This paper was presented at the 4th Micro and Nano Flows Conference (MNF2014), which was held at University College, London, UK. The conference was organised by Brunel University and supported by the Italian Union of Thermofluiddynamics, IPEM, the Process Intensification Network, the Institution of Mechanical Engineers, the Heat Transfer Society, HEXAG - the Heat Exchange Action Group, and the Energy Institute, ASME Press, LCN London Centre for Nanotechnology, UCL University College London, UCL Engineering, the International NanoScience Community, www.nanopaprika.eu.We present herein a first attempt to integrate large and small scale phenomena within an image-based computational domain. The aim of the present study is to highlight some of the underlying mechanisms that govern cellular interaction in the vascular wall, using a nonlinear model of vasomotion. We show that macroscopic rhythmic activity and emergent phenomena can indeed reflect ion movements at the level of the individual cell

4th Micro and Nano Flows Conference (MNF2014)

Boileau, E

Nithiarasu, P

Parthimos, D

English

Brunel University Research Archive

4th Micro and Nano Flows ConferenceUCL, London, UK, 7-10 September 2014Local Regulation of Arterial Tone: an Insight into Wall Dynamics Using Mathematical ModelsEtienne BOILEAU 1,*, Dimitris PARTHIMOS 2, Perumal NITHIARASU 1Corresponding author: Tel.: +44 (0)1792 604176; Email: e.boileau@swansea.ac.uk1 Swansea University, Computational Bioengineering and Rheology, Swansea SA2 8PP, UK2 Cardiff University School of Medicine, Institute of Molecular and ExperimentalMedicine, Wales Heart Research Institute, Cardiff CF14 4XN, UKAbstract We present herein a first attempt to integrate large and small scale phenomena within an image-based  computational  domain.  The  aim of  the  present  study is  to  highlight  some  of  the underlying  mechanisms  that  govern  cellular  interaction  in  the  vascular  wall,  using  a  nonlinear model of vasomotion. We show that macroscopic rhythmic activity and emergent phenomena can indeed reflect ion movements at the level of the individual cell.Keywords: Vasomotion, Smooth Muscle, Ion Calcium, Complex Dynamical System, Endothelium, Mass Transport, Computational Model1. IntroductionHealthy  vessels  are  characterised  by  an ability to adapt to the conditions imposed by the local environment, and respond to changes in blood flow by dilating or contracting.  The progressive impairment of this dilator function is known as endothelial dysfunction, and it has been shown to predict long-term development of arterial disease (Halcox et al., 2002; Halcox et al., 2009; Schachinger et al., 2000).The  formation  of  atheromatic  plaque  in stenosed vessels  is also an important  area of modelling investigation (El Khatib et al., 2007; Cilla et al., 2014). In addition to the effects of hæmodynamics  and  transport,  in  early atheroma,  there  exists  complex  interactions between the  arterial  wall  components,  which cause  inflammatory  signalling  that  leads  to monocyte  accumulation,  foam  cell degeneration  and  formation  of  atheromatous plaque.  Such  alterations  can  result  in significant  modification  of  the  arterial geometry.  Many  such  factors  can  be incorporated in mathematical models,  to gain insights  into  the  interplay  between  disease processes  and  arterial  function.  The  role  of mathematical and computational modelling in studying  cardiovascular  disease  as  an integrated  systemic  dysfunction  is  therefore very important.We propose a first attempt in this direction. A mathematical  description  of the ion transport systems  that  participate  in  the  regulation  of vascular  tone  is  presented  below. Computational  results  are  shown  for  a population of coupled smooth muscle cells on the  surface  of  an  image-based  domain. Modelling strategies are also discussed, with a view  to  add  additional  components  of  the arterial  wall,  including  flow-endothelium interactions.1.1 Arterial Structure and Function   The arterial wall consists of three layers. The first layer, the intima, is made up of a single lining  of  endothelial  cells  (ECs),  the endothelium, supported by an internal  elastic lamina  that  separates  the  intima  from  the second  layer,  the  media.  The  endothelium forms a selective permeable membrane, and is a  key  actor  in  mediating  the  interactions between  the  lumen  or  flow  domain  and  the - 1 -4th Micro and Nano Flows ConferenceUCL, London, UK, 7-10 September 2014smooth  muscle  (SM)  layer.  The  media  is composed  of  smooth  muscle  cells  (SMCs), embedded in a matrix of elastin and collagen fibres. In arterioles, it consists largely of SM. Changes in their contractile tension cause the vessel to dilate or constrict, thereby regulating vessel  diameter  and  blood  flow  through  the microcirculation.The  vascular  SM  is  responsible  for spontaneous  fluctuations  in  vessel  diameter that are not caused by changes in heart rate  or blood  pressure,  and  are  referred  to  as ‘vasomotion’.  Vasomotion  is  thought  to contribute  to  the  enhancement  of  lymphatic drainage and microcirculatory mass transport, thereby reflecting the critical role of resistance vessels in regulating interstitial tissue pressure and delivery of oxygen and nutrients (Griffith, 1996).The  Smooth  Muscle. The  excitation-contraction  coupling,  or  electrochemical coupling,  is  primarily  controlled  by movements of ion calcium Ca2+ into and out of the  cytoplasm.  Cytosolic  Ca2+ concentration ([Ca2+]i) is normally maintained at a basal level of  100nm by the plasma membrane Ca∼ 2+-ATPase  (PMCA),  the  Na+-Ca2+ exchanger (NCX)  and  the  sarco  endoplasmic  reticulum Ca2+-ATPase  (SERCA).  Vasoconstricting stimuli  initiate  SM contraction  by increasing the [Ca2+]i , to reach the μm range. The rise in [Ca2+]i  is  primarily  mediated  by  voltage- (VOCCs) and store-operated channels (SOCs). The intracellular oscillator is identified by the cyclic  Ca2+ -induced  Ca2+  release  from ryanodine-sensitive stores (RyR CICR)  of the sarco  endoplasmic  reticulum  (SR/ER) (Parthimos  et  al.,  1999;  Parthimos  et  al., 2007).  Membrane  potential  is  a  distinct  dynamic variable  determined  by  the  additive contribution  of  a  large  number  of  ionic transport  mechanisms,  such  as  ion  channels (Cl-, K+),  pumps, and exchangers. The system of  ordinary  differential  equations  describing the  coupling  between  the  store-mediated intracellular  Ca2+  oscillator  and  membrane potential  fluctuations,  due  to  ionic  transport mechanisms across the cellular  membrane,  is presented below. For a  detailed   description, see Parthimos  et  al.  (1999),  Parthimos  et  al. (2007).    The  Role  of  the  Endothelium. Although smooth  muscles  are  responsible  for  vascular tone  and  contractile  phenomena,  the endothelium act as an active interface between the  lumen  and  the  inner  layers  of  the  blood vessel wall. Electrical and chemical signalling between  endothelial  and  SM  cells  thus contribute to arterial function in various ways. The  role  of  the  endothelium  in  vasomotion nevertheless  remains  unclear.  Some  studies reported rhythmic contractions in the absence of  an  intact  endothelium,  whereas  others claimed  that  its  presence  is  needed  for synchronization (Peng et al., 2001; Sell et al., 2002; Okazaki et al., 2003). Although the results  presented  below do not include any coupling between endothelial and SM cells, future work will address this point, in addition to considering the effects of mass transport  on  mobilization  of  endothelial [Ca2+]i.  The  release  of  the  nucleotide adenosine-5’  triphosphate  (ATP)  by endothelial cells, via wall shear stress (WSS)-dependent  mechanisms,  is  thought  to  play  a key  role  in  endothelium-mediated vasodilation.  Its  influence  on  regional  blood flow is also seen through the effects of its end-products,  such  as  adenosine  diphosphate (ADP).  Flow  regulation  of  ATP-  and  ADP-Ca2+  coupling  in  the concentration  boundary-layer  depends  on  several  factors.   Small changes in the model parameters, such as the rate  constants  for  the  degradation  and/or production of ATP significantly influence the pattern  of  catalytic  reactions  at  the endothelium  (Boileau  et  al.,  2013).   As  we have shown earlier, and as illustrated in Fig. 1, even though the bulk concentration of secreted agonists is low and mostly unaffected by the flow,  variations  exist  at  the  endothelium, which  are  nevertheless  difficult  to  quantify numerically.  We  hypothesize  that  signal transduction  pathways  could  be  altered  by - 2 -4th Micro and Nano Flows ConferenceUCL, London, UK, 7-10 September 2014Figure 1: At two instants during diastole and  systole.  Intravascular  concentration  of  nucleotides remains mostly insensitive to the  reactions  occurring  at  the  endothelium.  Signalling pathways may however be affected  by the concentration boundary layer.mass  transport  phenomena  and  reaction kinetics occuring at the vessel wall. 2. Model FormulationLet x = [Ca2+]i represent the cytosolic free Ca2+ concentration,  y=[Ca2+]SR  the  Ca2+ concentration  in  the  sarcoplasmic  reticulum and  z the  cell  membrane  potential,  then  the system  of  ordinary  differential  equations characterising vasomotion is the following:where  the  electric  reversal  potentials  with respect to Ca2+ and Na+ are determined from the  Nernst  equation.  All  values  of  the coefficients are found in Table 1. A complete description of the mathematical model can be found  in   Parthimos  et  al.  (1999)  and Parthimos et al. (2007). The subscript r refers to  RyR-mediated  CICR, as  opposed to  InsP3 -induced  Ca2+  release  (not  included  in  the present formulation).3. Results of coupling SMCs-SMCsThe nature of coupling between adjacent cells remains  difficult  to  ascertain.  In  the  current formulation  we  explore  the  consequences  of either Ca2+  or electrical coupling, by assuming a  simple  gradient  driven  flux  between neighbouring cells coupled via gap junctions. The  actual  geometry  of  each  SMC  is  not modelled  explicitly.  The  mean  cell  surface area of is  143μm∼ 2.All cells are dynamically identical except for the  A parameter, which reflects influx of Ca2+ via  non-specific  cationic  channels  (NSCCs), and is  randomly distributed  around its  mean value given in Table 1. - 3 -4th Micro and Nano Flows ConferenceUCL, London, UK, 7-10 September 2014Hence the SMC population contains cells that are quiescent, in an under- or over-stimulated state,  in terms of Ca2+   availability,  and cells that are oscillating spontaneously. Within this range, the oscillatory regime is maintained at a physiological level.Weakly coupled cells are dominated by local dynamics  and  chaotic  patterns.  For  stronger Ca2+  coupling,  discrete  pacemaker  nodes become  prominent,  and  these  are  eventually dominated by one or two nodes,  resulting in clear wave fronts, as illustrated in Fig. 2.                                                                                                 This  behaviour  coincides  with  the synchronization  of  intracellular  Ca2+ oscillations,  showing  that  macroscopic rhythmic  activity  can  indeed  reflect  ion movements at the level of the individual cell.The interplay between extracellular Ca2+ influx via NSCCs and VOCCs was also investigated, although  limited  degree  of  synchronization was  observed  under  conditions  of  purely electrical  coupling.  Simulations  showed  how low or high influx through either channel leads to  the  dominance  of  Ca2+  and/or  electrical coupling, and to the loss of oscillatory activity.The  theoretical  consequences  of  local neighbour  interactions  through  electrical and/or  chemical  coupling  have  only  been briefly  presented,  and  will  be  further compared.   Additional  manifestations  of blood-wall and arterial dynamics coupling will be  investigated,  and  potential  mechanisms underlying flow-endothelium interactions will be discussed.   - 4 -Figure  2:  Computational  modelling  results  illustrating  the  emergence  of  propagating  wavefronts.  A  gradual  transition  of  uncorrelated oscillatory activity into coherent  patterns  is  observed  under  conditions  of  increased intercellular Ca2+ coupling from left  to right. Colours are coded as: red=cytosolic  free [Ca2+]i,  green=[Ca2+]SR,  blue=membrane potential. A magenta colour is seen where red  and  blue  strengths  are  equal,  and  therefore  indicative  of  membrane  depolarization  and  high Ca2+influx.4th Micro and Nano Flows ConferenceUCL, London, UK, 7-10 September 2014ReferencesBoileau  E.,  Bevan  RLT.,  Sazonov  I.,  Nithiarasu  P.  (2013)  Flow-induced  ATP  release  in  patient-specific  arterial  geometries:  a  comparative  study  of  computational  models.  Intl.  J.  Numer.  Method Biomed. Eng. 29, 1038–1056El Khatib N., Genieys V., Volpert V. (2007)  Atherosclerosis  initiation  modeled  as  an  inflammatory process. Math. Model. Nat.  Phenom. 2, 126-141. Cilla  M.,  Pena  E.,  Martinez  MA.  (2014)  Mathematical  modelling  of  atheroma  plaque  formation  and  development  in  coronary arteries. J. R. Soc. Interface. 11,  20130866.Griffith  TM. (1996)  Temporal  chaos  in  the  microcirculation.  Cardiovasc.  Res.  31,  342- 358.Halcox, JPJ.,  Donald,  AE.,  Ellins,  E.,  et  al.  (2009)  Endothelial  function  predicts  progression  of  carotid  intima-media  thickness. Circ. 119, 1005-1012. Halcox, JPJ., Schenke, WH., Zalos, G., et al.  (2002)  Prognostic  value  of  coronaryvascular endothelial dysfunction. Circ. 106, 653-658.Okazaki K., Seki S., Kanaya J., et al. (2003)  Role  for  the  endothelium-derivedhyperpolarizing  factor  in  phenylephrine-induced oscillatory vasomotion in rat small mesenteric  artery.  Anesthesiology.  98,  1164-1171.Parthimos,  D.,  Edwards.  DH.,  Griffith,  TM.  (1999)  Minimal  model  of  arterial  chaos  generated  by  coupled  intracellular  and  membrane  Ca2+  oscillators.  Am.  J.  Physiol. 277, H1119–H1144.Parthimos,  D.,  Haddock,  RE.,  Hill,  CE.,  Griffith, TM. (2007) Dynamics of a three-variable  nonlinear  model  of  vasomotion:  comparison  of  theory  and  experiments.  Biophys. J. 93, 1534–1556.Peng  H.,  Matchkov  A.,  Ivarsen  A.,  et  al.  (2001)  Hypothesis  for  the  initiation  of  vasomotion. Circ. Res. 88, 810-815.Schachinger,  V.,  Britten,  MB.,  Zeiher,  AM.  (2000)  Prognostic  impact  of  coronary  vasodilator  dysfunction  on  adverse  long-term outcome of coronary heart  disease.  Circ. 101, 1899-1906.Sell  M.,  Boldt  W.,  Markwardt  F.  (2002)   Desynchronising effect of the endothelium on  intracellular  Ca2+  concentration  dynamics in vascular smooth muscle cells of rat mesenteric arteries. Cell Calcium. 32, 105-120. - 5 -

Local Regulation of Arterial Tone: an Insight into Wall Dynamics Using Mathematical Models

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Local Regulation of Arterial Tone: an Insight into Wall Dynamics Using Mathematical Models

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