Mathematical methods for modeling the microcirculation

The microcirculation plays a major role in maintaining homeostasis in the body. Alterations or dysfunctions of the microcirculation can lead to several types of serious diseases. It is not surprising, then, that the microcirculation has been an object of intense theoretical and experimental study over the past few decades. Mathematical approaches offer a valuable method for quantifying the relationships between various mechanical, hemodynamic, and regulatory factors of the microcirculation and the pathophysiology of numerous diseases. This work provides an overview of several mathematical models that describe and investigate the many different aspects of the microcirculation, including geometry of the vascular bed, blood flow in the vascular networks, solute transport and delivery to the surrounding tissue, and vessel wall mechanics under passive and active stimuli. Representing relevant phenomena across multiple spatial scales remains a major challenge in modeling the microcirculation. Nevertheless, the depth and breadth of mathematical modeling with applications in the microcirculation is demonstrated in this work. A special emphasis is placed on models of the retinal circulation, including models that predict the influence of ocular hemodynamic alterations with the progression of ocular diseases such as glaucoma

Analysis of pattern dynamics for a nonlinear model of the human cortex via bifurcation theories

This thesis examines the bifurcations, i.e., the emergent behaviours, for the Waikato cortical model under the influence of the gap-junction inhibitory diffusion D₂ (identified as the Turing bifurcation parameter) and the time-to-peak for hyperpolarising GABA response γi (i.e., inhibitory rate-constant, identified as the Hopf bifurcation parameter). The cortical model simplifies the entire cortex to a cylindrical macrocolumn (∼ 1 mm³) containing ∼ 10⁵ neurons (85% excitatory, 15% inhibitory) communicating via both chemical and electrical (gap-junction) synapses. The linear stability analysis of the model equations predict the emergence of a Turing instability (in which separated areas of the cortex become activated) when gap-junction diffusivity is increased above a critical level. In addition, a Hopf bifurcation (oscillation) occurs when the inhibitory rate-constant is sufficiently small. Nonlinear interaction between these instabilities leads to spontaneous cortical patterns of neuronal activities evolving in space and time. Such model dynamics of delicately balanced interplay between Turing and Hopf instabilities may be of direct relevance to clinically observed brain dynamics such as epileptic seizure EEG spikes, deep-sleep slow-wave oscillations and cognitive gamma-waves. The relationship between the modelled brain patterns and model equations can normally be inferred from the eigenvalue dispersion curve, i.e., linear stability analysis. Sometimes we experienced mismatches between the linear stability analysis and the formed cortical patterns, which hampers us in identifying the type of instability corresponding to the emergent patterns. In this thesis, I investigate the pattern-forming mechanism of the Waikato cortical model to better understand the model nonlinearities. I first study the pattern dynamics via analysis of a simple pattern-forming system, the Brusselator model, which has a similar model structure and bifurcation phenomena as the cortical model. I apply both linear and nonlinear perturbation methods to analyse the near-bifurcation behaviour of the Brusselator in order to precisely capture the dominant mode that contributes the most to the final formed-patterns. My nonlinear analysis of the Brusselator model yields Ginzburg-Landau type amplitude equations that describe the dynamics of the most unstable mode, i.e., the dominant mode, in the vicinity of a bifurcation point. The amplitude equations at a Turing point unfold three characteristic spatial structures: honeycomb Hπ, stripes, and reentrant honeycomb H₀. A codimension-2 Turing–Hopf point (CTHP) predicts three mixed instabilities: stable Turing–Hopf (TH), chaotic TH, and bistable TH. The amplitude equations precisely determine the bifurcation conditions for these instabilities and explain the pattern-competition mechanism once the bifurcation parameters cross the thresholds, whilst driving the system into a nonlinear region where the linear stability analysis may not be applicable. Then, I apply the bifurcation theories to the cortical model for its pattern predictions. Analogous to the Brusselator model, I find cortical Turing pattens in Hπ, stripes and H₀ spatial structures. Moreover, I develop the amplitude equations for the cortical model, with which I derive the envelope frequency for the beating-waves of a stable TH mode; and propose ideas regarding emergence of the cortical chaotic mode. Apart from these pattern dynamics that the cortical model shares with the Brusselator system, the cortical model also exhibits “eye-blinking” TH patterns latticed in hexagons with localised oscillations. Although we have not found biological significance of these model pattens, the developed bifurcation theories and investigated pattern-forming mechanism may enrich our modelling strategies and help us to further improve model performance. In the last chapter of this thesis, I introduce a Turing–Hopf mechanism for the anaesthetic slow-waves, and predict a coherence drop of such slow-waves with the induction of propofol anaesthesia. To test this hypothesis, I developed an EEG coherence analysing algorithm, EEG coherence, to automatically examine the clinical EEG recordings across multiple subjects. The result shows significantly decreased coherence along the fronto-occipital axis, and increased coherence along the left- and right-temporal axis. As the Waikato cortical model is spatially homogenous, i.e., there are no explicit front-to-back or right-to-left directions, it is unable to produce different coherence changes for different regions. It appears that the Waikato cortical model best represents the cortical dynamics in the frontal region. The theory of pattern dynamics suggests that a mode transition from wave–Turing–wave to Turing–wave–Turing introduces pattern coherence changes in both positive and negative directions. Thus, a further modelling improvement may be the introduction of a cortical bistable mode where Turing and wave coexist

Real-Time Substrate Feed Optimization of Anaerobic Co-Digestion Plants

In anaerobic co-digestion plants a mix of organic materials is converted to biogas using the anaerobic digestion process. These organic materials, called substrates, can be crops, sludge, manure, organic wastes and many more. They are fed on a daily basis and significantly affect the biogas production process. In this thesis dynamic real-time optimization of the substrate feed for anaerobic co-digestion plants is developed. In dynamic real-time optimization a dynamic simulation model is used to predict the future performance of the controlled plant. Therefore, a complex simulation model for biogas plants is developed, which uses the famous Anaerobic Digestion Model No. 1 (ADM1). With this model the future economics as well as stability can be calculated resulting in a multi-objective performance criterion. Using multi-objective nonlinear model predictive control (NMPC) the model predictions are used to find the optimal substrate feed for the biogas plant. Therefore, NMPC solves an optimization problem over a moving horizon and applies the optimal substrate feed to the plant for a short while before recalculating the new optimal solution. The multi-objective optimization problem is solved using state-of-the-art methods such as SMS-EMOA and SMS-EGO. The performance of the proposed approach is validated in a detailed simulation studyAlgorithms and the Foundations of Software technolog

Biological denitrification : fundamental kinetic studies, and process analysis for sequencing batch reactor operation

This study dealt with a detailed investigation of biological denitrification of nitrate and nitrite by a pure culture of Pseudomonas denitrificans (ATCC 13867), under anaerobic conditions. In the first part of the study, the kinetics of denitrification were studied in serum-bottle experiments. It was found that reduction of both nitrate and nitrite follows inhibitory expressions of the Andrews type. It was also found that when nitrite is present at levels above 15 mg/L, nitrite and nitrate are involved in a cross-inhibitory, non-competitive, interaction pattern. Analysis of the kinetic data has shown that the culture used has severe maintenance requirements, which can be described by the model proposed by Herbert. Experiments at different temperatures have revealed that the optimum temperature is around 38 °C. Activation energies have been determined as 8.6 Kcal/mole for nitrate, and 7.21 Kcal/mole for nitrite reduction. Studies on the effect of pH have shown that the optimal value is about 7.5. Based on the detailed kinetic expressions determined in the first part of the study, denitrification of nitrite and nitrate/nitrite mixtures was theoretically analyzed and experimentally investigated in a continuously operated sequencing batch reactor. The theoretical analysis was based on the bifurcation theory for forced systems. The different types of the dynamical behavior of the system were found, and are presented in the form of bifurcation diagrams and two-dimensional operating diagrams. The analysis predicts that there are domains in the operating parameter space where the system can reach different periodic patterns which are determined by the conditions under which the process is started-up. The analysis also predicts that improper selection of operating parameters can lead to high nitrite accumulation in the reactor. The predictions of the theory were tested in experiments with a specially designed system. The unit involved a fully automated 2-liter reactor which operated under different inlet flowrate and concentration conditions. During the experiments the system was perfectly sealed and the medium kept under a helium atmosphere of pressure slightly higher than 1 atm. In all cases, a remarkably good agreement was found between theoretical predictions and experimental data. The experimentally validated model can be used in process optimization studies, and preliminary scale-up calculations

Investigating the role of fast-spiking interneurons in neocortical dynamics

PhD ThesisFast-spiking interneurons are the largest interneuronal population in neocortex. It is well documented that this population is crucial in many functions of the neocortex by subserving all aspects of neural computation, like gain control, and by enabling dynamic phenomena, like the generation of high frequency oscillations. Fast-spiking interneurons, which represent mainly the parvalbumin-expressing, soma-targeting basket cells, are also implicated in pathological dynamics, like the propagation of seizures or the impaired coordination of activity in schizophrenia. In the present thesis, I investigate the role of fast-spiking interneurons in such dynamic phenomena by using computational and experimental techniques. First, I introduce a neural mass model of the neocortical microcircuit featuring divisive inhibition, a gain control mechanism, which is thought to be delivered mainly by the soma-targeting interneurons. Its dynamics were analysed at the onset of chaos and during the phenomena of entrainment and long-range synchronization. It is demonstrated that the mechanism of divisive inhibition reduces the sensitivity of the network to parameter changes and enhances the stability and exibility of oscillations. Next, in vitro electrophysiology was used to investigate the propagation of activity in the network of electrically coupled fast-spiking interneurons. Experimental evidence suggests that these interneurons and their gap junctions are involved in the propagation of seizures. Using multi-electrode array recordings and optogenetics, I investigated the possibility of such propagating activity under the conditions of raised extracellular K+ concentration which applies during seizures. Propagated activity was recorded and the involvement of gap junctions was con rmed by pharmacological manipulations. Finally, the interaction between two oscillations was investigated. Two oscillations with di erent frequencies were induced in cortical slices by directly activating the pyramidal cells using optogenetics. Their interaction suggested the possibility of a coincidence detection mechanism at the circuit level. Pharmacological manipulations were used to explore the role of the inhibitory interneurons during this phenomenon. The results, however, showed that the observed phenomenon was not a result of synaptic activity. Nevertheless, the experiments provided some insights about the excitability of the tissue through scattered light while using optogenetics. This investigation provides new insights into the role of fast-spiking interneurons in the neocortex. In particular, it is suggested that the gain control mechanism is important for the physiological oscillatory dynamics of the network and that the gap junctions between these interneurons can potentially contribute to the inhibitory restraint during a seizure.Wellcome Trust

Model Order Reduction

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This three-volume handbook covers methods as well as applications. This third volume focuses on applications in engineering, biomedical engineering, computational physics and computer science