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At least 1,783 records · Page 99Linked to original sources
PubMed1989

Mathematical model of dependence of heart rate on tissue concentration of acetylcholine.

The change in sinus period elicited by vagal stimulation depends on the rate of acetylcholine (ACh) release from the nerve endings, the rate of ACh degradation in the nodal tissue, and the responsiveness of the sinus node to ACh. Vagal stimulation in anesthetized dogs prolonged sinus period. After cessation of vagal stimulation, the sinus period returned to the prestimulation period. We developed a mathematical model to analyze the dynamics of ACh degradation in the neuroeffector junction and the dependence of sinus period on the concentration of ACh. From the in vitro reaction kinetics of acetylcholinesterase, we derived an analytical expression for the rate of ACh degradation in the intact animal. Our model represents the electrical behavior of the sinus node by the electrical activity of one pacemaker cell with six membrane ionic currents. This model predicts the decline in sinus period of the intact anesthetized dog as acetylcholinesterase degrades ACh in the neuroeffector junction. The half-life of ACh after cessation of vagal stimulation was estimated to be 2.7 s. We conclude that following termination of vagal stimulation, the sinus node of the intact animal responds to ACh as if the sinus node were one oscillator.

Acetylcholine↗
PubMed1991

Mathematical model of geometry and fibrous structure of the heart.

We developed a mathematical representation of ventricular geometry and muscle fiber organization using three-dimensional finite elements referred to a prolate spheroid coordinate system. Within elements, fields are approximated using basis functions with associated parameters defined at the element nodes. Four parameters per node are used to describe ventricular geometry. The radial coordinate is interpolated using cubic Hermite basis functions that preserve slope continuity, while the angular coordinates are interpolated linearly. Two further nodal parameters describe the orientation of myocardial fibers. The orientation of fibers within coordinate planes bounded by epicardial and endocardial surfaces is interpolated linearly, with transmural variation given by cubic Hermite basis functions. Left and right ventricular geometry and myocardial fiber orientations were characterized for a canine heart arrested in diastole and fixed at zero transmural pressure. The geometry was represented by a 24-element ensemble with 41 nodes. Nodal parameters fitted using least squares provided a realistic description of ventricular epicardial [root mean square (RMS) error less than 0.9 mm] and endocardial (RMS error less than 2.6 mm) surfaces. Measured fiber fields were also fitted (RMS error less than 17 degrees) with a 60-element, 99-node mesh obtained by subdividing the 24-element mesh. These methods provide a compact and accurate anatomic description of the ventricles suitable for use in finite element stress analysis, simulation of cardiac electrical activation, and other cardiac field modeling problems.

Animals↗
PubMed1994

A mathematical model of sinus node function: validation by recording of sinus node electrograms.

We have applied a mathematical function to model sinus node recovery after atrial pacing at different cycle lengths in 13 autonomically denervated, anesthetized canines. This model provides "pure" indexes of sinoatrial conduction and of suppression of automaticity. We compared sinoatrial conduction computed from the model (SACTc) with sinoatrial conduction measured directly utilizing a catheter recording the sinus node electrogram (SACTd) to test the validity of the model. We used both the model and the sinus node electrogram to further define events after pacing. There was close agreement between SACTc and SACTd with correlation coefficients at pacing cycle lengths of 20 and 50% below resting of 0.86 (P < 0.0001) and 0.78 (P < 0.0001), respectively. We verified that the conventional sinus node recovery time is due both to suppression of automaticity and prolongation of sinoatrial conduction. Sinoatrial conduction time returned rapidly to normal over 1-3 beats, whereas automaticity recovered exponentially over 20-40 postpacing intervals. The model accurately describes postpacing phenomena and has the potential to provide a useful adjunct to the measurement of the sinus node electrogram for assessment of sinus node function in the experimental laboratory.

Animals↗
PubMed1994

Role of active changes in venous capacity by the carotid baroreflex: analysis with a mathematical model.

To elucidate the role of venous capacity active changes in short-term cardiovascular homeostasis, a mathematical model of the carotid-sinus baroreflex system has been developed. In the model the cardiovascular system is represented as the series arrangement of six lumped compartments, which synthesize the fundamental hemodynamic properties of the systemic arterial, systemic venous, pulmonary arterial, and pulmonary venous circulations as well as of the left and right cardiac volumes. Cardiac outputs from the left and right ventricles are computed as a function of both downstream arterial pressure (afterload) and upstream atrial pressure (preload). Four distinct feedback regulatory mechanisms, working on systemic arterial resistance, heart rate, systemic venous unstressed volume, and systemic venous compliance, are assumed to operate on the cardiovascular system in response to carotid sinus pressure changes. All model parameters, both in the cardiovascular system and in feedback regulatory mechanisms, have been assigned on the basis of physiological data now available. The model is used here to simulate the pattern of the main hemodynamic quantities in the short time period (1-2 min) after acute carotid sinus activation in vagotomized subjects. Simulation results indicate that the model can reproduce experimental data quite well, with reference both to open-loop experiments and to an acute blood hemorrhage performed in closed-loop conditions. Moreover, computer simulations indicate that active changes in venous unstressed volume are of primary importance in regulating cardiac output and systemic arterial pressure during activation of the carotid sinus baroreflex.

Animals↗
PubMed2000

Mathematical models of action potentials in the periphery and center of the rabbit sinoatrial node.

Mathematical models of the action potential in the periphery and center of the rabbit sinoatrial (SA) node have been developed on the basis of published experimental data. Simulated action potentials are consistent with those recorded experimentally: the model-generated peripheral action potential has a more negative takeoff potential, faster upstroke, more positive peak value, prominent phase 1 repolarization, greater amplitude, shorter duration, and more negative maximum diastolic potential than the model-generated central action potential. In addition, the model peripheral cell shows faster pacemaking. The models behave qualitatively the same as tissue from the periphery and center of the SA node in response to block of tetrodotoxin-sensitive Na(+) current, L- and T-type Ca(2+) currents, 4-aminopyridine-sensitive transient outward current, rapid and slow delayed rectifying K(+) currents, and hyperpolarization-activated current. A one-dimensional model of a string of SA node tissue, incorporating regional heterogeneity, coupled to a string of atrial tissue has been constructed to simulate the behavior of the intact SA node. In the one-dimensional model, the spontaneous action potential initiated in the center propagates to the periphery at approximately 0.06 m/s and then into the atrial muscle at 0.62 m/s.

Action Potentials↗
PubMed1975

Mathematical and mechanical modeling of stress-strain relationship of pericardium.

Several mathematical expressions (models) were compared for use in describing the stress-strain (sigma - epsilon) relationship of pericardium. The expression sigma = alpha[ebeta epsilon - 1] was preferred because of its simpler form, theoretical consistency, and "good fit" of experimental data. A method was developed for estimating the precisions of the estimates of the parameters alpha and beta. This approach can have general usefulness in assessing the significance of a change in stress-strain relationship of various soft tissues following different interventions. A mechanical model was formulated for the pericardium which consisted of springs representing the collagen and elastin fibers connected in parallel. It could be simulated by the above equation and could describe the behavior of the pericardium.

Elasticity↗
PubMed1979

A two-dimensional transient mathematical model of human thermoregulation.

Current models of human thermoregulation are limited in that they fail to account for temperature distribution in any spatial direction other than radially outward from the body centerline. They are therefore incapable of accounting for nonuniform environmental conditions or nonuniform heat generation from muscles or organs within the body. However, for many situations, these nonuniform conditions are commonplace and lead to disparate skin temperatures and heat loss rates on different sides of the same body compartement. A new mathematical model of human thermoregulation that has been developed and is presented here has the capability of predicting transient temperature variations in two spatial dimensions, both radially and angularly, as measured from the body centerline. In so doing, the model accounts for nonuniform environments and internal heat generation rates. Typical results from the model are demonstrated, and comparisons with available experimental data are also presented.

Blood Circulation↗
PubMed1980

Mathematical model of steroidogenesis in rat and rabbit testes.

The purpose of this study was to develop a mathematical model of testicular steroidogenesis which could not only predict steroid secretion but also could be implemented to test the validity of assumptions used in studies of testicular testosterone biosynthesis in rats and rabbits. Since the two predominant fates of testicular steroids are metabolism and secretion, we hypothesized that the data for construction of the model could be observed testicular steroid secretions and that these data could then be used to elucidate intratesticular steroid conversions. Equations based on steroid secretion by testes perfused in vitro were developed to estimate transition probabilities corresponding to steroid secretion or conversion. The model was tested by comparing the predicted steroid secretion rates with those observed for control testes perfused in vitro. In addition, the transition probabilities determined by the model were used to indicate the predominant series of reactions for converting pregnenolone to testosterone. The results presented herein confirm the capability of the model to predict steroid secretion rates and to predict preferred pathways for testosterone biosynthesis in maximally stimulated testes perfused in vitro. Moreover, the model construction required only algebra and steady-state measurements.

Animals↗
PubMed1983

A mathematics for physiology.

Nonlinear dynamics gives us a language within which to express the basic concepts of physiology. Form, qualitative change, oscillation, stability, and other biologically important notions find natural expressions in this new mathematical approach. An introduction to qualitative dynamics is sketched, its relevance to physiology is argued, and several examples of its usefulness in providing models and metaphors for describing brain activity are offered.

Animals↗
PubMed1986

A limit cycle mathematical model of the REM sleep oscillator system.

A limit cycle mathematical model of the rapid-eye-movement (REM) sleep oscillator system has been developed from a structural model of interaction of populations of REM-on and REM-off neurons. The marked differences in latency, amplitude, and duration of the first REM sleep period seen with circadian variation and depressive pathology are modeled by beginning the REM oscillation at different initial points relative to the final position in the limit cycle. Beginning from a point that is graphically interior to the limit cycle produces a long-latency, short-duration, and less intense first REM period. Beginning from a point graphically exterior to the limit cycle produces a short-latency, long-duration, and more intense first REM period. In the model the determinant of whether the oscillation begins exterior or interior to the limit cycle is the time course of decay of the REM-off population discharge activity at sleep onset. When this time course is made to depend on circadian phase, the model produces a very close match to the empirically observed large shifts between the first and second REM periods in duration (often a 50% change) and intensity and also closely mimics the empirically observed shifts in REM latency as human sleep begins at different circadian phases. Although this variation in limit cycle entry accounts for the major changes in REM sleep over the night, the model also postulates a continuous but small circadian variation (of the order of +/- 5% change in REM parameters) acting throughout the course of a night's sleep. Because the model is derived from actual physiological data, rather than being a purely ad hoc or phenomenological construct, it offers the possibility of direct tests of its postulates through neurobiological studies in animals, by circadian phase-related manipulations of the sleep cycle, and through perturbations of the system in humans by the use of drugs. Indeed, an explicit phase-response curve of the system to cholinergic agonists has been developed; this will permit experimental tests of the model in both animals and humans.

Animals↗
PubMed2005

A region-based mathematical model of the urine concentrating mechanism in the rat outer medulla. I. Formulation and base-case results.

We have developed a highly detailed mathematical model for the urine concentrating mechanism (UCM) of the rat kidney outer medulla (OM). The model simulates preferential interactions among tubules and vessels by representing four concentric regions that are centered on a vascular bundle; tubules and vessels, or fractions thereof, are assigned to anatomically appropriate regions. Model parameters, which are based on the experimental literature, include transepithelial transport properties of short descending limbs inferred from immunohistochemical localization studies. The model equations, which are based on conservation of solutes and water and on standard expressions for transmural transport, were solved to steady state. Model simulations predict significantly differing interstitial NaCl and urea concentrations in adjoining regions. Active NaCl transport from thick ascending limbs (TALs), at rates inferred from the physiological literature, resulted in model osmolality profiles along the OM that are consistent with tissue slice experiments. TAL luminal NaCl concentrations at the corticomedullary boundary are consistent with tubuloglomerular feedback function. The model exhibited solute exchange, cycling, and sequestration patterns (in tubules, vessels, and regions) that are generally consistent with predictions in the physiological literature, including significant urea addition from long ascending vasa recta to inner-stripe short descending limbs. In a companion study (Layton AT and Layton HE. Am J Physiol Renal Physiol 289: F1367-F1381, 2005), the impact of model assumptions, medullary anatomy, and tubular segmentation on the UCM was investigated by means of extensive parameter studies.

Animals↗
PubMed2005

A region-based mathematical model of the urine concentrating mechanism in the rat outer medulla. II. Parameter sensitivity and tubular inhomogeneity.

In a companion study (Layton AT and Layton HE. Am J Physiol Renal Physiol 289: F1346-F1366, 2005), a region-based mathematical model was formulated for the urine concentrating mechanism (UCM) in the outer medulla (OM) of the rat kidney. In the present study, we quantified the sensitivity of that model to several structural assumptions, including the degree of regionalization and the degree of inclusion of short descending limbs (SDLs) in the vascular bundles of the inner stripe (IS). Also, we quantified model sensitivity to several parameters that have not been well characterized in the experimental literature, including boundary conditions, short vasa recta distribution, and ascending vasa recta (AVR) solute permeabilities. These studies indicate that regionalization elevates the osmolality of the fluid delivered into the inner medulla via the collecting ducts; that model predictions are not significantly sensitive to boundary conditions; and that short vasa recta distribution and AVR permeabilities significantly impact concentrating capability. Moreover, we investigated, in the context of the UCM, the functional significance of several aspects of tubular segmentation and heterogeneity: SDL segments in the IS that are likely to be impermeable to water but highly permeable to urea; a prebend segment of SDLs that may be functionally like thick ascending limb (TAL); differing IS and outer stripe Na(+) active transport rates in TAL; and potential active urea secretion into the proximal straight tubules. Model calculations predict that these aspects of tubular of segmentation and heterogeneity generally enhance solute cycling or promote effective UCM function.

Animals↗
PubMed2007

Flow-dependent transport in a mathematical model of rat proximal tubule.

The mathematical model of rat proximal tubule has been extended to include calculation of microvillous torque and to incorporate torque-dependent solute transport in a compliant tubule. The torque calculation follows that of Du Z, Yan Q, Duan Y, Weinbaum S, Weinstein AM, and Wang T (Am J Physiol 290: F289-F296, 2006). In the model calculations, torque-dependent scaling of luminal membrane transporter density [either as an ensemble or just type 3 Na(+)/H(+) exchanger (NHE3) alone] had a relatively small impact on overall Na(+) reabsorption and could produce a lethal derangement of cell volume; coordinated regulation of luminal and peritubular transporters was required to represent the overall impact of luminal flow on Na(+) reabsorption. When the magnitude of torque-dependent Na(+) reabsorption in the model agrees with that observed in mouse proximal tubules, the model tubule shows nearly perfect perfusion-absorption balance at high luminal perfusion rates, but enhanced sensitivity of reabsorption at low flow. With a slightly lower coefficient for torque-sensitive transporter insertion, perfusion-absorption balance in the model tubule is closer to observations in the rat over a broader range of inlet flows. In simulation of hyperglycemia, torque-dependent transport attenuated the diuretic effect and brought the model tubule into closer agreement with experimental observation in the rat. The model was also extended to represent finite rates of hydration and dehydration of CO(2) and H(2)CO(3). With carbonic anhydrase inhibition, torque-dependent transport blunted the diuretic effect and enhanced the shift from paracellular to transcellular NaCl reabsorption. The new features of this model tubule are an important step toward simulation of glomerulotubular balance.

Animals↗
PubMed1988

Mathematical simulation of forced expiration.

Flow limitation during forced expiration is simulated by a mathematical model. This model draws on the pressure-area law obtained in the accompanying paper, and the methods of analysis for one-dimensional flow in collapsible tubes developed by Shapiro (Trans. ASME J. Biomech. Eng. 99: 126-147, 1977). These methods represent an improvement over previous models in that 1) the effects of changing lung volume and of parenchymal-bronchial interdependence are simulated; 2) a more realistic representation of collapsed airways is employed; 3) a solution is obtained mouthward of the flow-limiting site by allowing for a smooth transition from sub- to supercritical flow speeds, then matching mouth pressure by imposing an elastic jump (an abrupt transition from super- to subcritical flow speeds) at the appropriate location; and 4) the effects of levels of effort (or vacuum pressure) in excess of those required to produce incipient flow limitation are examined, including the effects of potential physiological limitation.

Airway Obstruction↗
PubMed1993

A mathematical model of the human ventilatory response to isocapnic hypoxia.

A mathematical model of the ventilatory response to a period of sustained isocapnic hypoxia in humans has been developed. After a step into hypoxia, there is an initial rapid increase in ventilation (on-transient) followed by a slow decline. At the relief of hypoxia, there is a rapid decrease in ventilation (off-transient); the magnitude of this off-transient is smaller than that of the on-transient. Previously, the asymmetry between the on- and off-transients has been dealt with by modeling the steps into and out of hypoxia separately. The current objective was to model the whole of the response by allowing the peripheral sensitivity to hypoxia to decline during the sustained exposure to hypoxia. The model was fitted to breath-by-breath data from 20-min periods of hypoxia (end-tidal oxygen 50 Torr) at two different levels of end-tidal carbon dioxide tension from five subjects. The model was able to describe the features of the ventilatory changes well, including the slow decline and the asymmetry.

Carbon Dioxide↗
PubMed1999

Kinetics of absorption atelectasis during anesthesia: a mathematical model.

Recent computed tomography studies show that inspired gas composition affects the development of anesthesia-related atelectasis. This suggests that gas absorption plays an important role in the genesis of the atelectasis. A mathematical model was developed that combined models of gas exchange from an ideal lung compartment, peripheral gas exchange, and gas uptake from a closed collapsible cavity. It was assumed that, initially, the lung functioned as an ideal lung compartment but that, with induction of anesthesia, the airways to dependent areas of lung closed and these areas of lung behaved as a closed collapsible cavity. The main parameter of interest was the time the unventilated area of lung took to collapse; the effects of preoxygenation and of different inspired gas mixtures during anesthesia were examined. Preoxygenation increased the rate of gas uptake from the unventilated area of lung and was the most important determinant of the time to collapse. Increasing the inspired O2 fraction during anesthesia reduced the time to collapse. Which inert gas (N2 or N2O) was breathed during anesthesia had minimal effect on the time to collapse.

Absorption↗
PubMed2000

Development of a mathematical model that predicts optimal muscle activation patterns by using brief trains.

Because muscles must be repetitively activated during functional electrical stimulation, it is desirable to identify the stimulation pattern that produces the most force. Previous experimental work has shown that the optimal pattern contains an initial high-frequency burst of pulses (i.e., an initial doublet or triplet) followed by a low, constant-frequency portion. Pattern optimization is particularly challenging, because a muscle's contractile characteristics and, therefore, the optimal pattern change under different physiological conditions and are different for each person. This work describes the continued development and testing of a mathematical model that predicts isometric forces from fresh and fatigued muscles in response to brief trains of electrical pulses. By use of this model and an optimization algorithm, stimulation patterns that produced maximum forces from each subject were identified.

Algorithms↗
PubMed1996

A mathematical model of the crustacean stretch receptor neuron. Biomechanics of the receptor muscle, mechanosensitive ion channels, and macrotransducer properties.

1. A mathematical model of the primary transduction process in a mechanoreceptor, the slowly adapting stretch receptor organ of the crayfish, has been developed taking into account the viscoelastic properties of the accessory structures of the receptor, i.e., the receptor muscle, the biophysical properties of the mechanosensitive channels (MSCs) and the passive electrical properties of the neuronal membrane (leak conductance and capacitative properties). The work is part of an effort to identify and characterize the mechanical and ionic mechanisms in a complex mechanoreceptor. The parameters of the model are based mainly on results of our own experiments and to some extent on results from other studies. The performance of the model has been compared with the performance of the slowly adapting receptor. 2. The model resulted in nonlinear differential equations that were solved by an iterative, fourth order Range-Kutta method. For the calculations of potential, the cell was treated as an idealized spherical body. The extension of the receptor muscle was 0-30%, which is within the physiological limits for this receptor. 3. The mechanical properties of the receptor muscle were modeled by a simple Voigt element (a spring in parallel with a dashpot) in series with a nonlinear spring. This element can describe resonably well the tension development in the receptor muscle at least for large extensions (> 12%). However, for small extensions (< 12%), the muscle seems to be more stiff than for large extensions. 4. The receptor current at different extensions of the receptor was computed using typical viscoelastic parameters for a receptor muscle together with a transformation of muscle tension to tension in the neuronal dendrites and finally the properties of the mechanosensitive channels. The model fit was satisfactory in the high extension range whereas in the low extension range the deviation from the experimental results could be explained partly by insufficient modeling of the nonlinear viscoelastic properties. The voltage dependence of the receptor current was also well predicted by the model. 5. If the parameters of the viscoelastic model were adjusted for each extension so that each tension response closely resembled the experimental values, the fit of the current responses was improved but still deviated from the experimental currents. One factor that might explain the difference is the possibility that the MSCs in the stretch receptor neuron might have intrinsic adaptive properties. Introducing an exponential adaptive behavior of individual MSCs increased the ability of the model to predict the receptor current. 6. The receptor potential was calculated by modeling the neuronal membrane by a lumped leak conductance and capacitance The calculated receptor potential was higher than the experimental receptor potential. However, the fit of the receptor potential was improved substantially by introducing an adaptation of the MSCs as outlined in the preceding paragraph. the remaining discrepancy might be explained by insufficient blocking of K+ channels in the experiment. 7. The model can predict a wide range of experimental data from the slowly adapting stretch receptor neuron including the mechanical response of the receptor muscle, the receptor current and its voltage dependence, and the receptor potential. It also describes accurately the passive electrical properties of the neuronal membrane.

Adaptation, Physiological↗