Coexistence between silent and bursting states in a biophysical Hodgkin-Huxley-type of model.

Classification of the dynamical mechanisms that support bistability between bursting oscillations and silence has not yet been clarified in detail. The purpose of this paper is to demonstrate that the coexistence of a stable equilibrium point with a state of continuous bursting can occur in a slightly modified, biophysical model that describe the dynamics of pancreatic beta-cells. To realize this form of coexistence, we have introduced an additional voltage-dependent potassium current that is activated in the region around the original, unstable equilibrium point. It is interesting to note that this modification also leads the model to display a blue-sky catastrophe in the transition region between chaotic and bursting states.

Onset of chaos in a single-phase power electronic inverter.

Supported by experiments on a power electronic DC/AC converter, this paper considers an unusual transition from the domain of stable periodic dynamics (corresponding to the desired mode of operation) to chaotic dynamics. The behavior of the converter is studied by means of a 1D stroboscopic map derived from a non-autonomous ordinary differential equation with discontinuous right-hand side. By construction, this stroboscopic map has a high number of border points. It is shown that the onset of chaos occurs stepwise, via irregular cascades of different border collisions, some of which lead to bifurcations while others do not.

Insulin aspart pharmacokinetics: an assessment of its variability and underlying mechanisms.

BACKGROUND: Insulin aspart (IAsp) is used by many diabetics as a meal-time insulin to control post-prandial glucose levels. As is the case with many other insulin types, the pharmacokinetics (PK), and consequently the pharmacodynamics (PD), is associated with clinical variability, both between and within individuals. The present article identifies the main physiological mechanisms that govern the PK of IAsp following subcutaneous administration and quantifies them in terms of their contribution to the overall variability. MATERIAL AND METHODS: CT scanning data from Thomsen et al. (2012) are used to investigate and quantify the properties of the subcutaneous depot. Data from Brange et al. (1990) are used to determine the effects of insulin chemistry in subcutis on the absorption rate. Intravenous (i.v.) bolus and infusion PK data for human insulin are used to understand and quantify the systemic distribution and elimination (Pørksen et al., 1997; Sjöstrand et al., 2002). PK and PD profiles for type 1 diabetics from Chen et al. (2005) are analyzed to demonstrate the effects of IAsp antibodies in terms of bound and unbound insulin. PK profiles from Thorisdottir et al. (2009) and Ma et al. (2012b) are analyzed in the nonlinear mixed effects software Monolix® to determine the presence and effects of the mechanisms described in this article. RESULTS: The distribution of IAsp in the subcutaneous depot show an initial dilution of approximately a factor of two in a single experiment. Injected insulin hexamers exist in a chemical equilibrium with monomers and dimers, which depends strongly on the degree of dilution in subcutis, the presence of auxiliary substances, and a variety of other factors. Sensitivity to the initial dilution in subcutis can thus be a cause of some of the variability. Temporal variations in the PK are explained by variations in the subcutaneous blood flow. IAsp antibodies are found to be a large contributor to the variability of total insulin PK in a study by Chen et al. (2005), since only the free fraction is eliminated via the receptors. The contribution of these and other sources of variability to the total variability is quantified via a population PK analysis and two recent clinical studies (Thorisdottir et al., 2009; Ma et al., 2012b), which support the presence and significance of the identified mechanisms. CONCLUSIONS: IAsp antibody binding, oligomeric transitions in subcutis, and blood flow dependent variations in absorption rate seem to dominate the PK variability of IAsp. It may be possible via e.g. formulation design to reduce some of these variability factors.

Complex patterns of metabolic and Ca²âº entrainment in pancreatic islets by oscillatory glucose.

Glucose-stimulated insulin secretion is pulsatile and driven by intrinsic oscillations in metabolism, electrical activity, and Ca(2+) in pancreatic islets. Periodic variations in glucose can entrain islet Ca(2+) and insulin secretion, possibly promoting interislet synchronization. Here, we used fluorescence microscopy to demonstrate that glucose oscillations can induce distinct 1:1 and 1:2 entrainment of oscillations (one and two oscillations for each period of exogenous stimulus, respectively) in islet Ca(2+), NAD(P)H, and mitochondrial membrane potential. To our knowledge, this is the first demonstration of metabolic entrainment in islets, and we found that entrainment of metabolic oscillations requires voltage-gated Ca(2+) influx. We identified diverse patterns of 1:2 entrainment and showed that islet synchronization during entrainment involves adjustments of both oscillatory phase and period. All experimental findings could be recapitulated by our recently developed mathematical model, and simulations suggested that interislet variability in 1:2 entrainment patterns reflects differences in their glucose sensitivity. Finally, our simulations and recordings showed that a heterogeneous group of islets synchronized during 1:2 entrainment, resulting in a clear oscillatory response from the collective. In summary, we demonstrate that oscillatory glucose can induce complex modes of entrainment of metabolically driven oscillations in islets, and provide additional support for the notion that entrainment promotes interislet synchrony in the pancreas.

Data-driven assessment of the association of polymorphisms in 5-Fluorouracil metabolism genes with outcome in adjuvant treatment of colorectal cancer.

A major challenge in the assessment of medicines, treatment options, etc., is to establish a framework for the comparison of risks and benefits of many different types and magnitudes, a framework that at the same time allows a clear distinction between the roles played by the statistical analyses of data and by judgements based on personal experience and expertise. The purpose of this study was to demonstrate how clinical data can be weighted, scored and presented by the use of an eight-step data-driven benefit-risk assessment method, where two genetic profiles are compared. Our aim was to present a comprehensive approach that is simple to apply, allows direct comparison of different types of risks and benefits, quantifies the clinical relevance of data and is tailored for the comparison of different options. We analysed a cohort of 302 patients with colorectal cancer treated with 5-Fluorouracil (5-FU). Endpoints were cure rate, survival rate, time-to-death (TTD), time-to-relapse (TTR) and main adverse drug reactions. Multifactor dimensionality reduction (MDR) was used to identify genetic interaction profiles associated with outcome. We have been able to demonstrate that a specific MDR-derived combination (the MDR-1 group) of dihydropyrimidine dehydrogenase and thymidylate synthase polymorphisms is associated with increased and clinically significant difference for cure and survival rates, TTD and probably also for TTR, which are seen as the most important endpoints. An inferior profile was observed for severe myocardial ischaemia. A probably inferior profile was seen for severe arthralgia/myalgia and severe infections. A clear superior profile was seen for severe mucositis/stomatitis. The proposed approach offers comprehensive, data-driven assessment that can facilitate decision processes, for example, in a clinical setting. It employs descriptive statistical methods to highlight the clinically relevant differences between options.

A comprehensive approach to benefit-risk assessment in drug development.

Major regulatory agencies, for example, FDA and EMA, have started to request comprehensive benefit-risk analyses of pharmaceutical products prior to approval or labelling expansion. The purpose of this study is to develop a generally applicable and reliable data-driven benefit-risk assessment method, where two or more drugs/doses can be compared. Our aim is to formulate an approach that is simple to apply, allows direct comparison of different types of risks and benefits, and is tailored for application in different disease areas both during clinical development and in the marketing approval phase. The proposed benefit-risk assessment method involves eight successive steps: (1) establishment of the decision context, (2) identification of benefit and risk criteria, (3) weighting, (4) scoring, (5) evaluation of uncertainty, (6) calculation of weighted scores, (7) visualization, and (8) discussion and formulation of an overall conclusion. To reduce the impact of subjective judgements, scores are assigned to each criterion on the basis of objective information (data) wherever possible. The proposed benefit-risk evaluation approach offers comprehensive, data-driven assessments that can facilitate decision processes. It employs descriptive statistical methods to highlight the clinically significant differences between drugs in clinical trials. The approach can be used in single as well as in multiple trials and provides clear diagrams as the basis for presentation and discussion of the results.

Bioavailability and variability of biphasic insulin mixtures.

Absorption of subcutaneously administered insulin is associated with considerable variability. Some of this variability was quantitatively explained for both soluble insulin and insulin suspensions in a recent contribution to this journal (Søeborg et al., 2009). In the present article, the absorption kinetics for mixtures of insulins is described. This requires that the bioavailability of the different insulins is considered. A short review of insulin bioavailability and a description of the subcutaneous depot thus precede the presentation of possible mechanisms associated with subcutaneous insulin degradation. Soluble insulins are assumed to be degraded enzymatically in the subcutaneous tissue. Suspended insulin crystals form condensed heaps that are assumed to be degraded from their surface by invading macrophages. It is demonstrated how the shape of the heaps affects the absorption kinetics. Variations in heap formation thus explain some of the additional variability associated with suspended insulins (e.g. NPH insulins) compared to soluble insulins. The heap model also describes how increasing concentrations of suspended insulins lead to decreasing bioavailability and lower values of Cmax. Together, the findings constitute a comprehensive, quantitative description of insulin absorption after subcutaneous administration. The model considers different concentrations and doses of soluble insulin, including rapid acting insulin analogues, insulin suspensions and biphasic insulin mixtures. The results can be used in both the development of novel insulin products and in the planning of the treatment of insulin dependent diabetic patients.

Hyperbolic chaotic attractor in amplitude dynamics of coupled self-oscillators with periodic parameter modulation.

The paper proposes an approach to constructing feasible examples of dynamical systems with hyperbolic chaotic attractors based on the successive transfer of excitation between two pairs of self-oscillators that are alternately active. An angular variable that measures the relations of the current amplitudes for the two oscillators of each pair undergoes a transformation in accordance with the expanding circle map during each cycle of the process. We start with equations describing the dynamics in terms of complex or real amplitudes and then examine two models based on van der Pol oscillators. One model corresponds to the situation of equality of natural frequencies of the partial oscillators, and another to a nonresonant ratio of the oscillation frequencies relating to each of the two pairs. Dynamics of all models are illustrated with diagrams indicating the transformation of the angular variables, portraits of attractors, Lyapunov exponents, etc. The uniformly hyperbolic nature of the attractor in the stroboscopic Poincaré map is confirmed for a real-amplitude version of the equations by computations of statistical distribution of angles between stable and unstable manifolds at a representative set of points on the attractor. In other versions of the equations the attractors relate presumably to the partially hyperbolic class.

Bistability in autoimmune diseases.

Autoimmune diseases damage host tissue, which, in turn, may trigger a stronger immune response. Systems characterized by such positive feedback loops can display co-existing stable steady states. In a mathematical model of autoimmune disease, one steady state may correspond to the healthy state and another to an autoimmune steady state characterized by widespread tissue damage and immune activation. We show how a triggering event may move the system from the healthy to the autoimmune state and how transient immunosuppressive treatment can move the system back to the healthy state.

C-type period-doubling transition in nephron autoregulation.

The functional units of the kidney, called nephrons, utilize mechanisms that allow the individual nephron to regulate the incoming blood flow in response to fluctuations in the arterial pressure. This regulation tends to be unstable and to generate self-sustained oscillations, period-doubling bifurcations, mode-locking and other nonlinear dynamic phenomena in the tubular pressures and flows. Using a simplified nephron model, the paper examines how the regulatory mechanisms react to an external periodic variation in arterial pressure near a region of resonance with one of the internally generated mode-locked cycles. We show how the stable and unstable resonance cycles generated in this response undergo interconnected cascades of period-doubling bifurcations and how each period doubling leads to the formation of a new pair of saddle-node bifurcation curves along the edges of the resonance zone. We also show how period doubling of the resonance cycles is accompanied by a torus-doubling process in the quasiperiodic regime that exists outside of the resonance zone.

Coupling-induced complexity in nephron models of renal blood flow regulation.

Tubular pressure and nephron blood flow time series display two interacting oscillations in rats with normal blood pressure. Tubuloglomerular feedback (TGF) senses NaCl concentration in tubular fluid at the macula densa, adjusts vascular resistance of the nephron's afferent arteriole, and generates the slower, larger-amplitude oscillations (0.02-0.04 Hz). The faster smaller oscillations (0.1-0.2 Hz) result from spontaneous contractions of vascular smooth muscle triggered by cyclic variations in membrane electrical potential. The two mechanisms interact in each nephron and combine to act as a high-pass filter, adjusting diameter of the afferent arteriole to limit changes of glomerular pressure caused by fluctuations of blood pressure. The oscillations become irregular in animals with chronic high blood pressure. TGF feedback gain is increased in hypertensive rats, leading to a stronger interaction between the two mechanisms. With a mathematical model that simulates tubular and arteriolar dynamics, we tested whether an increase in the interaction between TGF and the myogenic mechanism can cause the transition from periodic to irregular dynamics. A one-dimensional bifurcation analysis, using the coefficient that couples TGF and the myogenic mechanism as a bifurcation parameter, shows some regions with chaotic dynamics. With two nephrons coupled electrotonically, the chaotic regions become larger. The results support the hypothesis that increased oscillator interactions contribute to the transition to irregular fluctuations, especially when neighboring nephrons are coupled, which is the case in vivo.

Absorption kinetics of insulin after subcutaneous administration.

Many diabetic patients depend on regular and well-controlled administration of insulin to avoid unacceptable excursions in plasma glucose. A complicating factor is that the absorption of insulin shows a considerable variability, both between patients, and from administration to administration for the same patient. To understand the mechanisms that influence this variability we present a quantitative description of the absorption kinetics for both soluble insulin and insulin crystals. The concentration dependent distribution of insulin between different oligomers is first analysed and described. Next, the disappearance of soluble and crystalline insulin from subcutis is described and explained as a function of the administered dose, the insulin concentration and crystal specific parameters, but without diffusion. The effect of diffusion is then included, and the appearance of insulin in plasma following subcutaneous administration is simulated and discussed. Our results not only explain the observed variability, but they also explain how dose size, insulin concentration, insulin crystals etc. influence the absorption kinetics.

Formation and destruction of multilayered tori in coupled map systems.

The paper first illustrates how multilayered tori can arise through one or more pitchfork bifurcations of the saddle cycle on an ordinary resonance torus. The paper hereafter describes three different scenarios by which a multilayered torus can be destructed. One scenario involves a saddle-node bifurcation in which the middle layer of a three-layered torus disappears in an abrupt transition to chaos while the outer-layer manifolds and their associated saddle and unstable-focus cycles continue to exist and to control the transient dynamics. In a second scenario, the unstable focus cycles of the intermediate layers in a five-layered torus turn into unstable nodes, and closed loop connections are established between the unstable nodes and the points of the stable resonance node on the torus. Finally, a third scenario describes a transition in which homoclinic bifurcations destroy first the outer layers and thereafter also the inner layer. The paper also illustrates how the formation and destruction of multilayered tori can occur in the cluster dynamics of an ensemble of globally coupled maps. This leads to three additional scenarios for the destruction of multilayered tori.

Integrating systems approaches into pharmaceutical sciences.

During the first week of December 2007, the European Federation for Pharmaceutical Sciences (EUFEPS) and BioSim, the major European Network of Excellence on Systems Biology, held a challenging conference on the use of mathematical models in the drug development process. More precisely, the purpose of the conference was to promote the 'Integration of Systems Approaches into Pharmaceutical Sciences' in view of optimising the development of new effective drugs. And a challenge this is, considering both the high attrition rates in the pharmaceutical industry and the failure of finding definitive drug solutions for many of the diseases that plague mankind today. The conference was co-sponsored by the American College of Clinical Pharmacology, the European Center for Pharmaceutical Medicine, and the Swiss Society of Pharmaceutical Sciences and, besides representatives from the European Regulatory Agencies and FDA, the meeting was attended by 75 industrial and some 45 academic participants.

Phase-modulation laser interference microscopy: an advance in cell imaging and dynamics study.

We describe how phase-modulation laser interference microscopy and wavelet analysis can be applied to noninvasive nonstained visualization and study of the structural and dynamical properties of living cells. We show how phase images of erythrocytes can reveal the difference between various erythrocyte forms and stages of hemolysis and how phase images of neurons reveal their complex intracellular structure. Temporal variations of the refractive index are analyzed to detect cellular rhythmic activity on different time scales as well as to uncover interactions between the cellular processes.

Transitions from phase-locked dynamics to chaos in a piecewise-linear map.

Recent work has shown that torus formation in piecewise-smooth maps can take place through a special type of border-collision bifurcation in which a pair of complex conjugate multipliers for a stable cycle abruptly jump out of the unit circle. Transitions from an ergodic to a resonant torus take place via border-collision fold bifurcations. We examine the transition to chaos through torus destruction in such maps. Considering a piecewise-linear normal-form map we show that this transition, by virtue of the interplay of border-collision bifurcations with period-doubling and homoclinic bifurcations, can involve mechanisms that differ qualitatively from those described by Afraimovich and Shilnikov.

Synchronization among mechanisms of renal autoregulation is reduced in hypertensive rats.

We searched for synchronization among autoregulation mechanisms using wavelet transforms applied to tubular pressure recordings in nephron pairs from the surface of rat kidneys. Nephrons have two oscillatory modes in the regulation of their pressures and flows: a faster (100-200 mHz) myogenic mode, and a slower (20-40 mHz) oscillation in tubuloglomerular feedback (TGF). These mechanisms interact; the TGF mode modulates both the amplitude and the frequency of the myogenic mode. Nephrons also communicate with each other using vascular signals triggered by membrane events in arteriolar smooth muscle cells. In addition, the TGF oscillation changes in hypertension to an irregular fluctuation with characteristics of deterministic chaos. The analysis shows that, within single nephrons of normotensive rats, the myogenic mode and TGF are synchronized at discrete frequency ratios, with 5:1 most common. There is no distinct synchronization ratio in spontaneously hypertensive rats (SHR). In normotensive rats, full synchronization of both TGF and myogenic modes is the most probable state for pairs of nephrons originating in a common cortical radial artery. For SHR, full synchronization is less probable; most common in SHR is a state of partial synchronization with entrainment between neighboring nephrons for only one of the modes. Modulation of the myogenic mode by the TGF mode is much stronger in hypertensive than in normotensive rats. Synchronization among nephrons forms the basis for an integrated reaction to blood pressure fluctuations. Reduced synchronization in SHR suggests that the effectiveness of the coordinated response is impaired in hypertension.

Noise controlled synchronization in potassium coupled neural models.

The paper applies biologically plausible models to investigate how noise input to small ensembles of neurons, coupled via the extracellular potassium concentration, can influence their firing patterns. Using the noise intensity and the volume of the extracellular space as control parameters, we show that potassium induced depolarization underlies the formation of noise-induced patterns such as delayed firing and synchronization. These phenomena are associated with the appearance of new time scales in the distribution of interspike intervals that may be significant for the spatio-temporal oscillations in neuronal ensembles.

Vascular coupling induces synchronization, quasiperiodicity, and chaos in a nephron tree.

The paper presents a study of synchronization phenomena in a system of 22 nephrons supplied with blood from a common cortical radial artery. The nephrons are assumed to interact via hemodynamic and vascularly propagated coupling, both mediated by vascular connections. Using anatomic and physiological criteria, the nephrons are divided into groups: cortical nephrons and medullary nephrons with short, intermediate and long Henle loops. Within each of these groups the delay parameters of the internal feedback regulation are given a random component to represent the internephron variability. For parameters that generate simple limit cycle dynamics in the pressure and flow regulation of single nephrons, the ensemble of coupled nephrons showed steady state, quasiperiodic or chaotic dynamics, depending on the interaction strengths and the arterial blood pressure. When the solutions were either quasiperiodic or chaotic, cortical nephrons synchronized to a single frequency, but the longer medullary nephrons formed two clusters with different frequencies. Under no physiologically realistic combination of parameters did all nephrons assume a common frequency. Our results suggest a greater variability in the nephron dynamics than is apparent from measurements performed on cortical nephrons only. This variability may explain the development of chaotic dynamics in tubular pressure records from hypertensive rats.