Gelatine methacrylamide-based hydrogels : an alternative three-dimensional cancer cell culture system

Modern cancer research requires physiological, three-dimensional (3-D) cell culture platforms, wherein the physical and chemical characteristics of the extracellular matrix (ECM) can be modified. In this study, gelatine methacrylamide (GelMA)-based hydrogels were characterized and established as in vitro and in vivo spheroid-based models for ovarian cancer, reflecting the advanced disease stage of patients, with accumulation of multicellular spheroids in the tumour fluid (ascites). Polymer concentration (2.5-7% w/v) strongly influenced hydrogel stiffness (0.5±0.2kPa to 9.0±1.8kPa) but had little effect on solute diffusion. The diffusion coefficient of 70kDa fluorescein isothiocyanate (FITC)-labelled dextran in 7% GelMA-based hydrogels was only 2.3 times slower compared to water. Hydrogels of medium concentration (5% w/v GelMA) and stiffness (3.4kPa) allowed spheroid formation and high proliferation and metabolic rates. The inhibition of matrix metalloproteinases and consequently ECM degradability reduced spheroid formation and proliferation rates. The incorporation of the ECM components laminin-411 and hyaluronic acid further stimulated spheroid growth within GelMA-based hydrogels. The feasibility of pre-cultured GelMA-based hydrogels as spheroid carriers within an ovarian cancer animal model was proven and led to tumour development and metastasis. These tumours were sensitive to treatment with the anti-cancer drug paclitaxel, but not the integrin antagonist ATN-161. While paclitaxel and its combination with ATN-161 resulted in a treatment response of 33-37.8%, ATN-161 alone had no effect on tumour growth and peritoneal spread. The semi-synthetic biomaterial GelMA combines relevant natural cues with tunable properties, providing an alternative, bioengineered 3-D cancer cell culture in in vitro and in vivo model systems.

Detailed analysis of a conservative difference approximation for the time fractional diffusion equation

Diffusion equations that use time fractional derivatives are attractive because they describe a wealth of problems involving non-Markovian Random walks. The time fractional diffusion equation (TFDE) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α ∈ (0, 1). Developing numerical methods for solving fractional partial differential equations is a new research field and the theoretical analysis of the numerical methods associated with them is not fully developed. In this paper an explicit conservative difference approximation (ECDA) for TFDE is proposed. We give a detailed analysis for this ECDA and generate discrete models of random walk suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation. The stability and convergence of the ECDA for TFDE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.

A second order control-volume finite-element least-squares strategy for simulating diffusion in strongly anisotropic media

An unstructured mesh �nite volume discretisation method for simulating di�usion in anisotropic media in two-dimensional space is discussed. This technique is considered as an extension of the fully implicit hybrid control-volume �nite-element method and it retains the local continuity of the ux at the control volume faces. A least squares function recon- struction technique together with a new ux decomposition strategy is used to obtain an accurate ux approximation at the control volume face, ensuring that the overall accuracy of the spatial discretisation maintains second order. This paper highlights that the new technique coincides with the traditional shape function technique when the correction term is neglected and that it signi�cantly increases the accuracy of the previous linear scheme on coarse meshes when applied to media that exhibit very strong to extreme anisotropy ratios. It is concluded that the method can be used on both regular and irregular meshes, and appears independent of the mesh quality.

Implicit difference approximation for the time fractional diffusion equation

In this paper, we consider a time fractional diffusion equation on a finite domain. The equation is obtained from the standard diffusion equation by replacing the first-order time derivative by a fractional derivative (of order $0<\alpha<1$ ). We propose a computationally effective implicit difference approximation to solve the time fractional diffusion equation. Stability and convergence of the method are discussed. We prove that the implicit difference approximation (IDA) is unconditionally stable, and the IDA is convergent with $O(\tau+h^2)$, where $\tau$ and $h$ are time and space steps, respectively. Some numerical examples are presented to show the application of the present technique.

Numerical approximation of a fractional-in-space diffusion equation (II) - with nonhomogeneous boundary conditions

In this paper, a space fractional di®usion equation (SFDE) with non- homogeneous boundary conditions on a bounded domain is considered. A new matrix transfer technique (MTT) for solving the SFDE is proposed. The method is based on a matrix representation of the fractional-in-space operator and the novelty of this approach is that a standard discretisation of the operator leads to a system of linear ODEs with the matrix raised to the same fractional power. Analytic solutions of the SFDE are derived. Finally, some numerical results are given to demonstrate that the MTT is a computationally e±cient and accurate method for solving SFDE.