Ebook: Mathematical Modelling in Medicine

In the summer of 1997, the BioMath group at IMFUFA, Roskilde University, Denmark arranged the conference, “Mathematical Modelling in Medicine”. The goal of the conference was to present the state of the art on mathematical modelling in physiology and medicine. The speakers were all leading researchers educated in various traditional disciplines including mathematics, physics, engineering, physiology, medicine and cognitive engineering. The conference became a major success and the participants agreed to publish a proceeding of interests for graduate students as well as researchers in this interdisciplinary field.

The result is the current book. It is divided into four distinct parts which cover mathematical models of heart, arterial tree, baroreceptor control and applications for simulators. The mathematical models covering these four topics are contained in a number of articles in each part. In addition, historical reviews on the heart, arterial tree and baroreceptors are also included in the articles offering a broader view and understanding of the current physiological models. The models presented are all based on fundamental physiological principles. This common guideline may result in more solid models from which we can obtain new physiological insights. The articles can be read independently, but it may be an advantage to read some or all of the articles together, due to their close relationship.

In the last decade, we have observed a rapid increase in the popularity and success of mathematical models. This is not solely a consequence of the development and spread of fast computers, making easier access to simulations of complex systems. Another determining factor is the enormous increase in available clinical data. The precise continuous samplings of new data have generated experiments from which we can gain new insights into the dynamics of physiological systems and not only into their steady state behaviour patterns. Yet another important element is the attempts to focus on precise definitions of physiological concepts in order to avoid confusion, misunderstandings and waste of efforts. Most, if not all, concepts can only be clearly defined by use of mathematics. Otherwise obscurity and ambiguity will undoubtedly rise sooner or later. Mathematics may also provide a tool to structure thoughts, an area which have gained an increasing attention. Also, new important questions generated by the application of mathematical models, questions which could not be asked without such models, are believed to play a role in the interest of mathematical modelling. In addition to the importance of this interdisciplinary field in research, mathematical models have received an growing interest as an applied activity in industry, e.g. in the construction and use of simulators in training and education of medical doctors and nurses. Recently, the SIMA group in Denmark has developed a full scale anaesthesia simulator based on mathematical models. This simulator is currently in use at several hospitals in Denmark for the training of anaesthesiologists and anaesthetic nurses. Also, the simulator has become an obligatory part of their education.

Finally, we would like to thank the Moth-Lund foundation and the Danish Natural Science Research Council for supporting the conference, Mathematical Modelling in Medicine, as well as IMFUFA at Roskilde University for supporting this book. Also, we would like to thank the authors for their contributions and for being very co-operative, and our colleagues at IMFUFA, Roskilde University for many fruitful discussions.

JOHNNY T. OTTESEN

ROSKILDE UNIVERSITY

MICHAEL DANIELSEN

ROSKILDE UNIVERSITY

In human perception, the heart was not always part of the blood circulating system. It was later included as a suction pump until Harvey argued that the heart is actually a compression pump, the central organ of the circulation, and the only organ responsible for the motion of blood. Considered initially as an autonomous pump, the heart gradually became viewed as subservient to the needs of the peripheral organs it perfuses. Constant properties assigned to the heart had to be replaced, one after another, by adjustable parameters. Even the adequacy of the heart as the sole pump has been doubted, an issue that resurfaces today.

A recently developed model of the left ventricle, based on experimental data, has been shown to exhibit the main features of the heart's ability to pump. Two special cases during blood ejection, termed pressure deactivation and hyperactivation, were identified. This study proposes an ‘ejection effect’ correction to the model that addresses deactivation, hyperactivation and adjusts the shape of the computed ventricular ejection curve in late systole. Also, a new approach based on new animal experiments is proposed to identify the ejection effect mechanism(s).

A new analytical model of the left ventricle as a pump, developed from isolated canine experiments, was adapted to describe each of the four heart chambers in a complete human circulatory system model. Each chamber is embodied as a volume and time dependent isovolumic pressure source, after Otto Frank's classic experiments. Analytical results show that a small set of equations is sufficient to describe the main features of the heart as a pump, including isovolumic and ejecting beats for a wide range of ventricular and circulation conditions. This model allows interactive teaching of cardiovascular system dynamics. Computed results demonstrate that, with additional experiments, a quantitative description of the human circulation for research purposes may emerge from this approach.

Qualitative similarity is demonstrated between velocity fields predicted by C S. Peskin's mathematical model for blood flow in a plane section through the left side of the mammalian heart and human velocity data measured with MRI-technique. Also, the underlying muscle model for the heart model is investigated with respect to simulated ventricular pressure.

The arterial system can be modelled by means of distributed analogs and by lumped parameter representations, such as the Windkessel. The distributed models, together with biological data, lead to the explanation of arterial function in terms of pressure and flow. The lumped parameter models form the basis of arterial compliance and characteristic impedance estimation. They also make comparison between mammals possible and it could be shown that input impedance, when scaled, is similar in mammals.

The systemic arteries can be modeled as a bifurcating tree of compliant tapering vessels while blood flow and pressure can be predicted by solving Navier-Stokes equations for each of the branches. If all branches are included the computational cost will become prohibitively large. Therefore, the tree must be truncated after a limited number of generations and a suitable outflow boundary condition must be applied. To this end we propose a structured tree in which the root impedance is calculated using a semi-analytical approach. In the structured tree the fluid dynamic equations are linearized giving a wave equation, which can be solved analytically for each vessel. This provides a dynamical boundary condition based on physiological principles which is computationally feasible. It exhibits the actual phase lag between flow and pressure as well as accommodating the wave propagation effects for the entire systemic arterial tree. Finally, the model has been compared with a standard and well established model, where outflow at the terminals are determined by attaching a Windkessel type boundary condition.

Measurement of arterial compliance is of interest in evaluating patients with atherosclerosis and other diseases which affect the vessels.

Arterial compliance is the relation between changes in transmural pressure and volume of an arterial segment, where a high compliance signifies large changes in volume per change in transmural pressure. The relation between changes in transmural pressure and volume is far from linear as compliance increases progressively with decreases in blood pressure.

A change in compliance could indicate static changes in arterial wall composition, i.e. the relation between elastic and collagen fibres and accumulation of disease related deposits or dynamic changes caused by alterations in muscular tone.

The most used method reflecting arterial compliance is the measurement of pulse wave velocity. However, the pulse wave velocity method measures compliance at ambient transmural pressures and is affected both by the actual blood pressure and the rate of pressure change. Another commonly used method employs the echo-tracking technique to measure the arterial diameter simultaneously with continuous blood pressure monitoring. By this method it is possible to calculate arterial compliance for continuous pressure values between the diastole and the systole.

The volume-oscillometry method is based on the fact that the artery can be made to collapse at the end of the diastole by an occlusive cuff while it remains open in a pressure dependent manner during the rest of the cardiac cycle. Changes in the arterial volume is transmitted to the cuff, where it induces a measurable change in pressure, and hence the volume of the artery can be calculated at different values of transmural pressures. Using this method on normal subjects has shown that the arterial compliance decreases with increasing age and that females have lower compliance than males primarily due to a smaller diameter of their arteries. It has also been shown that patients with essential (diastolic) hypertension have compliances which are higher or equal to those of normal subjects, and that patients with systolic hypertension have lower arterial compliances than normal subjects. The former finding is in contrast with pulse wave velocity measurements, where diastolic hypertension was associated with low arterial compliance.

Cardiac output is largely controlled by venous return, the driving force of which is the energy remaining at the postcapillary venous site. This force is influenced by forces acting close to the right atrium, and internally or externally upon the veins along their course. Analogue models of the venous system require at least three elements: a resistor, a capacitor and an inductor, with the latter being of more importance in the venous than in the arterial system. Non-linearities must be considered in pressure/flow relations in the small venules, during venous collapse, or low flow conditions. The venous capacitance is also non-linear, but may be considered linear under certain conditions. The models have to include time varying pressure sources created by respiration and skeletal muscles, and if the description includes the upright position, the partly unidirectional flow through the venous valves has to be considered.

A systematic discussion on linear as well as non-linear compartmental models of the cardiovascular system and its various feedback mechanism is given. Most of the results are independent of explicit functional expressions. The topological structure of the model is essential for the response to a local change in peripheral resistance. Inclusion of Parallel paths versus serial paths gives qualitatively different response. Global asymptotic stability follows from the general theory.

A Mathematical model of the short-term arterial pressure control in humans is presented. It includes a six-compartment description of the vascular system, an elastance variable model of the pulsating heart, two groups of baroreceptors (high-pressure or sinoaortic baroreceptors and low-pressure or cardiopulmonary baroreceptors), the efferent activity in the sympathetic nerves and in the vagus, and the response of four distinct effectors (heart period, systemic peripheral resistance, systemic venous unstressed volume and heart contractility).

Several experimental results reported in the physiological literature can be reproduced with the model quite well. The examples presented in this work include the effect of combined sympathetic and vagal stimulation on heart rate, the baroreflex response to mild and severe acute haemorrhages, and the baroreflex response to ventricular pacing at different rates performed during atrio-ventricular block.

The results suggest that:

i) The sympathetic nerves and the vagus interact linearly in regulating heart period. The apparent negative interaction observed experimentally can be ascribed merely to the hyperbolic relationship which links heart rate to heart period.

ii) The cardiopulmonary baroafferents play a significant role in the control of systemic arterial pressure during mild haemorrhages (lower than 3 – 4% of the overall blood volume). In this range, they may allow arterial pressure to be maintained at its normal level without the intervention of the sinoaortic baroreceptors. In contrast, the sinoartic baroreceptors become the major responsible of the observed cardiovascular adjustments during more severe haemorrhages, when the role of cardiopulmonary baroreceptors becomes progressively exhausted.

iii) The stability margin of the closed-loop system is quite low. Increasing the static gain of the baroreceptors or reducing the rate-dependent component may result in self-sustained oscillations similar to Mayer waves.

A non-linear model of the human cardiovascular regulation system (CVS) has been developed, with the aim of studying the short term regulation process by the autonomic nervous system (ANS). This restricts the time interval under study to range from fractions of seconds to about 30 seconds. As a consequence, the only feedback mechanism taken into account is the baroreceptors loop. Our main guideline has been first to try to capture the main features of the system that have been observed and reported in the relevant publications in physiology - main non-linearities in the elementary components of the system, intrinsic periodicities - while keeping to a minimum the number of model parameters, with the question of identifiability from real data in mind. Also, we have tried to retain as much physiological significance to these parameters as possible, for forthcoming diagnosis purposes, while using non invasive measurements only. At the present time, no venous return has been taken into account but should be included later on. The long term objective of this research is to develop tools for helping in medical diagnosis. This may go through the detection of defaults in this closed loop system, within the framework of control systems theory, hereby using all the known concepts from that field, that have been much used for engineering systems (mechanical, electrical etc.): these include signal processing techniques as well as control algorithms or identification and detection methods.

Parameters based on heart rate variability alone or on both heart rate and blood pressure variability (baroreflex sensitivity) are becoming increasingly clinical relevant. Nevertheless a complete insight in physiological mechanisms underlying these parameters is still lacking. A computer model may help to fill up some of the gaps. We present a model which consists of a simple beat-to-beat hemodynamic part (Starling heart and Windkessel) linked to a detailed continuous modelled neural control part. The intermediate between continuous and beat-to-beat part is an integral pulse frequency modulator (IPFM) acting as cardiac pacemaker. Input for the IPFM is a “sympathovagal” balance signal, with different dynamics for sympathetic and vagal branches. Low-frequency variability is supposed to arise from resonance of existing noise, while high-frequency variability (respiration) is assumed to enter the closed loop at the hemodynamic (blood pressure) site. Results of three studies have been used for validation: (1) spontaneous variability in heart rate and blood pressure (baroreflex transfer functions), (2) vagal blockade with atropine, (3) a modified Valsalva manoeuvre performed in normal and quadriplegic man. Steady state as well as dynamic properties of the model reasonably well fitted to these experimental data.

The paper presents as an example the types of mathematical models that are used in a full scale anesthesia simulator. The models vary from detailed reference models, based on fundamental physiological laws and principles, to simple script based models. A few considerations on the implementation of the models are also given.

Empirical investigations in medical ethics are often questionnaire studies relying on the respondents' written responses about their attitudes and actions. This approach is valuable, but there may be quite large discrepancies between responses and actual behaviour. Unfortunately actual behaviour is often difficult to observe, especially where sensitive or very time dependent clinical decisions are concerned.

We have tried to circumvent this problem by using a micro-world computer simulation of a cardiac care unit to study treatment decisions in “life or death” situations. The task facing the physician in this simulation is to be in charge of a 6-bed cardiac care unit for a period of 14 “days” controlling admissions, discharges, and treatment of all patients. The simulation dynamically generates the disease states of the patients in the unit including physical findings, laboratory findings, ECG, X-ray descriptions etc., as well as the patients responses to treatment. The simulation also presents new patients from the emergency room, thereby forcing the physician to decide whether the new patient should be admitted, and whether one of the present patients should be discharged or moved to another ward if no bed is free.

The paper describes this simulator, the mathematical model underlying the simulation of patients with acute myocardial infarction, and the results of the initial studies using the simulator.