On the roles of vascular smooth muscle contraction in cerebral blood flow autoregulation – a modeling perspective Jin Yang1 and John W. Clark, Jr.2 Abstract— We here review existing models of vascular smooth muscle cell, endothelial cell and cell-cell communication, which have been developed to better understand vascular tone and blood flow autoregulation. In particular, we discuss models that intended to explain modulation of myogenic tone by intraluminal pressure in resistance arterioles. Modeling efforts in the recent past have witnessed a shift from empirical models to models with mechanistic details that underscore different physical aspects of vascular regulation. Future models should synthesize mechanistic interactions in a hierarchy, from molecular regulation of ion channels to whole organ blood flow control.
I. INTRODUCTION Whereas it is generally agreed that the topic of cerebral blood autoregulation (CBA) has at least three major components, myogenic, neurogenic and metabolic , this paper is centered on intrinsic myogenic aspects. The myogenic response, or Bayliss effect , describes vascular smooth muscle reaction to arterial pressure changes and plays a critical role in CBA. Excitation-contraction coupling in vascular smooth muscle cells (VSMCs) has been a particularly strong focus for research into vascular responses to blood pressure modulation. Many mathematical models with emphasis on different aspects have been developed to study the multiscale physiology of VSMCs and blood vessels. Here we further restrict our attention to those models that have studied endogenous and exogenous regulatory aspects of VSMC contraction. More generally, cellular signaling mechanisms associated with myogenic response and its physiological roles have been reviewed elsewhere , . II. MODELS OF SMOOTH MUSCLE AND ENDOTHELIAL CELLS The myogenic response originates from changes in arterial pressure that modulates mechanical vessel wall stress and strain, and which has been widely hypothesized to initiate responses by VSMC contraction, causing constrictions in blood vessels to limit and maintain the arterial blood flow. Mechanosensitive ion channels on the VSMC membrane sense vessel stress and strain. When activated, these channels lead a non-specific ionic influx that depolarizes the cell membrane, which subsequently activates the voltagegated L-type Ca2+ channels to allow extracellular Ca2+ to enter the VSMC and elevate the intracellular Ca2+ 1 CAS-MPG Partner Institute of Computational Biology, Shanghai Institutes for Biological Sciences, Shanghai, China 200031 [email protected]
2 Department of Electrical Engineering, Rice University, Houston, TX 77005, USA [email protected]
978-1-4244-9270-1/15/$31.00 ©2015 IEEE
concentration, [Ca2+ ]i . A positive feedback mechanism to boost transient [Ca2+ ]i , the ryanodine receptor (RyR) regulated calcium-induced calcium-release (CICR) from the sarcoplasmic reticulum (SR) stores further contributes to the build-up of [Ca2+ ]i . Since [Ca2+ ]i controls VSMC contraction, the topic of Ca2+ regulation has been central to most models that describe the cellular aspects of the VSMC. Increasing [Ca2+ ]i induces myosin-actin interactions, which are typically described by the widely-adopted myosin-phosphorylation model by Hai and Murphy . On the other hand, [Ca2+ ]i is balanced by counteracting ionic channels, including Na+ /Ca2+ exchanger and Ca2+ pump. More importantly, Ca2+ -regulated VSMC contraction is antagonized by exogenous factors such as nitric oxide (NO) released from nearby endothelial cells (ECs), which induces VSMC production of the second messenger cGMP by activation of soluble guanylate cyclase (sGC). Most models of VSMC membrane electrophysiology use Hodgkin-Huxley equations to describe the nonlinear dynamic relationships between the whole cell ionic membrane currents and the transmembrane potential. The gating mechanisms of different ionic channels are modeled by fitting to voltage-clamp data obtained under varied membrane ”holding” potentials and agonist concentrations. Existing models have been able to account for the effects on intracellular ionic concentrations (Na+ , K+ , Ca2+ and Cl− ) and transmembrane potential by utilizing descriptions of different types of ionic membrane currents. For an early example, GonzalezFernandez and Ermentrout  proposed a dynamic model of SMC contraction to explain the molecular and mechanical mechanisms of myogenic response, which connect SMC membrane electrophysiology (voltage-gated Ca2+ channel and Ca2+ -activated K+ channel), activation of contractile element (myosin phosphorylation) and wall stress. The model was able to reproduce decreasing vessel diameter in response to blood pressure change elevation observed in experiments. The model by Yang et al.  represented a first attempt of constructing a comprehensive model that incorporated behaviors of a series ion channels including the L-type Ca2+ channel, voltage-gated K+ channel, Ca2+ -activated K+ channel, the inward rectifier, non-specific stretch-sensitive channel, Na+ /Ca2+ exchanger, and Na+ /K+ and Ca2+ pumps. Besides the cross-membrane Ca2+ influxes through the voltage-gated Ca2+ channels, the model also considered intracellular Ca2+ dynamics influenced by CICR and internal Ca2+ buffers. The model then fed the intracellular interaction between Ca2+ and calmodulin to drive myosin phosphorylation and latch bridge formation kinetics. The
Ca2+ -regulated dynamics of myosin phosphorylation was later coupled to an active force-generation component in a modified Hill model of cell mechanics , similar to an early ”fraction clutch” mechanism proposed by Gestrelius and Borgstrom , to produce single-cell force-velocity and force length responses, which was a first model to connect cellular signaling to VSMC mechanics. This integrated cell model provides biophysically based explanations of electrical, chemical and mechanical phenomena in cerebrovascular smooth muscle under isometric and isotonic conditions. This model was subsequently embedded into a larger idealized model of a cannulated cerebrovascular artery. This later study coupled the time course of cellular contractile activity to macroscopic changes in the vessel diameter . The model provides biophysical insights into the myogenic mechanism as it responds to changes in transmural pressure, linking events at subcellular level to macroscopic changes in vessel diameter. As such, the model initiated a mechanistic approach to further investigation of the myogenic response, an approach that had not been taken previously by any other models. In another extension, the VMSC model was integrated with the NO-cGMP pathway to account for the effects of endothelial-derived VSMC relaxation, where NOinduced production of second messenger cGMP reduces myosin activity by antagonizing Ca2+ influx and desensitizing Ca2+ -activated myosin phosphorylation . This model can serve as a general modeling framework for studying NOmediated VSMC relaxation under various physiological and pathological conditions. To interpret measured data on the rat mesenteric smooth muscle cell, Kapela et al.  developed another detailoriented mathematical model of the mesenteric VSMC, which can simulate its responses to norepinephrine (NE) and NO stimulations. The model was also able to generate Ca2+ oscillations due to Ca2+ release and refilling dynamics from the SR. This model was one of a series of models developed by Tsoukias and coworkers to study vasoreactivity in the rat mesenteric arterioles. Models of individual VSMC components have been improved in recent years, in particular, in the area of cell mechanics modeling. These advances can be incorporated into a more comprehensive cellular VSMC model. For example, in a more general approach, Stalhand et al.  derived from first principles a thermodynamically-consistent mechanochemical model for smooth muscle contraction, which recovers the counterpart model by Yang et al. , in terms of a linear, small deformation approximation. In addition, modeling the contractile apparatus of myosinactin interaction has further refined parameterization to the classic Hai and Murphy four state phosphorylation model that describes a Markov chain dynamics of myosin light chain in terms of its (un)phosphorylated and actin filament (non)attachment states. The Ca2+ -regulated myosin phosphorylation model by Murtada et al.  adopted Hai and Murphy’s minimal independent myosin phosphorylation model and proposed a new coupling cell mechanics model based on strain energy function. Murtada et al.  model
was able to interpret measured data from intact smooth muscle in guinea pig taenia coli by Arner et al. , including active cell stress-Ca2+ relationship, dynamic isometric active cell stress response to Ca2+ stimulation, and isotonic active cell stress response. At this point, we would like to digress to note that historically, models of VSMCs have been developed for different vascular systems to help explain different physiological functions. Models specific for studying cerebral circulation are still rare. For example, the excellent model by Kapela et al  was tailored to study rat mesenteric arterioles. A model by Parthimos et al.  focus on studying vasomotion of rabbit ear arteries. The model by Edwards and Pallone  studied effects of Na+ in subplasmalemmal microdomains on [Ca2+ ]i of VSMC-like contractile cells, pericytes, in descending vasa recta. A model by Bursztyn et al.  was developed to study excitation-contraction coupling of myometrial SMCs. Despite the fact that different vascular systems of resistance arterioles do share common features, to quantitatively explain the myogenic response in cerebrovascular regulation and predict regulatory mechanisms, a model must account for characteristics observed in the cerebrovascular system. Specifically, it is important to recognize that the architecture of the arteriolar system in the cerebral circulation is quite different from that of many other organs. The cerebral circulation offers a segmental vascular resistance to blood flow provided by a segmental arteriolar network , . This is in contrast with the comparatively simple structure seen in the micro-circulation of most organs, wherein only the small arterioles contribute to vascular resistance. Investigators have shown that the segmental structure has functional consequences . In addition, compared to other tissues, the brain is a highly metabolic tissue that relies on a consistent oxygen supply from the CBF. Therefore, autoregulation of CBF across a wide range of perfusion blood pressures (e.g, 60-160 mmHg for human) is critical to the survival and normal function of the brain. Simple phenomenological models of myogenic response can be mathematically efficient when connecting experimental observables to underlying causal variables. For example, Carlson and Secomb  developed a wall mechanics model of resistance vessel to examine the dependence of myogenic response on circumferential tension or stress. The VSMC tone is coupled to wall tension or stress through an empirical sigmoidal function. The model was able to measured relationship between intravascular pressure and arteriolar calibers. One limitation of such an approach is obviously the lack of connections to cellular mechanisms, which makes the model nonpredictive to guide interventions that might target molecular components in VSMCs or ECs. The key to modeling myogenic tone is to convert the intravascular pressure change into responses in VSMC tone. In a cellular VSMC model, this transduction is mediated by the mechanosensitive ion channels. However, the molecular mechanism with regard to activation of the mechanosensitive channel in response to changing wall stress remains unresolved. It is unclear
whether VSMC stress, strain or even endothelial-derived factors directly regulate channel gating and via what mechanisms. Existing models have employed an empirical wallstress gating assumption that relates VSMC stress to the open probability of the mechanosensitive channel , . As a neighboring cell layer to the VSMC layer in the arterial wall, endothelial cells (ECs) act as an interface that plays an important role in maintaining and modulating vascular tone. ECs affect the VSMC contraction by releasing so called endothelial-derived hyperpolarizing factor (EDHF) or endothelial-derived relaxing factor (EDRF), such as nitric oxide, to ultimately induce VSMC relaxation and thus reduce the vascular resistance to blood flow. Models have considered EDHF effects on VSMC without explicitly modeling the ECs such as in the model by Yang et al.  that merely considers nitric oxide as an isolated external input factor to VSMC. Mathematical models have been developed in the past to explain experimental data and predict EC functions. The detailed electrophysiology model by Silva et al.  represents one of the most comprehensive model of vascular EC, which accounts for dynamics of multiple ionic species including K+ , Cl− , Na+ and Ca2+ through membrane channels and internal ER and SR stores. The model was later integrated with the VSMC model by Kapela et al.  to form a multicellular model. III. MODELS OF CELL-CELL COUPLING To date, most models of VSMC or EC are developed for studying intracellular signaling behaviors of individual cell types. Coarse-grained concatenation of a homogeneous population of cells along the vessel wall has also been considered , even though the cell-cell coupling (VSMCVSMC, EC-EC, and VMSC-EC coupling) is essential for signal transduction and myogenic tone propagation. Regulation of SMC contraction involves complex interplay among electrical, chemical and mechanical VSMC components as well as intercellular communication between VSMC and ECs. VSMCs lie circumferentially along the arterial wall, whereas the ECs lie along the longitudinal axis of the vessel wall. Each EC may cross multiple VSMCs. Electrical conduction among VSMCs are much less efficient than that among ECs, making them propagators who relay vascular signals and synchronize VSMCs to modulate myogenic tone . Besides electrical intercellular communications via myoendothelial gap junctions, EC-VSMC coupling is also mediated by factors such as NO and EDHF. An important feature of multicellular models is the ability to predict the spatial propagation of electrical and chemical signals. To study the collective behaviors of VMSCs that generate vessel motor responses, Koenigsberger et al.  modeled the VSMC-VSMC communication with a single sheet of VSMCs on a 2-dimensional grid. The authors extended the model by Parthimos et al.  to account for the electrophysiology of a single VSMC. Individual VSMCs are coupled to their neighboring cells through gap junctions that serve as gateways for electrical, Ca2+ and IP3 exchanges
between cells. The model demonstrated that arterial contraction and vasomotion are possible by the VSMC-VSMC coupling alone. In a later model, Koenigsberger et al.  further superposed a sheet of endothelial, a layer of ECs, spatially in parallel to that of the VSMCs, in which the electrophysiology of single EC was modeled to account for membrane potential, Ca2+ and IP3 dynamics, following an early simplistic Goldbeter et al. model  that studied Ca2+ oscillations by an IP3 -induced CICR mechanism. The Koenigsberger et al. model  considered EC-EC coupling via EC gap junctions, EC-VSMC electrical coupling, as well as EC-VSMC Ca2+ and IP3 exchanges through myoendothelial gap junctions. Model simulations demonstrated that the endothelium may act to induce or abolish vasomotion in response to changing vasoconstrictor levels according to a bifurcation analysis of the model. Based on a single VSMC model , Jacobsen et al.  also modeled the VSMC-VSMC coupling in the rat mesenteric arteries by a 2-dimensional wrapped sheet of VSMCs. The model predicted that cGMP could induce VSMC membrane potential oscillation by activating cGMP-sensitive Ca2+ -dependent Cl− channels, and sufficient intercellular exchanges may synchronize the cell population and cause arterial vasomotion. Like the model by Koenigsberger et al. , this model also demonstrated that VSMC-VSMC coupling in the absence of the endothelium may be adequate to produce synchronized VSMC Ca2+ and membrane potential oscillations, and thus vasomotion. Kapela et al. ,  took a slightly different approach to model the intercellular coupling. Integrating their previous models of EC  and VSMC , the authors first investigated the coupling between a single EC-VSMC pair, which represents a coarse-grained approximation to more complex spatial arrangement . However, Kapela et al. address this limitation with a follow-up model , which assumed that VSMCs circumferentially wrap around the vessel wall, whereas ECs line up in parallel to each other along the axial direction. Intercellular exchanges also proceeded through EC-EC, VSMC-VSMC and myoendothelial gap junctions that allow transductions of electrical, ionic and chemical signals. Both models ,  showed that EC stimulation may attenuate [Ca2+ ]i in the VSMCs. The signal conductivity along the vessel expectedly depended on the coupling strength at the gap junctions. IV. DISCUSSION The ultimate goal to simulate vascular responses is to integrate cell models with biomechanics models to account for dynamics of motor responses by the vascular system at different spatial and temporal scales. Integrative modeling is a promising direction to synthesize various individual vascular components to study tissue-level physiology. However, building a unified multiphysics model that can accurately predict influences of coordinated acts of cellular signaling , intercellular communication , , arterial responses , , and blood flow regulation ,  remains a formidable challenge.
Comprehensive models of systems biology with multiphysics details allow evaluations of functional influences by individual factors under the context of an interacting environment, which cannot be adequately represented by lumped systems that focus on isolated mechanisms. However, any complex model comes with a cost. A multiphysics model is limited by incomplete and inconsistent experimental characterizations of individual components and their interactions, making it difficult to accurately identify essential parameters due to lack of relevant data. In particular, quantitative data regarding membrane electrophysiology, contractile activity, cell mechanics and vessel responses in a specific vascular tissue are usually unavailable. Furthermore, comprehensive models, as opposed to simple models, typically do not have explicit solutions and therefore rely on numerical simulations to probe model behaviors. This is a considerable limitation. Nonetheless, as in the natural course of model development, building a multiphysics model is a process, as is well-illustrated in the field of cardiac modeling . For the multiscale physiological modeling of cerebrovascular functions, further incorporation of forthcoming experimental information will help to improve the model’s analytical and predictive power. R EFERENCES  M.J. Cipolla, J. Sweet, S-L. Chan, M.J. Tavares, N. Gokina and J.E. Brayden, ”Increased pressure-induced tone in rat parenchymal arterioles vs. middle cerebral arteries: roles of ion channels and calcium sensitivity,” J. Appl. Physiol. vol 117, pp 53-59, 2014  W.M. Bayliss, ”On the local reactions of the arterial wall to changes of internal pressure,” J Physiol., vol 28, pp 220-231, 1902  M.J. Davis, M.A. Hill, ”Signaling Mechanisms Underlying the Vascular Myogenic Response,” Physiol. Rev., vol. 79, pp. 387-423, 1999  M.J. Davis, ”Perspective: physiological role(s) of the vascular myogenic response,” Microcirculation, vol. 19, pp. 99-114, 2012.  C. Hai, R. A. Murphy, Am. J. Physiol. vol. 254, C99-C106, 1988  J. M. Gonzalez-Fernadez and B. Ermentrout, ”On the origin and dynamics of the vasomotion of small arteries,” Math. Biosci., vol. 112, pp. 127-167, 1994  J. Yang, J.W. Clark, R.M. Bryan, C.S. Robertson, ”The myogenic response in isolated rat cerebrovascular arteries: smooth muscle cell model,” Med. Eng. Phys., vol. 25, pp. 691-709, 2003  Y.C. Fung, Biomechnics: Mechanical Properties of Living Tissues, 2nd ed. Springer Verlag, 1993  S. Gestrelius, P. Borgstrom, ”A dynamic model of smooth muscle contraction,” Biophys. J., vol. 50, pp. 157-169, 1986  J. Yang, J.W. Clark, R.M. Bryan, C.S. Robertson, ”The myogenic response in isolated rat cerebrovascular arteries: vessel model,” Med. Eng. Phys., vol. 25, pp. 710-717, 2003  J. Yang, J.W. Clark, R.M. Bryan, C.S. Robertson, ”Mathematical modeling of the nitric oxide/cGMP pathway in the vascular smooth muscle cell,” Am. J. Physiol. Heart Circ. Physiol., vol. 289, pp. H886H897, 2005  A. Kapela, A. Bezerianos, N.M. Tsoukias, ”A mathematical model of Ca2+ dynamics in rate mesenteric smooth muscle cell: agonist and NO stimulation,” J. Theor. Biol., vol. 253, pp. 238-260, 2008  J. Stalhand, A. Klarbring, G.A. Holzapfel, ”Smooth muscle contraction: mechanochemical formulation for homogeneous finite strains,” Prog. Biophys. Mol. Biol., vol. 96, pp. 465-481, 2008  S. Murtada, M. Kroon, G.A. Holzapfel, ”A calcium-driven mechanochemical model for prediction of force generation in smooth muscle,” Biomech Model Mechanobiol, vol. 9, pp. 749-762, 2010  A. Arner, ”Mechanical characteristics of chemically skinned guineapig taenia coli,” Eur. J. Physiol., vol. 395, pp. 277-284, 1982  D. Parthimos, D.H. Edwards and T.M. Griffith, ”Minimal model of arterial chaos generated by coupled intracellular and membrane Ca2+ oscillators,” Am. J. Physiol. Heart Circ. Physiol., vol. 277, pp. H1119H1144, 1999
 A. Edwards, T.L. Pauline, ”Modification of cytosolic calcium signaling by subplasmalemmal microdomains,” Am. J. Physiol. Renal Physiol., vol. 292, pp. F1827-F1845, 2007  L. Bursztyn, O. Eytan, A.J. Jaffa, D. Elad, ”Mathematical model of excitation-contraction in a uterine smooth muscle cell,” Am. J. Physiol. Cell Physiol., vol. 292, pp. C1816-C1829, 2007  F.M. Faraci, W.G. Mayhan, D.D. Heistad, ”Segmental vascular responses to acute hypertension in cerebrum and brain stem,” Am. J. Physiol., vol. 252, pp. H738-H742, 1987  M.J.Cipolla, J.G. Sweet, N.I. Gokina, S.L. White and M.T. Nelson, ”Mechanisms of enhanced basal tone of brain parenchymal arterioles during early postischemic reperfusion: role of ET-1-induced peroxynitrite generation,” J. Cereb. Blood Flow Metab., vol. 33, pp. 1486-1492, 2013  B.E. Carlson, T.W. Secomb, ”A theoretical model for the myogenic response based on the length-tension characteristics of vascular smooth muscle,” Microcirculation, vol. 12, pp. 327-338, 2005  M. Koenigsberger, R. Sauser, J. Beny, J Meister, ”Effects of arterial wall stress on vasomotor,” Biophys. J. vol. 91, pp. 1663-1674, 2006  H.S. Silva, A. Kapela, N.M. Tsoukias, ”A mathematical model of plasma membrane electrophysiology and calcium dynamics in vascular endothelial cells,” Am. J. Physiol. Cell Physiol., vol. 293, pp. C277C293, 2007  R. E. Haddock, T.H. Grayson, T.D. Brackenbury, K.R. Meaney, C.B. Neylon, S.L. Sandow, C.E. Hill, ”Endothelial coordination of cerebral vasomotion via myoendothelial gap junctions containing connexins 37 and 40,” Am. J. Physiol. Heart Circ. Physiol., vol. 291, pp. H2047H2056, 2006  M. Koenigsberger,R. Sauser, M. Lamboley, J. Beny, J Meister, ”Ca2+ Dynamics in a population of smooth muscle cells: Modeling the recruitment and synchronization,” Biophys. J., vol. 87, pp. 92-104, 2004  M. Koenigsberger, R. Sauser, J. Beny and J. Meister, ”Role of the endothelium on arterial vasomotion,” Biophys. J., vol.88, pp.38453854, 2005  A. Goldbeter, G. Dupont, M.J. Berridge, ”Minimal model for signalinduced Ca2+ oscillations and for their frequency encoding through protein phosphorylation,” Proc. Natl. Acad. Sci. USA, vol. 87, pp. 1461-1465, 1990  J.C.B. Jacobsen, C. Aalkjaer, H. Nilsson, V.V. Matchkov, J. Freiberg, N. Holstein-Rathlou, ”Activation of a cGMP-sensitive calciumdependent chloride channel may cause transition from calcium waves to whole cell oscillations in smooth muscle cells,” Am. J. Physiol. Heart Circ. Physiol., vol. 293, pp. H215-H228, 2007  J.C.B. Jacobsen, C. Aalkaer, H. Nilsson, V.V. Matchkov, J. Freiberg, N. Holstein-Rathlou, ”A model of smooth muscle cell synchronization in the arterial wall,” Am. J. Physiol. Heart Circ. Physiol., vol. 293, pp. H229-H237, 2007  A. Kapela, A. Bezerianos, N.M. Tsoukias, ”A mathematical model of vasoreactivity in rat mesenteric arterioles: I. Myoendothelial communication,” Microcirculation, vol. 16, pp. 694-713, 2009  A. Kapela, S. Nagaraja, N.M. Tsoukias, ”A mathematical model of vasoreactivity in rat mesenteric arterioles: II. Conducted vasoreactivity,” Am. J. Physiol. Heart Circ., vol. 298, pp. H52-H65, 2010  N.M. Tsoukias, ”Calcium dynamics and signaling in vascular regulation: computational models”, Wiley Interdiscip Rev. Syst. Biol. Med., vol. 3, pp. 93-106, 2011  S. Nagaraja, A. Kapela, N.M. Tsoukias, ”Intercellular communication in the vascular wall: A modeling perspective,” Microcirculation, vol. 19, pp. 391-402, 2012  A. Kapela, S. Nagaraja, J. Parikh, N.M. Tsoukias, ”Modeling Ca2+ signaling in the microcirculation: intercellular communication and vasoreactivity”, Crit. Rev. Biomed. Eng. vol. 39, pp. 435-460, 2011  G.A. Holzapfel, R.W. Ogden, ”Constitutive modelling of arteries,” Proc. R. Soc. A, vol. 466, pp. 1551-1597, 2010  T.W. Secomb, ”Theoretical models for regulation of blood flow,” Microcirculation, vol. 15, pp.765-775, 2008  P. Kohl and D. Noble, ”Systems biology and the virtual physiological human”, Mol. Syst. Biol., vol. 5 pp.292, 2009  M. Ursino and C.A. Lodi, ”Interaction among autoregulation, CO2 reactivity, and intracranial pressure: a mathematical model”, Am. J. Physiol. Heart Circ. Physiol., vol. 274, pp.H1715-H1728, 1998