Research Collection

Master Thesis

Modelling and control of the human cardiovascular system

Author(s): Gisler, Stefan

Publication Date: 2011

Permanent Link: https://doi.org/10.3929/ethz-a-007207574

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ETH Library

Master Thesis

Modelling and control of the humancardiovascular system

Stefan Gisler

AdvisersMartin Wieser and Dr. Heike Vallery

andProf. Dr. Robert Riener

Sensory Motor Systems Lab (SMS)Swiss Federal Institute of Technology Zurich (ETH)

Submission: April 2011

Contents

1 Introduction 1

2 Human cardiovascular system 52.1 Hemodynamic system . . . . . . . . . . . . . . . . . . . . . . . 52.2 Blood pressure regulation . . . . . . . . . . . . . . . . . . . . 72.3 Orthostatic reaction and muscle pump . . . . . . . . . . . . . 92.4 Cardiovascular pathology . . . . . . . . . . . . . . . . . . . . . 102.5 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5.1 Cardiovascular responses to passive tilting . . . . . . . 112.5.2 Cardiovascular modelling . . . . . . . . . . . . . . . . . 13

3 Cardiovascular model 153.1 Hemodynamic system . . . . . . . . . . . . . . . . . . . . . . . 163.2 Blood pressure regulation . . . . . . . . . . . . . . . . . . . . 183.3 Influence of gravity . . . . . . . . . . . . . . . . . . . . . . . . 213.4 Influence of stepping . . . . . . . . . . . . . . . . . . . . . . . 223.5 Model simulations . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5.1 Fast tilt-up and tilt-down . . . . . . . . . . . . . . . . 243.5.2 Stepping . . . . . . . . . . . . . . . . . . . . . . . . . . 253.5.3 Quasi-static . . . . . . . . . . . . . . . . . . . . . . . . 26

3.6 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Control design 374.1 Model predictive control (MPC) design . . . . . . . . . . . . . 374.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5 Methods 495.1 Healthy subjects . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.1.1 Implementation . . . . . . . . . . . . . . . . . . . . . . 495.1.2 Blood pressure recording . . . . . . . . . . . . . . . . . 495.1.3 Experimental design . . . . . . . . . . . . . . . . . . . 50

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5.2 Patients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2.1 Implementation . . . . . . . . . . . . . . . . . . . . . . 535.2.2 Blood pressure recording . . . . . . . . . . . . . . . . . 535.2.3 Experimental design . . . . . . . . . . . . . . . . . . . 54

6 Results 556.1 Healthy subjects . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.1.1 Heart rate control . . . . . . . . . . . . . . . . . . . . . 556.1.2 Blood pressure control . . . . . . . . . . . . . . . . . . 556.1.3 Combined heart rate and blood pressure control . . . . 56

6.2 Patients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.3 Controller performance . . . . . . . . . . . . . . . . . . . . . . 60

7 Discussion 61

8 Conclusion and Outlook 65

A Model summary 73A.1 List of variables . . . . . . . . . . . . . . . . . . . . . . . . . . 75A.2 List of parameters . . . . . . . . . . . . . . . . . . . . . . . . . 76A.3 Model equations in non-linear state-space form . . . . . . . . . 78A.4 Steady-state equations in non-linear state-space form . . . . . 80A.5 Parameter identification . . . . . . . . . . . . . . . . . . . . . 82A.6 Model constraints . . . . . . . . . . . . . . . . . . . . . . . . . 85

B Summarised results 87

References 95

ii

Abstract

Bed-rest leads to cardiovascular deconditioning and may induce a decline instroke volume, cardiac output and oxygen uptake. Further, it increases therisk of orthostatic intolerance. In an early phase of rehabilitation, it is there-fore important to prevent the development of cardiovascular deconditioningwhich can be done by verticalisation and mobilisation. In the future, theenhanced ERIGO tilt-table will be able to control physiological signals andhence, stabilise the patient’s cardiovascular system.This thesis focuses on the control of heart rate and blood pressure by meansof verticalisation (tilting) and mobilisation (stepping). In a first step, a car-diovascular non-linear model with two inputs (tilting and stepping) and threeoutputs (heart rate, systolic and diastolic blood pressure) is developed basedon physiological principles and existing work. The model is then used fordesigning a model predictive controller which was found well suited for thegiven control problem.Five healthy subjects have been tested with three different configurations:isolated heart rate control, isolated blood pressure control and combinedcontrol. One patient has been tested with blood pressure control which yiel-ded promising results.

Keywords– Orthostatic intolerance, cardiovascular modelling, model pre-dictive control

iii

Acknowledgements

First, I want to thank Prof. Dr. Riener for being accepted to do this thesis atthe Sensory-Motor Systems Lab. Then I want to thank my advisers MartinWieser and Dr. Heike Vallery for their valuable support during the work.Special thanks go to Martin Wieser for his great efforts while testing anddebugging the system.This thesis would not have been possible without the probands and patients.A big thanks goes to all the probands, the “Zurcher Hohenklinik” in Wald,and all the patients that participated in this study. At this point, I also wantto thank Rafael Rust and Lilith Butler for their support during the patientmeasurements in Wald.Last but not least, I want to thank all the students in the student room forthe nice and inspiring atmosphere.

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Chapter 1

Introduction

One major problem with neurological patients suffering from stroke, trau-matic brain injury or paraplegia is the long bed rest after the accident. Itleads to deconditioning of the patients’ cardiovascular system and evokessecondary complications such as orthostatic intolerance. Further complica-tions can include venous thrombosis, muscle atrophy, joint contractures andosteoporosis [1], [2]. Therefore, early mobilisation of the patient is crucial asit can reduce the risk of cardiovascular deconditiong and improves the stateof health.

This thesis focuses on the cardiovascular aspects of bed-ridden patients, i.e.how the cardiovascular system can be prevented from deconditioning and be-coming unstable. Prolonged bed rest leads to a decrease in circulating bloodvolume, a decrease in stroke volume and pulse pressure, and an increasedheart rate. A direct result of these indications is the inability of the patient’scardiovascular system to regulate blood pressure when standing up (ortho-static intolerance). In the upright position, the patient suddenly starts tofeel dizzy or even faints due to excessive blood pooling in the lower extre-mities and reduced blood perfusion of the upper body. However, orthostaticintolerance is not only caused by prolonged bedrest but can also be a conse-quence of an impaired vegetative nervous system. In paraplegia patients,the sympathetic effector nerves to the heart and the smooth musculature aredisrupted or even broken. This leads to a malfunction of the baroreflex whichis responsible for regulating arterial blood pressure (see chapters 2.2, 2.4).As a consequence, the sudden decrease in arterial blood pressure cannot beregulated and the patient faints.

A tilt-table therapy is aimed at reconditioning the patient’s cardiovascularsystem by verticalising to an angle of about 80 degrees. Additional leg mo-

1

vements which can include stepping or cycling movements increase venousreturn due to the effects of the muscle pump and improve orthostatic tole-rance. The ERIGO device which has been used at the institute since thebeginning of the AwaCon project combines these therapies and allows for anoptimal treatment of patients with neurological disorders (Figure 1.1). Moreinformation 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 thethree inputs. Right: ERIGO during therapy session.

heart rate, respiration frequency, skin conductance, oxygen saturation, EEGand EMG can be recorded. However, for this thesis only blood pressure andheart rate need to be recorded, where EMG recordings may be helpful toanalyse muscle activity during mobilisation.

The goal of the project is to control and stabilise the cardiovascular systemof patients with neurological disorders by verticalisation, mobilisation andcyclic loading of the lower limbs (Figure 1.1). This will help to improve thecardiovascular status of these patients and will have the potential to reducemedication, enhance physiotherapy and shorten the duration of early reha-bilitation [3]. Furthermore, the risk of deconditioning of the cardiovascularsystem, and complications resulting from this, can be decreased. Additionalproject information is available on the homepage of the SMS Lab 2.In earlier projects at the SMS, isolated control of heart rate and diastolic

1http://www.hocoma.com/en/products/erigo/2http://www.sms.mavt.ethz.ch/research/projects/awacon

2

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

3

4

Chapter 2

Human cardiovascular system

This chapter will give a short introduction to physiology and pathophysio-logy of the human cardiovascular system and summarises some results fromliterature.

2.1 Hemodynamic system

Left lungRight lung

Left heartRight

heart

Head

Splanchnic &

renal circulation

Legs

Figure 2.1: Schematic representation of the human circulatory system. Adap-ted from: http://www.ionwave.ca

The major task of the hemodynamic system is to supply every single cell

5

of the organsim with oxygen and nutrients and carry away carbon dioxide(CO2) as well as metabolic waste products. In the circulation, the heart actsas a pump which produces a pressure gradient between arterial and venouscirculation. Driven by this pressure gradient, deoxygenated blood from thevenous circulation flows back to the right heart where it is pumped throughthe lung. In the lung the blood is enriched with oxygen and reenters systemiccirculation when pumped into the aorta by the left ventricle. The arterialtree then supplies the whole body with oxygen and nutrients. From theperipheral regions, where the oxygen and the nutrients are used, the bloodreturns to the right heart and the circulation is closed (Figure 2.1).Flows and pressures within the human hemodynamic system are characteri-sed by the following list of hemodynamic variables:

• Stroke volume (SV ) defines the amount of blood pumped into the aortawithin one beat.

• Cardiac output (CO) is calculated as the product of stroke volume andheart rate (HR)

CO = SV ·HR

• Systolic blood pressure (sBP ) is the maximal blood pressure that oc-curs during the contracting heart phase (systole).

• Diastolic blood pressure (dBP ) is the minimal blood pressure thatoccurs during the filling period of the heart, when the ventricles arerelaxed (diastole).

• Mean arterial pressure (MAP ) is defined as the integrated blood pres-sure over one heart period divided by the time of one heart period.

MAP =

∫ t+tRRt

BP (t)dt

tRR

where BP (t) is the continuous blood pressure and tRR is the time ofone heart period (R-R interval). A common approximation is given as

MAP =1

3· sBP +

2

3· dBP

• Central venous pressure (CV P ) is the pressure in the intrathoracicveins and the right atrium. Normal values range from 2 to 4 mmHG [7].

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• Total peripheral resistance (TPR) is a rather hypothetic measure ofvessel resistance in the systemic circulation. In duality to Ohm’s lawU = R · I, total peripheral resistance is defined as

TPR =MAP − CV P

CO

CV P is usually neglected in this calculation and we get TPR = MAPCO

.

2.2 Blood pressure regulation

Regulation mechanisms in the cardiovascular system are responsible for adap-ting the hemodynamic variables such as blood pressure and cardiac outputaccording to the body needs. During exercise for example, cardiac outputis strongly increased in order to cover the high oxygen need in the skeletalmuscles. Another situation where these regulation mechanisms are active iswhen the body adapts to changes in environmental conditions such as tem-perature differences. And last, these mechanisms are also active in responseto orthostatic stress what will be of interest for blood pressure and heart ratecontrol on the ERIGO.Hemodynamic variables can be influenced in several ways. Natural controlmechanisms include neurogenic (over the vegetative nervous sytem), hormo-nal (over circulating hormones), humoral (with locally formed substances)or myogenic regulation (vasoconstriction with smooth musculature). Forshort-term regulation the neurogenic mechanisms which include baroreflex,cardiopulmonary reflex and chemoreceptor reflex are most important. Thesethree types of neurogenic blood pressure regulation will now be described inmore 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 sinusare the sensors in this reflex mechanism. They transmit neural signals to thecentral nervous system or more precisely to the cardiovascular centre in themedulla oblongata. The impulse frequency of the afferent neurons is determi-ned by the course of the arterial blood pressure: Low arterial blood pressureleads to a high impulse frequency. However, impulse frequency is not onlydetermined by absolute value of the arterial blood pressure but also by itstime rate of change. This proportional-derivative (PD) sensor characteristicsenable the baroreceptors to send all relevant information about heart func-tion to the central nervous system. In the medulla oblongata the informationfrom the baroreceptors is transmitted to the efferent vegetative nervous sys-tem which determines heart rate, heart contractility and vasoconstriction of

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peripheral blood vessels, closing the reflex arch. It has to be added that aninhibitory interneuron in the medulla provokes negative feedback which isessential for regulating and stabilising arterial blood pressure.

Left lungRight lung

Left heartRight

heart

Head

Splanchnic &

renal circulation

Legs

Cardiovascular

centre

Baro-

reflex

Baroreceptors

Regulate heart rate &

cardiac contractility

Regulate peripheral resistance

Figure 2.2: Blood pressure regulation with the baroreflex loop.

The cardiopulmonary reflex is another blood pressure regulating mechanismthat works synergistically with the baroreflex. The cardiopulmonary recep-tors are located in the venous system, more precisely in the atria and A.pulmonalis. However, cardiopulmonary receptors are not only responsiblefor blood pressure regulation but also for volume regulation. Stimulation ofthe receptors by dilated atria leads to an inhibited production of the anti-diuretic hormone (ADH). As a consequence, urine secretion is increased andthe circulating blood volume can be reduced. Furthermore, activation of thecardiopulmonary receptors decreases sympathetic activity and inhibits Reninproduction in the kidneys. Renin promotes the formation of Angiotensin IIwhich has a direct vasoconstrictive effect on the smooth musculature in thevessels. Moreover, Angiotensin II stimulates the production of Aldosteronein the kidneys which increases reabsorption of sodium and water. In the longterm, this leads to a higher blood volume and an increased blood pressure.Hence, the Renin-Angiotensin-Aldosterone system (RAAS) is capable of in-creasing arterial blood pressure by the vasoconstrictive effect of AngiotensinII and the volume retention caused by Aldosterone. Note that volume re-gulation is a long-term regulation because it includes hormonal mechanisms

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and because it takes some time until body fluids have diffused through thecapillary walls.The chemoreceptor reflex is mainly responsible for respiration control, butcan also influence cardiovascular regulation if partial pressure of oxygen inthe blood decreases [7]. The reflex mechanism particularly becomes activeif blood pressure falls below 80 mmHg and once active, it acts in the samefeedback structure as the baroreflex. As a result, arterial blood pressure isincreased.In detail, myogenic regulation is also a kind of neural regulation mechanismif we consider the sympathetic effected vasoconstriction in the peripheral ar-terioles. However, there is also a mechanism called autoregulation that isattributed to myogenic regulation. Autoregulation is the ability of a bloodvessel to keep the blood flow constant under changing perfusion pressures.When perfusion pressures are increased, the smooth musculature is activa-ted 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 si-tuations. Normally, the neural regulation mechanisms as discussed above areable to maintain homeostasis quite fast. Nevertheless, there might be situa-tions where the regulation is incapable of keeping arterial blood pressure andcerebral perfusion at a safe level. Low blood volume or high temperaturesfor example are conditions that increase the risk of defective homeostasis.This can lead to a syncope which can be rather dangerous when the faintingperson 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 heal-thy person, up to half a litre of blood is shifted from the upper body to thelower extremities [7]. Arterial blood pressure falls immediately and the reflexmechanisms are activated. However, peripheral vasoconstriction caused bysympathetic regulation is usually too weak in order to lower venous bloodpooling effectively. Fortunately, there is another mechanism besides the neu-ral regulation which is capable of stabilising the cardiovascular system. Theprinciple behind this mechansim is that the contraction of the skeletal legmuscles efficiently compresses the venous compartments, decreases venouspooling and increases venous return to the heart. Because the venous valvesare closed, backflow is not possible and the blood is forced to return back tothe heart (figure 2.3). This “muscle pump” is always active when the skeletal

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leg musculature is active, for example during walking.

Figure 2.3: The muscle pump mechanism stabilises the cardiovascular systemby efficiently reducing venous blood pooling and increasing venous return byrepeated contractions of the skeletal leg musculature. Source: University ofMinnesota

2.4 Cardiovascular pathology

Cardiovascular instability and orthostatic hypotension are common deficitsin bed-ridden patients [1], [2]. In spinal cord injury (SCI) patients, forexample, one reason for these deficits are the disrupted efferent sympatheticpathways regulating heart rate, heart contractility and peripheral vasocons-triction. Therefore, neural regulation mechanisms can not work properlyand blood pressure often drops dramatically in reaction to orthostatic stress.The disturbed balance between sympathicus and parasympathicus leads toan exaggerated increase in heart rate as a compensatory reaction to the bloodpressure decrease. This happens because parasympathetic heart rate regu-lation is still intact in SCI patients as efferent parasympathetic nerves areconnected to the Vagus nerve and not to the spine. Naturally, sympatheticnervous system disfunction is not the only reason for orthostatic intolerancein neurological patients. Claydon et al. [8] summarise these factors for SCIpatients as follows:

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• 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 tightconnection with the sympathetic nervous system disfunction. As explainedabove, baroreflex regulation is severly damaged because of an impaired sym-pathetic nervous system.The lack of the skeletal muscle pump together with immobilisation and pro-longed bed-rest are the reason for cardiovascular deconditioning which inturn negatively affects the overall recovery. Lastly, Claydon et al. reportevidence that SCI patients have a decreased plasma volume as a result of animpaired salt and water balance. This leads to problems in volume regula-tion, i.e. hypovolemia and low resting blood pressure with a predispositionto orthostatic intolerance.

2.5 Literature review

2.5.1 Cardiovascular responses to passive tilting

Passive tilting leads to an immediate increase of blood volume in the legveins of about half a liter [7]. Venous return is decreased and because of theFrank-Starling mechanism stroke volume and pulse pressure are diminishedas well. To counter the blood pressure drop, neural reflexes are instantlyactivated and sympathetic action is increased. This has two consequences:Firstly, heart rate rises by approximately 20 % [7] and secondly, diastolicblood 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 topassive tilting according to standard physiological work of reference suchas [7]. Table 2.1 lists the outcome of several studies about cardiovascularresponses to passive tilting involving healthy subjects. Note that most ofthese experimental results conform with the “standard” physiological res-ponse.As the aim of the thesis and the whole project is to enhance therapy of neu-rological patients, a quick survey of typical pathophysiological cardiovascular

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year HR sBP dBP MAPHainsworth 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, ↑ meanssignificant increase, ↓ means significant decrease. (Adapted and completedwith HR from [5])

responses will be done. Table 2.2 presents standard cardiovascular responsesof SCI patients. All of these studies conform with the normal pathologicalreaction to orthostatic stress in SCI patients as described in section 2.4. Inaddition, on the basis of the work of Houtman [25] and Legramante [26] itcan be stated that the higher the lesion the bigger are the implications onthe cardiovascular system and the cardiovascular regulation.

year aetiology HR sBP dBP MAPCorbett 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, ↑ meanssignificant increase and ↓ means significant decrease

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2.5.2 Cardiovascular modelling

Computational models of the human cardiovascular system have been deve-loped for many different purposes. An elaborate cardiovascular model canbe used to identify aetiologies of cardiovascular diseases such as orthostaticintolerance (OI). Heldt et al. [28] have presented a complex mathematical mo-del which reproduces cardiovascular responses to orthostatic stress. In theirstudy the model was used to investigate the mechanisms that cause postspa-ceflight OI. Leaning et al. [29] formulated a detailed model intended to studyand predict the overall effects of an injected drug. However, a cardiovascularmodel can also be used to examine specific aspects of the cardiovascular sys-tem such as blood-pressure fluctuations and heart-rate variability [30], [31].Most of these models are aimed at explaining a certain cardiovascular phe-nomenon and are usually rather complex with a high model order. They arenormally based on a large number of compartments representing the differentparts of the circulation (heart chambers, ventricles, venous and arterial seg-ments). Each compartment or reservoir has a certain pressure Pj and volumeVj

Pj =Vj − Vj0Cj

(2.1)

where Cj is the compliance and Vj0 the unstressed or zero-pressure volume.Most models that describe the overall cardiovascular system incorporate someelements of nervous system regulation. The baroreflex plays an essential rolebecause it governs the short-term dynamics of blood pressure and heart rate.Long-term dynamics are most often less important than short-term effectsand can be neglected in the model description. Therefore, blood pressureregulation mechanisms such as RAAS do not need to be modelled.There are hardly any cardiovascular models in literature which incorporatean orthostatic component and are kept simple. One exception is in the workof Akkerman [32] who presented a mathematical beat-to-beat model designedfor tilt-table experiments. He analysed the dynamics of cardiovascular signalsafter fast tilt-up and tilt-down. The model forms the basis of the wholecontroller design and will be explained in detail in the following section.

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14

Chapter 3

Cardiovascular model1

In order to control physiological quantities such as blood pressure and heartrate, an appropriate model of the cardiovascular system is needed. Thismodel should have two inputs, namely the inclination angle α of the ERIGOdevice and the stepping frequency fstep. Based on these inputs the modelshould output heart rate, systolic and diastolic blood pressure (Figure 3.1).

Cardiovascular model

step

uf

α =

S

D

HR

y P

P

=

Figure 3.1: Inputs and outputs of the cardiovascular model

In order to develop a mathematical model of the human cardiovascular sys-tem for blood pressure regulation, the following two assumptions are made:

• For the internal blood pressure regulation, only the baroreflex is takeninto account. Other mechanisms such as the cardiopulmonary reflexand the RAAS system are not needed to explain the main blood pres-sure characteristics in tilt-up and tilt-down because they govern thelong-term dynamics.

1The material in this chapter is closely related to Akkerman’s work [32]

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• The blood volume is constant and fluid movements through the capil-lary walls are not considered.

So the model only contains the most important elements that are needed tosimulate orthostatic reactions, namely a closed hemodynamic system, a kindof internal blood pressure regulation system and the influence of gravity andstepping (Figure 3.2). These parts will be explained in detail in the followingsections.

3.1 Hemodynamic system

The hemodynamic system as explained in section 2.1 can be modelled as aconnected system of pipes representing blood vessels. The heart acts as apump, maintains systemic blood pressure and transports oxygen-poor bloodto the lung. The microcirculation in the peripheral parts of the body is thebottleneck in the pipe system and is therefore also called the peripheral re-sistance. The fact that blood vessels are not stiff tubes but compliant vesselsis accounted for by introducing a venous and an arterial reservoir which iscommon engineering practice. In fact, the flattening effect that arterial com-pliance has on the systolic blood pressure peaks is called the “Windkessel”effect which is in accordance with the above mentioned engineering principleof introducing reservoirs for the modelling of compliant tubes.In Akkerman’s model, only the lung, the arteries and the veins are modelledas proper compartments as defined by Equation 2.1. The volumes of thesecompartments are denoted by VP , VA and VV respectively. In addition, eachof these compartments is attributed a compliance (CP , CA and CV ) and azero-pressure volume (VP0, VA0 and VV 0). According to Equation 2.1 thecompartment pressures at heart beat k + 1 can then be expressed as:

PR(k + 1) =VV (k)− VV 0

CV(3.1)

PL(k + 1) =VP (k)− VP0

CP(3.2)

PD(k + 1) =VA(k)− VA0

CA(3.3)

where PR(k) is the right atrial pressure, PL(k) the left atrial pressure andPD(k) the diastolic blood pressure which directly depends on the arterialblood volume.The flow between these three reservoirs is characterised by the following setof equations where VPP (k) describes the volume in the pulmonary pipeline

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Lung

Venous reservoir

Baroreceptors

Peripheral circulation

Head

Right

heart

Left heart

Cardiovascular

centre

Baro-

reflex

Windkessel

,L L

P Q ,S D

P P

WQ R

I

B,

P PV C

,A A

V C

RP

RQ

, ,V V V

V P C

Figure 3.2: Simplified representation of the human cardiovascular systemused for model synthesis. Adapted from [32]

which is needed to model the delay between right and left atrium. ξP denotesthe number of right stroke volumes that are in the pulmonary pipeline.

VP (k) = VP (k − 1) +QR(k − ξP )−QL(k) (3.4)

VA(k) = VA(k − 1) +QL(k)−QW (k) (3.5)

VV (k) = VV (k − 1) +QW (k)−QR(k) (3.6)

VPP (k) = VPP (k − 1) +QR(k)−QR(k − ξP ) (3.7)

Based on the Frank-Starling law and the restitution properties of ventricularmyocardium, the left and right stroke volumes QL(k) and QR(k) depend onthe preload and the length of the previous R-R interval I(k− 1). Akkerman

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adapted these findings from [33]:

QR(k) = γRPR(k)I(k − 1) (3.8)

QL(k) = γLPL(k)I(k − 1) (3.9)

where γR and γL are constant factors called “Starling” factors. The periphe-ral flow QW (k) depends on the peripheral resistance R(k) and the pressuredifference between the arterial and the venous segment.

QW (k) = CA (PS(k)− PV (k))

(1− exp

−I(k)

R(k)CA

)(3.10)

where PV (k) denotes venous pressure evaluated just after systole when theright stroke volume has been ejected into the pulmonary pipeline:

PV (k) =VV (k − 1)− VV 0 −QR(k)

CV(3.11)

The equations for pulse pressure PP (k) and systolic blood pressure PS(k)complete the hemodynamic system:

PP (k) =QL(k)

CA(3.12)

PS(k) = PD(k) + PP (k) (3.13)

All the introduced variables are beat-to-beat variables which means that theyare updated at each heart beat. It is not clear, however, at which instantof the heart beat these variables are refreshed. The systolic blood pressurePS(k) for example is updated during the systole when the continuous bloodpressure curve peaks at its maximum value. In contrast, the diastolic bloodpressure PD(k) is updated at the end of the diastole. Each hemodynamicvariable has its natural physiological sampling instant. Another example arethe right and the left stroke volumes QR(k) and QL(k). These variables areupdated at the beginning of the systole when the stroke volumes are ejectedinto the pulmonary pipeline and the aorta respectively. Figure 3.3 graphicallysummarises the sampling instants of the introduced hemodynamic variables.

3.2 Blood pressure regulation

Blood pressure regulation is performed by different body mechanisms. Thereare short-term regulations (minutes, hours) and long-term regulations (days,

18

Figure 3.3: Hemodynamic timetable describing at which moment of the heartbeat each hemodynamic variable is evaluated. Source: Akkerman [32]

weeks 2) as mentioned above. For the purpose of blood pressure and heartrate control, only the short-term regulations have to be considered. There-fore, the modeling will focus on the baroreflex mechanism. Katona et al. [34]have developed a baroreflex model which is composed of a sympathetic anda parasympathetic branch (Figure 3.4) which is widely used in computatio-nal modelling of the human cardiovascular system. Based on a hypotheticbarosignal which is a function of arterial blood pressure and pulse pressure,the model outputs the heart period. The model is split in two parts becausesympathetic and parasympathetic dynamics are rather different. Parasym-pathetic activity leads to a fast decrease of heart rate which can be shown byelectrical stimulation of the Vagus nerve. In contrast, the sympathetic contri-bution on heart rate is slower. In Katona’s model there is a fixed boundarybetween sympathetic and parasympathetic regulation. Of course, in realitythere is a smooth transition between these two types of blood pressure regu-lation. However, it is not needed to map this behaviour to the model and thisintuitive simplification is very well applicable. It is even the case, that forapplications where these subtle dynamics are of minor importance, Katona’sbaroreflex model can be further simplified. In this thesis, the two branchesare merged to one neglecting the different dynamics of sympathetic and pa-rasympathetic regulation. More important is the extension of the model by

2Time specifications: http://homepages.uel.ac.uk/M.S.Meah/bs250page4clec3.

htm

19

Figure 3.4: Katona’s baroreflex model for heart rate regulation [34]. Theneural input signal f(t) is divided in a sympathetic (bottom) and parasym-pathetic part (top) where µ defines the borderline between sympathetic andparasympathetic regulation.

a branch for regulation of the peripheral resistance as proposed by Akker-man [32]. This regulation is based on the sympathetic part of the hypotheticneural barosignal and the implementation is straightforward (Figure 3.5).Put in equations, the baroreflex model can be stated as follows:

PT1

in

out

+

+

-

+

+

scaling

PT1

+

scaling

cB

β

cB

cB

SB

I

Rρ

PR

B

Figure 3.5: Simplified baroreflex model (based on Katona [34] and Akker-man [32]): the two branches regulating heart rate have been merged to one,a second branch has been added for regulation of peripheral resistance.

20

B(k) = PB(k) + kPPP (k)− kB (3.14)

=1

3PS(k) +

2

3PD(k)− σB sinα(k) + kPPP (k)− kB (3.15)

BS(k) = min(B(k), Bc) (3.16)

JBI(k) = e−1τBI JBI(k − 1) +

(1− e

−1τBI

)B(k) (3.17)

JBR(k) = e−1τBR JBR(k − 1) +

(1− e

−1τBR

)BS(k) (3.18)

I(k) = (JBI(k) +Bc)β (3.19)

R(k) = (Bc − JBR(k))ρ+RP (3.20)

Please note, that this baroreflex model is a simplification of Akkerman’smodel. 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 thetilt-table influences the cardiovascular system and the physiological variables.We are only interested in the gravity component Fg along the body axis whichis

Fg = g · sinα. (3.21)

Based on that, the first model input u1(k) can be stated as follows:

u1(k) = sinα (3.22)

Gravity acts on every single blood vessel in the cardiovascular system andcreates rather large hydrostatic pressure differences in a standing human.Arterial pressure is decreased by 25 mmHG at head level and increased by95 mmHG at leg level [7]. The question arises how gravitational forces can beintegrated into the existing hemodynamic model. It is chosen to let gravitydirectly affect the right and left atrial pressures PR(k) and PL(k) which isa mathematically convenient alternative to modelling the whole hydrostaticcolumn [32]. The atrial pressures then depend on the gravity factors ζR(k)and ζL(k):

PR(k) =(VV (k − 1)− VV 0)ζR(k)

CV(3.23)

PL(k) =(VP (k − 1)− VP0)ζL(k)

CP(3.24)

ζR(k) = 1− σR sinα(k) (3.25)

ζL(k) = 1 + σL sinα(k) (3.26)

21

Besides the atrial pressures, also the mean arterial pressure at the level ofthe baroreceptors PB(k) has to be corrected for the gravity influence. Thereason is the height difference between the baroreceptors and the heart, wherearterial pressure is evaluated.

PB(k) =1

3PS(k) +

2

3PD(k)− σB sinα(k) (3.27)

3.4 Influence of stepping

Akkerman’s model does not contain a component which describes the effectsof 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 themuscle pump through continuous leg movements. This has the followingthree immediate effects:

• Compression of the venous leg compartments leads to an increase ofperipheral resistance.

• The contracting skeletal muscles decrease expandability of the venousvessels 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 thehypothetic pressure level at which neural regulation is switched fromparasympathetic to sympathetic action is increased.

Although the stepping mechanism moves the legs passively and we can onlyspeak of a “passive” muscle pump, the stabilising effects on the cardiovascularsystem are still present, although diminished. Czell et al. [35] have conclu-ded after their pilot study with healthy adults, that passive leg movementsstabilises blood circulation and prevents from syncopes. So fortunately, thestabilising effects on the cardiovascular system are still there and can be ex-ploited in the early rehabilitation process of neurological patients.The above listed effects are transformed to mathematical equations so thatthey can take influence on the existing cardiovascular model of Akkerman.As stepping is the second input after the inclination angle, u2(k) will be theexpression for the normalised stepping frequency:

u2(k) =fstep(k)

fstep,max(3.28)

22

where fstep,max is normally 48 stepsmin

. As it takes some time for the cardiovas-cular system to adapt to the stepping movements, u2(k) has to be modelledas a first-order system with the time constant τstep which is usually chosenaround 40 beats. In addition, the stepping influence at supine position hasexperimentally been found to be very low (figure 3.11, first 10 minutes).Thus, it is easiest to make u2(k) linearly dependent on u1(k). The adaptedstepping input is denoted as κ(k):

κ(k + 1) = e−1τstep κ(k) +

(1− e

−1τstep

)u2(k)u1(k) (3.29)

κ(k) now operates in an additive nature on peripheral resistance, venouscompliance and the neural barosignal:

R(k) = (Bc −BS(k))ρ+RP + kSRκ(k) (3.30)

CV (k) = CV + kSCκ(k) (3.31)

B(k) = PB(k) + kPPP (k)− kB − kSBκ(k)

=1

3PS(k) +

2

3PD(k)− σB sinα(k) + kPPP (k)− kB − kSBκ(k)

(3.32)

23

3.5 Model simulations

The cardiovascular model can now be used to simulate and analyse heart rateand blood pressure in response to various inputs. In addition, it is possibleto investigate other cardiovascular signals such as stroke volume, peripheralresistance or cardiac output. In order to get an idea for what happens in thebody during a tilt manoeuvre, a standard fast tilt-up and tilt-down shouldbe examined first. Simulations have been done with standard steady-statevalues as given in table 3.1. These values were used in combination with a setof 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◦, fstep = 0); “+” stands for tilted position(α = 76◦, fstep = 0); “s” stands for stepping (α = 76◦, fstep = fstep,max)

Steady-state valueHR− 65HR+ 80P−S 120P+S 125P−D 80P+D 95

HRs 75P sS 130

P sD 95

3.5.1 Fast tilt-up and tilt-down

The adjective “fast” refers to the fact that the tilt-table angle α changes fromthe minimal angle of zero degrees to the maximal angle of 76 degrees in twoor three heart beats (vice versa for tilt-down). Of course, this is not feasiblein reality where a full tilt may take up to 30 seconds. However, it is a goodway to analyse the dynamics of such a fast tilt, which probably would notbe that pronounced when tilting at a slower rate.

Fast tilt-upModel responses with the most important physiological variables are depictedin 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

racteristics will be explained in the following paragraph about fast tilt-downsimulation. The reason is that tilt-down responses are usually much fasterthan tilt-up responses and that the dynamic features are easier to identifyand explain.

Fast tilt-downWhen a person is tilted from the initial upright position back to the supineposition, blood is shifted in the body under the influence of gravity. This hastwo immediate effects:

• Blood in the pulmonal pathways is shifted into the lung reservoir cau-sing a lack of blood in the left atrium.

• Blood from the venous reservoir is forced back to the right atrium andvenous return is increasing rapidly.

The first effect leads to a fast decrease in arterial blood pressure and leftstroke volume. As the blood supply in the left atrium is abruptly dimi-nished, the left stroke volume is immediately decreased according to theFrank-Starling law. This process is visible in the simulated model responsesas the initial negative peak in blood pressure, left stroke volume and leftatrial pressure.The second effect causes an immediate rise of right stroke volume in responseto the increased venous return. After some time, this extra blood volumehas made its way through the pulmonal pathways and ends up in the leftatrium. This in turn causes the left stroke volume to rise again and leadsto the positive blood pressure peak 7 to 8 seconds after the start of the tiltmanoeuvre.In a third phase the phyisological signals settle to their steady-state valueswhich is the case after approximately 20 seconds. Left and right strokevolume are balanced and the above description nicely shows how the Frank-Starling mechanism enables the adjustment of left and right stroke volumeaccording to respective ventricle load.

3.5.2 Stepping

As the stepping influence on the cardiovascular system is biggest when thetable is fully tilted, only the simulation results for α = 76◦ are shown (Fi-gure 3.8). The heart rate shows the expected non-minimum phase behaviouras described by [6], the diastolic blood pressure is hardly influenced and the

25

systolic blood pressure rises, as described by [5]. The barosignal shows the in-verse behaviour of the heart rate, which makes perfect sense as the barosignaldirectly determines heart rate. Peripheral resistance is decreased when step-ping is activated which can be compared to the adaptation of the peripheralresistance to exercise. The increase of the stroke volumes and the pulse pres-sure point out the stabilising effect of stepping on the cardiovascular system.

3.5.3 Quasi-static

The reason for a quasi-static simulation of the cardiovascular model is theanalysis of the steady-state behaviour of heart rate and blood pressure atall angles α in the admitted range. Only the angle input is considered forthis simulation because the stepping acts smoothly on the outputs whereasthe system is expected to show rather different behaviour in the sympatheticand the parasympathetic region respectively. Remember that although thebaroreflex regulation on heart rate is active over the whole range, periphe-ral resistance is only influenced by sympathetic regulation (see section 3.2).Figure 3.9 depicts the dependencies of the relevant cardiovascular variableson 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 baroreceptorsignal which is based on the arterial pressure at the level of the baro-receptors.

• Systolic blood pressure both increases and decreases at lower angles,and strictly increases at higher angles:Systolic blood pressure PS is calculated as the sum of diastolic bloodpressure PD and pulse pressure PP . At small angles, stroke volumesdon’t change much, but PD is increased. This leads to the increasein PS at small angles. However, as soon as the stroke volumes andsubsequently the pulse pressure is decreased, PS is decreased as well.In the sympathetic regulation domain, PS strictly increases because PDgrows faster than PP declines.

• Diastolic blood pressure strictly increases with α, but at a lower rateat lower angles:The reason is that at higher angles peripheral resistance is increasedby the baroreflex which leads to higher arterial pressures.

• Peripheral resistance by design only increases at higher angles, whensympathetic regulation becomes active.

26

• Stroke volumes are diminished when tilting.

Figure 3.10 compares the results from the quasi-static simulation with resultsfrom Hainsworth [10], Matalon [36], Heldt [28], Fisler [37] and Nguyen [5].It can be deduced that the accordance of the model results with literaturestudies and previous work at SMS is satisfying.

27

0 50 1000

20

40

60

80

Heart beats

Ang

le[d

eg]

0 50 10060

70

80

90

Heart beats

Hea

rt r

ate

[bpm

]

0 50 100

80

100

120

140

Heart beats

Blo

od p

ress

ure

[mm

HG

]

0 50 10050

60

70

80

90

Heart beats

Bar

osig

nal

[mm

HG

]

0 50 1002

3

4

5

Heart beats

Atr

ial p

ress

ures

[mm

HG

]

0 50 100

40

60

80

Heart beats

Str

oke

volu

mes

[ml]

0 50 1001000

1200

1400

1600

Heart beats

Per

iphe

ral r

esis

tanc

e[m

mH

G m

s/m

l]

0 50 1003.5

4

4.5

5

5.5

Heart beats

Car

diac

out

put

[l/m

in]

PS

PD

PL

PR Q

R

QL

Figure 3.6: Simulation of a fast tilt-up without stepping

28

0 50 1000

20

40

60

80

Heart beats

Ang

le[d

eg]

0 50 10060

70

80

90

Heart beatsH

eart

rat

e[b

pm]

0 50 100

80

100

120

140

Heart beats

Blo

od p

ress

ure

[mm

HG

]

0 50 10050

60

70

80

90

Heart beats

Bar

osig

nal

[mm

HG

]

0 50 1002

3

4

5

Heart beats

Atr

ial p

ress

ures

[mm

HG

]

0 50 100

40

60

80

Heart beats

Str

oke

volu

mes

[ml]

0 50 1001000

1200

1400

1600

Heart beats

Per

iphe

ral r

esis

tanc

e[m

mH

G m

s/m

l]

0 50 1003.5

4

4.5

5

5.5

Heart beats

Car

diac

out

put

[l/m

in]

PR

QR

PS

PD

PL

QL

Figure 3.7: Simulation of a fast tilt-down without stepping

29

0 50 100 1500

20

40

Heart beats

Ste

ppin

g[s

teps

/min

]

0 50 100 15074

76

78

80

82

Heart beats

Hea

rt r

ate

[bpm

]

0 50 100 150124

126

128

130

132

Heart beats

Sys

tolic

blo

od p

ress

ure

[mm

HG

]

0 50 100 15050

55

60

65

Heart beats

Bar

osig

nal

[mm

HG

]

0 50 100 15090

95

100

Heart beats

Dia

stol

ic b

lood

pre

ssur

e[m

mH

G]

0 50 100 15050

55

60

65

Heart beats

Str

oke

volu

mes

[ml]

0 50 100 1501400

1450

1500

1550

Heart beats

Per

iphe

ral r

esis

tanc

e[m

mH

G m

s/m

l]

0 50 100 1504

4.2

4.4

4.6

Heart beats

Car

diac

out

put

[l/m

in]

Figure 3.8: Simulation of an activation of the stepping mechanism (α = 76◦)

30

0 500 1000 15000

20

40

60

80

Heart beats

Ang

le[d

eg]

0 500 1000 1500

65

70

75

80

Heart beatsH

eart

rat

e[b

pm]

0 500 1000 1500118

120

122

124

126

Heart beats

Sys

tolic

blo

od p

ress

ure

[mm

HG

]

0 500 1000 150040

60

80

100

Heart beats

Bar

osig

nal

[mm

HG

]

0 500 1000 150070

80

90

100

Heart beats

Dia

stol

ic b

lood

pre

ssur

e[m

mH

G]

0 500 1000 150050

60

70

80

Heart beats

Str

oke

volu

mes

[ml]

0 500 1000 15001200

1300

1400

1500

Heart beats

Per

iphe

ral r

esis

tanc

e[m

mH

G m

s/m

l]

0 500 1000 15004

4.2

4.4

4.6

4.8

Heart beats

Car

diac

out

put

[l/m

in]

Figure 3.9: Quasi static simulation without stepping

31

0 20 40 60 80−2

0

2

4

6

8

10

12

14

16

18

20

Angle [deg]

∆ H

R [b

pm]

0 20 40 60 80−2

0

2

4

6

8

10

12

14

16

18

20

Angle [deg]

∆ dB

P [m

mH

G]

HainsworthMatalonFislerModel

HainsworthSmithModel

Figure 3.10: Comparison of steady-state behaviour. Left: HR as a functionof α. Right: Diastolic BP as a function of α

32

3.6 Model validation

The step of model validation will be performed using measurements fromthree healthy subjects (see chapter 5.1.3). Evaluation will be done in aqualitative way analysing each measurement separately. Although averagingover all subjects would probably yield better agreement between the modelsimulation and the measurement, interesting details from the individual caseswould be lost.The measurement was divided into an identification and a validation part.The according measurement protocol is illustrated in the lowermost plot offigure 3.11. Note that between the identification and the validation there wasa recalibration of the blood pressure measurement device. This can introduceoffsets in some cases whereas diastolic blood pressure seems to be affectedthe most. In figure 3.11 for example, this offset amounted to about 4 mmHgand has been corrected accordingly.Validation results for the first subject (MW) are satisfying and it demons-trates that it is possible to simulate or predict heart rate and blood pressuredynamics.

0 10 20 30 40 50 60 70

60

80

Time [min]

HR

[bpm

]

0 10 20 30 40 50 60 7080

100

120

Time [min]

sBP

[mm

HG

]

0 10 20 30 40 50 60 7040

60

80

Time [min]

dBP

[mm

HG

]

0 10 20 30 40 50 60 700

50

Time [min]

α [d

eg] /

f step

[ste

ps/m

in]

αfstep

Figure 3.11: Model validation with subject MW

33

However, the identified models for the other two subjects deviate more fromthe measured signal than it was the case for the first subject. This should beanalysed in more detail: For the second subject (figure 3.12) it can be saidthat heart rate and diastolic blood pressure were well reproduced by the mo-del. Systolic blood pressure, however, did not show clear trends: During theidentification phase, systolic blood pressure stayed constant when tilting butincreased in the end of the experiment during the slow ramp of the inclina-tion angle. Another issue are the calibration offsets, that have already beenmentioned above. For the second subject, diastolic blood pressure jumpedby about 10 mmHg which has been corrected for in the modelled diastolicblood pressure curve. Already the low values of about 40 mmHg after 17minutes are unrealistic compared to the baseline values at the beginning ofthe experiment which were around 55 mmHg. The worse thing however isthat after the recalibration the value is not set back to 55 mmHg but is evenincreased to about 65 mmHg. These huge jumps in the measured signalsare physiologically improbable in such a short timespan and it unveils theweaknesses of the blood pressure measurement device.

0 10 20 30 40 50 60 705060708090

Time [min]

HR

[bpm

]

0 10 20 30 40 50 60 7080

100

120

Time [min]

sBP

[mm

HG

]

0 10 20 30 40 50 60 7040

60

80

Time [min]

dBP

[mm

HG

]

0 10 20 30 40 50 60 700

50

Time [min]

α [d

eg] /

f step

[ste

ps/m

in]

αfstep

Figure 3.12: Model validation with subject MSW

34

The validation measurement for the third subject emphasises the above men-tioned problems: First, systolic blood pressure is hard to reproduce or model.Second, diastolic blood pressure measurement is tampered with calibrationoffsets. However, it has to be added that the last measurement is an extremeexample for what can happen with physiological signals.

0 10 20 30 40 50 60 70

60

80

100

Time [min]

HR

[bpm

]

0 10 20 30 40 50 60 7080

100120140160180

Time [min]

sBP

[mm

HG

]

0 10 20 30 40 50 60 70

60

80

Time [min]

dBP

[mm

HG

]

0 10 20 30 40 50 60 700

50

Time [min]

α [d

eg] /

f step

[ste

ps/m

in]

αfstep

Figure 3.13: Model validation with subject DH

Generally, it can be concluded that the model reproduces heart rate anddiastolic blood pressure with a satisfying accuracy. Problems occur, if repro-ducibility is not given, i.e. if the subject responds differently for the sameinputs. However, this is a more general issue as any deterministic modelwould struggle with low reproducibility.In contrast to heart rate and diastolic blood pressure, systolic blood pressureis more difficult to predict and model for healthy subjects. Still, we decidedto go on with the strategy of controlling all three variables, because in theend, the system will be used with patients. Patients usually react much bet-ter with systolic blood pressure to verticalisation because neural regulationis impaired.To counter the problem with calibration offsets, it will be important not to

35

do a calibration between identification and control or to reidentify the modelwhen a calibration is necessary. If the offset is only small, it may not be nee-ded to reidentify the model because for control only the relative responsesare important. However, if the offset is too high and the new values areout of the identified range, it gets impossible to start the control experimentwithout adapting the model.

36

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 beable to keep heart rate and blood pressure within reasonable bounds andminimise fluctuations by adjusting the inclination angle and the steppingfrequency. For the given MIMO system which is apparently strongly coupled,a SISO approach trying to control each output in an isolated manner seemsinfeasible. The fact that the system has less inputs than outputs makes iteven harder to do so. It is therefore advisable to choose a control strategywhich can handle these issues. A linear optimal control approach is frequentlyused in advanced control applications, and has been chosen for this thesis aswell. Two controllers have been developed, implemented and tested: a LinearQuadratic Regulator (LQR) and a Model Predictive Controller (MPC).The LQ controller which was augmented by an integral part in order toeliminate steady-state control errors was experimentally found to be veryhard to tune. The reason is that the system is likely to operate near or on theconstraints boundaries for the control inputs. As a result, control inputs areoften saturated and anti-windup strategies are therefore necessary. However,for a true MIMO system with strong couplings as has been developed in theprevious chapter it is difficult to apply standard anti-windup techniques. It istherefore desirable to have a technique which intrinsically accounts for inputconstraints. This explains why Model Predictive Control (MPC) is suited forthe 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 mostimportant features are that MPC takes account of actuator limitations and

37

that it is suited for multivariable control problems. The principle behindMPC is as follows: based on the system model the controller predicts futureoutputs and finds the optimal control inputs by minimising a certain costfunction. An intuitive analogon for MPC is driving a car [38]. Imaginethat the reference path is the lane, the plant is the car and the controller isrepresented by the driver. The control objective is to keep the car on thelane, while steering as little as possible, keeping a certain distance to thekurbs, obey speed limitations and so on. The driver now has an internalbelief or model of how the car reacts to his inputs. He uses this knowledgeto predict future behaviour of the car and give according control inputs inorder to stay on the reference path, minimise steering effort and meeting allgiven constraints.This control problem can generally be formulated with a cost function andaccording constraints [39].

min∆u

J = (rs − yp)TQ(rs − yp) + ∆uTR∆u (4.1)

M∆u ≤ γ (4.2)

where ∆u(mNc× 1) is the control input, yp(pNp × 1) the predicted output,rs(pNp × 1) the reference, Q(pNp × pNp) the output weighting matrix andR(mNc×mNc) the control input weighting matrix. The matrix M(4mNc×mNc) and the vector γ(4mNc × 1) define the constraints. The scalars n,m and p are the number of states, the number of inputs and the number ofoutputs of the MIMO system. The control horizon is denoted asNc. Table 4.1summarises these notations. As we will only need constraints on the controlinputs, state and output constraints are neglected in this formulation. Notethat the vectors rs and yp contain Np discrete samples over the predictionhorizon and the vector ∆u contains Nc discrete samples over the controlhorizon:

∆u =(∆u(k) ∆u(k + 1) ∆u(k + 2) · · · ∆u(k +Nc − 1)

)T(4.3)

yp =(yp(k + 1) yp(k + 2) yp(k + 3) · · · yp(k +Np)

)T(4.4)

rs =(rs(k + 1) rs(k + 2) rs(k + 3) · · · rs(k +Np)

)T(4.5)

Based on the above given description, a model predictive controller will nowbe developed for the given nonlinear cardiovascular model of chapter 3. Fi-gure 4.1 sketches the signal flows of the control system and shows the twomajor parts of the controller which are the optimisation routine and the stateobserver.

38

Table 4.1: MPC glossary.

Variable Value Description

Np 5 Prediction horizonNc 2 Control horizonTu 20 s Controller sample timeNu - Number of heart beats in Tun 10 Number of statesm 2 Number of inputsp 3 Number of outputsr - Reference signalQ Eq. 4.20 Weighting matrix for statesR Eq. 4.21 Weighting matrix for control actions

x,y,u - Non-linear state, output, inputxlin,ylin,ulin - State, output, input of linearised model

x, y, u - State, output, input of augmented linearised model

State observer

The state observer is needed because with the exception of the heart inter-val the system states are not measurable. The states are updated in eachtime step with the non-linear cardiovascular model equations and are thencorrected in a second step based on the error between the observed and mea-sured outputs. Basically, this is nothing else than a traditional Kalman filterdoing first a prediction update followed by a measurement update. The onlydifference is that the states are updated in a non-linear way. Algorithm 1describes the state observer in pseudo code.

Algorithm 1 State observer

1: if x is not defined (start of control experiment) then2: x = baseline values3: end if4: Save previous state estimate xold = x5: Nonlinear prediction update: [y x] = f(x,u)6: Measurement update: x = x+ Kob(y − y)7: Output ∆x = x− xold

The baseline values x describe the system state at the supine position withoutstepping (u = (0 0)T ). x is calculated during parameter identification asa byproduct (appendix A.5). f(x,u) denotes the non-linear cardiovascular

39

model (appendix A.3).The observer gain Kob is calculated in a stochastically optimal way based onthe linearisation about the current set-point where w(k) is the process noiseand v(k) the measurement noise (eq. 4.6).

xlin(k + 1) = Alinxlin(k) + Blin(ulin(k) +w(k)) (4.6)

ylin(k) = Clinxlin(k) + v(k)

Values for the entries of the diagonal covariance matrices W = E(wwT )and V = E(vvT ) are given in Table 4.2. Measurement noise was calculatedbased on the assumption that each output (HR, PS, PD) has a standarddeviation of 2 bpm or 2 mmHg respectively. Process noise was set basedon experimental findings, such that the estimated outputs converged to themeasured outputs and noise rejection was satisfying. Note that the high valueof V(1, 1) is explained by the fact that the first component of ylin is the R-Rinterval which has much higher nominal values than the other componentsof ylin.

Table 4.2: Entries of covariance matrices W and V

Entry ValueV(1, 1) 0.03252

V(2, 2) 0.01672

V(3, 3) 0.02502

W(1, 1) 0.042

W(2, 2) 0.042

MPC

Observer

Minimise cost

function

Cardiovascular model

-+ x∆

Sr

u y

Figure 4.1: MPC overview

40

Optimisation routine

The prediction or the optimisation is based on system 4.6 without noise:

xlin(k + 1) = Alinxlin(k) + Blinulin(k) (4.7)

ylin(k) = Clinxlin(k)

However, instead of equations 4.7 an augmented state-space model, contai-ning an additional integrator, will be used for the prediction (equations 4.8).This has the advantage that the current control error ylin(k) is included inthe description which penalises deviations from the set-point.

Augmented state-space model:(∆xlin(k + 1)ylin(k + 1)

)=

(Alin 0n×m

ClinAlin Ip

)︸ ︷︷ ︸

=:A

(∆xlin(k)ylin(k)

)+

(Blin

ClinBlin

)︸ ︷︷ ︸

=:B

∆u(k)

(4.8)

y(k) =(0p×n Ip

)︸ ︷︷ ︸=:C

(∆xlin(k)ylin(k)

)︸ ︷︷ ︸

=:x(k)

where I denotes the identity matrix and 0 the null matrix. We will now derivethe elements of equation 4.1 based on Wang [39]: The state x(k) developsaccording to the augmented state-space model. Note that ∆u(k) = ∆u(k).

x(k + 1) = Ax(k) + B∆u(k) (4.9)

x(k + 2) = Ax(k + 1) + B∆u(k + 1) (4.10)

= A2x(k) + AB∆u(k) + B∆u(k) (4.11)

...

x(k +Np) = ANpx(k) + ANp−1B∆u(k) + ANp−2B∆u(k + 1) + . . . (4.12)

+ ANp−NcB∆u(k) (4.13)

The output at time instant k + i then is:

y(k + i) = CAix(k) + CAi−1B∆u(k) + CAi−2B∆u(k + 1) + . . . (4.14)

+ CAi−NcB∆u(k), i = 1, . . . , Np

The predicted output y(pNp×1) can be written in vector form using F(pNp×

41

n) and Φ(pNp ×mNc):

y =

y(k + 1)y(k + 2)

...y(k +Np)

= Fx(k) + Φ∆u (4.15)

where

F =

CACA2

...CANp

(4.16)

Φ =

CB 0 0 0 · · · 0

CAB CB 0 0 · · · 0CA2B CAB CB 0 · · · 0

......

......

. . ....

CANp−1B CANp−2B CANp−3B CANp−4B · · · CANp−NcB

(4.17)

The cost function at time instant k can now be written as follows:

J(k) = yTQy + ∆uTR∆u (4.18)

= (Fx(k) + Φ∆u)TQ(Fx(k) + Φ∆u) + ∆uTR∆u (4.19)

The weighting matrices Q(pNp×pNp) and R(mNc×mNc) are defined basedon the maximal input and output values (umax, ymax):

Q =

q1 0 0 0 0 0 · · ·...

. . ....

......

... · · ·0 0 qp 0 0 0 · · ·0 0 0 q1 0 0 · · ·...

......

.... . .

... · · ·0 0 0 0 0 qp · · ·...

......

......

.... . .

(4.20)

42

R =

r1 0 0 0 0 0 · · ·...

. . ....

......

... · · ·0 0 rm 0 0 0 · · ·0 0 0 r1 0 0 · · ·...

......

.... . .

... · · ·0 0 0 0 0 rm · · ·...

......

......

.... . .

(4.21)

where

qi =1

y2i,maxi = 1, . . . , p (4.22)

rj =1

u2j,maxj = 1, . . . ,m (4.23)

The above mentioned cost function (eq. 4.18) has to be minimised undersome constraints on ∆u(k) and u(k).

∆umin <= ∆u(k) <= ∆umax (4.24)

umin <= u(k) <= umax (4.25)

These constraints can also be written in matrix form as a function of theoptimisation vector ∆u: (

M1

M2

)∆u ≤

(N1

N2

)(4.26)

where M1(2mNc × mNc) and N1(mNc × 1) define the constraints on theamplitude of the control signal:

M1 =

Im 0m 0m · · · 0m

Im Im 0m · · · 0m

......

.... . .

...Im Im Im Im Im

−Im 0m 0m · · · 0m

−Im −Im 0m · · · 0m

......

.... . .

...−Im −Im −Im −Im −Im

(4.27)

43

N1 =

umax − u(k − 1)umax − u(k − 1)

...−umin + u(k − 1)−umin + u(k − 1)

...

(4.28)

M2(2mNc×mNc) and N2(2mNc×1) define the constraints on the differenceof the control signal and can be written down similarly:

M2 =

(Im·Nc

−Im·Nc

)(4.29)

N2 =

∆umax

∆umax...

∆umin

∆umin...

(4.30)

Finally the objective is to minimise the cost function J(k) subject to thegiven constraints:

min∆u

J(k) = (Fx(k) + Φ∆u)TQ(Fx(k) + Φ∆u) + ∆uTR∆u (4.31)(M1

M2

)∆u ≤

(N1

N2

)In order to incorporate anticipative action or “look-ahead” functionality inthe MPC design, the objective can be reformulated:

min∆u

J(k) = (rs − (Fx(k) + Φ∆u+ ys))TQ(rs − (Fx(k) + Φ∆u+ ys)) + ∆uTR∆u

(4.32)(M1

M2

)∆u ≤

(N1

N2

)where ys(pNp × 1) denotes the non-linear output at the setpoint:

ys =

ys(k)ys(k)

...ys(k)

(4.33)

44

Note that the reference rs is also given with real physiological values for theR-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 pro-blem in the form of equation 4.34. It is left to the reader to verify thatequation 4.32 can be reformulated to equation 4.35 in order to meet thisrequirement.

min∆u

J =1

2∆uH∆u+ fT∆u (4.34)

M∆u ≤ γ

min∆u

J(k) =1

2∆u(ΦTQΦ + R)∆u+ (−ΦTQrs + ΦTQFx(k) + ΦTQys)∆u

(4.35)(M1

M2

)∆u ≤

(N1

N2

)

Adaptations for controller sample time

If the calculated control action is not sent to the plant in every time step, butonly every Nu time steps, the optimisation problem has to be reformulated.This is the case for our plant, where the subject on the ERIGO would feeluncomfortable if control commands were sent in every heart beat. Further,two much motion of both the tilt-table and the stepping mechanism wouldneedlessly activate the cardiovascular system. It is therefore necessary tochoose a controller sample time Tu which is higher than the time needed forone time step of the augmented system 4.8 (i.e. one heart beat). Note, thatthis change does not influence the observer, which still runs at the originalsample time.

The optimisation vector ∆u now has the form

∆u =

∆u(k)

∆u(k +Nu)...

∆u(k +NpNu)

(4.36)

45

The objective is the same as before (equation 4.32), but the prediction ma-trices and the reference vector rs change:

F =

CANu

CA2Nu

...CANpNu

(4.37)

Φ =

∑Nci=1 CANu−iB 0 · · · 0∑Nci=1 CA2Nu−iB

∑Nci=1 CANu−iB · · · 0∑Nc

i=1 CA3Nu−iB∑Nc

i=1 CA2Nu−iB · · · 0...

.... . .

...∑Nci=1 CANPNu−iB

∑Nci=1 CA(NP−1)Nu−iB · · ·

∑Nci=1 CA(NP−Nc+1)Nu−iB

(4.38)

rs =

rs(k)

rs(k +Nu)...

rs(k +NpNu)

(4.39)

Note that Nu is calculated at the beginning of the MPC calculation every Tuseconds based on the current heart rate, i.e. the current R-R interval y1 inmilliseconds:

Nu =1000 · Tu

y1(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-statevalues from table 3.1. Two setpoints around α = 60◦ and α = 30◦ withdeactived stepping have then been calculated. Linearisations and Kalmanobservers were obtained around these setpoints and have then been used forthe controller.Now, the first part of the simulation consists of a reference step from thefirst setpoint (α = 60◦) to the second setpoint (α = 30◦). The controlleranticipates the step as soon as it is included in the prediction horizon whichis 100 s (Np = 5, Controller sample time = 20 s). It can be seen that not onlythe angle is lowered, but also the stepping mechanism is activated because itquickly decreases heart rate.

46

The second part of the simulation depicts the reaction of the controller to adisturbance on the outputs: heart rate is suddenly increased by four beatsper minute, and systolic blood pressure is decreased by four mmHg. Thecontroller reacts by activating the stepping mechanism which is capable tocompensate such a disturbance. As a consequence, the outputs are regulatedback to the reference values.

0 1 2 3 4 5 6 7 8

75

80

Time [min]

HR

[bpm

]

0 1 2 3 4 5 6 7 8115

120

125

Time [min]

Sys

tolic

BP

[mm

HG

]

0 1 2 3 4 5 6 7 880

85

90

95

Time [min]

Dia

stol

ic B

P[m

mH

G]

0 1 2 3 4 5 6 7 80

50

Time [min]

α [d

eg] /

f step

[ste

ps/m

in]

α fstep

Figure 4.2: Controller simulation: The first part depicts the system responseto a step at minute 2. The second part shows the system response to adisturbance in heart rate and systolic blood pressure at minute 5.

47

48

Chapter 5

Methods

5.1 Healthy subjects

5.1.1 Implementation

The healthy subjects were measured at the SMS lab, where the ERIGO devicehas been customised for easy interfacing and data recording. No changes hadto be done on the hardware side. As for the software, the controller could beimplemented into an existing Matlab/Simulink environment that interfaceswith the ERIGO via an xPC target machine. From Matlab, the model istransformed to C code, compiled and loaded on to the target machine. Thisrequires all code in the model to be written in EML-code, which is basicallyMatlab m-code with some restrictions. The restrictions are such, that only Ccompatible code is allowed, which means that variable sizes need to be clearlydefined in advance for example. Furthermore, not all Matlab functions areavailable including the quadratic programming solver “quadprog” and allcontrol system related functions such as “ss” or “kalman” that are neededfor control design. The model predictive controller, which needs a quadraticprogramming solver in order to do the optimisation, has thus been realisedwith the open-source C++ implementation “qpOASES” 1which is based onthe active-set strategy [40]. Control design has been done offline.

5.1.2 Blood pressure recording

Continuous blood pressure recording was done with the non-invasive CNAPTMMonitor500 2 (Figure 5.1). The CNAPTMMonitor outputs the raw blood pressure

1http://www.kuleuven.be/optec/software/qpOASES2http://www.cnsystems.at/product-line/cnap-monitor-500/

49

wave as an analog signal which is fed over a galvanic separation to the xPCtarget. Setting up the measurement system takes a few minutes: when thearm and finger cuffs are adjusted properly, the device is ready to use aftera short calibration phase. The device has to be recalibrated after one hourof measurement, so that accuracy is warranted. Please refer to Kupke [6](p.5-6) for more details about the CNAPTMMonitor specifications.

Figure 5.1: CNAPTMMonitor 500 with arm and finger cuffs: non-invasivecontinuous blood pressure measurement device used for all measurementsand control experiments.

As the CNAP device only outputs the raw blood pressure signal, the systolicand the diastolic blood pressure have to be extracted separately. This is donewith an online peak detection routine extracting maxima (systolic bloodpressure) and minima (diastolic blood pressure) [6]. Once the peaks areidentified, heart rate calculation is performed.

5.1.3 Experimental design

Model validation

Three subjects (2 female, 1 male) aged between 20 and 35 years were measu-red in total for model validation (table 5.1). The measurement protocol wasdefined as illustrated in Figure 3.11. The identification part was designedsuch that both the influence of stepping and the influence of tilting couldbe analysed in supine as well as in tilted position. Note that that when thestepping is activated or deactivated, the waiting time is not only 3 but 5minutes. This is because of the slower dynamics of heart rate and bloodpressure in response to stepping.

50

Table 5.1: Healthy subjects participating in the model validation

MW MSW DHSex m f f

Weight 68 55 58

Control experiments

Five subjects (3 female, 2 male) aged between 20 and 35 years were measuredin total (table 5.2). For each of the five subjects three control experiments,lasting 20 minutes each, were done. The first experiment was isolated heartrate control, the second was blood pressure control (systolic and diastolic),and the last experiment was combined control of all three physiological si-gnals. The control experiments were preceded by an identification phase inorder 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 MESex m m f f f

Weight 90 68 65 63 62

This is done with an shortened identification measurement compared to mo-del validation which lasts 11 minutes in total. First, the baseline values atthe supine position are calculated which is done by taking the average overthe last two minutes before the tilt. Similarly, the steady-state values for thetilted position (α = 76◦) are calculated by taking the mean value of minutes 5and 6 to account for the transient behaviour. Finally, the steady-state valuesfor the stepping are calculated by taking the mean of minutes 10 and 11. Thereason for the longer duration of the stepping part is the slower dynamics ofheart rate and blood pressure in response to stepping, i.e. it takes more timeto 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)

ys =(Is P s

S P sD

)T(5.3)

Figure 5.2 exemplifies the identification protocol in the lowermost plot andthe measured signals together with the simulated model response for subjectPB in the upper three plots.

51

0 2 4 6 8 10 12 1460

70

80

90

Time [min]

HR

[bpm

]

0 2 4 6 8 10 12 14120

140

160

Time [min]

sBP

[mm

HG

]

0 2 4 6 8 10 12 14

60

80

Time [min]

dBP

[mm

HG

]

0 2 4 6 8 10 12 140

50

Time [min]

α [d

eg] /

f step

[ste

ps/m

in]

Figure 5.2: Model identification (subject PB). The lowermost plot shows theidentification protocol. The upper three plots depict the measured signals(green) together with the simulated model responses (blue).

52

5.2 Patients

5.2.1 Implementation

As the measurements were done at the hospital, the implementation had tobe adapted to local conditions. The ERIGO at the hospital, which is thestandard version delivered by HOCOMA, did not contain the same inter-faces as the ERIGO at the SMS lab. So, one major issue was that controlcommands had to be given by hand following the numbers on the Matlabdisplay. The inclination angle α was set using a water-level and the steppingfrequency was adjusted using the ERIGO touchscreen. Transferring the cus-tomised ERIGO from the SMS lab at ETH Zurich to Wald would probablyhave 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 bloodpressure data, extracting the physiological signals, and displaying the com-puted control inputs on the laptop screen.

5.2.2 Blood pressure recording

Blood pressure recording was also done with the CNAPTMmonitor just as itwas the case with healthy subjects. However, the blood pressure signal wasfed to a biosignal amplifier (USBamp from g.tec 3: figure 5.3), from where itcould be routed to the laptop over a standard USB connection.

Figure 5.3: Biosignal amplifier USBamp from g.tec.

3http://www.gtec.at

53

5.2.3 Experimental design

One patient was measured at the “Zurcher Hohenklinik” in Wald. Just likewith the healthy subjects, the unknown model parameters had to be calcula-ted after the initial identification phase. Afterwards, isolated control of eitherheart rate or blood pressure has been done. Different from the measurementswith healthy subjects, control with patients was done with only one setpoint.One reason is that patients usually have large drifts in physiological valuesand the model has to be readjusted every now and then. Another reason isthat two setpoints only make sense to test the step response of the controlsystem, but is meaningless for stabilising the cardiovascular system of a neu-rological patient. From a clinical point of view, it is only important to controlto one setpoint which may be defined by the attending doctor. Furthermore,the system’s step response has already been tested with healthy subjects.

54

Chapter 6

Results

This chapter describes the results of the control experiments done at theSMS lab with healthy people and with neurological patients at the “ZurcherHohenklinik” in Wald.

6.1 Healthy subjects

6.1.1 Heart rate control

Figure 6.1 shows the results from the heart rate control experiment. Thesubject showed the standard physiological reactions to tilting and stepping:heart rate has risen with increasing tilt angle α and decreased in the longrun with active stepping.At the beginning it took some time until the heart rate had risen to thedesired value. This is not only because of the heart rate dynamics but alsobecause the controller increased the inclination angle rather slowly due tohigh values in the weighting matrix R. The next interesting feature is theactivation of the stepping prior to the reference step. This happens becauseof the anticipative nature of the model predictive controller which detectsthe reference step in advance and includes it in the optimisation procedure.When the step is there after 10 minutes, the controller switches the state-space descriptions from the linearisation around the first set-point to thelinearisation around the second set-point. It is satisfying to see that thecontroller is robust to this immediate switch.

6.1.2 Blood pressure control

The blood pressure control experiment that is plotted in Figure 6.2 has beendone with a different subject than the heart rate control. However, this sub-

55

0 2 4 6 8 10 12 14 16 18 20

60

65

70

75

80

85

Time [min]

HR

[bpm

]

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

Time [min]

α [d

eg] /

f st

ep [s

teps

/min

]

αfstep

Figure 6.1: Healthy subjects (MW): heart rate control.

ject also showed the standard physiological reactions to tilting and stepping:Systolic and diastolic blood pressure increased with the tilting angle, thestepping mechanism mainly increased systolic blood pressure and had littleeffect on diastolic blood pressure.The slow rise of the tilting angle in the beginning of the experiment is analogto the previous experiment with heart rate control. After 3 minutes, thestepping was activated because systolic blood pressure was still too low andthe angle was already quite high. As a result, systolic blood pressure increa-sed and reached the desired reference value. After 5 minutes, there was ahuge drop in systolic blood pressure which is probably due to a detectionerror or a movement artefact 1. It is good that the controller was robust tothis drop and did not produce severe counteractions. In the second part ofthe experiment, control performance is very satisfying and combined controlof systolic and diastolic blood pressure seems feasible.

6.1.3 Combined heart rate and blood pressure control

Again, for the last experiment, a different subject has been chosen becausethe according results nicely illustrates performance of combined control. Thesubject showed normal physiological responses to tilting and stepping as it

1The artefact has been cut off for calculation of mean and standard deviation.

56

0 2 4 6 8 10 12 14 16 18 20

100

120

140

Time [min]

sBP

[mm

HG

]

0 2 4 6 8 10 12 14 16 18 20

50

60

70

80

Time [min]

dBP

[mm

HG

]

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

Time [min]

α [d

eg] /

f step

[ste

ps/m

in]

fstep

α

Figure 6.2: Healthy subjects (PB): blood pressure control.

has also been the case for the previous two subjects.Figure 6.3 reveals that combined control of heart rate and blood pressure isdifficult. It can be seen that after a few minutes a quite stable situation hasbeen built up where the heart rate was too low and systolic blood pressurewas too high. Such a situation can not be handled by the controller. Anincrease of the inclination angle α would make sense for heart rate, but alsofurther increases systolic blood pressure. Similarly, a decrease of the anglewould not lower the errors, either. The angle therefore settles at a valuewhere the errors in heart rate and systolic blood pressure are minimal. Anincrease of the stepping frequency makes even less sense because heart ratewould be further decreased and systolic blood pressure further increased.

6.2 Patients

One patient had the opportunity to experience the latest kind of ERIGOtherapy. Based on the above findings, priority was set to blood pressurecontrol because from a clinical point of view it is more important than heartrate. Figure 6.4 shows a blood pressure control experiment of a neurologicalpatient lasting 30 minutes. The patient showed the standard pathophysio-

57

0 2 4 6 8 10 12 14 16

60

80

Time [min]

HR

[bpm

]

0 2 4 6 8 10 12 14 16

100

150

Time [min]

sBP

[mm

HG

]

0 2 4 6 8 10 12 14 16

60

80

Time [min]

dBP

[mm

HG

]

0 2 4 6 8 10 12 14 160

50

Time [min]

α [d

eg] /

f step

[ste

ps/m

in]

α fstep

Figure 6.3: Healthy subjects (ME): combined heart rate and blood pressurecontrol.

logical reaction which means that blood pressure dropped with increasingtilting angle and rised with active stepping. These tendencies are visible infigure 6.4 during the initial identification phase.Already at the beginning of the experiment the tendency of the systolic bloodpressure to fall when tilting is clearly visible. The controller reacts to thisblood pressure drop by lowering the tilt angle and activating the steppingmechanism. After 20 to 21 minutes, systolic blood pressure has been sta-bilised again and even exceeds the reference value. That is the reason whythe stepping frequency is decreased a little in order to compensate for this.As the experiment went on, stepping frequency was steadily increased whichindicates that the the patient’s cardiovascular system got more unstable overtime and blood pressure would have fallen without intervention. At the endof the experiment, there is a sudden drop in blood pressure which is probablybecause of the doctor and therapists that entered the room at that time andset the patient under stress. The controller reacted accordingly and triedto compensate this by increasing the stepping frequency and lowering thetilting angle.

58

0 5 10 15 20 25 30 35 40 45

100

110

120

Time [min]

sBP

[mm

HG

]

0 5 10 15 20 25 30 35 40 4565

70

75

80

85

90

Time [min]

dBP

[mm

HG

]

0 5 10 15 20 25 30 35 40 450

20

40

60

80

Time [min]

α [d

eg] /

f step

[ste

ps/m

in]

α

fstep

Figure 6.4: Patients: blood pressure control. The first 14 minutes depict theidentification phase. The control experiment starts at minute 15.

59

6.3 Controller performance

Table 6.1 summarises mean errors µe and standard deviations σ for all controlexperiments with healthy subjects and table 6.2 presents the results for themeasured patient.

Table 6.1: Healthy subjects: controller performance

Heart rate [bpm] Blood pressure [mmHg]µe σ sBP: µe sBP: σ dBP: µe sBP: σ

MW -0.61 3.05 -1.35 1.92 -2.17 2.11PB -0.26 4.36 1.81 3.09 -0.98 3.15LB 0.21 3.18 7.37 2.76 -2.12 2.98RR -0.73 2.44 3.64 2.18 -2.26 2.44ME -2.26 1.98 5.76 3.63 -3.19 2.01

Mean -0.73 3.00 3.45 2.72 -2.14 2.54

Table 6.2: Patients: controller performance

Blood pressure [mmHg]sBP: µe sBP: σ dBP: µe dBP: σ

Patient -1.15 3.56 -2.53 2.73

60

Chapter 7

Discussion

The goal of this thesis was to develop a new physiological controller for heartrate, systolic and diastolic blood pressure. This controller has been testedin different configurations 1 with healthy subjects. However, the final goal isto use it both for patients with orthostatic hypotension in order to stabilisetheir cardiovascular system and minimally conscious patients in the contextof the AWACON project 2. So, the first step consisted of developing a modelof the human cardiovascular system that is not only capable of capturing thehealthy physiological response to tilting and stepping, but also the patho-physiological response. The pathophysiological response to tilting is usuallycharacterised by a systolic blood pressure decrease when tilting which can becompensated for by activating the stepping mechanism. This behaviour iscompatible with the introduced model. However, there are responses wherethe model can not be fitted to the identified steady-state values, not even byadjusting the fixed parameters described in table A.2. One example is whenheart rate decreases with rising tilting angle. Such a cardio-inhibitory res-ponse has indeed been measured with one of the patients, and consequentlyit was not possible to fit the model and do a control experiment. No othersituations have been encountered where a model fit would not have beenpossible. Nevertheless, in some cases the identified steady-state values hadto be slightly adjusted as not all model conditions (appendix A.6) were ful-filled. In the development phase of the model, some efforts have been madeto better adapt the model to the pathophysiological tilt response. One ideawas to lessen or even cut the sympathetic influence in the baroreflex modelin order to account for the diminished sympathetic actions on the efferentpathways. First simulation results were satisfying, but the idea was dropped

1heart rate control; blood pressure control (systolic and diastolic); combined control ofheart rate and blood pressure

2http://www.sms.mavt.ethz.ch/research/projects/awacon

61

for the sake of consistency and because the standard model already capturedthe described pathophysiological response.

Based on the model, an appropriate controller has been developed. Onlylinear control has been considered, although there has once been the idea oflinearising around the current state and not about the current setpoint whichwould have been a kind of non-linear control (exact linearisation). Due tostability reasons, this thought was not pursued. Furthermore, as the systemis expected to operate near the setpoint, the linearisation will not deviatemuch from the non-linear model.A more important issue is observer performance. It was experimentally foundthat the observer reliably converges to the actual values of heart rate andblood pressure. This has been verified by comparing the estimated out-put values with the measured ones. Despite of that, problems can occur,if one or more of the three output signals leave the identified range. Smallovershoots are tolerated. However, if these overshoots become too large thenon-linear model can become unstable and subsequently, the non-linear pre-diction update of the observer diverges as well and can not be stabilised bythe measurement update anymore (see chapter 4.1). In order to avoid thatthe simulations aborts, the estimated states are reset when the system movesout of bounds. It is clear, however, that this does not prevent the system todiverge again unless heart rate and blood pressure come back into the iden-tified range. In addition, the state resetting can lead to jumps in the controlsignals which is definitely unwanted. Thus, if the measured blood pressuredrifts away too much, the best thing to do is aborting the simulation, recali-brating the blood pressure device and identify the model with new baselinevalues. Note that the response, i.e. the difference between maximal andminimal values, usually stays the same. So for the new model identification,only the baseline values need to be updated.

The experiments with isolated control of heart rate yielded acceptable re-sults based on the performance values in table 6.1 and visual inspection ofthe results. Figure 6.1 shows that especially during the second part of the ex-periment, controller performance for isolated heart rate control is satisfyingalthough the inclination angle α was in average quite low and sometimes insaturation. Still the controller was able to regulate the heart rate to thedesired value because the control bandwith is successively enlarged by thestepping control input.Based on the control experiment with subject PB (figure 6.2) it can be statedthat isolated blood pressure control worked well. However, the performancevalues of table 6.1 reveal that in average the control performance was worse.

62

Still, it must not have been expected that systolic and diastolic blood pres-sure control will work as the results from model validation, concluding thatsystolic blood pressure is hard to predict and model, were not encouraging.Fortunately, for most healthy subjects, there is a general increasing tendencyof systolic blood pressure in response to tilting and stepping. As long as thesetendencies stay the same during the control experiment and systolic bloodpressure stays in the identified range, blood pressure control seems possiblenot only for patients but also for healthy subjects.Combined control of all three physiological signals was shown to be diffi-cult in the previous chapter. In our opinion, this is not believed to onlyoriginate from the fact that systolic blood pressure is hard to model as fullblood pressure control has been shown to work properly. Rather, the pro-blem must be attributed to the control system structure: the smaller theinput to output ratio, the harder it becomes to control the system. Imagine,that if there was a third control input lowering systolic blood pressure forexample, combined control would be easier. Further, it has to be emphasisedthat the plant is a biological system which is definitely subject to many moreinfluences than just tilting and stepping. A higher plant-model mismatchmust therefore be assumed in comparison to technical systems which behavemore deterministically. Hence, it actually had to be expected that for oursystem the input to output ratio of 2 to 3 was already too small causing thecontrol system to struggle. As already mentioned above, control performancecould be improved by adding control inputs which ideally act separately onthe three outputs. It will have to be investigated to what extent heart rateand blood pressure can be influenced by auditory, visual, gustatory and ol-factory stimuli. A second idea is to control heart rate and mean arterialpressure instead of all three physiological signals. Small adaptations of themodel will allow to test if such a control yields acceptable results. Thirdly,as an alternative, only blood pressure can be controlled, whereas heart rateis observed by the controller. Only if the heart rate goes beyond predefinedboundaries, the according control error is considered by the controller in theoptimisation function. This enhanced blood pressure control strategy wouldprobably yield better performance results compared to standard combinedcontrol. Furthermore, it will still be guaranteed that heart rate is kept nearor inside the predefined safety band.

63

64

Chapter 8

Conclusion and Outlook

Based on the experimental results it can be stated that isolated control ofheart rate and blood pressure (systolic and diastolic) for healthy subjects ispossible. In addition, blood pressure control has also shown to be feasiblewith neurological patients at the hospital. Combined control of heart rate andblood pressure has been tested with healthy subjects, but did not producesatisfying results. Further investigations have to be taken in order to see ifcombined control of mean arterial blood pressure and heart rate is possible.Another possibility would be to adapt the control strategy. One approachfor example is to only control blood pressure, but still monitor heart rate. Ifheart rate then exceeds certain bounds, the controller will take appropriateactions.In the future, it is hoped, that an intelligent system capable of monitoringand manipulating physiological signals will be realised. Such a system wouldbe helpful for stabilising the cardiovascular system of bed-ridden patients,has the potential to reduce medication and improves the overall quality ofrehabilitation.

65

66

List of Figures

1.1 Left: Schematic representation of the ERIGO device with thethree inputs. Right: ERIGO during therapy session. . . . . . . 2

2.1 Schematic representation of the human circulatory system.Adapted from: http://www.ionwave.ca . . . . . . . . . . . . 5

2.2 Blood pressure regulation with the baroreflex loop. . . . . . . 82.3 The muscle pump mechanism stabilises the cardiovascular sys-

tem by efficiently reducing venous blood pooling and increa-sing venous return by repeated contractions of the skeletal legmusculature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1 Inputs and outputs of the cardiovascular model . . . . . . . . 153.2 Simplified representation of the human cardiovascular system

used for model synthesis. Adapted from [32] . . . . . . . . . . 173.3 Hemodynamic timetable describing at which moment of the

heart beat each hemodynamic variable is evaluated. Source:Akkerman [32] . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.4 Katona’s baroreflex model for heart rate regulation [34]. Theneural input signal f(t) is divided in a sympathetic (bottom)and parasympathetic part (top) where µ defines the borderlinebetween sympathetic and parasympathetic regulation. . . . . . 20

3.5 Simplified baroreflex model (based on Katona [34] and Akker-man [32]): the two branches regulating heart rate have beenmerged to one, a second branch has been added for regulationof peripheral resistance. . . . . . . . . . . . . . . . . . . . . . 20

3.6 Simulation of a fast tilt-up without stepping . . . . . . . . . . 283.7 Simulation of a fast tilt-down without stepping . . . . . . . . . 293.8 Simulation of an activation of the stepping mechanism (α = 76◦) 303.9 Quasi static simulation without stepping . . . . . . . . . . . . 313.10 Comparison of steady-state behaviour. Left: HR as a function

of α. Right: Diastolic BP as a function of α . . . . . . . . . . 323.11 Model validation with subject MW . . . . . . . . . . . . . . . 33

67

3.12 Model validation with subject MSW . . . . . . . . . . . . . . 343.13 Model validation with subject DH . . . . . . . . . . . . . . . . 35

4.1 MPC overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2 Controller simulation: The first part depicts the system res-

ponse to a step at minute 2. The second part shows the sys-tem response to a disturbance in heart rate and systolic bloodpressure at minute 5. . . . . . . . . . . . . . . . . . . . . . . . 47

5.1 CNAPTMMonitor 500 with arm and finger cuffs: non-invasivecontinuous blood pressure measurement device used for allmeasurements and control experiments. . . . . . . . . . . . . . 50

5.2 Model identification (subject PB). The lowermost plot showsthe identification protocol. The upper three plots depict themeasured signals (green) together with the simulated modelresponses (blue). . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.3 Biosignal amplifier USBamp from g.tec. . . . . . . . . . . . . . 53

6.1 Healthy subjects (MW): heart rate control. . . . . . . . . . . . 566.2 Healthy subjects (PB): blood pressure control. . . . . . . . . . 576.3 Healthy subjects (ME): combined heart rate and blood pres-

sure control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586.4 Patients: blood pressure control. The first 14 minutes depict

the identification phase. The control experiment starts at mi-nute 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

B.1 Healthy subject MW: heart rate control . . . . . . . . . . . . . 87B.2 Healthy subject PB: heart rate control . . . . . . . . . . . . . 88B.3 Healthy subject LB: heart rate control . . . . . . . . . . . . . 88B.4 Healthy subject RR: heart rate control . . . . . . . . . . . . . 89B.5 Healthy subject ME: heart rate control . . . . . . . . . . . . . 89B.6 Healthy subject MW: blood pressure control . . . . . . . . . . 90B.7 Healthy subject PB: blood pressure control . . . . . . . . . . . 90B.8 Healthy subject LB: blood pressure control . . . . . . . . . . . 91B.9 Healthy subject RR: blood pressure control . . . . . . . . . . . 91B.10 Healthy subject ME: blood pressure control . . . . . . . . . . 92B.11 Healthy subject MW: combined heart rate and blood pressure

control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92B.12 Healthy subject PB: combined heart rate and blood pressure

control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93B.13 Healthy subject LB: combined heart rate and blood pressure

control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

68

B.14 Healthy subject RR: combined heart rate and blood pressurecontrol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

B.15 Healthy subject ME: combined heart rate and blood pressurecontrol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

69

70

List of Tables

2.1 Literature summary. ≈ means no significant change, ↑ meanssignificant increase, ↓ means significant decrease. (Adaptedand completed with HR from [5]) . . . . . . . . . . . . . . . . 12

2.2 Literature summary. ≈ means no significant change, ↑ meanssignificant increase and ↓ means significant decrease . . . . . . 12

3.1 Standard steady-state values used for the model simulations:“−” stands for supine position (α = 0◦, fstep = 0); “+” standsfor tilted position (α = 76◦, fstep = 0); “s” stands for stepping(α = 76◦, fstep = fstep,max) . . . . . . . . . . . . . . . . . . . . 24

4.1 MPC glossary. . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Entries of covariance matrices W and V . . . . . . . . . . . . 40

5.1 Healthy subjects participating in the model validation . . . . . 515.2 Healthy subjects participating in the control experiments . . . 51

6.1 Healthy subjects: controller performance . . . . . . . . . . . . 606.2 Patients: controller performance . . . . . . . . . . . . . . . . . 60

A.1 Cardiovascular model variables. . . . . . . . . . . . . . . . . . 75A.2 Fixed cardiovascular model parameters. . . . . . . . . . . . . . 76A.3 Unknown cardiovascular model parameters. . . . . . . . . . . 77

71

72

73

Appendix A

Model summary

PR(k) =(VV (k − 1)− VV 0)ζR(k)

CV + kSCκ(k)(A.1)

PL(k) =(VP (k − 1)− VP0)ζL(k)

CP(A.2)

PD(n) =VA(k − 1)− VA0

CA(A.3)

VP (k) = VP (k − 1) +QR(k − ξP )−QL(k) (A.4)

VPP (k) = VPP (k − 1) +QR(k)−QR(k − ξP ) (A.5)

VA(k) = VA(k − 1) +QL(k)−QW (k) (A.6)

VV (k) = VV (k − 1) +QW (k)−QR(k) (A.7)

VT = VP (k) + VPP (k) + VA(k) + VV (k) (A.8)

QR(k) = γRPR(k)I(k − 1) (A.9)

QL(k) = γLPL(k)I(k − 1) (A.10)

QW (k) = CA(PS(k)− PV (k))

(1− exp

−I(k)

R(k)CA

)(A.11)

PV (k) =VV (k − 1)− VV 0 −QR(k)

CV + kSCκ(k)(A.12)

PP (k) =QL(k)

CA(A.13)

PS(k) = PD(k) + PP (k) (A.14)

ζR(k) = 1− σR sinα(k) (A.15)

ζL(k) = e−1τL ζL(k) +

(1− e

−1τL

)(1 + σL sinα(k)) (A.16)

PB(k) =1

3PS(k) +

2

3PD(k)− σB sinα(k) (A.17)

74

B(k) = PB(k) + kPPP (k)− kB − kSBκ(k) (A.18)

BS(k) = min(B(k), Bc) (A.19)

JBI(k) = e−1τBI JBI(k − 1) +

(1− e

−1τBI

)B(k) (A.20)

JBR(k) = e−1τBR JBR(k − 1) +

(1− e

−1τBR

)BS(k) (A.21)

I(k) = (JBI(k) +Bc)β + kSRκ(k) (A.22)

R(k) = (Bc − JBR(k))ρ+RP (A.23)

A.1 List of variables

Table A.1: Cardiovascular model variables.

Variable Unit Description

B(k) [mmHg] BarosignalBS(k) [mmHg] Sympathetic barosignalI(k) [ms] R-R intervalJBI [mmHg] Filtered barosignal for regulation of I(k)JBR [mmHg] Filtered symp. barosignal for regulation of R(k)PB(k) [mmHg] MAP at the level of the baroreceptorsPD(k) [mmHg] Diastolic blood pressurePS(k) [mmHg] Systolic blood pressurePP (k) [mmHg] Pulse pressurePR(k) [mmHg] Right atrial pressurePL(k) [mmHg] Left atrial pressurePV (k) [mmHg] Venous pressureQL(k) [ml] Left stroke volumeQR(k) [ml] Right stroke volumeQW (k) [ml] Transported blood volume in peripheryR(k) [mmHg ms/ml] Peripheral resistanceVA(k) [ml] Arterial blood volumeVP (k) [ml] Blood volume in the lungVV (k) [ml] Venous blood volumeζL(k) [-] Gravitational stress factor for PLζR(k) [-] Gravitational stress factor for PRα(k) [deg] Tilt angleκ(k) [-] Influence of stepping

75

A.2 List of parameters

The values in table A.2 are adapted from Akkerman [32]. The followingadjustments with regard to Akkerman have been made:

1. Vact,VV 0,VP0 and VA0 are scaled by weight of the subject (kV = m · 705105

for male subjects, kV = m · 655105

for female subjects.

2. CV , CP and CA have been adjusted to scale stroke volumes, left atrialpressure and right atrial pressure to normal physiological values basedon [7] (p.419, p.420) (Q−

L = Q−R = 70 ml, P−

L = 4 mmHg and P−R =

3 mmHg)

Table A.2: Fixed cardiovascular model parameters.

Parameter Value Unit Description

Vact 1500 · kV [ml] Total active blood volumeVV 0 2950 · kV [ml] Zero pressure volume of veinsVP0 160 · kV [ml] Zero pressure volume of lungVA0 495 · kV [ml] Zero pressure volume of arteriesCV 156.75 [ml/mmHg] Vein complianceCP 174 [ml/mmHg] Lung complianceCA 1.75 [ml/mmHg] Arteries complianceξP 2 [-] Stroke volumes in pulmonary pipelineσB 19 [mmHg] Grav. stress factor for PBσR 0.4 [-] Grav. influence on right atrial pressureσL 0.4 [-] Grav. influence on left atrial pressureβ 6 [ms/mmHg] Baroreflex sensitivity on R-R interval

Bcr, Bc 0.85 [-, mmHg] Baroreflex boundaryτstep 40 [heart beats] Time const.: steppingτL 6 [heart beats] Time const.: left atriumτBI 4 [heart beats] Time const.: B(k)→ I(k)τBR 3 [heart beats] Time const.: BS(k)→ R(k)

Table A.3 summarises the unknown parameters that are calculated duringmodel identification.

76

Table A.3: Unknown cardiovascular model parameters.

Parameter Unit Description

γR [ml/(mmHg ms)] Starling factor rightγL [ml/(mmHg ms)] Starling factor leftRP [mmHg ms/ml] R in parasympathetic regionρ [ms/ml] Baroreflex sensitivity on peripheral resist.kP [-] Pulse pressure contribution to barosignalkB [mmHg] Threshold value for barosignalkSC [ml/mmHg] Stepping: venous compl.kSR [mmHG ms/ml] Stepping: peripheral resist.kSB [mmHg] Stepping: baroreflex resetting

77

A.3 Model equations in non-linear state-space

form

x1(k + 1) = e−1τstep x1(k) +

(1− e

−1τstep

)u2(k)u1(k) (A.24)

x2(k + 1) = e−1τL x2(k) +

(1− e

−1τL

)(1 + σLu1(k)) (A.25)

x3(k + 1) = x3(k) + x9(k)− x2(k) (x3(k)− VP0)γLCP

x5(k) (A.26)

x4(k + 1) = x4(k) + x2(k) (x3(k)− VP0)γLCP

x5(k)−QW (k) (A.27)

x5(k + 1) = x7(k)β +Bcβ (A.28)

x6(k + 1) = x6(k) +QW (k)− (x6(k)− VV 0)(1− σRu1(k))γRx5(k)

CV + kSCx1(k)(A.29)

x7(k + 1) = x7(k)e−1/τBI +

{kP

(x2(k)(x3(k)− VP0)

γLx5(k)

CPCA

)− kB+

(A.30)

1

3

(x2(k)(x3(k)− VP0)

γLx5(k)

CPCA+ (x4(k)− VA0)

1

CA

)+

2

3

((x4(k)− VA0)

1

CA

)− σBu1(k)− kSBx1(k)

}·(1− e−1/τBI

)x8(k + 1) = x8(k)e−1/τSR +

{kP

(x2(k)(x3(k)− VP0)

γLx5(k)

CPCA

)− kB+

(A.31)

1

3

(x2(k)(x3(k)− VP0)

γLx5(k)

CPCA+ (x4(k)− VA0)

1

CA

)+

2

3

((x4(k)− VA0)

1

CA

)− σBu1(k)− kSBx1(k)

}·(1− e−1/τSR

)x9(k + 1) = x10(k) (A.32)

x10(k + 1) = (x6(k)− VV 0)(1− σRu1(k))γRx5(k)

CV + kSCx1(k)(A.33)

78

The peripheral flow QW (k) is defined as follows:

QW (k) = {PS(k)− PV (k)}CA{

1− exp

(−x5(k + 1)

R(k)CA

)}=

{x2(k)x5(k)(x3(k)− VP0)

γLCPCA

+ (x4(k)− VA0)1

CA−[

(x6(k)− VV 0)− (x6(k)− VV 0)(1− σRu1(k))γRx5(k)

CV + kSCx1(k)

]1

CV + kSCx1(k)

}CA{

1− exp

(−x7(k)β −Bcβ

[(Bc − x8(k)) ρ+RP + kSRx1(k)]CA

)}(A.34)

Output equations:

y1(k) = x5(k) (A.35)

y2(k) =1

CA

(x2(k) (x3(k)− VP0) γLx5(k)

CP+ (x4(k)− VA0)

)(A.36)

y3(k) =1

CA(x4(k)− VA0) (A.37)

79

A.4 Steady-state equations in non-linear state-

space form

Steady-state equations: xi(n+ 1) = xi(n)!

= xi

x1 : x1 = u2u1 (A.38)

x2 : x2 = 1 + σLu1 (A.39)

x3 : 0 = x9 − x2 (x3 − VP0)γLCP

x5 (A.40)

x4 : 0 = x2 (x3 − VP0)γLCP

x5 −QW (A.41)

x5 : x5 = x7β +Bcβ (A.42)

x6 : 0 = QW − (x6 − VV 0)(1− σRu1)γRx5

CV + kSCx1(A.43)

x7 : x7 =

{kP

(x2(x3 − VP0)

γLx5CPCA

)− kB+

1

3

(x2(x3 − VP0)

γLx5CPCA

+ (x4 − VA0)1

CA

)+

2

3

((x4 − VA0)

1

CA

)− σBu1 − kSBx1

}(A.44)

x8 : x8 = min(Bc, x7) (A.45)

x9 : x9 = x10 (A.46)

x10 : x10 = (x6 − VV 0)(1− σRu1)γRx5

CV + kSCx1(A.47)

Output steady-state equations:

y1 : y1 = I = x5 (A.48)

y2 : y2 = PS =1

CA·(x2(x3 − VP0)

γLx5CP

+ (x4 − VA0))

(A.49)

y3 : y3 = PD =1

CA(x4 − VA0) (A.50)

80

For x4 and x6 we define the peripheral flow QW :

QW ={P S − P V

}CA

{1− exp

(−x5RCA

)}=

{x2x5(x3 − VP0)

γLCPCA

+ (x4 − VA0)1

CA−[

(x6 − VV 0)− (x6 − VV 0)(1− σRu1)γRx5

CV + kSCx1

]1

CV + kSCx1

}CA{

1− exp

(−x7β −Bcβ

[(Bc − x8) ρ+RP + kSRx1]CA

)}(A.51)

81

A.5 Parameter identification

Part A

Part A is implemented in Matlab as “biq.m”. Procedure is adapted fromAkkerman [32].Vector of unknowns: z = [γR γL kB kP RP ρ]The parameter vector z will be identified from two steady-state measure-ments y+ (tilted) and y− (supine).

ζ+/−R = 1− σR sin(α+/−) (A.52)

ζ+/−L = 1 + σL sin(α+/−) (A.53)

V+/−A = CAP

+/−D + VA0 (A.54)

P+/−P = P

+/−S − P+/−

D (A.55)

Q+/−L = CAP

+/−P (A.56)

Q+/−L = CAP

+/−P (A.57)

Q+/− != Q

+/−W = Q

+/−R = Q

+/−L (A.58)

ε =Q+LI

−

Q−LI

+(A.59)

V+/−PP = ξPQ

+/−R (A.60)

Now, determine V+/−V and V

+/−P :

1 − ζ−Rζ+Rε 0 0

0 0 1 − ζ−Lζ+Lε

1 0 1 00 1 0 1

·V +V

V −V

V +P

V −P

=

VV 0 − VV 0

ζ−Rζ+Rε

VP0 − VP0ζ−Lζ+Lε

Vtot − V +PP − V

+A

Vtot − V −PP − V

−A

(A.61)

P+/−R =

(V+/−V − VV 0)ζ

+/−R

CV(A.62)

82

P+/−L =

(V+/−P − VP0)ζ

+/−L

CP(A.63)

γR =Q−R

P−R I

− (A.64)

γL =Q−L

P−L I

− (A.65)

P+/−V =

(V+/−V − VV 0 −Q+/−

R )

CV(A.66)

P+/−B =

1

3P

+/−S +

2

3P

+/−D − σB sin(α+/−) (A.67)

Baroreflex:

B− =I−

β(1 +Bcr)(A.68)

Bc = BcrB− (A.69)

B+ =I+

β−Bc (A.70)

R+/− =−I+/−

CA· ln

(1− Q

+/−W

CA(P+/−S − P+/−

V )

)−1

(A.71)

RP = R− (A.72)

ρ =R+ −RP

Bc −B+S

(A.73)

where B+S = min(Bc, B

+)

kP =B+ −B− − P+

B + P−B

P+P − P

−P

(A.74)

kB = −B+ + P+B + kPP

+P (A.75)

83

Part B

Part B is implemented in Matlab as “id2.m”. The procedure is based on thenon-linear steady-state equations (appendix A.4).In part B, the stepping parameters kSC , kSR and kSB are identified. Thesteady-state measurement of the output variable ys at maximum tilt angleand with active stepping will be needed. First the steady-state variables x1,x2, x3, x4 and x5 can directly be inferred from the inputs u = (sin(76◦) 1)T

and outputs ys.

x1 : x1 = us2us1 (A.76)

x2 : x2 = 1 + σLus1 (A.77)

y1 : x5 = ys1 (A.78)

y2 : x3 =CACP (ys2 − ys3)(1 + σLus1)γLy

s1

+ VP0 (A.79)

y3 : x4 = CAys3 + VA0 (A.80)

The other steady-state variables and the unknown stepping parameters canbe calculated using the remaining steady-state equations and the total bloodvolume constraint.

x3 : 0 = x9 − CA(ys2 − ys3) (A.81)

x4 : 0 = CA(ys2 − ys3)−QW (A.82)

x5 : ys1 = x7β +Bcβ (A.83)

x7 : x7 = kP (ys2 − ys3)− kB +1

3ys2 +

2

3ys3 − σBus1 − kSBx1 (A.84)

x8 : x8 = min(Bc, x7) (A.85)

x10 : x10 = (x6 − VV 0)(1− σRus1)γRy

s1

CV + kSCx1(A.86)

Volume : Vtot = V V + V P + V PP + V A

= x6 + x3 + ξPx9 + x4 (A.87)

With the definition of QW (A.34), (A.82) becomes

(ys2 − ys3) =

{ys2 −

[(x6 − VV 0)− (x6 − VV 0)(1− σRus1)

γRys1

CV + kSCx1

]1

CV + kSCx1

}·

{1− exp

(−x7β −Bcβ

[(Bc − x8)ρ+RP + kSRx1]CA

)}(A.88)

84

The unknown stepping parameters can now easily be calculated from (A.84), (A.86)and (A.88):

kSB =

(−x7 + kP (ys2 − ys3)− kB +

1

3ys2 +

2

3ys3 − σBus1

)1

x1(A.89)

kSC =

((x6 − VV 0)(1− σRus1)γRys1

x9− CV

)1

x1(A.90)

kSR =

−x7β −Bcβ

ln(

1− (ys2−ys3)ys2−PV

)CA− ((Bc − x8)ρ+RP )

1

x1(A.91)

where

P V =

[(x6 − VV 0)− (x6 − VV 0)(1− σRus1)

γRx5CV + kSCx1

]1

CV + kSCx1(A.92)

A.6 Model constraints

The model only produces meaningful results if certain constraints on themodel parameters are fulfilled. A basic assumption for example is that heartrate will always increase when tilting. The following conditions need to bechecked before doing any simulation or control experiment in order to avoidunexpected results.

• Constraint 1: R+ > R−

The peripheral resistance is bigger in the tilted position than in thesupine position.

R+

R− =I+ ln 1−Q−

CA(P−S −P−

V )

I− ln 1−Q+

CA(P+S −P+

V )

> 1 (A.93)

• Constraint 2: kP > 0Negative values for kP are unphysiological.

kP =

I−−I+β− 1

3(P−

S − P+S )− 2

3(P−

D − P+D )− σB sinα+

(P−S − P

−D )− (P+

S − P+D )

> 0 (A.94)

• Constraint 3: B− > Bc, B+ < Bc and B+ ≥ 0

The borderline for parasympathetic and sympathetic regulation Bc

85

must be between the extreme values B− and B+.

Bcr < 1 (A.95)

Bcr >I+

2I− − I+(A.96)

Bcr ≤I+

I− − I+(A.97)

• Constraint 4: P+/−R > 0 and P

+/−L > 0

σL >ε1 − 1

sinα+(A.98)

σR >1− ε1sinα+

(A.99)

where

ε1 =(P+

S − P+D )I−(VT − VP0 − VV 0 − VA0 − CAP−

D − ξPCA(P−S − P

−D ))

(P−S − P

−D )I+(VT − VP0 − VV 0 − VA0 − CAP+

D − ξPCA(P+S − P

+D ))

(A.100)

• Constraint 5: P+R < P−

R and P+L < P−

L

The atrial pressures are bigger in the tilted position than in the supineposition.

(P+S − P

+D )I−

(P−S − P

−D )I+

< 1 (A.101)

• Constraint 6: ρ > 0Negative values for ρ, the sympathetic sensitivity factor on peripheralresistance, is unphysiological.

(R+ −R−)β

2Bcβ − I+> 0 (A.102)

86

Appendix B

Summarised results

0 2 4 6 8 10 12 14 16 18 2060

70

80

90

Time [min]

HR

[bpm

]

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

Time [min]

α [d

eg] /

f st

ep [s

teps

/min

]

α

fstep

Figure B.1: Healthy subject MW: heart rate control

87

0 2 4 6 8 10 12 14 16 18 2060

70

80

90

Time [min]

HR

[bpm

]

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

Time [min]

α [d

eg] /

f st

ep [s

teps

/min

]

α

fstep

Figure B.2: Healthy subject PB: heart rate control

0 2 4 6 8 10 12 14 16 18 2060

70

80

Time [min]

HR

[bpm

]

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

Time [min]

α [d

eg] /

f st

ep [s

teps

/min

]

αfstep

Figure B.3: Healthy subject LB: heart rate control

88

0 2 4 6 8 10 12 14 16 18 2060

70

80

Time [min]

HR

[bpm

]

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

Time [min]

α [d

eg] /

f st

ep [s

teps

/min

]

fstep

α

Figure B.4: Healthy subject RR: heart rate control

0 2 4 6 8 10 12 14 16 18 20

60

70

80

Time [min]

HR

[bpm

]

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

Time [min]

α [d

eg] /

f st

ep [s

teps

/min

]

fstep

α

Figure B.5: Healthy subject ME: heart rate control

89

0 2 4 6 8 10 12 14 16 18 20

90

100

Time [min]

sBP

[mm

HG

]

0 2 4 6 8 10 12 14 16 18 2050607080

Time [min]

dBP

[mm

HG

]

0 2 4 6 8 10 12 14 16 18 200

50

Time [min]

α [d

eg] /

f st

ep [s

teps

/min

]

α fstep

Figure B.6: Healthy subject MW: blood pressure control

0 2 4 6 8 10 12 14 16 18 20100

120

140

Time [min]

sBP

[mm

HG

]

0 2 4 6 8 10 12 14 16 18 2050607080

Time [min]

dBP

[mm

HG

]

0 2 4 6 8 10 12 14 16 18 200

50

Time [min]

α [d

eg] /

f st

ep [s

teps

/min

]

αfstep

Figure B.7: Healthy subject PB: blood pressure control

90

0 2 4 6 8 10 12 14 16 18 20100110120130

Time [min]

sBP

[mm

HG

]

0 2 4 6 8 10 12 14 16 18 2060

80

Time [min]

dBP

[mm

HG

]

0 2 4 6 8 10 12 14 16 18 200

50

Time [min]

α [d

eg] /

f st

ep [s

teps

/min

]

α fstep

Figure B.8: Healthy subject LB: blood pressure control

0 2 4 6 8 10 12 14 16 18 20

110120130

Time [min]

sBP

[mm

HG

]

0 2 4 6 8 10 12 14 16 18 2060

80

Time [min]

dBP

[mm

HG

]

0 2 4 6 8 10 12 14 16 18 200

50

Time [min]

α [d

eg] /

f st

ep [s

teps

/min

]

αfstep

Figure B.9: Healthy subject RR: blood pressure control

91

0 2 4 6 8 10 12 14 16 18 20

80100120

Time [min]

sBP

[mm

HG

]

0 2 4 6 8 10 12 14 16 18 2040

60

80

Time [min]

dBP

[mm

HG

]

0 2 4 6 8 10 12 14 16 18 200

50

Time [min]

α [d

eg] /

f st

ep [s

teps

/min

]

α fstep

Figure B.10: Healthy subject ME: blood pressure control

0 2 4 6 8 10 12 14 16 18 2060708090

Time [min]

HR

[bpm

]

0 2 4 6 8 10 12 14 16 18 208090

100110

Time [min]

sBP

[mm

HG

]

0 2 4 6 8 10 12 14 16 18 2050607080

Time [min]

dBP

[mm

HG

]

0 2 4 6 8 10 12 14 16 18 200

50

Time [min]

α [d

eg] /

f step

[ste

ps/m

in]

fstepα

Figure B.11: Healthy subject MW: combined heart rate and blood pressurecontrol

92

0 2 4 6 8 10 12 14 16 18 2060708090

Time [min]

HR

[bpm

]

0 2 4 6 8 10 12 14 16 18 20

120

140

Time [min]

sBP

[mm

HG

]

0 2 4 6 8 10 12 14 16 18 2050607080

Time [min]

dBP

[mm

HG

]

0 2 4 6 8 10 12 14 16 18 200

50

Time [min]

α [d

eg] /

f step

[ste

ps/m

in]

αfstep

Figure B.12: Healthy subject PB: combined heart rate and blood pressurecontrol

0 2 4 6 8 10 12 14 16 18 2060

70

80

Time [min]

HR

[bpm

]

0 2 4 6 8 10 12 14 16 18 20

110120130

Time [min]

sBP

[mm

HG

]

0 2 4 6 8 10 12 14 16 18 2060708090

Time [min]

dBP

[mm

HG

]

0 2 4 6 8 10 12 14 16 18 200

50

Time [min]

α [d

eg] /

f step

[ste

ps/m

in]

fstep

α

Figure B.13: Healthy subject LB: combined heart rate and blood pressurecontrol

93

0 2 4 6 8 10 12 14 16 18 2050607080

Time [min]

HR

[bpm

]

0 2 4 6 8 10 12 14 16 18 2090

100110120

Time [min]

sBP

[mm

HG

]

0 2 4 6 8 10 12 14 16 18 2050607080

Time [min]

dBP

[mm

HG

]

0 2 4 6 8 10 12 14 16 18 200

50

Time [min]

α [d

eg] /

f step

[ste

ps/m

in]

αfstep

Figure B.14: Healthy subject RR: combined heart rate and blood pressurecontrol

0 2 4 6 8 10 12 14 16

60

80

Time [min]

HR

[bpm

]

0 2 4 6 8 10 12 14 16

100

150

Time [min]

sBP

[mm

HG

]

0 2 4 6 8 10 12 14 16

60

80

Time [min]

dBP

[mm

HG

]

0 2 4 6 8 10 12 14 160

50

Time [min]

α [d

eg] /

f step

[ste

ps/m

in]

α fstep

Figure B.15: Healthy subject ME: combined heart rate and blood pressurecontrol

94

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