Scott A. Stevens

A mathematical model of idiopathic intracranial hypertension incorporating increased arterial inflow and variable venous outflow collapsibility

OBJECT A collapsible segment in the venous outflow has been noted in many patients with idiopathic intracranial hypertension (IIH). Mathematical modeling has shown that these collapsible segments can account for the elevated cerebrospinal fluid (CSF) pressures associated with IIH. However, the model required an elevated outflow resistance of up to 10 times normal to predict the CSF pressures actually found clinically. Measurement of blood flow in patients with IIH has shown that inflow rates vary, with higher rates noted in patients with lesser outflow stenoses. The aim of this work was to extend a simple model of cerebral hydrodynamics to accommodate a collapsible sinus and elevations in cerebral blood flow in accordance with in vivo measurements. METHODS Forty patients with IIH underwent MR imaging in which the degree of stenosis on MR venography was compared with the total blood inflow by using MR flow quantification. The relative outflow resistance in IIH was estimated using the CSF opening pressure. The patients were compared with 14 age-matched control individuals. RESULTS Patients were divided into 3 groups based on MR venography appearances (minimal stenosis, stenosis of 40-70% and > 70% stenosis). In vivo measurements suggested a relative resistance elevation of 2.5 times normal, 4.2 times normal, and 4.8 times normal in the 3 groups, respectively. There was an increased inflow of 1.56 times normal, 1.28 times normal, and 1.19 times normal in these groups. CONCLUSIONS The model correctly predicted the CSF pressures noted in vivo, suggesting that high arterial inflow is required for patients with low-grade stenoses to be symptomatic.

Modeling intracranial pressures in microgravity: the influence of the blood-brain barrier.

INTRODUCTION A majority of astronauts experience symptoms of headache, vomiting, nausea, lethargy, and gastric discomfort during the first few hours or days after entering a microgravity environment. Due to similarities in symptoms and their time evolution, it has been hypothesized that some of these conflicts are related to the development of benign intracranial hypertension in these individuals in microgravity. METHODS This hypothesis was tested using a validated mathematical model that embeds the intracranial system in whole-body physiology. This model was used to predict steady-state intracranial pressures in response to various cardiovascular stimuli associated with microgravity, including changes in arterial pressure, central venous pressure, and blood colloid osmotic pressure. The model also allowed alterations of the blood-brain barrier due to factors such as gravitational unloading and increased exposure to radiation in space to be considered. RESULTS Simulations predicted that intracranial pressure will increase significantly if, combined with a drop in blood colloid osmotic pressure, there is a reduction in the integrity of the blood-brain barrier in microgravity. DISCUSSION These results suggest that in some otherwise healthy individuals microgravity environments may elevate intracranial pressure to levels associated with benign intracranial hypertension, producing symptoms that can adversely affect crew health and performance.

A Model for Idiopathic Intracranial Hypertension and Associated Pathological ICP Wave-Forms

Idiopathic intracranial hypertension (IIH) is a syndrome of unknown cause characterized by elevated intracranial pressure (ICP). While imaging often reveals a stenosis of the transverse sinuses, the role of this feature in IIH has been in dispute. Many patients with chronic daily headache have been found to actually be suffering from a milder form of IIH without papilledema (IIHWOP). These patients often demonstrate hypertensive B-waves and plateau-like waves upon continuous ICP monitoring. Recently, we presented modeling studies which suggest that the sinus stenosis and hypertension of IIH are physiological manifestations of a stable state of elevated pressures that exists when the transverse sinus is sufficiently collapsible. Many of the features of IIH were explained by this model but the prevalence of pathological ICP wave-forms observed in IIHWOP remained unresolved. The model presented here is a modified version of a previous model with a semi-collapsible sinus represented by a refined downstream Starling-like resistor based on experimental data. The qualitative behavior of this model is presented in terms of the collapsibility of the transverse sinus. For a sufficiently rigid sinus, there is a unique stable state of normal pressures. As the degree of collapsibility increases, there is a Hopf bifurcation, the normal state becomes unstable, low-frequency, high-amplitude ICP waves prevail, and small perturbations can lead to hypertensive ICP spikes. As the collapsibility increases further, so does the duration of the waves, until they are replaced by two stable states: one of normal pressures and one of elevated pressures. In this parameter domain, temporary perturbations can now cause permanent transitions between states. The model presented here retains the capability of our previous model to elucidate many features of IIH and additionally provides insight into the prevalence of the low-frequency, high-amplitude waves observed in IIHWOP.

Mean pressures and flows in the human intracranial system as determined by mathematical simulations of a steady-state infusion test.

Abstract In order to understand the fluid dynamics within the human intracranial system, the relatively small flow of extracellular fluid into, and out of, the interstitial brain tissue must be determined. Due to the magnitude of these flows, it is difficult to measure them clinically. Through a steady-state infusion simulation run on a mathematical model, values for these small flows may be calculated based on clinical data regarding the conductance of cerebrospinal fluid outflow. In this way, the mathematical model allows information to be obtained regarding these small mean flows, as well as the remaining mean flows and mean pressures throughout the Intracranial space, with minimal reliance on data from intrusive procedures. [Neurol Res 2000; 22: 809-814]

A modeling study of idiopathic intracranial hypertension: etiology and diagnosis.

Abstract Objective: To investigate the relationship between idiopathic intracranial hypertension (IIH) and transverse sinus stenosis through experiments performed on a validated mathematical model. Methods: A mathematical model of intracranial pressure (ICP) dynamics has been extended to accommodate venous sinus compression through the introduction of a Starling-like resistor between the sagittal and transverse sinuses. Results: In the absence of this type of resistor, the sinuses are rigid, and the model has only a unique, stable steady state with normal pressures. With resistance a function of the external pressure on the sinus, a second stable steady state may exist. This state is characterized by elevated ICP concurrent with a compressed transverse sinus. Simulations predict that a temporary perturbation that causes a transient elevation of ICP can induce a permanent transition from the normal to the higher steady state. Comparisons to clinical data from IIH patients provide supporting evidence for the validity of the models predictions. Simulations suggest a possible clinical diagnostic technique to determine if an individual has a compressible transverse sinus and is at risk for developing IIH. Conclusions: Results of the model experiments suggest that the primary cause of IIH may be a compressible, as opposed to rigid, transverse sinus, and that the observed stenosis is a necessary characteristic of the elevated pressure state.

A differentiable, periodic function for pulsatile cardiac output based on heart rate and stroke volume

Many mathematical models of human hemodynamics, particularly those which describe pressure and flow pulses throughout the circulatory system, require as specified input a modeling function which describes cardiac output in terms of volume per unit time. To be realistic, this cardiac output function should capture, to the greatest extent possible, all relevant features observed in measured physical data. For model analysis purposes, it is also highly desirable to have a model function that is continuous, differentiable, and periodic. This paper addresses both classes of needs by developing such a function. Physically, the present function provides an accurate model for flow into the ascending aorta. It is completely specified by a minimal number of standard input parameters associated with left ventricle dynamics, including heart rate, mean cardiac output, and an estimation of the peak-to-mean flow ratio. Analytically, it can be expressed as a product of two continuous, differentiable and periodic factors. Further, the Fourier expansion of this model function is shown to be a finite Fourier series, and explicit closed-form expressions are given for the non-zero coefficients in this series.

Kinking of the atrioventricular plane during the cardiac cycle.

Systolic descent of the atrioventricular plane toward the relatively stationary left ventricular apex is well described. As the atrioventricular plane includes two separate valvular units, systolic atrioventricular plane displacement should not be homogenous. In 6 sheep, sonomicrometric crystals were implanted at the base of the right coronary sinus, anterolateral and posteromedial fibrous trigones, posterior mitral annulus, left ventricular apex, and the tips of the anterior and posterior mitral leaflets. The aortomitral angle was calculated and related to simultaneous left ventricular and aortic pressures and mitral valve movement. The aortomitral angle was largest at end diastole (150.73° ± 15.48°). During isovolumic contraction, it narrowed rapidly to 144.90° ± 16.64°, followed by a slower narrowing during ejection until it reached its smallest angle at end systole (139.66° ± 16.78°). During isovolumic relaxation, the aortomitral angle increased to 143.66° ± 16.02° at the beginning of diastole. During the first third of diastole, it narrowed again to 141° ± 16.24° before re-expanding to maximum at end diastole. During systole, the atrioventricular plane descended non-homogeneously toward the apex, with kinking at the hinge between the aortic and mitral annulus plane. This deformation of the atrioventricular plane has relevance in valve surgery.

A mathematical model of the systemic circulatory system with logistically defined nervous system regulatory mechanisms

A mathematical model is developed that accurately describes the pressure, volume and flow dynamics of the systemic circulatory system over the full physiological range of human pressures and volumes. At the heart of this model are mathematical representations for the autonomic and central nervous system reflexes which maintain arterial pressure, cardiac output and cerebral blood flow. These representations involve functions in which a maximum effect and a minimum effect are smoothly connected by a logistic transition. A new approach to modelling the pressure – volume relationship in a vessel with smooth muscle contraction is also presented. To test the model, simulations of cardiac arrest and various haemorrhagic situations were conducted, and predicted results were compared with clinical observations. Near-perfect agreement was obtained between predicted and observed values of the mean circulatory filling pressure, cardiac output and arterial pressure decay in the face of significant haemorrhage, and the critical values delineating progressive from non-progressive hypovolaemic shock.

Modelling the Response of Intracranial Pressure to Microgravity Environments

A majority of astronauts experience symptoms of headache, vomiting, nausea, lethargy, and gastric discomfort during the first few hours or days after entering a microgravity environment. It has been hypothesised that some of these symtoms are related to the development of benign intracranial hypertension as a result of the cephalic fluid shifts and relative venous congestion that occur in microgravity. This hypothesis is tested here using a mathematical model of lumped-parameter type that embeds the intracranial system in whole-body physiology. In addition to considering microgravity environments, this model is used to examine the response of intracranial pressures to head-down tilt (HDT), a ground-based experimental procedure often used to simulate the cardiovascular effects of microgravity. Predicted pressures in these simulations include those in the cerebral vasculature, ventricular and extra-ventricular cerebrospinal fluid (CSF), and the brain tissue extracellular fluid. Various cardiovascular stimuli associated with microgravity, including changes in arterial pressure, central venous pressure, and blood colloid osmotic pressure, are considered both individually and in concert. Small alterations of the blood-brain barrier in space due to factors such as gravitational unloading and increased exposure to radiation are also allowed. Simulation results predict that in a healthy individual the upward fluid shifts and changes in central venous pressure in microgravitycannot, by themselves, produce a large elevation in ICP so long as the blood-brain barrier remains intact. Indeed, in this case the simulations suggest that ICP in microgravity is significantly less than that in long-term HDT, and may even be less than that in the supine position on Earth. However, simulations predict that ICP can increase significantly if, combined with a drop in blood colloid osmotic pressure, there is even a slight reduction in the integrity of the blood-brain barrier. These results suggest that in some otherwise healthy individuals, microgravity environments may elevate ICP to levels associated with benign intracranial hypertension, producing symptoms that can adversely affect crew performance.