|
Hypothesis-based modelling
|
|
Introduction The purpose of a hypothesis-driven model is to test alternative assumptions regarding the mechanisms that produce a given phenomenon. The idea is to generate insights and understanding, and for this purpose the modeling process may be just as important as the model itself. A reference behavior is used to calibrate, test and understand the phenomenon studied. As such, the model is not expected to reproduce the phenomenon in detail, but to isolate those mechanisms that are directly involved in the way the phenomenon in question behaves. The model is validated by its ability to explain different aspects of this phenomenon and predict the response of the system under conditions not previously examined. Using meaningful parameter values, the model must be able to generate the essential dynamical characteristics of the observed phenomenon such as the period of oscillation, the phase relationships and amplitudes of the various variables, etc. We demonstrate the reasoning with an example from biology, where the modeler tries to understand the oscillations of insuline. Background The complexity of hormonal secretion is evident from a large number of experimental studies . Most hormones exhibit a characteristic circadian (24 h) rhythm with the secretion of growth hormone, for example, being markedly increased during early periods of sleep. Hormonal secretion also provides amble evidence of faster temporal components, and the secretion of luteinizing hormone, follicle stimulating hormone, and many other hormones display pronounced ultradian (2-5 h) oscillations. The mechanisms underlying these rhythms can often be traced back to a cyclical activity in the central nervous system. Interaction between the rhythmic modes will produce more complicated dynamic phenomena such as synchronization or mutual modulation where the frequency and amplitude of one mode depends on the phase of a slower mode. Changes of the hormonal release patterns may be part of normal physiological regulation or they may signal a state of disease. The secretion of insulin basically follows the intake of food. It is reasonable to suppose, however, that both the insulin secretion rate and the tissue insulin sensitivity display a circadian rhythm that affects the food-dependent variations. As illustrated in Fig. 1, experiments performed with continuous enteral nutrition or constant intravenous glucose infusion , demonstrate that the secretion of insulin can also exhibit ultradian rhythms with periods in the 80-150 min range. To further complicate the picture, measurements performed on blood from the portal vein (that connects the pancreas to the liver) reveal rapid insulin secretion pulses with 5-12 min periods, and the insulin producing pancreatic cells show even faster pulses of insulin release, in this case associated with variations in the cellular membrane potentials. Purpose While relatively detailed models already exist of the processes in and among the pancreatic cells, the possible role of and the mechanisms behind the 80-150 min oscillations in insulin secretion still require significant efforts to be clarified. The purpose of the present contribution is to use these oscillations as the basis for a discussion of the steps and ideas involved in the formulation of a so-called hypothesis-driven model. Our analysis takes its point of departure in experimental time series obtained through sampling of blood glucose, insulin and C-peptide concentrations for normal young men over a 24-h period with a temporal resolution of 10 min ,. The experiments were conducted under conditions of constant intravenous glucose infusion and, to minimize the effect of initial transients, the constant glucose infusion was started 8 h before the beginning of the measurements. Since molecules of C-peptide and insulin are released from the pancreas at the same rate, but C-peptide follows a much simpler elimination path, the peptide concentration is used to calculate the insulin secretion rate (ISR). Discriminatory experiments Insulin primarily serves to regulate the transport of glucose across the cellular membranes of muscle and fat cells, and availability of glucose in the blood is the main factor controlling the rate of insulin secretion. Together these mechanisms constitute a negative feedback regulation in which an increasing blood glucose concentration promotes insulin secretion, and an increasing insulin concentration promotes glucose uptake by the cells. Since the lifetime of insulin in the organism is relatively short (5-10 min), the system is strongly dissipative. This implies that it quickly absorbs external perturbations and returns to its stationary state. In the presence of a constant glucose infusion most existing models will show a stable equilibrium point of node type (i.e., with purely exponential transients). This, however, does not concur with the appearance of ultradian oscillations. With the applied experimental conditions, the ultradian insulin oscillations cannot be related to a varying supply of glucose. Nor do the oscillations appear to involve interactions with counter-regulatory hormones, since analyses of simultaneous variations in glucagon and cortisol concentrations fail to show correlation with the insulin oscillations. Experimental results also show that the insulin oscillations persist after transplantation of the pancreas , indicating that the oscillations do not depend directly on signals from the central nervous system. Two fundamentally different types of explanation then remain: (1) the ultradian oscillations could reflect the activity of a pancreatic pacemaker, or (2) the oscillations could arise from instability in the insulin-glucose feedback regulation. Experimentally one can distinguish between these two possibilities by means of a glucose-clamp experiment in which the blood glucose concentration is maintained constant by continuously adjusting the rate of glucose infusion. Under these conditions, the pulsatile secretion of insulin was found to no longer occur . If the insulin oscillations were caused by a pancreatic pacemaker, on the other hand, one would expect them to persist even when the glucose concentration is kept constant. Dynamic hypothesis: model assumptions After inspection of a considerable number of experimental time series, we hypothesize that the insulin-glucose regulation, when the ultradian oscillations occur, is operating close to a Hopf bifurcation. This type of instability, in which pair of complex conjugate eigenvalues for the equilibrium point cross into the positive half plane, typically arises in negative feedback systems when the feedback gain and/or delay become too large. In the present system, the feedback gain is controlled by the form of the relationships from blood glucose concentration to rate of insulin secretion and from insulin concentration to rate of cellular glucose utilization. Both of these relationships can be obtained from independent measurements and hence cannot be freely adjusted. Therefore, to make the model oscillate we need to introduce some form of delay in the system. One source of delay is associated with the finite equilibration time between the insulin stores in the blood plasma and the intercellular fluid. The characteristic time constant for the diffusion of insulin across the capillary wall and into the tissue is estimated to be of the order of 15 min. A somewhat less obvious delay is hypothesized to exist between variations in the blood insulin concentration and the rate of glucose release from the liver. Based on experimentally observed phase shifts, this delay is taken to be 20 min. Model formulation The model can now be formulated in terms of continuation equations for the material components insulin and glucose. With two insulin compartments (the blood plasma and the intercellular space) and a single compartment for the distribution of glucose, the model consists of three coupled differential equations: (i) The amount of glucose in the considered distribution volume changes at a rate determined by the rate of glucose infusion plus the rate of hepatic glucose release and minus the rate of glucose utilization. Glucose utilization is specified in terms of an insulin-dependent and an insulin-independent part, (ii) The amount of insulin in the plasma volume changes at a rate given by the rate of pancreatic insulin secretion minus the rates of hepatic insulin degradation and of insulin diffusion into the interstitial volume, (iii) The amount of insulin in the interstitial volume changes through the diffusion of insulin into this volume and through degradation of insulin in the tissue. By virtue of the S-shaped form of the relationships between insulin secretion and blood glucose concentration and between glucose utilization and interstitial insulin concentration these equations are nonlinear. Such nonlinearities are necessary to limit the excursions generated by the instability of the equilibrium point. A linear model will not be able to cope with the unstable behaviors that in many ways are the hallmark of living organisms. The model also includes three differential equations describing the delayed reaction of the hepatic glucose release. Except for the delay in hepatic glucose release, all parameters and nonlinear relations are physiologically justified through specific and independent measurements. A complete version of the computer program may be found here. Simulations with the model Figure 2 shows a typical example of the results obtained when simulating the model with a constant rate of glucose infusion of 6 mg/min per kg body weight. After a relatively short transient (not shown), the model settles into a stationary dynamics in the form of self-sustained oscillations in the plasma glucose and insulin concentrations with constant amplitudes and with a period of 110-120 min. Extensive simulations have shown that this dynamics is quite robust to parameter variations, and that the model can reproduce many of the experimentally observed characteristics of the ultradian insulin oscillations. In particular, the model can reproduce the characteristic ‘ringing’ phenomenon in response to a meal observed in some of the early experiments , and the calculated phase shift between the glucose and insulin oscillations is also in accordance with experimental observations. If the rate of glucose infusion is changed, the amplitudes of the glucose and insulin oscillations also change as the working points on the nonlinear characteristics move. It is interesting to note that the internally generated oscillations arise in the intermediate range of glucose supply. Due to the S-shaped characteristics for insulin production and glucose utilization, the feedback gain decreases both for high and low glucose infusion rates, and the equilibrium point regains its stability. The oscillations also become damped if the lifetime of insulin in the organism is reduced. In both cases, the transition occurs via a reverse Hopf bifurcation . Figure 2. Simulation of the model of ultradian insulin-glucose oscillations. With independently determined and physiologically realistic parameter values the model reproduces experimental observations under a variety of different conditions. The model maybe used for instance to predict how the internally generated oscillation in insulin secretion respond to an external sinusoidal forcing of the glucose infusion rate , or to examine the possible interaction of the oscillatory insulin secretion with receptor regeneration dynamics . Model improvements The most important task for future work is to determine the conditions under which self-sustained ultradian oscillations in the insulin-glucose regulation occur. Although such oscillations have been observed in several recent experiments ,, there is still a general reluctance to accept their existence. The delayed response of the hepatic glucose release is a crucial structural assumption in our model. This assumption obviously has to be supported in detail, and the simple description in terms of a third order delay must be replaced by a physiologically more satisfactory formulation. At the same time, a readjustment of the nonlinear characteristics that controls the release of glucose from the liver must be performed. This is presumably the most sensitive relation in the model. The two S-shaped characteristics for insulin production, respectively glucose utilization should be fitted to new and improved data and, perhaps, recast into biochemically more acceptable Hill function. Hereafter, one should reexamine the model’s response to a meal response and to an insulin bolus injection. The interaction between the ultradian and the circadian rhythms could be examined by forcing appropriate model parameters with a 24-h rhythm. Finally, the model could be used to follow different phases in the development of type II diabetes. It is important, however, that this work is performed in close collaboration with experimentally or clinically oriented experts, who can discuss and check the various hypotheses. 
|
| < Prev | Next > |
|---|
Main Menu
| Home |
| Articles |
| Search |
| Submit Article |
| Contact Us |
| Donate to Wikibin |
