Modeling steady state SO2-dependent changes in capillary ATP concentration using novel O2 micro-delivery methods

Adenosine triphosphate (ATP) is known to be released from the erythrocyte in an oxygen (O2) dependent manner. Since ATP is a potent vasodilator, it is proposed to be a key regulator in the pathway that mediates micro-vascular response to varying tissue O2 demand. We propose that ATP signaling mainly originates in the capillaries due to the relatively long erythrocyte transit times in the capillary and the short ATP diffusion distance to the electrically coupled endothelium. We have developed a computational model to investigate the effect of delivering or removing O2 to limited areas at the surface of a tissue with an idealized parallel capillary array on total ATP concentration. Simulations were conducted when exposing full surface to perturbations in tissue O2 tension (PO2) or locally using a circular micro-outlet (~100 μm in diameter), a square micro-slit (200 × 200 μm), or a rectangular micro-slit (1000 μm wide × 200 μm long). Results indicated the rectangular micro-slit has the optimal dimensions for altering hemoglobin saturations (SO2) in sufficient number capillaries to generate effective changes in total [ATP]. This suggests a threshold for the minimum number of capillaries that need to be stimulated in vivo by imposed tissue hypoxia to induce a conducted micro-vascular response. SO2 and corresponding [ATP] changes were also modeled in a terminal arteriole (9 μm in diameter) that replaces 4 surface capillaries in the idealized network geometry. Based on the results, the contribution of terminal arterioles to the net change in [ATP] in the micro-vascular network is minimal although they would participate as O2 sources thus influencing the O2 distribution. The modeling data presented here provide important insights into designing a novel micro-delivery device for studying micro-vascular O2 regulation in the capillaries in vivo.


INTRODUCTION
The microcirculation plays the important role of delivering and regulating the exchange of oxygen (O 2 ) and nutrients to surrounding live metabolic tissue. The transport processes in the microcirculation are tightly controlled and highly integrated. Since proper O 2 supply to tissue is critical for cellular function and survival, the mechanisms underlying O 2 transport and distribution have been under thorough investigation. The microvasculature has to continuously adjust erythrocyte distribution and hence O 2 supply to meet the varying demand of metabolic tissue. During exercise, erythrocyte supply rate increases delivering more O 2 carrying erythrocytes to the microvasculature. The highly regulated system implies the presence of signaling components that link tissue O 2 demand with blood flow and microvascular function.
A great amount of evidence suggests the involvement of the erythrocyte as a sensor and a key player in this regulation mechanism (Stein and Ellsworth, 1993;Ellsworth et al., 1995Ellsworth et al., , 2008. Erythrocytes are the carriers of O 2 , bound to hemoglobin, in the microcirculation. Erythrocytes also contain large amounts of adenosine triphosphate (ATP) (Miseta et al., 1993), a potent vasodilator, and are known to release it under hypoxic conditions (Bergfeld and Forrester, 1992;Jagger et al., 2001;González-Alonso et al., 2002). Once ATP is released, it binds to purinergic receptors (P2Y) on the vascular endothelium eliciting a vasodilatory signal which is conducted upstream in the arteriolar tree (Ellsworth et al., 2008). The resulting vaso-relaxation of smooth muscle cells (SMCs) surrounding upstream arterioles increases erythrocyte supply rate to meet the metabolic demand of the hypoxic region downstream that initiated the release of ATP from erythrocytes.
For a long time, arterioles have been investigated as a major site of microvascular signaling (Duling and Berne, 1970;Duling, 1974;Jackson, 1987). This has been assumed, mainly, due to the large longitudinal PO 2 gradients that exist at the arteriolar level. In terms of ATP mediated signaling, the presence of SMCs implies that the released ATP will act locally and instantaneously elicit a signal. However, the relatively short erythrocyte transit times in arterioles are anticipated to largely compromise the localization of this ATP signal, while the parabolic flow profile in the arteriole means only those cells closest to the wall experience the largest change in O 2 saturation (SO 2 ) and hence contribute to the signal. Cells flowing in the centerline will be experiencing a lesser drop in SO 2 and any released ATP will be carried downstream .
Venules may also be involved in the regulation of O 2 supply since they act as the collectors of large populations of deoxygenated ATP-releasing erythrocytes. However, the diversity in the erythrocyte SO 2 levels as they drain from various upstream capillaries indicates that venules may only contribute to the overall vaso-dilatory signal . Fine-tune regulation of O 2 distribution to specific capillaries or microvascular units in the microcirculation demands the signal be highly localized. This may only be achieved at the capillary level. Erythrocytes traverse capillaries with long transit times and are in almost direct contact with the capillary endothelium. Hence, released ATP, mediated by erythrocyte deoxygenation, will be effectively transferred to purinergic receptors on the endothelium. Many studies have shown that the capillary endothelium is conductive when locally stimulated by vasodilators (Dietrich, 1989;Dietrich and Tyml, 1992a,b;Song and Tyml, 1993;Collins et al., 1998;Bagher and Segal, 2011). Therefore, we hypothesize that the capillary bed is the major site for O 2 regulation in the microcirculation .
To test this hypothesis, we have been examining the microvascular response to local perturbations in tissue O 2 tension (PO 2 ) using a novel O 2 micro-delivery tool (Ghonaim et al., 2011). We have created an O 2 micro-delivery (and removal) system that allows for altering local tissue PO 2 and hence erythrocyte SO 2 in a few selected capillaries at the surface of the Extensor Digitorum Longus (EDL) muscle of the rat (Figure 1). This system replaces the gas exchange chamber originally used in our group to alter surface tissue PO 2 of the entire bottom surface of the muscle (Ghonaim et al., 2011;Ellis et al., 2012). The chamber is positioned in the platform of an inverted microscope and is connected to computer controlled gas flow meters which allows for capturing video images of the microvascular response to PO 2 perturbations while simultaneously controlling chamber PO 2 levels. Erythrocyte SO 2 values are calculated based on a dual-wavelength image capture system and video sequences are post-processed to extract functional images and hemodynamic information as previously described (Ellis et al., 1990(Ellis et al., , 1992Japee et al., 2004Japee et al., , 2005a. In our novel O 2 micro-delivery setup, ultrathin plastic/glass sheet patterned with an O 2 delivery micro-outlet replaces the gas permeable membrane in the original chamber (Ghonaim et al., 2011;Ellis et al., 2012). Data presented earlier (Ghonaim et al., 2011) show that circular micro-delivery outlets (100 μm in diameter) can alter SO 2 in single capillaries flowing directly over the outlet. However, in order to elicit microvascular responses, the optimal outlet dimensions should allow for a sufficient number of capillaries within a network to be stimulated to produce a large enough ATP signal. This should be accomplished while ensuring the high localization of the stimulus to affect only the desired capillaries. This requires testing with various O 2 outlet sizes and dimensions. Combining the possible technical challenges involved FIGURE 1 | (A) The novel O 2 micro-delivery approach. Extensor Digitorum Longus (EDL) rat muscle is surgically exposed and positioned on the viewing platform of an inverted microscope. O 2 is delivered to the surface of the muscle through a micro-outlet patterned in ultrathin glass/plastic sheet (Ghonaim et al., 2011). O 2 levels in the gas exchange chamber near the muscle surface are oscillated using computer controlled flow meters. Real-time videos of the trans-illuminated tissue are monitored and recorded using a dual-wavelength video microscopy system (Ellis et al., 1990(Ellis et al., , 1992Japee et al., 2004Japee et al., , 2005a) (B) Three designs of the oxygen micro-delivery outlet are tested: circular micro-outlet (∼100 μm in diameter) (Ghonaim et al., 2011), a square micro-slit (200 × 200 μm), and a rectangular micro-slit (1000 μm wide × 200 μm long).
in creating multiple designs of the O 2 micro-delivery device with the inherent complexities of the O 2 regulation system led us to develop a computational model for the system under investigation.
Recently, Goldman et al. (2012) presented a theoretical mathematical model based on previous work by Goldman and Popel (1999) and Arciero et al. (2008) to describe O 2 and ATP transport in the rat EDL microcirculation when using the original O 2 exchange chamber. In this study we employ the same approach to calculate SO 2 and ATP changes in selected capillaries flowing over an O 2 delivery outlet of specific dimensions. Three designs of the O 2 delivery micro-outlet were tested: circular outlet (100 μm in diameter), square outlet (200 × 200μm), and rectangular slit (200 μm long × 1000 μm wide). Average capillary SO 2 and ATP level at steady-state were calculated at various chamber PO 2 levels (15, 40, and 150 mmHg) relative to a zero flux boundary condition. In order to simplify the system under investigation, an idealized three dimensional (3D) parallel array capillary geometry has been used. Simulations were also run on a 3D idealized array geometry in which a terminal arteriole (9 μm in diameter) replaced 4 capillaries and was positioned 30 μm from the bottom tissue surface. These simulations allowed for investigating the potential role of the terminal arteriole in O 2 regulation. Confirming previous findings (Ghonaim et al., 2011), the results indicated that radial O 2 diffusion from an O 2 delivery micro-outlet regardless of its dimensions is limited to ∼50 μm, while axial diffusion affects ∼100 μm of tissue. The rectangular slit has the important property of ensuring that capillaries surrounding the network of interest are all experiencing the same PO 2 drop, which minimizes re-oxygenation and emphasizes the ATP signal. This design also produces sufficient ATP release in multiple capillaries that it should be able to consistently elicit micro-vascular responses, although this remains to be confirmed experimentally. The results presented here also predict minimal contribution of terminal arterioles to the net magnitude of ATP emerging from capillary network although they would participate as O 2 sources and hence influence the O 2 distribution. In the future, 3D capillary networks reconstructed from experimental data can be modeled which will provide more realistic data and help more closely predict changes in various parameters.

OXYGEN TRANSPORT MODEL
In this work, O 2 transport and ATP transport were modeled in an idealized 3D capillary network consisting of an array of parallel capillaries (oriented in the y direction). The computational model of O 2 transport was based on a finite-difference model described by Goldman and Popel (1999. In the model, the reaction-diffusion equation below was used to describe time-dependent tissue PO 2 P(x,y,z,t): where D is the tissue O 2 diffusion coefficient, α is the tissue O 2 solubility, and M(P) is consumption rate of O 2 in tissue ( Table 1). O 2 transport in tissue was facilitated by the presence of myoglobin where c Mb is myoglobin concentration, D Mb is the myoglobin diffusion coefficient, and S Mb (P) = P/(P + P 50 , Mb ) is the myoglobin SO 2 . Convective transport of O 2 in the microvessels at each axial location y was described using the following time-dependent mass balance equation for capillary SO 2 , S(y,t): where u is the mean blood velocity, R is the capillary radius, j is the O 2 flux at (y,θ) out of the capillary, C andC are blood O 2 -binding capacities, respectively, directly related to hematocrit: where H T is tube (volume-weighted) hematocrit, H D is discharge (flow-averaged) hematocrit, and C Hb is the binding capacity of hemoglobin (Table 1) (Goldman and Popel, 2001). P b is the blood PO 2 , and α b andα b are volume-and flow-weighted blood O 2 solubilities, respectively (Goldman and Popel, 2001), where, where α cell and α pl are the O 2 solubilities inside the erythrocyte and in the plasma (Goldman and Popel, 2001). The O 2 flux at the capillary-tissue interface was given by: where κ is the mass transfer coefficient and P w is the tissue PO 2 at the capillary surface. κ is a function of hematocrit as it describes the effect of RBC spacing on O 2 diffusion and exchange between capillary and tissue (Eggleton et al., 2000). At the capillary surface, the boundary condition was specified as: where n is the unit vector normal to the capillary surface and j is given by equation (3). In the model presented here, zero O 2 flux conditions (no O 2 exchange across tissue boundary) were specified at the tissue boundaries, except where PO 2 was fixed on part or all of the bottom surface to represent the effect of the O 2 exchange chamber (see below). As in the model described by Goldman and Popel (1999), Michaelis-Menten consumption kinetics, M = M 0 P/(P + P cr ), and the Hill equation for the oxyhemoglobin saturation curve, S(P) = P n /(P n + P n 50 ), were used along with the above O 2 transport equations to calculate tissue O 2 transport.
Hemodynamic parameters (erythrocyte mean velocity, v rbc , and hematocrit, H T ) were determined from in vivo experimental measurements in the EDL muscle of the rat. The capillary network consisted of 72 parallel capillaries, each of which was discretized into 50 cylindrical segments, and the tissue domain surrounding the capillaries had dimensions of 216 × 532× 500 μm and was discretized into 7,304,853 computational nodes using a grid spacing of approximately 2 μm (Figure 2A). Capillary entrance SO 2 (65%) and the tissue O 2 consumption rate (1.5 × 10 −4 ml O 2 /ml/s) were set based on previous experimental data .
For simulations that included a terminal arteriole in the 3D network geometry, the arteriole (9 μm in diameter) was positioned ∼30 μm from the bottom tissue surface and replaced 4 capillaries in the original parallel array capillary geometry ( Figure 2B). Simulations including the arteriole were run at both 65 and 80% arteriolar entrance SO 2 .

ATP TRANSPORT MODEL
ATP transport in the idealized 3D capillary network was modeled as described by Goldman et al. (2012), based on the O 2 transport mathematical model described above (Goldman and Popel, 2000). Using a capillary entrance ATP concentration of zero, plasma [ATP] was calculated by using a finite-difference method to solve the following continuum partial differential equation : where u is the mean blood velocity at axial location y, H D , and H T are the discharge and tube hematocrit, respectively, and R is capillary radius. C 0 and C 1 ( Table 1) are constants used to linearly approximate the ATP release rate as a function of SO 2, while k d provides an approximation of ATP degradation by the endothelium as previously described (Arciero et al., 2008).
To calculate the steady-state distributions of tissue PO 2 and capillary SO 2 and [ATP], time-dependent O 2 transport and ATP transport simulations were run, using zero initial conditions for all variables, until there were minimal changes in tissue O 2 consumption and PO 2 , and capillary O 2 flux, SO 2 and [ATP] between consecutive time steps.

TISSUE PO 2 BOUNDARY CONDITIONS USED TO MODEL OXYGEN EXCHANGE CHAMBER
For the idealized capillary geometry, 3D tissue PO 2 distribution and capillary [ATP] at steady state were calculated for O 2 delivery using full gas exchange chamber, circular micro-outlet (100 μm in diameter), square micro-outlet (200 × 200 μm), or a rectangular micro-slit (1000 μm wide × 200 μm long). For each chamber type, simulations were run at 3 PO 2 boundary conditions either over full surface (with full gas exchange chamber) or only at the micro-slit opening: 15, 40, and 150 mmHg. For the cases in which the PO 2 boundary condition is altered only at the microslit opening, the rest of the tissue surrounding the micro-slit is set to zero O 2 flux boundary condition. The results from all simulations were compared to a fourth control case in which full surface is set to zero O 2 flux boundary condition.
For the idealized capillary geometry that includes the terminal arteriole, O 2 diffusion was modeled for full chamber or a rectangular micro-slit (1000 μm wide × 200 μm long) at the 4 PO 2 boundary conditions discussed above. Each set of simulations was run with arteriolar entrance SO 2 of 65% or to 80%. Table 2 lists

MATHEMATICAL MODELING OF SO 2 -DEPENDENT ATP RELEASE IN CAPILLARY NETWORKS IN RESPONSE TO LOCALIZED TISSUE PO 2 PERTURBATIONS
In this study, the release of ATP in capillaries mediated by tissue hypoxia and the de-saturation of hemoglobin was modeled in a 3D idealized parallel capillary network. The dependence of the magnitude of total ATP release on the number of deoxygenated capillaries was also examined. Based on our previously described experimental work (Ghonaim et al., 2011), we mathematically simulated O 2 delivery to and removal from selected capillaries on the surface of skeletal muscle tissue (rat EDL) using three designs of O 2 exchange micro-outlets used in our in vivo experiments (Figure 1). In order to compare local O 2 perturbations using the micro-outlets to global perturbations using the full gas exchange chamber (Ghonaim et al., 2011;Ellis et al., 2012), O 2 delivery to and removal from the entire bottom tissue surface was also modeled. For each set of simulations, 3D tissue PO 2 distribution profiles and corresponding 3D capillary [ATP] maps were generated. Plots of calculated SO 2 and [ATP] along the length of selected capillaries (21,18,17,54) at steady state were also created. All simulations were run using software written in Fortran, and the results were analyzed and the plots were produced using MATLAB.

Full surface gas exchange chamber
In this set of simulations, the 3D PO 2 distribution in the tissue and corresponding SO 2 and [ATP] distribution along capillary length were modeled for the control scenario in which the full bottom tissue surface is exposed to PO 2 perturbations. This would experimentally simulate using the full gas exchange chamber. As shown in Figure 3 Figure 3C). This was clearly depicted in the corresponding vessel map ( Figure 3B). The deeper capillary (54) was less affected with ∼30% lower hemoglobin SO 2 and ∼12% increase in ATP release relative to zero flux. Exposing the full tissue surface to relatively high chamber PO 2 (150 mmHg) had the most significant impact on [ATP] in the capillary network. At 150 mmHg, hemoglobin SO 2 in both surface and deep tissue capillaries converged to ∼100% with ∼70% decrease in steady state [ATP] relative to no-flux ( Figure 3C). The depth of the PO 2 perturbation into the tissue when using the full gas exchange chamber was ∼100 μm as shown in the 3D PO 2 profiles ( Figure 3A).

Circular O 2 delivery micro-outlet
To investigate the effect of limiting the number of capillaries stimulated by local tissue PO 2 perturbations, we started by modeling capillary SO 2 and [ATP] changes when using a circular O 2 micro-outlet (100 μm in diameter, see Figure 1). Similar to previously discussed data (Ghonaim et al., 2011), substantive changes in local tissue PO 2 due to diffusion outwards from the circular outlet is limited to less than ∼50 μm, as shown in the 3D tissue PO 2 profiles ( Figure 4A). Also, the hypoxic and hyperoxic stimuli were highly localized to only those capillaries directly over the micro-outlet (17, 18, 21) as shown in the vessel maps (Figure 4B). At 40 mmHg chamber PO 2 level, calculated capillary SO 2 and [ATP] were in close agreement with the no-flux control for both surface and deep tissue capillaries with values being within ∼1 and ∼3%, respectively ( Figure 4C). Under imposed hypoxia, the capillary SO 2 dropped as capillaries crossed the micro-outlet region reaching a minimum value ∼40 μm downstream of the outlet after which SO 2 levels increased slightly due to re-oxygenation by surrounding capillaries.

Square O 2 delivery micro-outlet
Next, we simulated the effect of increasing the area of O 2 exchange, and hence perturbing a greater number of capillaries, by simulating an O 2 delivery micro-outlet 200 × 200 μm square. Similar to the circular micro-outlet design and as previously described (Ghonaim et al., 2011), the change of local tissue PO 2 surrounding the square outlet is limited to less than ∼50 μm, as shown in the 3D tissue PO 2 profiles ( Figure 5A).
In the case of the square micro-outlet, a larger number of surface capillaries experience the PO 2 perturbations, 7 of which were directly over the micro-outlet ( Figure 5B). Also, capillaries at both sides of those directly over the outlet seemed to be slightly affected by the PO 2 perturbations. At 40 mmHg, calculated SO 2 and capillary [ATP] distributions were similar to the no-flux control with surface capillaries having 15% higher SO 2 and 10% lower [ATP] values relative to zero flux condition ( Figure 5C). As observed with the circular micro-outlet, reoxygenation of stimulated capillaries following imposed hypoxia (15 mmHg) was at ∼40 μm downstream of the square microoutlet (Figure 5C)

Rectangular O 2 delivery micro-slit
The largest dimensions for an O 2 delivery micro-outlet currently being tested in our in vivo studies are for a rectangular micro-slit (1000 μm wide × 200 μm long). Since the 3D tissue dimensions in our computational model are less than the In these simulations, we are modeling O 2 delivery through a circular oxygen micro-delivery outlet (100 μm in diameter) to bottom tissue surface using novel micro-delivery approach (see Figure 1). Tissue surface directly on top of the micro-delivery outlet is exposed to 15, 40, or 150 mmHg chamber PO 2 level relative to a zero flux control boundary condition (A) Spatial 3D tissue PO 2 distribution (mmHg) at capillary entrance perspective ( width of the experimental micro-slit, the effect of the slit extends to both edges of the tissue allowing us to visualize the depth of the PO 2 distribution into the tissue. As shown in the 3D PO 2 plots (Figure 6A), the PO 2 perturbations extended ∼100 μm into the tissue with local tissue PO 2 changes similar to what was observed with other outlet designs. All surface capillaries shown on the vessel map are affected by the PO 2 perturbation as the outlet covers the entire surface width (Figure 6B).

FIGURE 5 | Simulations of 3D PO 2 and capillary [ATP] distribution in a tissue with idealized parallel capillary arrangement (72 hexagonally packed capillaries).
In these simulations, we are modeling O 2 delivery through a square oxygen micro-delivery outlet (200 × 200μm) to bottom tissue surface using our previously described novel micro-delivery approach (see Figure 1). Tissue surface directly on top of the micro-delivery outlet is exposed to 15, 40, or 150 mmHg chamber PO 2 level relative to a zero flux control boundary condition (

FIGURE 6 | Simulations of 3D PO 2 and capillary [ATP] distribution in a tissue with idealized parallel capillary arrangement (72 hexagonally packed capillaries).
In these simulations, we are modeling O 2 delivery through a rectangular oxygen micro-delivery outlet (1000 μm wide × 200 μm long) to bottom tissue surface using our previously described novel micro-delivery approach (see Figure 1). Tissue surface directly on top of the micro-delivery outlet is exposed to 15, 40, or 150 mmHg chamber PO 2 level relative to a zero flux control boundary condition (A) Spatial 3D tissue PO 2 distribution (mmHg) at capillary entrance perspective

Comparing change in relative ATP magnitude in response to varying the area of imposed tissue hypoxia
The change in the total magnitude of ATP (ATPtot) in the modeled network relative when imposing a hypoxic challenge (15 mmHg boundary condition) was calculated as percent increase above ATPtot at zero flux (Figure 7). Percent increase in ATP magnitude in the network was compared when exposing all of the bottom tissue surface to hypoxia (full chamber) or locally using the three micro-outlet designs discussed above. As shown in Figure 7, the total ATP magnitude increased with increase in micro-outlet dimensions and essentially the number of capillaries experiencing the PO 2 drop. The percent increase in ATPtot was more than doubled when locally perturbing tissue PO 2 using the rectangular micro-slit compared to the other micro-outlet designs. The total ATP magnitude calculated when limiting the area of tissue hypoxia using the rectangular micro-slit was only 38% lower relative to full exposed surface (Figure 7). The increase in the total ATP magnitude in a network exposed to local hypoxia was minimal (∼2%) when using the circular micro-outlet or and only 6% above that zero flux when using the square micro-outlet.

MATHEMATICAL MODELING OF ARTERIOLAR SO 2 AND ATP CONCENTRATION IN RESPONSE TO LOCALIZED TISSUE PO 2 PERTURBATIONS
In order to investigate the role terminal arterioles play in regulating SO 2 -mediated ATP signaling in capillary networks, particularly in the EDL muscle where larger arterioles are located much deeper in the tissue, the 3D idealized capillary geometry was modified to include a terminal arteriole, 9 μm in diameter, positioned 30 μm away from bottom tissue surface. The 3D tissue PO 2 distribution as well as SO 2 and [ATP] in the arteriole (vessel 69) and in the surrounding surface (capillaries 14 and 17) and deep tissue capillaries (represented by capillary 50) were modeled. Simulations were run for the case in which the full tissue surface is exposed to PO 2 perturbations (original gas exchange chamber) and for the case of spatially limited O 2 delivery using the rectangular O 2 delivery micro-slit. Also, the effect of varying arteriolar entrance SO 2 on the overall magnitude of ATP in response to altered tissue PO 2 was examined.

Full surface gas exchange chamber at 65 and 80% arteriolar entrance SO 2
In the 3D tissue PO 2 profiles and [ATP] vessel maps generated for these simulations, the PO 2 perturbations were shown to affect the terminal arteriole to a much lesser extent than the surface capillaries (Figures 8A,B, 9A,B). Also, these simulations showed the influence of the arteriole as an O 2 source on the SO 2 levels of nearby surface capillaries. For instance, the steady state SO 2 level in capillary 14, positioned right next to the arteriole, was ∼25% higher than the zero flux control condition when exposed to 40 mmHg chamber PO 2 and identical to the SO 2 level of the terminal arteriole (Figures 8, 9). However, the SO 2 level of the deeper tissue capillary (50), which was located at the same depth as capillary 54, was unchanged relative to zero flux. In general, the different arteriolar entrance SO 2 has no effect on the surface or deep tissue capillaries and had minimal influence on the arteriolar SO 2 at steady state. At 15 mmHg, the SO 2 level of the terminal arteriole entering at 65% dropped by 60% relative to zero flux condition corresponding to 44% increases in [ATP]. A smaller drop in SO 2 was calculated (52% decrease) for the arteriole entering at 80% corresponding to 40% increase in [ATP]. The SO 2 level in the surrounding surface capillaries as well as deeper tissue capillaries dropped by ∼70 and 35%, respectively, corresponding to ∼45 and 16% higher steady state capillary [ATP] relative to zero flux (Figures 8C, 9C). At 150 mmHg, hemoglobin SO 2 levels in surface and deep tissue capillaries as well as in the arteriole converged to ∼100% with ∼77% decrease in steady state [ATP] in the capillaries and 75% decrease in [ATP] in the arteriole relative to zero flux control (Figures 8C, 9C).

Rectangular oxygen delivery micro-slit at 65 and 80% arteriolar entrance SO 2
In these simulations, the capillary array that includes the terminal arteriole is exposed to local perturbations in tissue PO 2 through the rectangular micro-slit. At 40 mmHg chamber PO 2, the calculated steady state SO 2 and [ATP] levels at the venular end of surface and deep tissue capillaries as well as in the arteriole are within 5% of those at zero flux condition and uninfluenced by the arteriolar entrance SO 2 (Figures 10, 11). Under imposed hypoxia (15 mmHg), the calculated arteriolar SO 2 values at steady state were 50% higher than the case in which the full surface is exposed to the PO 2 perturbations and identical to those of deeper tissue capillaries. Hence, a minimal drop in SO 2 (38% decrease) was calculated in the arteriole relative to zero flux. These arteriolar steady state SO 2 values were uninfluenced by the different arteriolar entrance SO 2 . The influence of the arteriole as an O 2 source to nearby capillaries downstream of the micro-slit can be clearly observed in the 3D PO 2 profiles at 15 mmHg (Figures 10A, 11A). However, the surface capillaries (14, 17) experienced a sharper drop in SO 2 in response to the imposed hypoxia with 53% drop in SO 2 and a corresponding 39% increase in [ATP]. As observed when locally stimulating surface capillaries in the absence of the arteriole, capillaries were re-oxygenated ∼40 μm downstream of the hypoxic micro-slit region. At 150 mmHg, the steady state SO 2 levels in surface capillaries and in the arteriole converged to ∼88% while the SO 2 level of capillary 50 was slightly lower at 83% which corresponded to 65% and 57% decrease in [ATP], respectively, relative to zero flux (Figures 10C, 11C).

Estimating relative arteriolar ATP magnitude in response to tissue PO 2 perturbations
In order to estimate the contribution of the terminal arteriole to ATP mediated signaling in capillary networks, the In these simulations, we are modeling O 2 delivery to bottom tissue surface using the full gas exchange chamber (Ghonaim et al., 2011;Ellis et al., 2012). Full bottom tissue surface is exposed to 15, 40, or 150 mmHg chamber PO 2 level relative to a zero flux control boundary condition (A) Spatial 3D tissue PO 2 distribution (mmHg) at capillary entrance perspective ( steady state magnitude of ATP in the arteriole (ATPart) at various tissue PO 2 conditions was calculated and normalized against total ATP magnitude in the network (ATPtot) under zero flux condition (Figure 12). The relative arteriolar ATP magnitudes were calculated when full tissue surface is exposed to the PO 2 perturbations (full gas exchange chamber) or to local perturbations using the rectangular O 2 delivery micro-slit. As shown in Figure 12, the arteriolar ATP magnitude decreased with increase in chamber PO 2 level. However, the model suggested that under hypoxic conditions (15 mmHg), the terminal arteriole would contribute less than 3% of the total ATP signal originating from a capillary network. Also, although the percent decrease in ATP magnitude in the arteriole is similar to that calculated www.frontiersin.org September 2013 | Volume 4 | Article 260 | 11 FIGURE 9 | Simulations of 3D PO 2 and capillary [ATP] distribution in a tissue with idealized parallel capillary arrangement (68 hexagonally packed capillaries), which includes a traversing terminal arteriole (vessel 69) at an entrance SO 2 of 80%. In these simulations, we are modeling O 2 delivery to bottom tissue surface using the full gas exchange chamber (Ghonaim et al., 2011;Ellis et al., 2012). Full bottom tissue surface is exposed to 15, 40, or 150 mmHg chamber PO 2 level relative to a zero flux control boundary condition (A) Spatial 3D tissue PO 2 distribution (mmHg) at capillary entrance perspective ( for the total network when increasing chamber PO 2 from 15 to 150 mmHg, the absolute change in ATP magnitude (moles) in the arteriole is ∼95% less. Finally, it should be noted that [ATP] in the arteriole is ∼20% lower when limiting area of PO 2 perturbations using the rectangular micro-slit.

DISCUSSION
In the microcirculation, ATP is released from the erythrocytes in an SO 2 dependent manner. Released ATP would bind to purinergic receptors on the vascular endothelium which activates a signaling pathway leading to the opening of Ca 2+ gated K + channels and the hyperpolarization of the endothelial cell FIGURE 10 | Simulations of 3D PO 2 and capillary [ATP] distribution in a tissue with idealized parallel capillary arrangement (68 hexagonally packed capillaries), which includes a traversing terminal arteriole (vessel 69) at an entrance SO 2 of 65%. In these simulations, we are modeling O 2 delivery through a rectangular oxygen micro-delivery outlet (1000 μm wide × 200 μm long) to bottom tissue surface using our previously described novel micro-delivery approach (see Figure 1). Tissue surface directly on top of the micro-delivery outlet is exposed to 15, 40, or 150 mmHg chamber PO 2 level relative to a zero flux control boundary condition (A) Spatial 3D tissue PO 2 distribution ( changes along vessel length in selected vessels (14, 17, 69-arteriole, 50) marked by arrows on the vessel map. Capillary 17 is 16 μm from tissue surface, the arteriole and capillary 14 are 33 μm from tissue surface, and capillary 50 is deeper in the tissue, at 133 μm, yet is shown adjacent to the arteriole in the current perspective of the vessel map. (Ellsworth et al., 2008;Tran et al., 2012). The hyperpolarization signal is then conducted upstream through gap junctions. At the arteriolar wall, the incoming hyperpolarization signal is conducted to the SMC layer through myo-endothelial gap junctions resulting in vaso-relaxation and increase in erythrocyte supply rate (Ellsworth et al., 2008;Tran et al., 2012). The magnitude of the hyperpolarization signal would depend on the number of endothelial cells activated along the capillary and on the total number of capillaries stimulated within a network under hypoxic conditions. This understanding of how erythrocyte-released ATP controls micro-vascular O 2 delivery is consistent with the modeling results presented in this paper. The net increase in total ATP magnitude in the network with increase in the area exposed to hypoxia is the summative contribution of additional stimulated FIGURE 11 | Simulations of 3D PO 2 and capillary [ATP] distribution in a tissue with idealized parallel capillary arrangement (68 hexagonally packed capillaries), which includes a traversing terminal arteriole (vessel 69) at an entrance SO 2 of 80%. In these simulations, we are modeling O 2 delivery through a rectangular oxygen micro-delivery outlet (1000 μm wide × 200 μm long) to bottom tissue surface using our previously described novel micro-delivery approach (see Figure 1). Tissue surface directly on top of the micro-delivery outlet is exposed to 15, 40, or 150 mmHg chamber PO 2 level relative to a zero flux control boundary condition (A) Spatial 3D tissue PO 2 distribution (mmHg) at capillary capillaries (Figures 3-7). Also, these results help explain our observations of no vascular response when experimentally testing the effect of O 2 delivery through a circular micro-outlet (100 μm in diameter) in vivo (Ghonaim et al., 2011). Although this design maybe optimal for locally altering SO 2 in single capillaries, the stimulus would probably not be sufficient to elicit a micro-vascular response. Increasing the dimensions of the microoutlet would be necessary to stimulate a large enough number of capillaries, thus amplifying total magnitude of ATP release and signal. Also, as our modeling data suggest, increasing the microoutlet dimensions minimizes the effect of stimulated capillary FIGURE 12 | Total ATP magnitude (moles) at steady state calculated for entire network (ATPtot from all 68 capillaries) or in the terminal arteriole only (ATPart) normalized against ATPtot calculated at no-flux condition. Relative ATP magnitudes are calculated for an idealized 3D parallel capillary array network with a terminal arteriole (9 μm in diameter) positioned 30 μm from tissue surface. Simulations were run with entire bottom tissue surface being exposed to PO 2 perturbations using full gas exchange chamber or locally using a rectangular O 2 delivery micro-slit (1000 μm wide × 200 μm long). For both conditions, relative ATP magnitudes are calculated for the case in which the terminal arteriole has an entrance SO 2 of (A) 65% or (B) 80%.
re-oxygenation downstream of the micro-outlet. This is because the capillaries of interest would be surrounded by capillaries experiencing the same drop in PO 2 . This is more representative of the situation in vivo as the outlet physiologically simulates an arteriole crossing the capillary bed acting as an O 2 source or a venule withdrawing O 2 , which would affect multiple capillaries. In terms of the signaling response, delayed re-oxygenation following hypoxic stimulation ensures the ATP signal persists longer distances downstream thus stimulating a larger number of endothelial cells. Since each endothelial cell in skeletal muscle is ∼100 μm long, using the rectangular slit is estimated to activate at least 3.5 endothelial cells in each stimulated capillary. In comparison with the square micro-outlet, which has the same length (200 μm) as the rectangular micro-slit, ∼1 more endothelial cell is activated per capillary with the latter design. It should be noted that in the modeled geometry, which lacks realistic capillary branching and has an idealized, uniform capillary density, we are examining relative changes in the total magnitude of ATP when using various outlet designs. During in vivo experiments, a maximum of two micro-vascular units ∼10 capillaries may be positioned along the entire width of the rectangular micro-slit, while only one or two capillaries with a branching point could be positioned over the circular microoutlet (Ghonaim et al., 2011). Hence a 1000 μm wide × 200 μm long outlet might cover the threshold number of capillaries needed to elicit a micro-vascular response. This indicates the rectangular micro-slit would be optimal for stimulating enough capillaries by imposed hypoxia to generate high enough ATP signal.
The limited amount of change in tissue PO 2 due to diffusion (∼50 μm), as measured from the 3D tissue PO 2 profiles, beyond the edge of the micro-outlets (Figures 4A-6A and 4B-6B) was consistent with our previous observations (Ghonaim et al., 2011). The simulations indicated that the PO 2 perturbations are highly localized to only those capillaries directly over the micro-outlet region. Experimentally, the results suggest that the micro-outlet should be positioned at least 50 μm downstream of the terminal feeding arteriole to ensure that micro-vascular responses are only elicited from the capillaries positioned directly over the outlet. The extent of axial O 2 diffusion in the tissue when using the rectangular micro-slit was 50% deeper than that previously modeled for the circular microoutlet (Figures 6B, 10B, 11B) (Ghonaim et al., 2011) and similar to that of the full surface model (Ghonaim et al., 2011;Ellis et al., 2012). Due to the shape of the PO 2 profile, the maximal axial diffusion distance is estimated from the center of the outlet. The increase in the axial diffusion distance might be a compromise when using larger O 2 micro-outlets. With our current microscopic techniques we are unable to resolve vessels deeper than 60 μm.
Since in our experiments, we examine micro-vascular signaling from selected capillaries, it was critical that we assess the possible contribution of arterioles beyond our ability to focus. Since arterioles have relatively higher erythrocyte velocities than in the capillaries, they are anticipated to experience a much lesser change in SO 2 in response to PO 2 perturbations. This was supported by our simulation data (Figures 8-11). The main effect of a nearby terminal arteriole on a capillary within 50 μm is that it would act as an O 2 source. As shown in our modeling data (Figures 8D-11D), higher measured SO 2 in a capillary relative to other capillaries with comparable flow rates in the same preparation might imply the presence of a nearby arteriole. Since arterioles in the EDL muscle preparation are deeper in the tissue, their contribution to the total magnitude of ATP in a locally stimulated capillary network is probably negligible. The contribution of a terminal arteriole positioned directly over the micro-slit ∼30 μm from bottom surface was calculated to be less than 3% of the total magnitude of the ATP (Figure 12). Hence, when locally stimulating capillaries, even in the presence of an underlying arteriole, the observed micro-vascular responses mediated by intra-luminal ATP would be primarily due to ATP released in the stimulated capillaries.
In conclusion, we have modeled SO 2 -dependant changes in [ATP] at steady state in 3D idealized parallel capillary networks in response to local PO 2 perturbations. As the number of affected capillaries increases, the total magnitude ATP in the network increases. The results indicated that O 2 delivery or removal to overlaying tissue through a rectangular micro-slit (1000 μm wide × 200 μm long) would be optimal relative to other micro-outlet designs of smaller dimensions or a full surface classical exchange type chamber. Using the rectangular micro-slit it is anticipated that a sufficient number of capillaries will be stimulated to produce a large enough magnitude of ATP to elicit micro-vascular responses. This would be accomplished while maintaining the stimulus localized to the selected capillaries. The results also indicated that terminal arterioles have minimal influence on the total magnitude of ATP in the network under hypoxic condition. Hence, when locally stimulating the capillary bed, the majority of the signal elicited by ATP release would originate in the capillaries. The computational model presented provides valuable insights into how to study the ATP release mechanism and signaling in capillary networks in vivo. The modeling data help guide us in the design of an optimal tool for studying SO 2 -dependent ATP release in capillaries in vivo. In the future, we aim to model time-dependent ATP release to local PO 2 perturbations in a realistic capillary network geometry reconstructed from experimental data. Combining our in vivo experimental observations with computational modeling of the dynamics of SO 2 -dependent ATP release will help provide a more comprehensive understanding of O 2 mediated blood flow regulation in micro-vascular networks.