Modelling and control of the human cardiovascular system

Transcription

1 Research Collection Master Thesis Modelling and control of the human cardiovascular system Author(s): Gisler, Stefan Publication Date: 2011 Permanent Link: Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library

2 Master Thesis Modelling and control of the human cardiovascular system Stefan Gisler Advisers Martin Wieser and Dr. Heike Vallery and Prof. Dr. Robert Riener Sensory Motor Systems Lab (SMS) Swiss Federal Institute of Technology Zurich (ETH) Submission: April 2011

3 Contents 1 Introduction 1 2 Human cardiovascular system Hemodynamic system Blood pressure regulation Orthostatic reaction and muscle pump Cardiovascular pathology Literature review Cardiovascular responses to passive tilting Cardiovascular modelling Cardiovascular model Hemodynamic system Blood pressure regulation Influence of gravity Influence of stepping Model simulations Fast tilt-up and tilt-down Stepping Quasi-static Model validation Control design Model predictive control (MPC) design Simulation Methods Healthy subjects Implementation Blood pressure recording Experimental design i

4 5.2 Patients Implementation Blood pressure recording Experimental design Results Healthy subjects Heart rate control Blood pressure control Combined heart rate and blood pressure control Patients Controller performance Discussion 61 8 Conclusion and Outlook 65 A Model summary 73 A.1 List of variables A.2 List of parameters A.3 Model equations in non-linear state-space form A.4 Steady-state equations in non-linear state-space form..... A.5 Parameter identification A.6 Model constraints B Summarised results 87 References 95 ii

5 Abstract Bed-rest leads to cardiovascular deconditioning and may induce a decline in stroke volume, cardiac output and oxygen uptake. Further, it increases the risk of orthostatic intolerance. In an early phase of rehabilitation, it is therefore important to prevent the development of cardiovascular deconditioning which can be done by verticalisation and mobilisation. In the future, the enhanced ERIGO tilt-table will be able to control physiological signals and hence, stabilise the patient s cardiovascular system. This thesis focuses on the control of heart rate and blood pressure by means of verticalisation (tilting) and mobilisation (stepping). In a first step, a cardiovascular non-linear model with two inputs (tilting and stepping) and three outputs (heart rate, systolic and diastolic blood pressure) is developed based on physiological principles and existing work. The model is then used for designing a model predictive controller which was found well suited for the given control problem. Five healthy subjects have been tested with three different configurations: isolated heart rate control, isolated blood pressure control and combined control. One patient has been tested with blood pressure control which yielded promising results. Keywords Orthostatic intolerance, cardiovascular modelling, model predictive control iii

6 Acknowledgements First, I want to thank Prof. Dr. Riener for being accepted to do this thesis at the Sensory-Motor Systems Lab. Then I want to thank my advisers Martin Wieser and Dr. Heike Vallery for their valuable support during the work. Special thanks go to Martin Wieser for his great efforts while testing and debugging the system. This thesis would not have been possible without the probands and patients. A big thanks goes to all the probands, the Zürcher Höhenklinik in Wald, and all the patients that participated in this study. At this point, I also want to thank Rafael Rüst and Lilith Bütler for their support during the patient measurements in Wald. Last but not least, I want to thank all the students in the student room for the nice and inspiring atmosphere. iv

7 Chapter 1 Introduction One major problem with neurological patients suffering from stroke, traumatic brain injury or paraplegia is the long bed rest after the accident. It leads to deconditioning of the patients cardiovascular system and evokes secondary complications such as orthostatic intolerance. Further complications can include venous thrombosis, muscle atrophy, joint contractures and osteoporosis [1], [2]. Therefore, early mobilisation of the patient is crucial as it can reduce the risk of cardiovascular deconditiong and improves the state of health. This thesis focuses on the cardiovascular aspects of bed-ridden patients, i.e. how the cardiovascular system can be prevented from deconditioning and becoming unstable. Prolonged bed rest leads to a decrease in circulating blood volume, a decrease in stroke volume and pulse pressure, and an increased heart rate. A direct result of these indications is the inability of the patient s cardiovascular system to regulate blood pressure when standing up (orthostatic intolerance). In the upright position, the patient suddenly starts to feel dizzy or even faints due to excessive blood pooling in the lower extremities and reduced blood perfusion of the upper body. However, orthostatic intolerance is not only caused by prolonged bedrest but can also be a consequence of an impaired vegetative nervous system. In paraplegia patients, the sympathetic effector nerves to the heart and the smooth musculature are disrupted or even broken. This leads to a malfunction of the baroreflex which is responsible for regulating arterial blood pressure (see chapters 2.2, 2.4). As a consequence, the sudden decrease in arterial blood pressure cannot be regulated and the patient faints. A tilt-table therapy is aimed at reconditioning the patient s cardiovascular system by verticalising to an angle of about degrees. Additional leg mo- 1

8 vements which can include stepping or cycling movements increase venous return due to the effects of the muscle pump and improve orthostatic tolerance. The ERIGO device which has been used at the institute since the beginning of the AwaCon project combines these therapies and allows for an optimal treatment of patients with neurological disorders (Figure 1.1). More information about the ERIGO device can be found on the homepage of HO- COMA AG 1. On the ERIGO, physiological signals such as blood pressure, Figure 1.1: Left: Schematic representation of the ERIGO device with the three inputs. Right: ERIGO during therapy session. heart rate, respiration frequency, skin conductance, oxygen saturation, EEG and EMG can be recorded. However, for this thesis only blood pressure and heart rate need to be recorded, where EMG recordings may be helpful to analyse muscle activity during mobilisation. The goal of the project is to control and stabilise the cardiovascular system of patients with neurological disorders by verticalisation, mobilisation and cyclic loading of the lower limbs (Figure 1.1). This will help to improve the cardiovascular status of these patients and will have the potential to reduce medication, enhance physiotherapy and shorten the duration of early rehabilitation [3]. Furthermore, the risk of deconditioning of the cardiovascular system, and complications resulting from this, can be decreased. Additional project information is available on the homepage of the SMS Lab 2. In earlier projects at the SMS, isolated control of heart rate and diastolic

9 blood pressure with the inclination angle α as the only control input has been done [4], [5]. In a next step, combined control of heart rate and diastolic blood pressure has been succesfully tested with healthy subjects [6]. This latest version also contained another technical innovation: the idea was to not only use α as a control input, but also the stepping frequency f step which enables the controller to operate over an enlarged bandwith. For this project, the described line of innovation is continued: the goal of this thesis is to control heart rate, systolic and diastolic blood pressure with the two control inputs α and f step. It is a fact, that the control strategy from the isolated control problem, which consisted of an ordinary PI controller can not be adopted for the new more complex control problem. The challenge is that with an increasing number of inputs and outputs, there are more couplings inside the system and PI control is not suitable anymore. For a multi-input multi-output (MIMO) system, other control strategies have to be applied. The first step consists of developing a cardiovascular model which is the topic of chapter 3 which directly follows after the subsequent chapter about human cardiovascular physiology (chapter 2). Chapter 4 continues with the control design, followed by the results, the discussion and the conclusion (chapters 6, 7 and 8). 3

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11 Chapter 2 Human cardiovascular system This chapter will give a short introduction to physiology and pathophysiology of the human cardiovascular system and summarises some results from literature. 2.1 Hemodynamic system Head Right lung Left lung Right heart Left heart Splanchnic & renal circulation Figure 2.1: Schematic representation of the human circulatory system. Adapted from: The major task of the hemodynamic system is to supply every single cell 5 Legs

12 of the organsim with oxygen and nutrients and carry away carbon dioxide (CO 2 ) as well as metabolic waste products. In the circulation, the heart acts as a pump which produces a pressure gradient between arterial and venous circulation. Driven by this pressure gradient, deoxygenated blood from the venous circulation flows back to the right heart where it is pumped through the lung. In the lung the blood is enriched with oxygen and reenters systemic circulation when pumped into the aorta by the left ventricle. The arterial tree then supplies the whole body with oxygen and nutrients. From the peripheral regions, where the oxygen and the nutrients are used, the blood returns to the right heart and the circulation is closed (Figure 2.1). Flows and pressures within the human hemodynamic system are characterised by the following list of hemodynamic variables: Stroke volume (SV ) defines the amount of blood pumped into the aorta within one beat. Cardiac output (CO) is calculated as the product of stroke volume and heart rate (HR) CO = SV HR Systolic blood pressure (sbp ) is the maximal blood pressure that occurs during the contracting heart phase (systole). Diastolic blood pressure (dbp ) is the minimal blood pressure that occurs during the filling period of the heart, when the ventricles are relaxed (diastole). Mean arterial pressure (MAP ) is defined as the integrated blood pressure over one heart period divided by the time of one heart period. MAP = t+trr t BP (t)dt t RR where BP (t) is the continuous blood pressure and t RR is the time of one heart period (R-R interval). A common approximation is given as MAP = 1 3 sbp dbp Central venous pressure (CV P ) is the pressure in the intrathoracic veins and the right atrium. Normal values range from 2 to 4 mmhg [7]. 6

13 Total peripheral resistance (T P R) is a rather hypothetic measure of vessel resistance in the systemic circulation. In duality to Ohm s law U = R I, total peripheral resistance is defined as T P R = MAP CV P CO CV P is usually neglected in this calculation and we get T P R = MAP CO. 2.2 Blood pressure regulation Regulation mechanisms in the cardiovascular system are responsible for adapting the hemodynamic variables such as blood pressure and cardiac output according to the body needs. During exercise for example, cardiac output is strongly increased in order to cover the high oxygen need in the skeletal muscles. Another situation where these regulation mechanisms are active is when the body adapts to changes in environmental conditions such as temperature differences. And last, these mechanisms are also active in response to orthostatic stress what will be of interest for blood pressure and heart rate control on the ERIGO. Hemodynamic variables can be influenced in several ways. Natural control mechanisms include neurogenic (over the vegetative nervous sytem), hormonal (over circulating hormones), humoral (with locally formed substances) or myogenic regulation (vasoconstriction with smooth musculature). For short-term regulation the neurogenic mechanisms which include baroreflex, cardiopulmonary reflex and chemoreceptor reflex are most important. These three types of neurogenic blood pressure regulation will now be described in more detail: The baroreflex plays a central role in short-term blood pressure regulation. The baroreceptors which are located in the aortic arch and the carotid sinus are the sensors in this reflex mechanism. They transmit neural signals to the central nervous system or more precisely to the cardiovascular centre in the medulla oblongata. The impulse frequency of the afferent neurons is determined by the course of the arterial blood pressure: Low arterial blood pressure leads to a high impulse frequency. However, impulse frequency is not only determined by absolute value of the arterial blood pressure but also by its time rate of change. This proportional-derivative (PD) sensor characteristics enable the baroreceptors to send all relevant information about heart function to the central nervous system. In the medulla oblongata the information from the baroreceptors is transmitted to the efferent vegetative nervous system which determines heart rate, heart contractility and vasoconstriction of 7

14 peripheral blood vessels, closing the reflex arch. It has to be added that an inhibitory interneuron in the medulla provokes negative feedback which is essential for regulating and stabilising arterial blood pressure. Head Baroreceptors Cardiovascular centre Right lung Left lung Right heart Left heart Regulate heart rate & cardiac contractility Baroreflex Splanchnic & renal circulation Regulate peripheral resistance Legs Figure 2.2: Blood pressure regulation with the baroreflex loop. The cardiopulmonary reflex is another blood pressure regulating mechanism that works synergistically with the baroreflex. The cardiopulmonary receptors are located in the venous system, more precisely in the atria and A. pulmonalis. However, cardiopulmonary receptors are not only responsible for blood pressure regulation but also for volume regulation. Stimulation of the receptors by dilated atria leads to an inhibited production of the antidiuretic hormone (ADH). As a consequence, urine secretion is increased and the circulating blood volume can be reduced. Furthermore, activation of the cardiopulmonary receptors decreases sympathetic activity and inhibits Renin production in the kidneys. Renin promotes the formation of Angiotensin II which has a direct vasoconstrictive effect on the smooth musculature in the vessels. Moreover, Angiotensin II stimulates the production of Aldosterone in the kidneys which increases reabsorption of sodium and water. In the long term, this leads to a higher blood volume and an increased blood pressure. Hence, the Renin-Angiotensin-Aldosterone system (RAAS) is capable of increasing arterial blood pressure by the vasoconstrictive effect of Angiotensin II and the volume retention caused by Aldosterone. Note that volume regulation is a long-term regulation because it includes hormonal mechanisms 8

15 and because it takes some time until body fluids have diffused through the capillary walls. The chemoreceptor reflex is mainly responsible for respiration control, but can also influence cardiovascular regulation if partial pressure of oxygen in the blood decreases [7]. The reflex mechanism particularly becomes active if blood pressure falls below mmhg and once active, it acts in the same feedback structure as the baroreflex. As a result, arterial blood pressure is increased. In detail, myogenic regulation is also a kind of neural regulation mechanism if we consider the sympathetic effected vasoconstriction in the peripheral arterioles. However, there is also a mechanism called autoregulation that is attributed to myogenic regulation. Autoregulation is the ability of a blood vessel to keep the blood flow constant under changing perfusion pressures. When perfusion pressures are increased, the smooth musculature is activated and prohibits further expansion of the vessel walls (myogenic reaction: Bayliss effect). 2.3 Orthostatic reaction and muscle pump Everybody knows the dizzy feeling after standing up too fast in the morning. The body s internal regulation mechanisms are strongly challenged in such situations. Normally, the neural regulation mechanisms as discussed above are able to maintain homeostasis quite fast. Nevertheless, there might be situations where the regulation is incapable of keeping arterial blood pressure and cerebral perfusion at a safe level. Low blood volume or high temperatures for example are conditions that increase the risk of defective homeostasis. This can lead to a syncope which can be rather dangerous when the fainting person falls down on the floor or hits a hard object. One cause of such a syncope is the venous blood pooling in the legs. In a healthy person, up to half a litre of blood is shifted from the upper body to the lower extremities [7]. Arterial blood pressure falls immediately and the reflex mechanisms are activated. However, peripheral vasoconstriction caused by sympathetic regulation is usually too weak in order to lower venous blood pooling effectively. Fortunately, there is another mechanism besides the neural regulation which is capable of stabilising the cardiovascular system. The principle behind this mechansim is that the contraction of the skeletal leg muscles efficiently compresses the venous compartments, decreases venous pooling and increases venous return to the heart. Because the venous valves are closed, backflow is not possible and the blood is forced to return back to the heart (figure 2.3). This muscle pump is always active when the skeletal 9

16 leg musculature is active, for example during walking. Figure 2.3: The muscle pump mechanism stabilises the cardiovascular system by efficiently reducing venous blood pooling and increasing venous return by repeated contractions of the skeletal leg musculature. Source: University of Minnesota 2.4 Cardiovascular pathology Cardiovascular instability and orthostatic hypotension are common deficits in bed-ridden patients [1], [2]. In spinal cord injury (SCI) patients, for example, one reason for these deficits are the disrupted efferent sympathetic pathways regulating heart rate, heart contractility and peripheral vasoconstriction. Therefore, neural regulation mechanisms can not work properly and blood pressure often drops dramatically in reaction to orthostatic stress. The disturbed balance between sympathicus and parasympathicus leads to an exaggerated increase in heart rate as a compensatory reaction to the blood pressure decrease. This happens because parasympathetic heart rate regulation is still intact in SCI patients as efferent parasympathetic nerves are connected to the Vagus nerve and not to the spine. Naturally, sympathetic nervous system disfunction is not the only reason for orthostatic intolerance in neurological patients. Claydon et al. [8] summarise these factors for SCI patients as follows: 10

17 Sympathetic nervous system disfunction Altered baroreceptor sensitivity Lack of skeletal muscle pump Cardiovascular deconditioning Altered salt and water balance Baroreceptor sensitivity which is typically reduced in SCI patients is in tight connection with the sympathetic nervous system disfunction. As explained above, baroreflex regulation is severly damaged because of an impaired sympathetic nervous system. The lack of the skeletal muscle pump together with immobilisation and prolonged bed-rest are the reason for cardiovascular deconditioning which in turn negatively affects the overall recovery. Lastly, Claydon et al. report evidence that SCI patients have a decreased plasma volume as a result of an impaired salt and water balance. This leads to problems in volume regulation, i.e. hypovolemia and low resting blood pressure with a predisposition to orthostatic intolerance. 2.5 Literature review Cardiovascular responses to passive tilting Passive tilting leads to an immediate increase of blood volume in the leg veins of about half a liter [7]. Venous return is decreased and because of the Frank-Starling mechanism stroke volume and pulse pressure are diminished as well. To counter the blood pressure drop, neural reflexes are instantly activated and sympathetic action is increased. This has two consequences: Firstly, heart rate rises by approximately 20 % [7] and secondly, diastolic blood pressure rises because of increased peripheral resistance. In contrast, systolic blood pressure is normally rather constant [7], [9]. The above description is considered the healthy cardiovascular response to passive tilting according to standard physiological work of reference such as [7]. Table 2.1 lists the outcome of several studies about cardiovascular responses to passive tilting involving healthy subjects. Note that most of these experimental results conform with the standard physiological response. As the aim of the thesis and the whole project is to enhance therapy of neurological patients, a quick survey of typical pathophysiological cardiovascular 11

18 year HR sbp dbp MAP Hainsworth and Al-Shamma [10] 1988 Mukai et al. [11] 1995 Tanaka et al. [12] 1996 Cooke et al. [13] 1999 Yokoi and Aoki [14] 1999 Petersen et al. [15] 2000 Tulppo et al. [16] 2000 Toska and Walloe [17] 2002 n/a n/a Heldt et al. [18, 19] 2003/04 Colombo et al. [20] 2005 n/a Masuki et al. [21, 22] 2007 Chi et al. [23] 2008 Ramirez et al. [24] 2008 Table 2.1: Literature summary. means no significant change, means significant increase, means significant decrease. (Adapted and completed with HR from [5]) responses will be done. Table 2.2 presents standard cardiovascular responses of SCI patients. All of these studies conform with the normal pathological reaction to orthostatic stress in SCI patients as described in section 2.4. In addition, on the basis of the work of Houtman [25] and Legramante [26] it can be stated that the higher the lesion the bigger are the implications on the cardiovascular system and the cardiovascular regulation. year aetiology HR sbp dbp MAP Corbett et al. [27] 1971 Tetrapl. Houtman et al. [25] 2000 Normal n/a n/a Parapl. n/a n/a Tetrapl. n/a n/a Legramante et al. [26] 2001 Normal Parapl. Tetrapl. Table 2.2: Literature summary. means no significant change, means significant increase and means significant decrease 12

19 2.5.2 Cardiovascular modelling Computational models of the human cardiovascular system have been developed for many different purposes. An elaborate cardiovascular model can be used to identify aetiologies of cardiovascular diseases such as orthostatic intolerance (OI). Heldt et al. [28] have presented a complex mathematical model which reproduces cardiovascular responses to orthostatic stress. In their study the model was used to investigate the mechanisms that cause postspaceflight OI. Leaning et al. [29] formulated a detailed model intended to study and predict the overall effects of an injected drug. However, a cardiovascular model can also be used to examine specific aspects of the cardiovascular system such as blood-pressure fluctuations and heart-rate variability [30], [31]. Most of these models are aimed at explaining a certain cardiovascular phenomenon and are usually rather complex with a high model order. They are normally based on a large number of compartments representing the different parts of the circulation (heart chambers, ventricles, venous and arterial segments). Each compartment or reservoir has a certain pressure P j and volume V j P j = V j V j0 C j (2.1) where C j is the compliance and V j0 the unstressed or zero-pressure volume. Most models that describe the overall cardiovascular system incorporate some elements of nervous system regulation. The baroreflex plays an essential role because it governs the short-term dynamics of blood pressure and heart rate. Long-term dynamics are most often less important than short-term effects and can be neglected in the model description. Therefore, blood pressure regulation mechanisms such as RAAS do not need to be modelled. There are hardly any cardiovascular models in literature which incorporate an orthostatic component and are kept simple. One exception is in the work of Akkerman [32] who presented a mathematical beat-to-beat model designed for tilt-table experiments. He analysed the dynamics of cardiovascular signals after fast tilt-up and tilt-down. The model forms the basis of the whole controller design and will be explained in detail in the following section. 13

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21 Chapter 3 Cardiovascular model 1 In order to control physiological quantities such as blood pressure and heart rate, an appropriate model of the cardiovascular system is needed. This model should have two inputs, namely the inclination angle α of the ERIGO device and the stepping frequency f step. Based on these inputs the model should output heart rate, systolic and diastolic blood pressure (Figure 3.1). α u = fstep y HR = PS P D Cardiovascular model Figure 3.1: Inputs and outputs of the cardiovascular model In order to develop a mathematical model of the human cardiovascular system for blood pressure regulation, the following two assumptions are made: For the internal blood pressure regulation, only the baroreflex is taken into account. Other mechanisms such as the cardiopulmonary reflex and the RAAS system are not needed to explain the main blood pressure characteristics in tilt-up and tilt-down because they govern the long-term dynamics. 1 The material in this chapter is closely related to Akkerman s work [32] 15

22 The blood volume is constant and fluid movements through the capillary walls are not considered. So the model only contains the most important elements that are needed to simulate orthostatic reactions, namely a closed hemodynamic system, a kind of internal blood pressure regulation system and the influence of gravity and stepping (Figure 3.2). These parts will be explained in detail in the following sections. 3.1 Hemodynamic system The hemodynamic system as explained in section 2.1 can be modelled as a connected system of pipes representing blood vessels. The heart acts as a pump, maintains systemic blood pressure and transports oxygen-poor blood to the lung. The microcirculation in the peripheral parts of the body is the bottleneck in the pipe system and is therefore also called the peripheral resistance. The fact that blood vessels are not stiff tubes but compliant vessels is accounted for by introducing a venous and an arterial reservoir which is common engineering practice. In fact, the flattening effect that arterial compliance has on the systolic blood pressure peaks is called the Windkessel effect which is in accordance with the above mentioned engineering principle of introducing reservoirs for the modelling of compliant tubes. In Akkerman s model, only the lung, the arteries and the veins are modelled as proper compartments as defined by Equation 2.1. The volumes of these compartments are denoted by V P, V A and V V respectively. In addition, each of these compartments is attributed a compliance (C P, C A and C V ) and a zero-pressure volume (V P 0, V A0 and V V 0 ). According to Equation 2.1 the compartment pressures at heart beat k + 1 can then be expressed as: P R (k + 1) = V V (k) V V 0 C V (3.1) P L (k + 1) = V P (k) V P 0 C P (3.2) P D (k + 1) = V A(k) V A0 C A (3.3) where P R (k) is the right atrial pressure, P L (k) the left atrial pressure and P D (k) the diastolic blood pressure which directly depends on the arterial blood volume. The flow between these three reservoirs is characterised by the following set of equations where V P P (k) describes the volume in the pulmonary pipeline 16

23 Head Lung V, C P P Baroreceptors B Cardiovascular centre P R Q R Right heart Windkessel P, Q L L Left heart P, S P VV, PV, CV V, C A A D I Baroreflex Venous reservoir QW R Peripheral circulation Figure 3.2: Simplified representation of the human cardiovascular system used for model synthesis. Adapted from [32] which is needed to model the delay between right and left atrium. ξ P denotes the number of right stroke volumes that are in the pulmonary pipeline. V P (k) = V P (k 1) + Q R (k ξ P ) Q L (k) (3.4) V A (k) = V A (k 1) + Q L (k) Q W (k) (3.5) V V (k) = V V (k 1) + Q W (k) Q R (k) (3.6) V P P (k) = V P P (k 1) + Q R (k) Q R (k ξ P ) (3.7) Based on the Frank-Starling law and the restitution properties of ventricular myocardium, the left and right stroke volumes Q L (k) and Q R (k) depend on the preload and the length of the previous R-R interval I(k 1). Akkerman 17

24 adapted these findings from [33]: Q R (k) = γ R P R (k)i(k 1) (3.8) Q L (k) = γ L P L (k)i(k 1) (3.9) where γ R and γ L are constant factors called Starling factors. The peripheral flow Q W (k) depends on the peripheral resistance R(k) and the pressure difference between the arterial and the venous segment. ( Q W (k) = C A (P S (k) P V (k)) 1 exp I(k) ) (3.10) R(k)C A where P V (k) denotes venous pressure evaluated just after systole when the right stroke volume has been ejected into the pulmonary pipeline: P V (k) = V V (k 1) V V 0 Q R (k) C V (3.11) The equations for pulse pressure P P (k) and systolic blood pressure P S (k) complete the hemodynamic system: P P (k) = Q L(k) C A (3.12) P S (k) = P D (k) + P P (k) (3.13) All the introduced variables are beat-to-beat variables which means that they are updated at each heart beat. It is not clear, however, at which instant of the heart beat these variables are refreshed. The systolic blood pressure P S (k) for example is updated during the systole when the continuous blood pressure curve peaks at its maximum value. In contrast, the diastolic blood pressure P D (k) is updated at the end of the diastole. Each hemodynamic variable has its natural physiological sampling instant. Another example are the right and the left stroke volumes Q R (k) and Q L (k). These variables are updated at the beginning of the systole when the stroke volumes are ejected into the pulmonary pipeline and the aorta respectively. Figure 3.3 graphically summarises the sampling instants of the introduced hemodynamic variables. 3.2 Blood pressure regulation Blood pressure regulation is performed by different body mechanisms. There are short-term regulations (minutes, hours) and long-term regulations (days, 18

25 Figure 3.3: Hemodynamic timetable describing at which moment of the heart beat each hemodynamic variable is evaluated. Source: Akkerman [32] weeks 2 ) as mentioned above. For the purpose of blood pressure and heart rate control, only the short-term regulations have to be considered. Therefore, the modeling will focus on the baroreflex mechanism. Katona et al. [34] have developed a baroreflex model which is composed of a sympathetic and a parasympathetic branch (Figure 3.4) which is widely used in computational modelling of the human cardiovascular system. Based on a hypothetic barosignal which is a function of arterial blood pressure and pulse pressure, the model outputs the heart period. The model is split in two parts because sympathetic and parasympathetic dynamics are rather different. Parasympathetic activity leads to a fast decrease of heart rate which can be shown by electrical stimulation of the Vagus nerve. In contrast, the sympathetic contribution on heart rate is slower. In Katona s model there is a fixed boundary between sympathetic and parasympathetic regulation. Of course, in reality there is a smooth transition between these two types of blood pressure regulation. However, it is not needed to map this behaviour to the model and this intuitive simplification is very well applicable. It is even the case, that for applications where these subtle dynamics are of minor importance, Katona s baroreflex model can be further simplified. In this thesis, the two branches are merged to one neglecting the different dynamics of sympathetic and parasympathetic regulation. More important is the extension of the model by htm 2 Time specifications: 19

26 Figure 3.4: Katona s baroreflex model for heart rate regulation [34]. The neural input signal f(t) is divided in a sympathetic (bottom) and parasympathetic part (top) where µ defines the borderline between sympathetic and parasympathetic regulation. a branch for regulation of the peripheral resistance as proposed by Akkerman [32]. This regulation is based on the sympathetic part of the hypothetic neural barosignal and the implementation is straightforward (Figure 3.5). Put in equations, the baroreflex model can be stated as follows: B PT1 B c + + β scaling I out B c in B S PT1 B c - + ρ R P scaling + + R Figure 3.5: Simplified baroreflex model (based on Katona [34] and Akkerman [32]): the two branches regulating heart rate have been merged to one, a second branch has been added for regulation of peripheral resistance. 20

27 B(k) = P B (k) + k P P P (k) k B (3.14) = 1 3 P S(k) P D(k) σ B sin α(k) + k P P P (k) k B (3.15) B S (k) = min(b(k), B c ) (3.16) ) (1 e 1 τ BI B(k) (3.17) J BI (k) = e 1 τ BI J BI (k 1) + J BR (k) = e 1 τ BR J BR (k 1) + (1 e 1 τ BR ) B S (k) (3.18) I(k) = (J BI (k) + B c )β (3.19) R(k) = (B c J BR (k))ρ + R P (3.20) Please note, that this baroreflex model is a simplification of Akkerman s model. Please refer to Akkerman [32] for the original work. 3.3 Influence of gravity The modelling of the orthostatic component describes how the angle α of the tilt-table influences the cardiovascular system and the physiological variables. We are only interested in the gravity component F g along the body axis which is F g = g sin α. (3.21) Based on that, the first model input u 1 (k) can be stated as follows: u 1 (k) = sin α (3.22) Gravity acts on every single blood vessel in the cardiovascular system and creates rather large hydrostatic pressure differences in a standing human. Arterial pressure is decreased by 25 mmhg at head level and increased by 95 mmhg at leg level [7]. The question arises how gravitational forces can be integrated into the existing hemodynamic model. It is chosen to let gravity directly affect the right and left atrial pressures P R (k) and P L (k) which is a mathematically convenient alternative to modelling the whole hydrostatic column [32]. The atrial pressures then depend on the gravity factors ζ R (k) and ζ L (k): P R (k) = (V V (k 1) V V 0 )ζ R (k) C V (3.23) P L (k) = (V P (k 1) V P 0 )ζ L (k) C P (3.24) ζ R (k) = 1 σ R sin α(k) (3.25) ζ L (k) = 1 + σ L sin α(k) (3.26) 21

28 Besides the atrial pressures, also the mean arterial pressure at the level of the baroreceptors P B (k) has to be corrected for the gravity influence. The reason is the height difference between the baroreceptors and the heart, where arterial pressure is evaluated. P B (k) = 1 3 P S(k) P D(k) σ B sin α(k) (3.27) 3.4 Influence of stepping Akkerman s model does not contain a component which describes the effects of the muscle pump when the stepping mechanism is activated. Therefore, these effects were analysed and subsequently added to the model. The stepping mechanism acts on the cardiovascular system by activating the muscle pump through continuous leg movements. This has the following three immediate effects: Compression of the venous leg compartments leads to an increase of peripheral resistance. The contracting skeletal muscles decrease expandability of the venous vessels and hence, venous compliance is decreased. The muscle pump alters the functionality of the baroreflex mechanism. Similar to the situation of exercise, a resetting takes place and the hypothetic pressure level at which neural regulation is switched from parasympathetic to sympathetic action is increased. Although the stepping mechanism moves the legs passively and we can only speak of a passive muscle pump, the stabilising effects on the cardiovascular system are still present, although diminished. Czell et al. [35] have concluded after their pilot study with healthy adults, that passive leg movements stabilises blood circulation and prevents from syncopes. So fortunately, the stabilising effects on the cardiovascular system are still there and can be exploited in the early rehabilitation process of neurological patients. The above listed effects are transformed to mathematical equations so that they can take influence on the existing cardiovascular model of Akkerman. As stepping is the second input after the inclination angle, u 2 (k) will be the expression for the normalised stepping frequency: u 2 (k) = f step(k) f step,max (3.28) 22

29 where f step,max is normally 48 steps. As it takes some time for the cardiovascular system to adapt to the stepping movements, u 2 (k) has to be modelled min as a first-order system with the time constant τ step which is usually chosen around 40 beats. In addition, the stepping influence at supine position has experimentally been found to be very low (figure 3.11, first 10 minutes). Thus, it is easiest to make u 2 (k) linearly dependent on u 1 (k). The adapted stepping input is denoted as κ(k): ( ) κ(k + 1) = e 1 τ step κ(k) + 1 e 1 τ step u 2 (k)u 1 (k) (3.29) κ(k) now operates in an additive nature on peripheral resistance, venous compliance and the neural barosignal: R(k) = (B c B S (k))ρ + R P + k SR κ(k) (3.30) C V (k) = C V + k SC κ(k) (3.31) B(k) = P B (k) + k P P P (k) k B k SB κ(k) = 1 3 P S(k) P D(k) σ B sin α(k) + k P P P (k) k B k SB κ(k) (3.32) 23

30 3.5 Model simulations The cardiovascular model can now be used to simulate and analyse heart rate and blood pressure in response to various inputs. In addition, it is possible to investigate other cardiovascular signals such as stroke volume, peripheral resistance or cardiac output. In order to get an idea for what happens in the body during a tilt manoeuvre, a standard fast tilt-up and tilt-down should be examined first. Simulations have been done with standard steady-state values as given in table 3.1. These values were used in combination with a set of fixed parameters (table A.2) for identification of the unknown parameters (appendix A.5). Table 3.1: Standard steady-state values used for the model simulations: stands for supine position (α = 0, f step = 0); + stands for tilted position (α = 76, f step = 0); s stands for stepping (α = 76, f step = f step,max ) Steady-state value HR 65 HR + P S 120 P + S 125 P D P + D 95 HR s 75 PS s 130 PD s Fast tilt-up and tilt-down The adjective fast refers to the fact that the tilt-table angle α changes from the minimal angle of zero degrees to the maximal angle of 76 degrees in two or three heart beats (vice versa for tilt-down). Of course, this is not feasible in reality where a full tilt may take up to 30 seconds. However, it is a good way to analyse the dynamics of such a fast tilt, which probably would not be that pronounced when tilting at a slower rate. Fast tilt-up Model responses with the most important physiological variables are depicted in Figure 3.6. It can be seen that these responses are in accordance with the standard physiological response of tilt-up. Details about the dynamic cha- 24

31 racteristics will be explained in the following paragraph about fast tilt-down simulation. The reason is that tilt-down responses are usually much faster than tilt-up responses and that the dynamic features are easier to identify and explain. Fast tilt-down When a person is tilted from the initial upright position back to the supine position, blood is shifted in the body under the influence of gravity. This has two immediate effects: Blood in the pulmonal pathways is shifted into the lung reservoir causing a lack of blood in the left atrium. Blood from the venous reservoir is forced back to the right atrium and venous return is increasing rapidly. The first effect leads to a fast decrease in arterial blood pressure and left stroke volume. As the blood supply in the left atrium is abruptly diminished, the left stroke volume is immediately decreased according to the Frank-Starling law. This process is visible in the simulated model responses as the initial negative peak in blood pressure, left stroke volume and left atrial pressure. The second effect causes an immediate rise of right stroke volume in response to the increased venous return. After some time, this extra blood volume has made its way through the pulmonal pathways and ends up in the left atrium. This in turn causes the left stroke volume to rise again and leads to the positive blood pressure peak 7 to 8 seconds after the start of the tilt manoeuvre. In a third phase the phyisological signals settle to their steady-state values which is the case after approximately 20 seconds. Left and right stroke volume are balanced and the above description nicely shows how the Frank- Starling mechanism enables the adjustment of left and right stroke volume according to respective ventricle load Stepping As the stepping influence on the cardiovascular system is biggest when the table is fully tilted, only the simulation results for α = 76 are shown (Figure 3.8). The heart rate shows the expected non-minimum phase behaviour as described by [6], the diastolic blood pressure is hardly influenced and the 25

32 systolic blood pressure rises, as described by [5]. The barosignal shows the inverse behaviour of the heart rate, which makes perfect sense as the barosignal directly determines heart rate. Peripheral resistance is decreased when stepping is activated which can be compared to the adaptation of the peripheral resistance to exercise. The increase of the stroke volumes and the pulse pressure point out the stabilising effect of stepping on the cardiovascular system Quasi-static The reason for a quasi-static simulation of the cardiovascular model is the analysis of the steady-state behaviour of heart rate and blood pressure at all angles α in the admitted range. Only the angle input is considered for this simulation because the stepping acts smoothly on the outputs whereas the system is expected to show rather different behaviour in the sympathetic and the parasympathetic region respectively. Remember that although the baroreflex regulation on heart rate is active over the whole range, peripheral resistance is only influenced by sympathetic regulation (see section 3.2). Figure 3.9 depicts the dependencies of the relevant cardiovascular variables on the inclination angle α. The following observations can be made: Heart rate strictly increases with α and shows an S shape: The heart rate characteristics directly follow from the baroreceptor signal which is based on the arterial pressure at the level of the baroreceptors. Systolic blood pressure both increases and decreases at lower angles, and strictly increases at higher angles: Systolic blood pressure P S is calculated as the sum of diastolic blood pressure P D and pulse pressure P P. At small angles, stroke volumes don t change much, but P D is increased. This leads to the increase in P S at small angles. However, as soon as the stroke volumes and subsequently the pulse pressure is decreased, P S is decreased as well. In the sympathetic regulation domain, P S strictly increases because P D grows faster than P P declines. Diastolic blood pressure strictly increases with α, but at a lower rate at lower angles: The reason is that at higher angles peripheral resistance is increased by the baroreflex which leads to higher arterial pressures. Peripheral resistance by design only increases at higher angles, when sympathetic regulation becomes active. 26

33 Stroke volumes are diminished when tilting. Figure 3.10 compares the results from the quasi-static simulation with results from Hainsworth [10], Matalon [36], Heldt [28], Fisler [37] and Nguyen [5]. It can be deduced that the accordance of the model results with literature studies and previous work at SMS is satisfying. 27

34 Angle [deg] Blood pressure [mmhg] Peripheral resistance [mmhg ms/ml] Atrial pressures [mmhg] Heart beats P S Heart beats P L P R P D Heart beats Heart beats Heart rate [bpm] Barosignal [mmhg] Stroke volumes [ml] Cardiac output [l/min] Heart beats Heart beats Q L QR Heart beats Heart beats Figure 3.6: Simulation of a fast tilt-up without stepping 28

35 Angle [deg] Blood pressure [mmhg] Peripheral resistance [mmhg ms/ml] Atrial pressures [mmhg] Heart beats Heart beats P R P D P L P S Heart beats Heart beats Heart rate [bpm] Barosignal [mmhg] Stroke volumes [ml] Cardiac output [l/min] Heart beats Heart beats Q R Q L Heart beats Heart beats Figure 3.7: Simulation of a fast tilt-down without stepping 29

36 82 Stepping [steps/min] Heart rate [bpm] Systolic blood pressure [mmhg] Diastolic blood pressure [mmhg] Peripheral resistance [mmhg ms/ml] Heart beats Heart beats Heart beats Heart beats Barosignal [mmhg] Stroke volumes [ml] Cardiac output [l/min] Heart beats Heart beats Heart beats Heart beats Figure 3.8: Simulation of an activation of the stepping mechanism (α = 76 ) 30

37 Angle [deg] Systolic blood pressure [mmhg] Diastolic blood pressure [mmhg] Peripheral resistance [mmhg ms/ml] Heart beats Heart beats Heart beats Heart beats Heart rate [bpm] Barosignal [mmhg] Stroke volumes [ml] Cardiac output [l/min] Heart beats Heart beats Heart beats Heart beats Figure 3.9: Quasi static simulation without stepping 31

38 Hainsworth Matalon Fisler Model Hainsworth Smith Model HR [bpm] 10 8 dbp [mmhg] Angle [deg] Angle [deg] Figure 3.10: Comparison of steady-state behaviour. Left: HR as a function of α. Right: Diastolic BP as a function of α 32

39 3.6 Model validation The step of model validation will be performed using measurements from three healthy subjects (see chapter 5.1.3). Evaluation will be done in a qualitative way analysing each measurement separately. Although averaging over all subjects would probably yield better agreement between the model simulation and the measurement, interesting details from the individual cases would be lost. The measurement was divided into an identification and a validation part. The according measurement protocol is illustrated in the lowermost plot of figure Note that between the identification and the validation there was a recalibration of the blood pressure measurement device. This can introduce offsets in some cases whereas diastolic blood pressure seems to be affected the most. In figure 3.11 for example, this offset amounted to about 4 mmhg and has been corrected accordingly. Validation results for the first subject (MW) are satisfying and it demonstrates that it is possible to simulate or predict heart rate and blood pressure dynamics. HR [bpm] sbp [mmhg] dbp [mmhg] α [deg] / f [steps/min] step 50 f step α Figure 3.11: Model validation with subject MW 33

40 However, the identified models for the other two subjects deviate more from the measured signal than it was the case for the first subject. This should be analysed in more detail: For the second subject (figure 3.12) it can be said that heart rate and diastolic blood pressure were well reproduced by the model. Systolic blood pressure, however, did not show clear trends: During the identification phase, systolic blood pressure stayed constant when tilting but increased in the end of the experiment during the slow ramp of the inclination angle. Another issue are the calibration offsets, that have already been mentioned above. For the second subject, diastolic blood pressure jumped by about 10 mmhg which has been corrected for in the modelled diastolic blood pressure curve. Already the low values of about 40 mmhg after 17 minutes are unrealistic compared to the baseline values at the beginning of the experiment which were around 55 mmhg. The worse thing however is that after the recalibration the value is not set back to 55 mmhg but is even increased to about 65 mmhg. These huge jumps in the measured signals are physiologically improbable in such a short timespan and it unveils the weaknesses of the blood pressure measurement device. HR [bpm] sbp [mmhg] dbp [mmhg] α [deg] / f step [steps/min] 50 f step α Figure 3.12: Model validation with subject MSW 34

41 The validation measurement for the third subject emphasises the above mentioned problems: First, systolic blood pressure is hard to reproduce or model. Second, diastolic blood pressure measurement is tampered with calibration offsets. However, it has to be added that the last measurement is an extreme example for what can happen with physiological signals. 100 HR [bpm] sbp [mmhg] dbp [mmhg] α [deg] / f [steps/min] step 50 f step α Figure 3.13: Model validation with subject DH Generally, it can be concluded that the model reproduces heart rate and diastolic blood pressure with a satisfying accuracy. Problems occur, if reproducibility is not given, i.e. if the subject responds differently for the same inputs. However, this is a more general issue as any deterministic model would struggle with low reproducibility. In contrast to heart rate and diastolic blood pressure, systolic blood pressure is more difficult to predict and model for healthy subjects. Still, we decided to go on with the strategy of controlling all three variables, because in the end, the system will be used with patients. Patients usually react much better with systolic blood pressure to verticalisation because neural regulation is impaired. To counter the problem with calibration offsets, it will be important not to 35

42 do a calibration between identification and control or to reidentify the model when a calibration is necessary. If the offset is only small, it may not be needed to reidentify the model because for control only the relative responses are important. However, if the offset is too high and the new values are out of the identified range, it gets impossible to start the control experiment without adapting the model. 36

43 Chapter 4 Control design Starting from the non-linear MIMO system derived in the previous chapter, an appropriate controller will now be developed. This controller has to be able to keep heart rate and blood pressure within reasonable bounds and minimise fluctuations by adjusting the inclination angle and the stepping frequency. For the given MIMO system which is apparently strongly coupled, a SISO approach trying to control each output in an isolated manner seems infeasible. The fact that the system has less inputs than outputs makes it even harder to do so. It is therefore advisable to choose a control strategy which can handle these issues. A linear optimal control approach is frequently used in advanced control applications, and has been chosen for this thesis as well. Two controllers have been developed, implemented and tested: a Linear Quadratic Regulator (LQR) and a Model Predictive Controller (MPC). The LQ controller which was augmented by an integral part in order to eliminate steady-state control errors was experimentally found to be very hard to tune. The reason is that the system is likely to operate near or on the constraints boundaries for the control inputs. As a result, control inputs are often saturated and anti-windup strategies are therefore necessary. However, for a true MIMO system with strong couplings as has been developed in the previous chapter it is difficult to apply standard anti-windup techniques. It is therefore desirable to have a technique which intrinsically accounts for input constraints. This explains why Model Predictive Control (MPC) is suited for the application at hand and why it is to prefer to a common LQ regulator. 4.1 Model predictive control (MPC) design The advantages of Model Predictive Control are manifold. Two of the most important features are that MPC takes account of actuator limitations and 37

44 that it is suited for multivariable control problems. The principle behind MPC is as follows: based on the system model the controller predicts future outputs and finds the optimal control inputs by minimising a certain cost function. An intuitive analogon for MPC is driving a car [38]. Imagine that the reference path is the lane, the plant is the car and the controller is represented by the driver. The control objective is to keep the car on the lane, while steering as little as possible, keeping a certain distance to the kurbs, obey speed limitations and so on. The driver now has an internal belief or model of how the car reacts to his inputs. He uses this knowledge to predict future behaviour of the car and give according control inputs in order to stay on the reference path, minimise steering effort and meeting all given constraints. This control problem can generally be formulated with a cost function and according constraints [39]. min s y p ) T Q(r s y p ) + u T R u u (4.1) M u γ (4.2) where u(mn c 1) is the control input, y p (pn p 1) the predicted output, r s (pn p 1) the reference, Q(pN p pn p ) the output weighting matrix and R(mN c mn c ) the control input weighting matrix. The matrix M(4mN c mn c ) and the vector γ(4mn c 1) define the constraints. The scalars n, m and p are the number of states, the number of inputs and the number of outputs of the MIMO system. The control horizon is denoted as N c. Table 4.1 summarises these notations. As we will only need constraints on the control inputs, state and output constraints are neglected in this formulation. Note that the vectors r s and y p contain N p discrete samples over the prediction horizon and the vector u contains N c discrete samples over the control horizon: u = ( u(k) u(k + 1) u(k + 2) u(k + N c 1) ) T y p = ( y p (k + 1) y p (k + 2) y p (k + 3) y p (k + N p ) ) T r s = ( r s (k + 1) r s (k + 2) r s (k + 3) r s (k + N p ) ) T (4.3) (4.4) (4.5) Based on the above given description, a model predictive controller will now be developed for the given nonlinear cardiovascular model of chapter 3. Figure 4.1 sketches the signal flows of the control system and shows the two major parts of the controller which are the optimisation routine and the state observer. 38

45 Table 4.1: MPC glossary. Variable Value Description N p 5 Prediction horizon N c 2 Control horizon T u 20 s Controller sample time N u - Number of heart beats in T u n 10 Number of states m 2 Number of inputs p 3 Number of outputs r - Reference signal Q Eq Weighting matrix for states R Eq Weighting matrix for control actions x, y, u - Non-linear state, output, input x lin, y lin, u lin - State, output, input of linearised model x, ỹ, ũ - State, output, input of augmented linearised model State observer The state observer is needed because with the exception of the heart interval the system states are not measurable. The states are updated in each time step with the non-linear cardiovascular model equations and are then corrected in a second step based on the error between the observed and measured outputs. Basically, this is nothing else than a traditional Kalman filter doing first a prediction update followed by a measurement update. The only difference is that the states are updated in a non-linear way. Algorithm 1 describes the state observer in pseudo code. Algorithm 1 State observer 1: if ˆx is not defined (start of control experiment) then 2: ˆx = baseline values 3: end if 4: Save previous state estimate ˆx old = ˆx 5: Nonlinear prediction update: [ŷ ˆx] = f(ˆx, u) 6: Measurement update: ˆx = ˆx + K ob (y ŷ) 7: Output ˆx = ˆx ˆx old The baseline values ˆx describe the system state at the supine position without stepping (u = (0 0) T ). ˆx is calculated during parameter identification as a byproduct (appendix A.5). f(ˆx, u) denotes the non-linear cardiovascular 39

46 model (appendix A.3). The observer gain K ob is calculated in a stochastically optimal way based on the linearisation about the current set-point where w(k) is the process noise and v(k) the measurement noise (eq. 4.6). x lin (k + 1) = A lin x lin (k) + B lin (u lin (k) + w(k)) (4.6) y lin (k) = C lin x lin (k) + v(k) Values for the entries of the diagonal covariance matrices W = E(ww T ) and V = E(vv T ) are given in Table 4.2. Measurement noise was calculated based on the assumption that each output (HR, P S, P D ) has a standard deviation of 2 bpm or 2 mmhg respectively. Process noise was set based on experimental findings, such that the estimated outputs converged to the measured outputs and noise rejection was satisfying. Note that the high value of V(1, 1) is explained by the fact that the first component of y lin is the R-R interval which has much higher nominal values than the other components of y lin. Table 4.2: Entries of covariance matrices W and V Entry Value V(1, 1) V(2, 2) V(3, 3) W(1, 1) W(2, 2) r S + - ˆx Minimise cost function Observer u y MPC Cardiovascular model Figure 4.1: MPC overview 40

47 Optimisation routine The prediction or the optimisation is based on system 4.6 without noise: x lin (k + 1) = A lin x lin (k) + B lin u lin (k) (4.7) y lin (k) = C lin x lin (k) However, instead of equations 4.7 an augmented state-space model, containing an additional integrator, will be used for the prediction (equations 4.8). This has the advantage that the current control error y lin (k) is included in the description which penalises deviations from the set-point. Augmented state-space model: ( ) xlin (k + 1) = y lin (k + 1) ( ) Alin 0 n m C lin A lin }{{ I p } =:A ỹ(k) = ( ( ) 0 p n I p) xlin (k) }{{} y lin (k) =:C }{{} =: x(k) ( ) ( xlin (k) Blin + y lin (k) C lin B lin } {{ } =:B ) ũ(k) (4.8) where I denotes the identity matrix and 0 the null matrix. We will now derive the elements of equation 4.1 based on Wang [39]: The state x(k) develops according to the augmented state-space model. Note that u(k) = ũ(k). x(k + 1) = A x(k) + B u(k) (4.9) x(k + 2) = A x(k + 1) + B u(k + 1) (4.10). = A 2 x(k) + AB u(k) + B u(k) (4.11) x(k + N p ) = A Np x(k) + A Np 1 B u(k) + A Np 2 B u(k + 1) +... (4.12) + A Np Nc B u(k) (4.13) The output at time instant k + i then is: ỹ(k + i) = CA i x(k) + CA i 1 B u(k) + CA i 2 B u(k + 1) +... (4.14) + CA i Nc B u(k), i = 1,..., N p The predicted output ỹ(pn p 1) can be written in vector form using F(pN p 41

48 n) and Φ(pN p mn c ): ỹ(k + 1) ỹ(k + 2) ỹ = = F x(k) + Φ u (4.15). ỹ(k + N p ) where CA CA 2 F =. CA Np (4.16) CB CAB CB Φ = CA 2 B CAB CB CA Np 1 B CA Np 2 B CA Np 3 B CA Np 4 B CA Np Nc B (4.17) The cost function at time instant k can now be written as follows: J(k) = ỹ T Qỹ + u T R u (4.18) = (F x(k) + Φ u) T Q(F x(k) + Φ u) + u T R u (4.19) The weighting matrices Q(pN p pn p ) and R(mN c mn c ) are defined based on the maximal input and output values (u max, y max ): q q p Q = q q p (4.20)

49 r r m R = r r m (4.21) where q i = 1 yi,max 2 r j = 1 u 2 j,max i = 1,..., p (4.22) j = 1,..., m (4.23) The above mentioned cost function (eq. 4.18) has to be minimised under some constraints on u(k) and ũ(k). u min <= u(k) <= u max (4.24) ũ min <= ũ(k) <= ũ max (4.25) These constraints can also be written in matrix form as a function of the optimisation vector u: ( M1 M 2 ) u ( N1 N 2 ) (4.26) where M 1 (2mN c mn c ) and N 1 (mn c 1) define the constraints on the amplitude of the control signal: I m 0 m 0 m 0 m I m I m 0 m 0 m I M 1 = m I m I m I m I m I m 0 m 0 m 0 m I m I m 0 m 0 m I m I m I m I m I m 43 (4.27)

50 ũ max ũ(k 1) ũ max ũ(k 1). N 1 = ũ min + ũ(k 1) ũ min + ũ(k 1). (4.28) M 2 (2mN c mn c ) and N 2 (2mN c 1) define the constraints on the difference of the control signal and can be written down similarly: ( ) I m N c M 2 = (4.29) N 2 = I m Nc u max u max. u min u min. (4.30) Finally the objective is to minimise the cost function J(k) subject to the given constraints: min u J(k) = (F x(k) + Φ u) T Q(F x(k) + Φ u) + u T R u (4.31) ( ) ( ) M1 N1 u M 2 N 2 In order to incorporate anticipative action or look-ahead functionality in the MPC design, the objective can be reformulated: min u J(k) = (r s (F x(k) + Φ u + y s )) T Q(r s (F x(k) + Φ u + y s )) + u T R u ( M1 M 2 ) u ( N1 N 2 ) (4.32) where y s (pn p 1) denotes the non-linear output at the setpoint: y s (k) y s (k) y s =. y s (k) (4.33) 44

51 Note that the reference r s is also given with real physiological values for the R-R interval, systolic and diastolic blood pressure. The minimisation can be done by standard quadratic programming routines. Note that such routines generally assume the quadratic programming problem in the form of equation It is left to the reader to verify that equation 4.32 can be reformulated to equation 4.35 in order to meet this requirement. min u J = 1 2 uh u + f T u (4.34) M u γ min u J(k) = 1 2 u(φt QΦ + R) u + ( Φ T Qr s + Φ T QF x(k) + Φ T Qy s ) u ( M1 M 2 ) u ( N1 N 2 ) (4.35) Adaptations for controller sample time If the calculated control action is not sent to the plant in every time step, but only every N u time steps, the optimisation problem has to be reformulated. This is the case for our plant, where the subject on the ERIGO would feel uncomfortable if control commands were sent in every heart beat. Further, two much motion of both the tilt-table and the stepping mechanism would needlessly activate the cardiovascular system. It is therefore necessary to choose a controller sample time T u which is higher than the time needed for one time step of the augmented system 4.8 (i.e. one heart beat). Note, that this change does not influence the observer, which still runs at the original sample time. The optimisation vector u now has the form u(k) u(k + N u ) u =. u(k + N p N u ) (4.36) 45

52 The objective is the same as before (equation 4.32), but the prediction matrices and the reference vector r s change: CA Nu CA 2Nu F = (4.37). Φ = Nc Nc i=1 CA2Nu i B Nc i=1 CA3Nu i B CA NpNu i=1 CANu i B 0 0 Nc i=1 CANu i B 0 Nc i=1 CA2Nu i B Nc i=1 CAN P N u i B N c i=1 CA(N P 1)N u i B Nc r s (k) r s (k + N u ) r s =. r s (k + N p N u ) i=1 CA(N P N c+1)n u i B (4.38) (4.39) Note that N u is calculated at the beginning of the MPC calculation every T u seconds based on the current heart rate, i.e. the current R-R interval y 1 in milliseconds: N u = 1000 T u y 1 (4.40) 4.2 Simulation In order to test the controller behaviour a simple test case has been set up. The cardiovascular model has been identified with the standard steady-state values from table 3.1. Two setpoints around α = and α = 30 with deactived stepping have then been calculated. Linearisations and Kalman observers were obtained around these setpoints and have then been used for the controller. Now, the first part of the simulation consists of a reference step from the first setpoint (α = ) to the second setpoint (α = 30 ). The controller anticipates the step as soon as it is included in the prediction horizon which is 100 s (N p = 5, Controller sample time = 20 s). It can be seen that not only the angle is lowered, but also the stepping mechanism is activated because it quickly decreases heart rate. 46

53 The second part of the simulation depicts the reaction of the controller to a disturbance on the outputs: heart rate is suddenly increased by four beats per minute, and systolic blood pressure is decreased by four mmhg. The controller reacts by activating the stepping mechanism which is capable to compensate such a disturbance. As a consequence, the outputs are regulated back to the reference values. HR [bpm] 75 Systolic BP [mmhg] Diastolic BP [mmhg] α [deg] / f step [steps/min] α f step Figure 4.2: Controller simulation: The first part depicts the system response to a step at minute 2. The second part shows the system response to a disturbance in heart rate and systolic blood pressure at minute 5. 47

54 48

55 Chapter 5 Methods 5.1 Healthy subjects Implementation The healthy subjects were measured at the SMS lab, where the ERIGO device has been customised for easy interfacing and data recording. No changes had to be done on the hardware side. As for the software, the controller could be implemented into an existing Matlab/Simulink environment that interfaces with the ERIGO via an xpc target machine. From Matlab, the model is transformed to C code, compiled and loaded on to the target machine. This requires all code in the model to be written in EML-code, which is basically Matlab m-code with some restrictions. The restrictions are such, that only C compatible code is allowed, which means that variable sizes need to be clearly defined in advance for example. Furthermore, not all Matlab functions are available including the quadratic programming solver quadprog and all control system related functions such as ss or kalman that are needed for control design. The model predictive controller, which needs a quadratic programming solver in order to do the optimisation, has thus been realised with the open-source C++ implementation qpoases 1 which is based on the active-set strategy [40]. Control design has been done offline Blood pressure recording Continuous blood pressure recording was done with the non-invasive CNAP TM Monitor (Figure 5.1). The CNAP TM Monitor outputs the raw blood pressure

56 wave as an analog signal which is fed over a galvanic separation to the xpc target. Setting up the measurement system takes a few minutes: when the arm and finger cuffs are adjusted properly, the device is ready to use after a short calibration phase. The device has to be recalibrated after one hour of measurement, so that accuracy is warranted. Please refer to Kupke [6] (p.5-6) for more details about the CNAP TM Monitor specifications. Figure 5.1: CNAP TM Monitor 500 with arm and finger cuffs: non-invasive continuous blood pressure measurement device used for all measurements and control experiments. As the CNAP device only outputs the raw blood pressure signal, the systolic and the diastolic blood pressure have to be extracted separately. This is done with an online peak detection routine extracting maxima (systolic blood pressure) and minima (diastolic blood pressure) [6]. Once the peaks are identified, heart rate calculation is performed Experimental design Model validation Three subjects (2 female, 1 male) aged between 20 and 35 years were measured in total for model validation (table 5.1). The measurement protocol was defined as illustrated in Figure The identification part was designed such that both the influence of stepping and the influence of tilting could be analysed in supine as well as in tilted position. Note that that when the stepping is activated or deactivated, the waiting time is not only 3 but 5 minutes. This is because of the slower dynamics of heart rate and blood pressure in response to stepping. 50

57 Table 5.1: Healthy subjects participating in the model validation MW MSW DH Sex m f f Weight Control experiments Five subjects (3 female, 2 male) aged between 20 and 35 years were measured in total (table 5.2). For each of the five subjects three control experiments, lasting 20 minutes each, were done. The first experiment was isolated heart rate control, the second was blood pressure control (systolic and diastolic), and the last experiment was combined control of all three physiological signals. The control experiments were preceded by an identification phase in order to identify the unknown parameters and fit the model to the subject. Table 5.2: Healthy subjects participating in the control experiments PB MW LB RR ME Sex m m f f f Weight This is done with an shortened identification measurement compared to model validation which lasts 11 minutes in total. First, the baseline values at the supine position are calculated which is done by taking the average over the last two minutes before the tilt. Similarly, the steady-state values for the tilted position (α = 76 ) are calculated by taking the mean value of minutes 5 and 6 to account for the transient behaviour. Finally, the steady-state values for the stepping are calculated by taking the mean of minutes 10 and 11. The reason for the longer duration of the stepping part is the slower dynamics of heart rate and blood pressure in response to stepping, i.e. it takes more time to reach the steady-state. The identified values can be stated as follows y = ( I P S P D ) T (5.1) y + = ( I + P + S P + D ) T (5.2) y s = ( I s P s S P s D) T (5.3) Figure 5.2 exemplifies the identification protocol in the lowermost plot and the measured signals together with the simulated model response for subject PB in the upper three plots. 51

58 90 HR [bpm] 70 sbp [mmhg] dbp [mmhg] α [deg] / f step [steps/min] Figure 5.2: Model identification (subject PB). The lowermost plot shows the identification protocol. The upper three plots depict the measured signals (green) together with the simulated model responses (blue). 52

59 5.2 Patients Implementation As the measurements were done at the hospital, the implementation had to be adapted to local conditions. The ERIGO at the hospital, which is the standard version delivered by HOCOMA, did not contain the same interfaces as the ERIGO at the SMS lab. So, one major issue was that control commands had to be given by hand following the numbers on the Matlab display. The inclination angle α was set using a water-level and the stepping frequency was adjusted using the ERIGO touchscreen. Transferring the customised ERIGO from the SMS lab at ETH Zurich to Wald would probably have been possible, but was not an option due to ethical reasons. In addition, the manual control worked well and no further actions had to be taken. The controller was implemented using a Simulink model reading raw blood pressure data, extracting the physiological signals, and displaying the computed control inputs on the laptop screen Blood pressure recording Blood pressure recording was also done with the CNAP TM monitor just as it was the case with healthy subjects. However, the blood pressure signal was fed to a biosignal amplifier (USBamp from g.tec 3 : figure 5.3), from where it could be routed to the laptop over a standard USB connection. Figure 5.3: Biosignal amplifier USBamp from g.tec