There is an unacceptably high complication rate after reimplantation of the autologous bone following DC in pediatric TBI patients, especially in young children up to seven years of age. Artificial or synthetic cranioplasties may be considered as alternatives to initial bone flap reimplantation in the growing child. Despite the fact that DC is an effective treatment in TBI with persistent intracranial hypertension, it is important to realize that DC is not only combined with replacement of the autologous bone flap but also with a high rate of additional complications especially in pediatric patients.

Two algorithms that combine Brownian dynamics (BD) simulations with mean-field partial differential equations (PDEs) are presented. This PDE-assisted Brownian dynamics (PBD) methodology provides exact particle tracking data in parts of the domain, whilst making use of a mean-field reaction-diffusion PDE description elsewhere. The first PBD algorithm couples BD simulations with PDEs by randomly creating new particles close to the interface which partitions the domain and by reincorporating particles into the continuum PDE-description when they cross the interface. The second PBD algorithm introduces an overlap region, where both descriptions exist in parallel. It is shown that to accurately compute variances using the PBD simulation requires the overlap region. Advantages of both PBD approaches are discussed and illustrative numerical examples are presented.

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Hybrid models of chemotaxis combine agent-based models of cells with partial differential equation models of extracellular chemical signals. In this paper, travelling wave properties of hybrid models of bacterial chemotaxis are investigated. Bacteria are modelled using an agent-based (individualbased) approach with internal dynamics describing signal transduction. In addition to the chemotactic behaviour of the bacteria, the individual-based model also includes cell proliferation and death. Cells consume the extracellular nutrient field (chemoattractant) which is modelled using a partial differential equation. Mesoscopic and macroscopic equations representing the behaviour of the hybrid model are derived and the existence of travelling wave solutions for these models is established. It is shown that cell proliferation is necessary for the existence of non-transient (stationary) travelling waves in hybrid models. Additionally, a numerical comparison between the wave speeds of the continuum models and the hybrid models shows good agreement in the case of weak chemotaxis and qualitative agreement for the strong chemotaxis case. In the case of slow cell adaptation, we detect oscillating behaviour of the wave, which cannot be explained by mean-field approximations.

We investigate the effect of turning delays on the behaviour of groups of differential wheeled robots and show that the group-level behaviour can be described by a transport equation with a suitably incorporated delay. The results of our mathematical analysis are supported by numerical simulations and experiments with E-Puck robots. The experimental quantity we compare to our revised model is the mean time for robots to find the target area in an unknown environment. The transport equation with delay better predicts the mean time to find the target than the standard transport equation without delay.

Group-level behavior of particles undergoing a velocity-jump process with hard-sphere interactions is investigated. We derive N-particle transport equations that include the possibility of collisions between particles and apply different approximation techniques to get expressions for the dependence of the collective diffusion coefficient on the number of particles and their diameter. The derived approximations are compared with numerical results obtained from individual-based simulations. The theoretical results compare well with Monte Carlo simulations providing the excluded-volume fraction is small.

Mathematical models of dispersal in biological systems are often written in terms of partial differential equations (PDEs) which describe the time evolution of population-level variables (concentrations, densities). A more detailed modelling approach is given by individual-based (agent-based) models which describe the behaviour of each organism. In recent years, an intermediate modelling methodology -hybrid modelling -has been applied to a number of biological systems. These hybrid models couple an individual-based description of cells/animals with a PDEmodel of their environment. In this chapter, we overview hybrid models in the literature with the focus on the mathematical challenges of this modelling approach. The detailed analysis is presented using the example of chemotaxis, where cells move according to extracellular chemicals that can be altered by the cells themselves. In this case, individual-based models of cells are coupled with PDEs for extracellular chemical signals. Travelling waves in these hybrid models are investigated. In particular, we show that in contrary to the PDEs, hybrid chemotaxis models only develop a transient travelling wave.

Low-passage cancer cell lines are versatile tools to study tumor cell biology. Here, we have employed four such cell lines, established from primary tumors of colorectal cancer (CRC) patients, to evaluate effects of the small molecule kinase inhibitors (SMI) vemurafenib, trametinib, perifosine, and regorafenib in an in vitro setting. The mutant BRAF (V600E/V600K) inhibitor vemurafenib, but also the MEK1/2 inhibitor trametinib efficiently inhibited DNA synthesis, signaling through ERK1/2 and expression of genes downstream of ERK1/2 in BRAF mutant cells only. In case of the AKT inhibitor perifosine, three cell lines showed a high or intermediate responsiveness to the drug while one cell line was resistant. The multikinase inhibitor regorafenib inhibited proliferation of all CRC lines with similar efficiency and independent of the presence or absence of KRAS, BRAF, PIK3CA, and TP53 mutations. Regorafenib action was associated with broad-range inhibitory effects at the level of gene expression but not with a general inhibition of AKT or MEK/ERK signaling. In vemurafenib-sensitive cells, the antiproliferative effect of vemurafenib was enhanced by the other SMI. Together, our results provide insights into the determinants of SMI efficiencies in CRC cells and encourage the further use of low-passage CRC cell lines as preclinical models.