Biophysical properties of microvascular endothelium: Requirements for initiating and conducting electrical signals

OBJECTIVE

Electrical signaling along the endothelium underlies spreading vasodilation and blood flow control. We use mathematical modeling to determine the electrical properties of the endothelium and gain insight into the biophysical determinants of electrical conduction.

METHODS

Electrical conduction data along endothelial tubes (40 μm wide, 2.5 mm long) isolated from mouse skeletal muscle resistance arteries were analyzed using cable equations and a multicellular computational model.

RESULTS

Responses to intracellular current injection attenuate with an axial length constant (λ) of 1.2-1.4 mm. Data were fitted to estimate the axial (r_a ; 10.7 MΩ/mm) and membrane (r_m ; 14.5 MΩ∙mm) resistivities, EC membrane resistance (R_m ; 12 GΩ), and EC-EC coupling resistance (R_gj ; 4.5 MΩ) and predict that stimulation of ≥30 neighboring ECs is required to elicit 1 mV of hyperpolarization at distance = 2.5 mm. Opening Ca²⁺ -activated K⁺ channels (K_Ca ) along the endothelium reduced λ by up to 55%.

CONCLUSIONS

High R_m makes the endothelium sensitive to electrical stimuli and able to conduct these signals effectively. Whereas the activation of a group of ECs is required to initiate physiologically relevant hyperpolarization, this requirement is increased by myoendothelial coupling and K_Ca activation along the endothelium inhibits conduction by dissipating electrical signals.

The yin and yang of KV channels in cerebral small vessel pathologies

Cerebral SVDs encompass a group of genetic and sporadic pathological processes leading to brain lesions, cognitive decline, and stroke. There is no specific treatment for SVDs, which progress silently for years before becoming clinically symptomatic. Here, we examine parallels in the functional defects of PAs in CADASIL, a monogenic form of SVD, and in response to SAH, a common type of hemorrhagic stroke that also targets the brain microvasculature. Both animal models exhibit dysregulation of the voltage-gated potassium channel, K_V 1, in arteriolar myocytes, an impairment that compromises responses to vasoactive stimuli and impacts CBF autoregulation and local dilatory responses to neuronal activity (NVC). However, the extent to which this channelopathy-like defect ultimately contributes to these pathologies is unknown. Combining experimental data with computational modeling, we describe the role of K_V 1 channels in the regulation of myocyte membrane potential at rest and during the modest increase in extracellular potassium associated with NVC. We conclude that PA resting membrane potential and myogenic tone depend strongly on K_V 1.2/1.5 channel density, and that reciprocal changes in K_V channel density in CADASIL and SAH produce opposite effects on extracellular potassium-mediated dilation during NVC.

Can endothelial hemoglobin-α regulate nitric oxide vasodilatory signaling?

We used mathematical modeling to investigate nitric oxide (NO)-dependent vasodilatory signaling in the arteriolar wall. Detailed continuum cellular models of calcium (Ca²⁺) dynamics and membrane electrophysiology in smooth muscle and endothelial cells (EC) were coupled with models of NO signaling and biotransport in an arteriole. We used this theoretical approach to examine the role of endothelial hemoglobin-α (Hbα) as a modulator of NO-mediated myoendothelial feedback, as previously suggested in Straub et al. (Nature 491: 473-477, 2012). The model considers enriched expression of inositol 1,4,5-triphosphate receptors (IP₃Rs), endothelial nitric oxide synthase (eNOS) enzyme, Ca²⁺-activated potassium (K_Ca) channels and Hbα in myoendothelial projections (MPs) between the two cell layers. The model suggests that NO-mediated myoendothelial feedback is plausible if a significant percentage of eNOS is localized within or near the myoendothelial projection. Model results show that the ability of Hbα to regulate the myoendothelial feedback is conditional to its colocalization with eNOS near MPs at concentrations in the high nanomolar range (>0.2 μM or 24,000 molecules). Simulations also show that the effect of Hbα observed in in vitro experimental studies may overestimate its contribution in vivo, in the presence of blood perfusion. Thus, additional experimentation is required to quantify the presence and spatial distribution of Hbα in the EC, as well as to test that the strong effect of Hbα on NO signaling seen in vitro, translates also into a physiologically relevant response in vivo.NEW & NOTEWORTHY Mathematical modeling shows that although regulation of nitric oxide signaling by hemoglobin-α (Hbα) is plausible, it is conditional to its presence in significant concentrations colocalized with endothelial nitric oxide synthase in myoendothelial projections. Additional experimentation is required to test that the strong effect of Hbα seen in vitro translates into a physiologically relevant response in vivo.

Inherent rhythm of smooth muscle cells in rat mesenteric arterioles: An eigensystem formulation

On the basis of experimental data and mathematical equations in the literature, we remodel the ionic dynamics of smooth muscle cells (SMCs) as an eigensystem formulation, which is valid for investigating finite variations of variables from the equilibrium such as in common experimental operations. This algorithm provides an alternate viewpoint from frequency-domain analysis and enables one to probe functionalities of SMCs' rhythm by means of a resonance-related mechanism. Numerical results show three types of calcium oscillations of SMCs in mesenteric arterioles: spontaneous calcium oscillation, agonist-dependent calcium oscillation, and agonist-dependent calcium spike. For simple single and double SMCs, we demonstrate properties of synchronization among complex signals related to calcium oscillations, and show different correlation relations between calcium and voltage signals for various synchronization and resonance conditions. For practical cell clusters, our analyses indicate that the rhythm of SMCs could (1) benefit enhancements of signal communications among remote cells, (2) respond to a significant calcium peaking against transient stimulations for triggering globally oscillating modes, and (3) characterize the globally oscillating modes via frog-leap (non-molecular-diffusion) calcium waves across inhomogeneous SMCs.

Endothelial mitochondria regulate the intracellular Ca2+ response to fluid shear stress

Shear stress is known to stimulate an intracellular free calcium concentration ([Ca(2+)]i) response in vascular endothelial cells (ECs). [Ca(2+)]i is a key second messenger for signaling that leads to vasodilation and EC survival. Although it is accepted that the shear-induced [Ca(2+)]i response is, in part, due to Ca(2+) release from the endoplasmic reticulum (ER), the role of mitochondria (second largest Ca(2+) store) is unknown. We hypothesized that the mitochondria play a role in regulating [Ca(2+)]i in sheared ECs. Cultured ECs, loaded with a Ca(2+)-sensitive fluorophore, were exposed to physiological levels of shear stress. Shear stress elicited [Ca(2+)]i transients in a percentage of cells with a fraction of them displaying oscillations. Peak magnitudes, percentage of oscillating ECs, and oscillation frequencies depended on the shear level. [Ca(2+)]i transients/oscillations were present when experiments were conducted in Ca(2+)-free solution (plus lanthanum) but absent when ECs were treated with a phospholipase C inhibitor, suggesting that the ER inositol 1,4,5-trisphosphate receptor is responsible for the [Ca(2+)]i response. Either a mitochondrial uncoupler or an electron transport chain inhibitor, but not a mitochondrial ATP synthase inhibitor, prevented the occurrence of transients and especially inhibited the oscillations. Knockdown of the mitochondrial Ca(2+) uniporter also inhibited the shear-induced [Ca(2+)]i transients/oscillations compared with controls. Hence, EC mitochondria, through Ca(2+) uptake/release, regulate the temporal profile of shear-induced ER Ca(2+) release. [Ca(2+)]i oscillation frequencies detected were within the range for activation of mechanoresponsive kinases and transcription factors, suggesting that dysfunctional EC mitochondria may contribute to cardiovascular disease by deregulating the shear-induced [Ca(2+)]i response.

Agmatine induced NO dependent rat mesenteric artery relaxation and its impairment in salt-sensitive hypertension

l-Arginine and its decarboxylated product, agmatine are important mediators of NO production and vascular relaxation. However, the underlying mechanisms of their action are not understood. We have investigated the role of arginine and agmatine in resistance vessel relaxation of Sprague-Dawley (SD) and Dahl salt-sensitive hypertensive rats. Second or 3rd-order mesenteric arterioles were cannulated in an organ chamber, pressurized and equilibrated before perfusing intraluminally with agonists. The vessel diameters were measured after mounting on the stage of a microscope fitted with a video camera. The gene expression in Dahl rat vessel homogenates was ascertained by real-time PCR. l-Arginine initiated relaxations (EC50, 5.8±0.7mM; n=9) were inhibited by arginine decarboxylase (ADC) inhibitor, difluoromethylarginine (DFMA) (EC50, 18.3±1.3mM; n=5) suggesting that arginine-induced vessel relaxation was mediated by agmatine formation. Agmatine relaxed the SD rat vessels at significantly lower concentrations (EC50, 138.7±12.1μM; n=22), which was compromised by l-NAME (l-N(G)-nitroarginine methyl ester, an eNOS inhibitor), RX821002 (α-2 AR antagonist) and pertussis toxin (G-protein inhibitor). The agmatine-mediated vessel relaxation from high salt Dahl rats was abolished as compared to that from normal salt rats (EC50, 143.9±23.4μM; n=5). The α-2A AR, α-2B AR and eNOS mRNA expression was downregulated in mesenteric arterioles of high-salt treated Dahl hypertensive rats. These findings demonstrate that agmatine facilitated the relaxation via activation of α-2 adrenergic G-protein coupled receptor and NO synthesis, and this pathway is compromised in salt-sensitive hypertension.

Role of microprojections in myoendothelial feedback--a theoretical study

We investigated the role of myoendothelial projections (MPs) in endothelial cell (EC) feedback response to smooth muscle cell (SMC) stimulation using mathematical modelling. A previously developed compartmental EC-SMC model is modified to include MPs as subcellular compartments in the EC. The model is further extended into a 2D continuum model using a finite element method (FEM) approach and electron microscopy images to account for MP geometry. The EC and SMC are coupled via non-selective myoendothelial gap junctions (MEGJs) which are located on MPs and allow exchange of Ca(2+), K(+), Na(+) and Cl(-) ions and inositol 1,4,5-triphosphate (IP3). Models take into consideration recent evidence for co-localization of intermediate-conductance calcium-activated potassium channels (IKCa) and IP3 receptors (IP3Rs) in the MPs. SMC stimulation causes an IP3-mediated Ca(2+) transient in the MPs with limited global spread in the bulk EC. A hyperpolarizing feedback generated by the localized IKCa channels is transmitted to the SMC via MEGJs. MEGJ resistance (Rgj) and the density of IKCa and IP3R in the projection influence the extent of EC response to SMC stimulation. The predicted Ca(2+) transients depend also on the volume and geometry of the MP. We conclude that in the myoendothelial feedback response to SMC stimulation, MPs are required to amplify the SMC initiated signal. Simulations suggest that the signal is mediated by IP3 rather than Ca(2+) diffusion and that a localized rather than a global EC Ca(2+) mobilization is more likely following SMC stimulation.

Bone marrow transplantation improves endothelial function in hypertensive Dahl salt-sensitive rats

Bone marrow-derived endothelial progenitor cells (EPCs) constitute an important endogenous system in the maintenance of endothelial integrity and vascular homeostasis. Cardiovascular risk factors are associated with a reduced number and functional capacity of EPCs. Here we investigated the effect of transplantation of bone marrow-derived cells from Dahl salt-resistant rat into age-matched Dahl salt-sensitive (DS) rat on blood pressure, endothelial function, and circulating EPC number. The recipient DS rats were fed a normal (0.5% NaCl, NS) or high-salt (4% NaCl, HS) diet for 6 weeks after bone marrow transplantation (BMT). DS rats on a NS or a HS diet without BMT were used as controls. Hypertensive DS (HS-DS) rat (systolic blood pressure: 213 ± 4 mm Hg vs. 152 ± 4 mm Hg in NS, P < .05) manifested impaired endothelium-dependent relaxation to acetylcholine (EDR), increased gene expression of vascular oxidative stress and proinflamamtory cytokines, and decreased eNOS expression. BMT on HS-DS rat significantly improved EDR and eNOS expression, reduced oxidative stress without reduction in SBP (206 ± 6 mm Hg). Flow cytometry analysis showed that there was no difference in the number of circulating EPCs, demonstrated by expression of EPC markers CD34, cKit, and vascular endothelial growth factor, between hypertensive and normotensive rats. Surprisingly, BMT resulted in a 5- to 10-fold increase in the previously mentioned EPC markers in hypertensive, but not normotensive rat. These results suggest that DS rat has an impaired ability to increase bone marrow-derived EPCs in response to HS diet challenge, which may contribute to endothelial dysfunction.

Glutathiyl radical as an intermediate in glutathione nitrosation

Nitrosation of thiols is thought to be mediated by dinitrogen trioxide (N(2)O(3)) or by nitrogen dioxide radical (()NO(2)). A kinetic study of glutathione (GSH) nitrosation by NO donors in aerated buffered solutions was undertaken. S-nitrosoglutathione (GSNO) formation was assessed spectrophotometrically and by chemiluminescence. The results suggest an increase in the rate of GSNO formation with an increase in GSH with a half-maximum constant EC(50) that depends on NO concentration. Our observed increase in EC(50) with NO concentration suggests a significant contribution of ()NO(2)-mediated nitrosation with the glutathiyl radical as an intermediate in the production of GSNO.

Intercellular communication in the vascular wall: a modeling perspective

Movement of ions (Ca(2+) , K(+) , Na(+) , and Cl(-) ) and second messenger molecules like inositol 1, 4, 5-trisphosphate inside and in between different cells is the basis of many signaling mechanisms in the microcirculation. In spite of the vast experimental efforts directed toward evaluation of these fluxes, it has been a challenge to establish their roles in many essential microcirculatory phenomena. Recently, detailed theoretical models of calcium dynamics and plasma membrane electrophysiology have emerged to assist in the quantification of these intra and intercellular fluxes and enhance understanding of their physiological importance. This perspective reviews selected models relevant to estimation of such intra and intercellular ionic and second messenger fluxes and prediction of their relative significance to a variety of vascular phenomena, such as myoendothelial feedback, conducted responses, and vasomotion.

Kinetic analysis of DAF-FM activation by NO: toward calibration of a NO-sensitive fluorescent dye

Nitric oxide (NO) research in biomedicine has been hampered by the absence of a method that will allow quantitative measurement of NO in biological tissues with high sensitivity and selectivity, and with adequate spatial and temporal resolution. 4-amino-5-methylamino-2',7'-difluorofluorescein (DAF-FM) is a NO sensitive fluorescence probe that has been used widely for qualitative assessment of cellular NO production. However, calibration of the fluorescent signal and quantification of NO concentration in cells and tissues using fluorescent probes, have provided significant challenge. In this study we utilize a combination of mathematical modeling and experimentation to elucidate the kinetics of NO/DAF-FM reaction in solution. Modeling and experiments suggest that the slope of fluorescent intensity (FI) can be related to NO concentration according to the equation: ddtFI=2αk(1)NO(2)O(2)DAF-FMkNO+DAF-FM where α is a proportionality coefficient that relates FI to unit concentration of activated DAF-FM, k(1) is the NO oxidation rate constant, and k was estimated to be 4.3±0.6. The FI slope exhibits saturation kinetics with DAF-FM concentration. Interestingly, the effective half-maximum constant (EC(50)) increases proportionally to NO concentration. This result is not in agreement with the proposition that N(2)O(3) is the NO oxidation byproduct that activates DAF-FM. Kinetic analysis suggests that the reactive intermediate should exhibit NO-dependent consumption and thus NO(2)() is a more likely candidate. The derived rate law can be used for the calibration of DAF-FM fluorescence and the quantification of NO concentration in biological tissues.

Multiple factors influence calcium synchronization in arterial vasomotion

The intercellular synchronization of spontaneous calcium (Ca(2+)) oscillations in individual smooth muscle cells is a prerequisite for vasomotion. A detailed mathematical model of Ca(2+) dynamics in rat mesenteric arteries shows that a number of synchronizing and desynchronizing pathways may be involved. In particular, Ca(2+)-dependent phospholipase C, the intercellular diffusion of inositol trisphosphate (IP(3), and to a lesser extent Ca(2+)), IP(3) receptors, diacylglycerol-activated nonselective cation channels, and Ca(2+)-activated chloride channels can contribute to synchronization, whereas large-conductance Ca(2+)-activated potassium channels have a desynchronizing effect. Depending on the contractile state and agonist concentrations, different pathways become predominant, and can be revealed by carefully inhibiting the oscillatory component of their total activity. The phase shift between the Ca(2+) and membrane potential oscillations can change, and thus electrical coupling through gap junctions can mediate either synchronization or desynchronization. The effect of the endothelium is highly variable because it can simultaneously enhance the intercellular coupling and affect multiple smooth muscle cell components. Here, we outline a system of increased complexity and propose potential synchronization mechanisms that need to be experimentally tested.

Modeling Ca2+ signaling in the microcirculation: intercellular communication and vasoreactivity

A network of intracellular signaling pathways and complex intercellular interactions regulate calcium mobilization in vascular cells, arteriolar tone, and blood flow. Different endothelium-derived vasoreactive factors have been identified and the importance of myoendothelial communication in vasoreactivity is now well appreciated. The ability of many vascular networks to conduct signals upstream also is established. This phenomenon is critical for both short-term changes in blood perfusion as well as long-term adaptations of a vascular network. In addition, in a phenomenon termed vasomotion, arterioles often exhibit spontaneous oscillations in diameter. This is thought to improve tissue oxygenation and enhance blood flow. Experimentation has begun to reveal important aspects of the regulatory machinery and the significance of these phenomena for the regulation of local perfusion and oxygenation. Mathematical modeling can assist in elucidating the complex signaling mechanisms that participate in these phenomena. This review highlights some of the important experimental studies and relevant mathematical models that provide the current understanding of these mechanisms in vasoreactivity.

Endothelial Ca2+ wavelets and the induction of myoendothelial feedback

When arteries constrict to agonists, the endothelium inversely responds, attenuating the initial vasomotor response. The basis of this feedback mechanism remains uncertain, although past studies suggest a key role for myoendothelial communication in the signaling process. The present study examined whether second messenger flux through myoendothelial gap junctions initiates a negative-feedback response in hamster retractor muscle feed arteries. We specifically hypothesized that when agonists elicit depolarization and a rise in second messenger concentration, inositol trisphosphate (IP(3)) flux activates a discrete pool of IP(3) receptors (IP(3)Rs), elicits localized endothelial Ca(2+) transients, and activates downstream effectors to moderate constriction. With use of integrated experimental techniques, this study provided three sets of supporting observations. Beginning at the functional level, we showed that blocking intermediate-conductance Ca(2+)-activated K(+) channels (IK) and Ca(2+) mobilization from the endoplasmic reticulum (ER) enhanced the contractile/electrical responsiveness of feed arteries to phenylephrine. Next, structural analysis confirmed that endothelial projections make contact with the overlying smooth muscle. These projections retained membranous ER networks, and IP(3)Rs and IK channels localized in or near this structure. Finally, Ca(2+) imaging revealed that phenylephrine induced discrete endothelial Ca(2+) events through IP(3)R activation. These events were termed recruitable Ca(2+) wavelets on the basis of their spatiotemporal characteristics. From these findings, we conclude that IP(3) flux across myoendothelial gap junctions is sufficient to induce focal Ca(2+) release from IP(3)Rs and activate a discrete pool of IK channels within or near endothelial projections. The resulting hyperpolarization feeds back on smooth muscle to moderate agonist-induced depolarization and constriction.

A mathematical model of vasoreactivity in rat mesenteric arterioles: I. Myoendothelial communication

To study the effect of myoendothelial communication on vascular reactivity, we integrated detailed mathematical models of Ca(2+) dynamics and membrane electrophysiology in arteriolar smooth muscle (SMC) and endothelial (EC) cells. Cells are coupled through the exchange of Ca(2+), Cl(-), K(+), and Na(+) ions, inositol 1,4,5-triphosphate (IP(3)), and the paracrine diffusion of nitric oxide (NO). EC stimulation reduces intracellular Ca(2+) ([Ca(2+)](i)) in the SMC by transmitting a hyperpolarizing current carried primarily by K(+). The NO-independent endothelium-derived hyperpolarization was abolished in a synergistic-like manner by inhibition of EC SK(Ca) and IK(Ca) channels. During NE stimulation, IP(3) diffusing from the SMC induces EC Ca(2+) release, which, in turn, moderates SMC depolarization and [Ca(2+)](i) elevation. On the contrary, SMC [Ca(2+)](i) was not affected by EC-derived IP(3). Myoendothelial Ca(2+) fluxes had no effect in either cell. The EC exerts a stabilizing effect on calcium-induced calcium release-dependent SMC Ca(2+) oscillations by increasing the norepinephrine concentration window for oscillations. We conclude that a model based on independent data for subcellular components can capture major features of the integrated vessel behavior. This study provides a tissue-specific approach for analyzing complex signaling mechanisms in the vasculature.

A mathematical model of vasoreactivity in rat mesenteric arterioles. II. Conducted vasoreactivity

This study presents a multicellular computational model of a rat mesenteric arteriole to investigate the signal transduction mechanisms involved in the generation of conducted vasoreactivity. The model comprises detailed descriptions of endothelial (ECs) and smooth muscle (SM) cells (SMCs), coupled by nonselective gap junctions. With strong myoendothelial coupling, local agonist stimulation of the EC or SM layer causes local changes in membrane potential (V(m)) that are conducted electrotonically, primarily through the endothelium. When myoendothelial coupling is weak, signals initiated in the SM conduct poorly, but the sensitivity of the SMCs to current injection and agonist stimulation increases. Thus physiological transmembrane currents can induce different levels of local V(m) change, depending on cell's gap junction connectivity. The physiological relevance of current and voltage clamp stimulations in intact vessels is discussed. Focal agonist stimulation of the endothelium reduces cytosolic calcium (intracellular Ca(2+) concentration) in the prestimulated SM layer. This SMC Ca(2+) reduction is attributed to a spread of EC hyperpolarization via gap junctions. Inositol (1,4,5)-trisphosphate, but not Ca(2+), diffusion through homocellular gap junctions can increase intracellular Ca(2+) concentration in neighboring ECs. The small endothelial Ca(2+) spread can amplify the total current generated at the local site by the ECs and through the nitric oxide pathway, by the SMCs, and thus reduces the number of stimulated cells required to induce distant responses. The distance of the electrotonic and Ca(2+) spread depends on the magnitude of SM prestimulation and the number of SM layers. Model results are consistent with experimental data for vasoreactivity in rat mesenteric resistance arteries.

Nitric oxide bioavailability in the microcirculation: insights from mathematical models

Over the last 30 years nitric oxide (NO) has emerged as a key signaling molecule involved in a number of physiological functions, including in the regulation of microcirculatory tone. Despite significant scientific contributions, fundamental questions about NO's role in the microcirculation remain unanswered. Mathematical modeling can assist in investigations of microcirculatory NO physiology and address experimental limitations in quantifying vascular NO concentrations. The number of mathematical models investigating the fate of NO in the vasculature has increased over the last few years, and new models are continuously emerging, incorporating an increasing level of complexity and detail. Models investigate mechanisms that affect NO availability in health and disease. They examine the significance of NO release from nonendothelial sources, the effect of transient release, and the complex interaction of NO with other substances, such as heme-containing proteins and reactive oxygen species. Models are utilized to test and generate hypotheses for the mechanisms that regulate NO-dependent signaling in the microcirculation.

A mathematical model of Ca2+ dynamics in rat mesenteric smooth muscle cell: agonist and NO stimulation

A mathematical model of calcium dynamics in vascular smooth muscle cell (SMC) was developed based on data mostly from rat mesenteric arterioles. The model focuses on (a) the plasma membrane electrophysiology; (b) Ca2+ uptake and release from the sarcoplasmic reticulum (SR); (c) cytosolic balance of Ca2+, Na+, K+, and Cl ions; and (d) IP3 and cGMP formation in response to norepinephrine(NE) and nitric oxide (NO) stimulation. Stimulation with NE induced membrane depolarization and an intracellular Ca2+ ([Ca2+]i) transient followed by a plateau. The plateau concentrations were mostly determined by the activation of voltage-operated Ca2+ channels. NE causes a greater increase in [Ca2+]i than stimulation with KCl to equivalent depolarization. Model simulations suggest that the effect of[Na+]i accumulation on the Na+/Ca2+ exchanger (NCX) can potentially account for this difference.Elevation of [Ca2+]i within a concentration window (150-300 nM) by NE or KCl initiated [Ca2+]i oscillations with a concentration-dependent period. The oscillations were generated by the nonlinear dynamics of Ca2+ release and refilling in the SR. NO repolarized the NE-stimulated SMC and restored low [Ca2+]i mainly through its effect on Ca2+-activated K+ channels. Under certain conditions, Na+-K+-ATPase inhibition can result in the elevation of [Na+]i and the reversal of NCX, increasing resting cytosolic and SR Ca2+ content, as well as reactivity to NE. Blockade of the NCX's reverse mode could eliminate these effects. We conclude that the integration of the selected cellular components yields a mathematical model that reproduces, satisfactorily, some of the established features of SMC physiology. Simulations suggest a potential role of intracellular Na+ in modulating Ca2+ dynamics and provide insights into the mechanisms of SMC constriction, relaxation, and the phenomenon of vasomotion. The model will provide the basis for the development of multi-cellular mathematical models that will investigate microcirculatory function in health and disease.

A mathematical model of plasma membrane electrophysiology and calcium dynamics in vascular endothelial cells

Vascular endothelial cells (ECs) modulate smooth muscle cell (SMC) contractility, assisting in vascular tone regulation. Cytosolic Ca2+ concentration ([Ca2+]i) and membrane potential ( Vm) play important roles in this process by controlling EC-dependent vasoactive signals and intercellular communication. The present mathematical model integrates plasmalemma electrophysiology and Ca2+ dynamics to investigate EC responses to different stimuli and the controversial relationship between [Ca2+]i and Vm. The model contains descriptions for the intracellular balance of major ionic species and the release of Ca2+ from intracellular stores. It also expands previous formulations by including more detailed transmembrane current descriptions. The model reproduces Vm responses to volume-regulated anion channel (VRAC) blockers and extracellular K+ concentration ([K+]o) challenges, predicting 1) that Vm changes upon VRAC blockade are [K+]o dependent and 2) a biphasic response of Vm to increasing [K+]o. Simulations of agonist-induced Ca2+ mobilization replicate experiments under control and Vm hyperpolarization blockade conditions. They show that peak [Ca2+]i is governed by store Ca2+ release while Ca2+ influx (and consequently Vm) impacts more the resting and plateau [Ca2+]i. The Vm sensitivity of rest and plateau [Ca2+]i is dictated by a [Ca2+]i “buffering” system capable of masking the Vm-dependent transmembrane Ca2+ influx. The model predicts plasma membrane Ca2+-ATPase and Ca2+ permeability as main players in this process. The heterogeneous Vm impact on [Ca2+]i may elucidate conflicting reports on how Vm influences EC Ca2+. The present study forms the basis for the development of multicellular EC-SMC models that can assist in understanding vascular autoregulation in health and disease.

A computational model of oxygen delivery by hemoglobin-based oxygen carriers in three-dimensional microvascular networks

A detailed computational model is developed to simulate oxygen transport from a three-dimensional (3D) microvascular network to the surrounding tissue in the presence of hemoglobin-based oxygen carriers. The model accounts for nonlinear O(2) consumption, myoglobin-facilitated diffusion and nonlinear oxyhemoglobin dissociation in the RBCs and plasma. It also includes a detailed description of intravascular resistance to O(2) transport and is capable of incorporating realistic 3D microvascular network geometries. Simulations in this study were performed using a computer-generated microvascular architecture that mimics morphometric parameters for the hamster cheek pouch retractor muscle. Theoretical results are presented next to corresponding experimental data. Phosphorescence quenching microscopy provided PO(2) measurements at the arteriolar and venular ends of capillaries in the hamster retractor muscle before and after isovolemic hemodilution with three different hemodilutents: a non-oxygen-carrying plasma expander and two hemoglobin solutions with different oxygen affinities. Sample results in a microvascular network show an enhancement of diffusive shunting between arterioles, venules and capillaries and a decrease in hemoglobin's effectiveness for tissue oxygenation when its affinity for O(2) is decreased. Model simulations suggest that microvascular network anatomy can affect the optimal hemoglobin affinity for reducing tissue hypoxia. O(2) transport simulations in realistic representations of microvascular networks should provide a theoretical framework for choosing optimal parameter values in the development of hemoglobin-based blood substitutes.

A computational model of oxygen transport in skeletal muscle for sprouting and splitting modes of angiogenesis

Oxygen transport from capillary networks in muscle at a high oxygen consumption rate was simulated using a computational model to assess the relative efficacies of sprouting and splitting modes of angiogenesis. Efficacy was characterized by the volumetric fraction of hypoxic tissue and overall heterogeneity of oxygen distribution at steady state. Oxygen transport was simulated for a three-dimensional vascular network using parameters for rat extensor digitorum longus (EDL) muscle when oxygen consumption by tissue reached 6, 12, and 18 times basal consumption. First, a control network was generated by using straight non-anastomosed capillaries to establish baseline capillarity. Two networks were then constructed simulating either abluminal lateral sprouting or intraluminal splitting angiogenesis such that capillary surface area was equal in both networks. The sprouting network was constructed by placing anastomosed capillaries between straight capillaries of the control network with a higher probability of placement near hypoxic tissue. The splitting network was constructed by splitting capillaries from the control network into two branches at randomly chosen branching points. Under conditions of moderate oxygen consumption (6 times basal), only minor differences in oxygen delivery resulted between the sprouting and splitting networks. At higher consumption levels (12 and 18 times basal), the splitting network had the lowest volume of hypoxic tissue of the three networks. However, when total blood flow in all three networks was made equal, the sprouting network had the lowest volume of hypoxic tissue. This study also shows that under the steady-state conditions the effect of myoglobin (Mb) on oxygen transport was small.

A model of nitric oxide capillary exchange

OBJECTIVE

Our aim was to develop a mathematical model that describes the nitric oxide (NO) transport in and around capillaries. The model is used to make quantitative predictions for (1) the contribution of capillary endothelium to the nitric oxide flux into the parenchymal tissue cells; (2) the scavenging of arteriolar endothelium-derived NO by capillaries in the surrounding tissue; and (3) the role of myoglobin in tissue cells and plasma-based hemoglobin on NO diffusion in and around capillaries.

METHODS

We used a finite element model of a capillary and surrounding tissue with discrete parachute-shape red blood cells (RBCs) moving inside the capillary to obtain the NO concentration distribution. An intravascular mass transfer coefficient is estimated as a function of RBC membrane permeability and capillary hematocrit. A continuum model of the capillary is also formulated, in which blood is treated as a homogeneous fluid; it uses the mass transfer coefficient and provides a closed-form analytic solution for the average exchange rate of NO in a capillary-perfused region.

RESULTS

The NO concentration in the parenchymal cells depends on parameters such as RBC membrane permeability and capillary hematocrit; the concentration is predicted for a wide range of parameters. In the absence of myoglobin or plasma-based hemoglobin, the average tissue concentration generally ranges between 20 and 300 nM. In the presence of myoglobin or after transfusion of a hemoglobin-based blood substitute, there is minimal NO penetration into the tissue from the capillary endothelium.

CONCLUSIONS

The model suggests that NO originating from the capillary wall can diffuse toward the parenchymal cells and potentially sustain physiologically significant concentrations. The model provides estimates of NO exchange and concentration level in capillary-perfused tissue, and it can be used in models of NO transport around arterioles or other NO sources.

A theoretical model of nitric oxide transport in arterioles: frequency- vs. amplitude-dependent control of cGMP formation

Nitric oxide (NO) plays many important physiological roles, including the regulation of vascular smooth muscle tone. In response to hemodynamic or agonist stimuli, endothelial cells produce NO, which can diffuse to smooth muscle where it activates soluble guanylate cyclase (sGC), leading to cGMP formation and smooth muscle relaxation. The close proximity of red blood cells suggests, however, that a significant amount of NO released will be scavenged by blood, and thus the issue of bioavailability of endothelium-derived NO to smooth muscle has been investigated experimentally and theoretically. We formulated a mathematical model for NO transport in an arteriole to test the hypothesis that transient, burst-like NO production can facilitate efficient NO delivery to smooth muscle and reduce NO scavenging by blood. The model simulations predict that 1) the endothelium can maintain a physiologically significant amount of NO in smooth muscle despite the presence of NO scavengers such as hemoglobin and myoglobin; 2) under certain conditions, transient NO release presents a more efficient way for activating sGC and it can increase cGMP formation severalfold; and 3) frequency-rather than amplitude-dependent control of cGMP formation is possible. This suggests that it is the frequency of NO bursts and perhaps the frequency of Ca(2+) oscillations in endothelial cells that may limit cGMP formation and regulate vascular tone. The proposed hypothesis suggests a new functional role for Ca(2+) oscillations in endothelial cells. Further experimentation is needed to test whether and under what conditions in silico predictions occur in vivo.

Model of nitric oxide diffusion in an arteriole: impact of hemoglobin-based blood substitutes

Administration of hemoglobin-based oxygen carriers (HBOCs) frequently results in vasoconstriction that is primarily attributed to the scavenging of endothelium-derived nitric oxide (NO) by cell-free hemoglobin. The ensuing pressor response could be caused by the high NO reactivity of HBOC in the vascular lumen and/or the extravasation of hemoglobin molecules. There is a need for quantitative understanding of the NO interaction with HBOC in the blood vessels. We developed a detailed mathematical model of NO diffusion and reaction in the presence of an HBOC for an arteriolar-size vessel. The HBOC reactivity with NO and degree of extravasation was studied in the range of 2-58 x 10(6) M(-1) x s(-1) and 0-100%, respectively. The model predictions showed that the addition of HBOC reduced the smooth muscle (SM) NO concentration in the activation range (12-28 nM) for soluble guanylate cyclase, a major determinant of SM contraction. The SM NO concentration was significantly reduced when the extravasation of HBOC molecules was considered. The myoglobin present in the parenchymal cells scavenges NO, which reduces the SM NO concentration.

Erythrocyte consumption of nitric oxide in presence and absence of plasma-based hemoglobin

Experimental measurements have suggested a consumption rate of nitric oxide (NO) by red blood cells (RBCs) that is orders of magnitude smaller than that of an equivalent concentration of free hemoglobin in solution. This difference has been attributed to external diffusion limitations in the transport of NO from the plasma to the surface of the RBC or to resistance in the transport through the erythrocytic membrane. A detailed mathematical model is developed to quantify the resistance to NO transport around a single RBC and to predict the consumption rate in the presence and absence of extracellular hemoglobin. We provide a description for the NO consumption rate as a function of hematocrit, RBC radius, membrane permeability, and extracellular hemoglobin concentration. We predict a first-order rate constant for NO consumption in blood between 7.5 x 10(2) and 6.5 x 10(3) s(-1) at a hematocrit of 45% for membrane permeability values between 0.1 and 40 cm/s. Our results suggest that the difference in NO uptake by RBCs and free hemoglobin is smaller than previously reported and it is hematocrit dependent.

Impact of volume-dependent alveolar diffusing capacity on exhaled nitric oxide concentration

Exhaled endogenous nitric oxide (NO) holds promise as a potential biomarker of pulmonary inflammation. Previous experimental and theoretical work has concluded that the alveolar concentration approaches a constant steady state value at end exhalation due to both a constant maximum flux or release of NO (J(max,alv)) and a constant diffusing capacity (D(NO,alv)) in the alveolar region. We have recently demonstrated that D(NO,alv) is not constant, but increases with alveolar volume (VA) given by the following average relationship: D(NO,alv) =48*VA(2/3) ml/min/mmHg (where VA is expressed in liters, STPD). We investigated the potential impact of a variable D(NO,alv) on exhaled concentration by incorporating the volume dependence into the currently accepted two-compartment model for NO exchange dynamics. Our results suggest that the mechanism underlying the plateau in exhaled concentration is a constant ratio J(max,alv)/D(NO,alv) This constant ratio requires a volume dependence of J(max,alv) similar to D(NO,alv), and is likely due to a decreasing alveolar surface area during exhalation.

A single-breath technique with variable flow rate to characterize nitric oxide exchange dynamics in the lungs

Current techniques to estimate nitric oxide (NO) production and elimination in the lungs are inherently nonspecific or are cumbersome to perform (multiple-breathing maneuvers). We present a new technique capable of estimating key flow-independent parameters characteristic of NO exchange in the lungs: 1) the steady-state alveolar concentration (C(alv,ss)), 2) the maximum flux of NO from the airways (J(NO,max)), and 3) the diffusing capacity of NO in the airways (D(NO,air)). Importantly, the parameters were estimated from a single experimental single-exhalation maneuver that consisted of a preexpiratory breath hold, followed by an exhalation in which the flow rate progressively decreased. The mean values for J(NO,max), D(NO,air), and C(alv,ss) do not depend on breath-hold time and range from 280-600 pl/s, 3.7-7.1 pl. s(-1). parts per billion (ppb)(-1), and 0.73-2.2 ppb, respectively, in two healthy human subjects. A priori estimates of the parameter confidence intervals demonstrate that a breath hold no longer than 20 s may be adequate and that J(NO,max) can be estimated with the smallest uncertainty and D(NO,air) with the largest, which is consistent with theoretical predictions. We conclude that our new technique can be used to characterize flow-independent NO exchange parameters from a single experimental single-exhalation breathing maneuver.

Effect of alveolar volume and sequential filling on the diffusing capacity of the lungs: II. Experiment

The diffusing capacity of the lung, DL, is a critical physiological parameter, yet the currently accepted clinical model (Jones-Meade) assumes a well-mixed alveolar region, and a constant DL independent of alveolar volume, VA, despite experimental evidence to the contrary. We have formulated a new mathematical model [Tsoukias, N.M, Wilson, A.F., George, S.C., 2000. Respir. Physiol. 120, 231-249] that considers variable alveolar mixing through a single parameter, k (0<k<1), and a DL that is a positive function of VA (DL=a+bVA or DL=alphaVA(beta)). The goal of this study is to determine the suitability of this model to determine the unknown parameters a, b, alpha, beta, and k from experimental data in normal subjects. The model predicts that the normal lung fills, in part, sequentially (k=0.51+/-0.35). The following average values in all seven subjects were obtained: DLNO=48.VA(2/3) ml/min/mmHg and DLCO=20+0.7.VA ml/min/mmHg (STPD) where VA is expressed in L (STPD). We conclude that the mathematical model is suitable for identifying the unknown parameters and thus can be used to characterize the degree of alveolar mixing (or sequential filling) as well as the volume dependence of DL.

Effect of alveolar volume and sequential filling on the diffusing capacity of the lungs: I. theory

The diffusing capacity, DL, is a critical physiological parameter of the lung used to assess gas exchange clinically. Most models developed to analyze experimental data from a single breath maneuver have assumed a well-mixed or uniform alveolar region, including the clinically accepted Jones-Meade method. In addition, all previous models have assumed a constant DL, which is independent of alveolar volume, VA. In contrast, experimental data provide evidence for a non-uniform alveolar region coupled with sequential filling of the lung. In addition, although the DL for carbon monoxide is a weak function of VA, the DL of nitric oxide depends strongly on VA. We have developed a new mathematical model of the single breath maneuver that considers both a variable degree of sequential filling and a variable DL. Our model predicts that the Jones-Meade method overestimates DL when the exhaled gas sample is collected late in the exhalation, but underestimates DL if the exhaled gas sample is collected early in the exhalation phase due to the effect of sequential filling. Utilizing a prolonged constant exhalation method, or a three-equation method, will also produce erroneous predictions of DL. We conclude that current methods may introduce significant error in the estimation of DL by ignoring the sequential filling of the lung, and the dependence of DL on VA.

A two-compartment model of pulmonary nitric oxide exchange dynamics

The relatively recent detection of nitric oxide (NO) in the exhaled breath has prompted a great deal of experimentation in an effort to understand the pulmonary exchange dynamics. There has been very little progress in theoretical studies to assist in the interpretation of the experimental results. We have developed a two-compartment model of the lungs in an effort to explain several fundamental experimental observations. The model consists of a nonexpansile compartment representing the conducting airways and an expansile compartment representing the alveolar region of the lungs. Each compartment is surrounded by a layer of tissue that is capable of producing and consuming NO. Beyond the tissue barrier in each compartment is a layer of blood representing the bronchial circulation or the pulmonary circulation, which are both considered an infinite sink for NO. All parameters were estimated from data in the literature, including the production rates of NO in the tissue layers, which were estimated from experimental plots of the elimination rate of NO at end exhalation (ENO) vs. the exhalation flow rate (VE). The model is able to simulate the shape of the NO exhalation profile and to successfully simulate the following experimental features of endogenous NO exchange: 1) an inverse relationship between exhaled NO concentration and VE, 2) the dynamic relationship between the phase III slope and VE, and 3) the positive relationship between ENO and VE. The model predicts that these relationships can be explained by significant contributions of NO in the exhaled breath from the nonexpansile airways and the expansile alveoli. In addition, the model predicts that the relationship between ENO and VE can be used as an index of the relative contributions of the airways and the alveoli to exhaled NO.

Single-exhalation profiles of NO and CO2 in humans: effect of dynamically changing flow rate

Endogenous production of nitric oxide (NO) in the human lungs has many important pathophysiological roles and can be detected in the exhaled breath. An understanding of the factors that dictate the shape of the NO exhalation profile is fundamental to our understanding of normal and diseased lung function. We collected single-exhalation profiles of NO and CO2 from normal human subjects after inhalation of ambient air (approximately 15 parts/billion) and examined the effect of a 15-s breath hold and exhalation flow rate (VE) on the following features of the NO profile: 1) series dead space, 2) average concentration in phase III with respect to time and volume, 3) normalized slope of phase III with respect to time and volume, and 4) elimination rate at end exhalation. The dead space is approximately 50% smaller for NO than for CO2 and is substantially reduced after a breath hold. The concentration of exhaled NO is inversely related to VE, but the average NO concentration with respect to time has a stronger inverse relationship than that with respect to volume. The normalized slope of phase III NO with respect to time and that with respect to volume are negative at a constant VE but can be made to change signs if the flow rate continuously decreases during the exhalation. In addition, NO elimination at end exhalation vs. VE produces a nonzero intercept and slope that are subject dependent and can be used to quantitate the relative contribution of the airways and the alveoli to exhaled NO. We conclude that exhaled NO has an airway and an alveolar source.