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Modelling the Role of Nitric Oxide in Cerebral Autoregulation

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Modelling the Role of Nitric Oxide in Cerebral Autoregulation Modelling the Role of Nitric Oxide in Cerebral Autoregulation Mark Catherall University College Supervisor: Prof. S. J. Payne PUMMA Research Group Department of Engineering Science University of Oxford Thesis submitted for the degree of Doctor of Philosophy October 2014 arXiv:1512.09160v1 [q-bio.TO] 30 Dec 2015 Abstract Malfunction of the system which regulates the bloodflow in the brain is a major cause of stroke and dementia, costing many lives and many billions of pounds each year in the UK alone. This regulatory system, known as cerebral autoregulation, has been the subject of much experimental and mathematical investigation yet our understanding of it is still quite limited. One area in which our understanding is particularly lacking is that of the role of nitric oxide, understood to be a potent vasodilator. The interactions of nitric oxide with the better understood myogenic response remain un-modelled and poorly understood. In this thesis we present a novel model of the arteriolar control mechanism, comprising a mixture of well-established and new models of individual processes, brought together for the first time. We show that this model is capable of reproducing experimentally observed behaviour very closely and go on to investigate its stability in the context of the vasculature of the whole brain. In conclusion we find that nitric oxide, although it plays a central role in determining equilibrium vessel radius, is unimportant to the dynamics of the system and its responses to variation in arterial blood pressure. We also find that the stability of the system is very sensitive to the dynamics of Ca2+ within the muscle cell, and that self-sustaining Ca2+ waves are not necessary to cause whole-vessel radius oscillations consistent with vasomotion. 2 Nomenclature A area ABP arterial blood pressure Ca calcium CBFV cerebral blood flow velocity cGMP cyclic guanosine monophosphate CICR calcium induced calcium release C compliance Cw concentration of NO in vessel wall ∆X difference in X between two places error between target and response η dimensionless variance eNOS endothelial nitric oxide synthase f viscous friction coefficient F force hw NO wall transport coefficient IP3 inositol 1,4,5-triphosphate k constant or iteration counter K rate constant λ normalized cell length L length mmHg millimetres of mercury µb viscosity of blood n number or exponent constant NO nitric oxide P pressure pX log10(X) q flow or constant Q flow r radius or decay rate R resistance to flow σ stress section x sw NO production rate in vessel wall SERCA sarco-(endo)plasmic reticulum calcium ATPase SR sarcoplasmic reticulum τ shear stress τx time constant for variable denoted by subscript x tw vessel wall thickness v speed Vx volume of compartment x VSMC vascular smooth muscle cell wc cell width !n natural frequency 3 x¯ equilibrium value of x dx x_ first time derivative of x ( dt ) d2x x¨ second time derivative of x ( dt2 ) XL Lower value of X XU Upper value of X z X ± ion of element X with charge z ± [X] concentration of X 4 Contents 1 Introduction 9 1.1 The Problem . 9 1.2 The Brain and its Blood Supply . 9 1.3 The Value of Mathematical Modelling . 10 1.4 Objective . 10 1.5 Overview . 11 2 Literature Review 13 2.1 The Cardiovascular System . 13 2.1.1 Blood Vessels . 13 2.1.2 Blood Pressure . 14 2.1.3 Blood Flow . 15 2.2 Cells . 15 2.2.1 Cell Structure . 15 2.2.2 Smooth Muscle Cells . 16 2.3 Autoregulation . 17 2.3.1 Myogenic Response . 17 2.3.2 The Role of Nitric Oxide . 18 2.3.3 Interactions of NO and Myogenic Mechanisms . 18 2.3.4 Other Autoregulatory Mechanisms . 20 2.4 Vasomotion . 20 2.5 Mathematical Models in the Literature . 21 2.5.1 Hodgkin and Huxley . 21 2.5.2 DeBoer . 22 2.5.3 Yang . 23 2.5.4 Stalhand . 24 2.5.5 Abatay . 24 2.5.6 Rowley-Payne . 25 2.5.7 Jacobsen . 26 2.5.8 Cornelissen and Carlson . 27 2.5.9 Koenigsberger . 28 2.6 Conclusions . 28 3 A Model of Vascular Regulation 31 3.1 Introduction . 31 3.2 Model Equations . 33 6 CONTENTS 3.2.1 NO model . 33 3.2.2 [NO] { [cGMP] relationship . 34 3.2.3 Myogenic Response Model . 35 3.2.4 4-state Kinetic Model . 36 3.2.5 Mechanical Model . 37 3.3 Solution of the Model in the Steady-State . 39 3.3.1 NO . 39 3.3.2 Myogenic Response . 40 3.3.3 4-state Kinetic . 40 3.3.4 Mechanical . 41 3.4 Solution of the Dynamic Model . 41 3.5 Assumptions . 41 3.6 Conclusions . 43 4 Fitting and Analysis of Vascular Regulation Model 45 4.1 Introduction . 45 4.2 Model Fitting Data . 45 4.2.1 Fitting the 4-State Kinetic Model . 46 4.2.2 Data to Fit the Model Overall . 47 4.3 Steady-State Response . 49 4.3.1 Fitting the Passive Response . 50 4.3.2 Fitting the Active Response . 53 4.3.3 Discussion . 55 4.4 Dynamic Responses . 56 4.4.1 Hypothesis/Justification for a Variable Ca2+ Time Constant . 61 4.5 Sensitivity Analysis . 62 4.5.1 Current Methods . 62 4.5.2 A Novel Method . 64 4.5.3 Additive Results . 65 4.5.4 Multiplicative Results . 65 4.5.5 Dynamic Results . 66 4.5.6 Discussion . 69 4.6 Conclusions . 70 5 A Model of Autoregulation 71 5.1 Introduction . 71 5.2 Integration of Arteriole Model Into Existing Whole-Brain Model . 71 5.2.1 Solution in the Steady-State . 73 5.2.2 Solution in the Dynamic Case . 74 5.3 Fitting and Analysis . 75 5.3.1 Steady-State Response . 75 5.3.2 Dynamic Responses . 77 5.3.3 Downstream Transport of NO and the Possibility of NO Signalling 80 5.3.4 Discussion . 82 5.3.5 Conclusions . 84 5.4 Inclusion of Passive Arterial Compliance . 85 CONTENTS 7 5.4.1 Conclusions . 88 5.5 Conclusions . 88 6 Vasomotion and Spontaneous Oscillations 89 6.1 Introduction . 89 6.2 Stability Analysis . 89 6.2.1 Discussion . 91 6.3 Replicating Vasomotion . 92 6.3.1 Adding Intracranial Compliance . 92 6.3.2 Adding Flow Inertia . 94 6.3.3 Making Ca2+ Dynamics 2nd Order . 96 6.4 Analysis of Instability . 99 6.4.1 Sensitivity to ζ . 104 6.4.2 Sensitivity to !n . 105 6.4.3 Role of Non-regulating Vasculature . 106 6.4.4 Effect on Flow . 108 6.4.5 Comparison to Vasomotion . 108 6.5 Discussion . 109 6.6 Conclusions . 110 7 Conclusions and Future Work 113 7.1 Summary of Results . 113 7.2 Discussion and Conclusions . 114 7.2.1 The Role of NO . 114 7.2.2 The Role of Ca2+ . 115 7.2.3 Interpretation of Ide Results . 115 7.3 Future Work . 116 8 CONTENTS Chapter 1 Introduction 1.1 The Problem Stroke and dementia are between them responsible for approximately 68,000 deaths per year in the England and Wales alone [76, 4]. The cost to the UK of stroke and dementia combined is an estimated $24.6 billion per year [5, 2]. Stroke can be either haemorrhagic (caused by bleeding in the brain), or ischaemic (caused by insufficient blood flow to a part of the brain). There are several different types of dementia; one of them, called vascular dementia, occurs due to insufficient blood supply within the brain, just as in ischaemic stroke, only on a much smaller scale. Ischaemia damages the brain because when brain tissue does not receive enough blood it is not supplied with enough oxygen to respire (a condition known as hypoxia), this in turn leads.
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