Brain oedema is thought to form and to clear through the use of water-protein channels, aquaporin-4 (AQP4), which are found in the astrocyte endfeet. The model developed here is used to study the function of AQP4 in the formation and elimination of oedema fluid in ischaemia-reperfusion injury. The cerebral space is assumed to be made of four fluid compartments: astrocyte, neuron, ECS and blood microvessels, and a solid matrix for the tissue, and this is modelled using multiple-network poroelastic theory. AQP4 allows the movement of water between astrocyte and the ECS and the microvessels. It is found that the presence of AQP4 may help in reducing vasogenic oedema shown by a decrease in brain tissue extracellular pressure. However, the astrocyte pressure will increase to compensate for this decrease, which may lead to cytotoxic oedema. In addition, the swelling will also depend on the ionic concentrations in the astrocyte and extracellular space, which may change after ischaemic stroke. Understanding the role of AQP4 in oedema may thus help the development of a treatment plan in reducing brain swelling after ischaemia-reperfusion.
Restoration of an adequate cerebral blood supply after an ischemic attack is a primary clinical goal. However, the blood-brain barrier may break down after a prolonged ischemia causing the fluid in the blood plasma to filtrate and accumulate into the cerebral tissue interstitial space. Accumulation of this filtration fluid causes the cerebral tissue to swell, a condition known as vasogenic oedema. Tissue swelling causes the cerebral microvessels to be compressed, which may further obstruct the blood flow into the tissue, thus leading to the no-reflow phenomenon or a secondary ischemic stroke. The actual mechanism of this however is still not fully understood. A new model is developed here to study the effect of reperfusion on the formation of vasogenic oedema and cerebral microvessel collapse. The formation of vasogenic oedema is modelled using the capillary filtration equation while vessel collapse is modelled using the tube law of microvessel. Tissue swelling is quantified in terms of displacement, which is modelled using poroelastic theory. The results show that there is an increase in tissue displacement and interstitial pressure after reperfusion. In addition, the results also show that vessel collapse can occur at high value of reperfusion pressure, low blood osmotic pressure, high cerebral capillary permeability and low cerebral capillary stiffness. This model provides insight on the formation of ischemia-reperfusion injury by tissue swelling and vessel collapse.
Minimally invasive tumor ablations (MITAs) are an increasingly important tool in the treatment of solid tumors across multiple organs. The problems experienced in modeling different types of MITAs are very similar, but the development of mathematical models is mostly performed in isolation according to modality. Fundamental research into the modeling of specific types of MITAs is indeed required, but to choose the optimal treatment for an individual the primary clinical requirement is to have reliable predictions for a range of MITAs. In this review of the mathematical modeling of MITAs 4 modalities are considered: radiofrequency ablation, microwave ablation, cryoablation, and irreversible electroporation. The similarities in the mathematical modeling of these treatments are highlighted, and the analysis of the models within a general framework is discussed. This will aid in developing a deeper understanding of the sensitivity of MITA models to physiological parameters and the impact of uncertainty on predictions of the ablation zone. Through robust validation and analysis of the models it will be possible to choose the best model for a given application. This is important because many different models exist with no objective comparison of their performance. The collection of relevant in vivo experimental data is also critical to parameterize such models accurately. This approach will be necessary to translate the field into clinical practice.