustained–delivery implants are a promising treatment methodology over conventional single free–drug intravenous or intrantumoural injections. A hypothetical comparison of drug concentrations at the tumour site under these two protocols is pictured. Systemic injections of anti–cancer drugs often result in only a fraction of the drug arriving at the tumour site followed by a rapid decrease of drug concentration at the tumour site. In comparison, sustained–release mechanisms deliver drug over a prolonged period resulting in a durable drug presence at the tumour site. Created using biorender.com.
Fig 2.
The main components of the VCBM–PDE model.
In the VCBM, we model the interactions between healthy cells (grey), cancer cells (orange), MCC (blue) and dead cells (read) in a 2–dimensional tumour microenvironment cross section. (A) The concentration of drug in the TME was modelled in a 2 dimensional domain bounded by B, where C(x,y,t) was the concentration in the TME at position (x,y). The fibre implant was then placed at a position x = xF and modelled as a line source. To capture the diffusion of drug from the fibre, we modelled the concentration of gemcitabine inside the fibre F(r,y,t) at radial position r and domain position y where the continuity condition in Eq (3) required equal concentrations at the fibre boundary and at the immedicate local microenvironment, i.e. F(rtotal,y,t) = C(xF,y,t). (B) The concentration of gemcitabine inside the polymeric fibres was modelled by radially symmetric diffusion Eq (2) using a finite volume method (FVM) discretisation and considering the 2D cylindrical cross section of the fibres which have length L and radius rtotal. The fibre was discretised into concentric annuli Fm,j at annulus m and cross section j, (i = 0,1,…,M) and the concentration of drug in each annulus Fm,j was modelled by considering drug diffusion across the bounadaries (e.g. Fm−1,j and Fm+1,j flow into Fm,j and vice vera). The full discretisation is presented in S1 Technical Supplementary Information. (C) Modelling assumptions for the VCBM were that cancer cells (orange) proliferate and some are able to cause epithelial to mesenchymal transtion and become invasive. We model this transition by assuming cells differentiate into a mesenchymal cancer cell (MCC) with one daughter cell placed on a neighbouring healthy cell. These MCCs cause the break down of surrounding tissue (i.e. replace healthy neighbouring cells with their progeny). Cancer cells can then die through gemcitabine uptake from their local environment. For more details see Fig B in S1 Text. (D) Individual cells were modelled as cell centres connected by springs [57]. The proliferation of a cell introduced a new cell into the lattice network which caused the rearrangement of the cells in the lattice with movement governed by Hooke’s law. (E) To simulate the gemcitabine concentration in the TME, Eq (1), we introduced a FVM discretisation, where the gemcitabine concentration was defined at discrete volumes centered around points in the discretisation. Cells could take up drug from the nearest grid point to their centre, and this concentration was used to determine their likelihood of drug–induced cell–death.
Fig 3.
Calibration of model parameters to in vitro experiments.
(A) Drug release profiles for gemcitabine with 3% alginate 15% PCL were measured by placing the gemcitabine–loaded alginate fibre in a solution bath and measuring the released drug concentration over time. (B) The drug concentration in the solution bath (black) was used to fit model parameters for the drug release from the fibre (green). Resulting parameters are in Table A in S1 Text. (C) The drug–induced death rate of pancreatic cancer cells was determined by simplifying the full model assumptions to consider a homogeneous model for live cancer cells PL(t) that were proliferating and dying (become dead cells PD(t)) through the effect of the drug gemcitabine C(t), Eqs (9)–(11). (D) Fitting an exponential growth curve to Mia–PaCa–2 cell proliferation in vitro [91] gave the growth rate of cells ϕ. Values are the mean±std. (E) To measure the efficacy of the protocol, the cell viability was determined after aliquots from drug released from gemcitabine–loaded fibre were placed in a well with proliferating Mia–PaCa–2 cells at 24, 48 and 72 hours. (F) The resulting cell viability at 72 hours from the experiment depicted in (E) was used to fit the drug–induced cell death rate (Eqs (9)–(11)). The data is plotted as a box and whisker plot. Resulting parameters for (D) and (F) are in Table B in S1 Text.
Fig 4.
Using the VCBM to model control tumour growth.
(A) Snapshots of the model simulation at 0, 5 and 10 days with cancer cells in orange, MCCs in blue and healthy cells in grey (a zoomed in version is in Fig I in S1 Text). (B) Mia–PaCa–2 tumour volume over 33 days measured in vivo in mice (black, n = 4). Overlaid is the tumour volume from the VCBM simulation (pink, n = 500) with parameters from Table C in S1 Text. (C) MCC counts in the VCBM simulations (n = 500). (D) Sensitivity analysis of control tumour growth. Maximum tumour volume over 33 days for perturbations of parameters with weights of 0.25, 0.75, 1.25, 1.75 and 2.25, and spatial plots of large and small tumours simulated using the depicted weightings. In the heatmap, each pixel represents 500 averaged simulations with two parameters. In the boxes, the parameters vertically and horizontally in the grid are the weightings in ascending order, with each pixel being a “coordinate” representing the weighting for each parameter and the result from 500 averaged tests. Diagonal pixels only use individual parameters with different weightings. Legend for cell colouring: cancer cell (orange) healthy cell (grey), MCC (blue).
Fig 5.
Impact of intratumoural free–drug point injections on tumour cell eradication.
(A) Tumour growth was investigated under different gemcitabine single free–drug injections: central, 0.9 mm from centre, 1.7 mm from centre, 2.5 mm from centre, 3.5 mm from centre. Locations of injections on the tumour surface for a single free–drug injection or four free–drug injections is depicted schematically. (B) VCBM with a single central injection and the drug concentration at 24h. (C) The tumour volume with four injections placed 30μm from the centre, and the drug concentration at each location at 6h. (D) Maximum tumour volume over 33 days for ±50% perturbations in parameter values compared to the normal value (i.e. baseline parameter values). (E) Maximum tumour volume over 33 days for different perturbations of IC50 compared to the normal volume. (F) The tumour volume over 33 days with each injection protocol, averaged over 500 simulations. Legend for cell colouring: cancer cell (orange) healthy cell (grey), MCC (blue), dead cell (read). Note, the white space in the simulation images represents ‘empty space’ in the tumour where dead cells have been previously, but have since disintegrated.
Fig 6.
Comparison of different fibre release and placement options.
(A) Tumour growth was investigated under different gemcitabine–loaded fibre placements dm: central, 0.9 mm from centre, 1.7 mm from centre, 2.5 mm from centre, 3.5 mm from centre and 4.3 mm from centre. Locations of fibres on tumour surface for single implantations is depicted schematically. (B) A heatmap for the averaged final state of a tumour after 33 days of simulation for different initial injection concentrations and fibre placements. “Eradicate” denotes a tumour volume below 1mm3, “stabilise” denotes a tumour volume less than the initial tumour volume, and “growth” denotes a tumour volume greater than the initial tumour volume. (C) The mean (solid lines) and standard deviation (shades areas) of the tumour volume over 33 days for different fibre placement options with corresponding values highlighted in (B). (D) The tumour volume on day 33 for different release rates (indicated by the γ value) and release profiles with a central fibre placement. (E) The tumour volume on day 33 for different release rates (indicated by the γ value) and release profiles with a fibre placed on the edge of the tumour (3.5 mm away from centre). See Section TS3 in S1 Technical Supplementary Information for more details on these release functions. Legend for cell colouring: cancer cell (orange).