Increased Oxygen Supply Results In

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Effects of Fiber Type and Size on the Heterogeneity of Oxygen Distribution in Exercising Skeletal Muscle

• Gang Liu,
• Feilim Mac Gabhann,
• Aleksander Southward. Popel

x

• Published: September eighteen, 2012
• https://doi.org/ten.1371/periodical.pone.0044375

Figures

Abstruse

The process of oxygen commitment from capillary to muscle fiber is essential for a tissue with variable oxygen demand, such as skeletal muscle. Oxygen distribution in exercising skeletal muscle is regulated by convective oxygen transport in the blood vessels, oxygen diffusion and consumption in the tissue. Spatial heterogeneities in oxygen supply, such as microvascular architecture and hemodynamic variables, had been observed experimentally and their marked effects on oxygen exchange had been confirmed using mathematical models. In this study, we investigate the effects of heterogeneities in oxygen demand on tissue oxygenation distribution using a multiscale oxygen send model. Muscles are composed of different ratios of the diverse cobweb types. Each fiber type has feature values of several parameters, including fiber size, oxygen consumption, myoglobin concentration, and oxygen diffusivity. Using experimentally measured parameters for different fiber types and applying them to the rat extensor digitorum longus musculus, we evaluated the effects of heterogeneous fiber size and fiber type properties on the oxygen distribution profile. Our simulation results suggest a marked increase in spatial heterogeneity of oxygen due to fiber size distribution in a mixed muscle. Our simulations too suggest that the combined effects of fiber blazon backdrop, except size, do not contribute significantly to the tissue oxygen spatial heterogeneity. However, the incorporation of the difference in oxygen consumption rates of different fiber types solitary causes college oxygen heterogeneity compared to control cases with uniform fiber backdrop. In contrast, incorporating variation in other fiber type-specific properties, such as myoglobin concentration, causes little change in spatial tissue oxygenation profiles.

Citation: Liu G, Mac Gabhann F, Popel As (2012) Effects of Fiber Blazon and Size on the Heterogeneity of Oxygen Distribution in Exercising Skeletal Muscle. PLoS ONE 7(nine): e44375. https://doi.org/10.1371/journal.pone.0044375

Editor: Timothy W. Secomb, University of Arizona, United states of America

Received: January 2, 2012; Accepted: Baronial 6, 2012; Published: September 18, 2012

Copyright: © Liu et al. This is an open-access commodity distributed nether the terms of the Creative Commons Attribution License, which permits unrestricted utilise, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: This work was supported by the National Institutes of Health (NIH) grants R01 HL18292 and HL101200 (ASP) and R00 HL093219 (FMG). The funders had no part in report design, data drove and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

Introduction

Oxygen send from capillaries to muscle fibers plays an essential role in the maintenance of physiological functions of skeletal muscle across a wide range of conditions and various forms of practice. Oxygen ship is regulated past convection in claret vessels and past diffusion across the vessel walls and into the interstitial infinite and parenchymal cells. Hemoglobin transports oxygen in the blood, while myoglobin serves every bit an oxygen storage and facilitates its diffusion inside myocytes. At resting conditions, oxygen consumption is less than the available oxygen delivered past the microvasculature, with nearly one-half of the oxygen returned to the venous circulation; thus, musculus at rest is typically oversupplied. During exercise, the oxygen consumption rate in skeletal muscle can increase as much as 50 fold compared to resting conditions [1]. The increased oxygen need is partially compensated by the increased blood flow in tissue microcirculation (10 to 25 fold) [2], and the increase of oxygen extraction by the tissue. The mismatch betwixt oxygen demand and oxygen supply may lead to regional tissue hypoxia, and prolonged hypoxia can consequence in angiogenesis (capillary growth from pre-existing vasculature), an adaptive response that leads to a decrease of oxygen diffusion distances [3]. Angiogenesis may as well consequence from the elevated shear stress often associated with functional hyperemia [4]. Bereft oxygen supply is a major downstream pathological effector of chronic ischemic diseases such as coronary artery disease and peripheral artery disease [five]. In those ischemic diseases, the obstruction of upstream blood vessels limits blood catamenia and convective oxygen transport.

Theoretical aspects of oxygen send processes have been extensively studied. The goal has been to describe the non-compatible oxygen distribution in skeletal muscle tissue. In the pioneering work by August Krogh [6], an oxygen transport model (besides called the Krogh-Erlang model) was adult to describe a single capillary supplying oxygen to a cylindrical volume of surrounding tissue. It was the first theoretical model used to understand oxygen send to tissue and was built on a number of idealized assumptions including: constant oxygen consumption rate; uniform oxygen diffusivity; and homogeneous capillary distribution. Since then, many alternative oxygen transport models have been developed (reviewed in [vii]–[nine]) with a number of boosted physiological features neglected in the Krogh-Erlang model, including: variable oxygen consumption rate [10]; myoglobin-facilitated diffusion [11]; menses redistribution [12]; centric oxygen diffusion in the tissue [13]; intravascular oxygen transport resistance [14], and pre- and postal service-capillary transport [fifteen], [sixteen]. In add-on, oxygen transport models have been extended to include complex microvascular network geometry with capillary tortuosity and anastomoses [17]–[nineteen]. The Krogh-Erlang model is informative; however because of the complexity and heterogeneity of tissue microarchitecture, it would exist difficult to plant a single Krogh-type parameter to describe the fiber types or the vessels adjacent to them. Instead, we construct a spatially detailed heterogeneous model based on the experimentally reported values of capillary∶fiber ratio for specific fibers and study the distribution of oxygen across the tissue being false.

Skeletal musculus is composed of an array of fibers with capillaries running primarily parallel to the fiber management. Muscles in the torso take dissimilar functions and unlike fiber compositions; fiber type composition within a muscle is closely correlated to the office of that muscle. Fibers are unremarkably categorized equally type I (slow oxidative twitch) and blazon 2 (fast twitch) based on contractile properties and oxidative chapters [20]–[22]. Type I fibers are mainly involved in aerobic activities and endurance do; they contain aplenty amounts of mitochondria and utilize more often than not oxidative phosphorylation, making them fatigue-resistant. In addition, they are myoglobin-rich with a cerise appearance. Blazon II fibers, appearing whiter than the 'red' Type I fibers due to lower myoglobin expression, can exist classified into three subtypes, Type IIa, IIb and IId/x. Type IIa (or oxidative fast twitch) fibers generate ATP through the glycolytic cycle but also have a loftier mitochondrial count, allowing them to obtain ATP through oxidative metabolism. They take higher contraction velocities compared to Blazon I fibers but are non fatigue-resistant. Type IIb (or glycolytic fast twitch) fibers have far fewer mitochondria and thus depend on glycolytic metabolism to generate ATP. They are activated when short and powerful bursts of contraction are required. Type IId/ten fibers take properties intermediate between Type IIa and Blazon IIb. Blazon I and Blazon IIa fibers have higher oxidative capacities and therefore eat more oxygen than Blazon IId/ten and Blazon IIb. These muscle fibers are distributed throughout the skeletal muscle tissue in varying proportions, depending on the muscle group involved. In add-on, there is a correlation between the cobweb blazon and the number of adjacent blood vessels: Type IIb fibers by and large accept fewer adjacent blood capillaries compared to Type I and Type IIa. In addition to histological and metabolic differences, musculus fibers also differ in their size and their components, with variations in the proportions of aqueous cytosol and lipid-rich membranes and droplets, allowing varying oxygen diffusivity in unlike fiber types [9].

The objective of our study is to assess the touch on of fiber blazon composition on the heterogeneity of oxygen distribution in tissue using a computational oxygen transport model. Our working hypothesis was that fiber-type dependent characteristics may balance with the number of nearby capillaries to locally 'match' oxygen supply and demand. We farther hypothesized that this matching may be more relevant in exercise atmospheric condition. One goal of the study was to use computational modeling techniques every bit a hypothesis testing tool to understand the role of each parameter. The computational model is an extension of our previously adult oxygen model (see Methods). We began past edifice several tissue geometries that contain fiber type-specific backdrop: at the tissue level this includes the proportion of each type of fiber nowadays; at the fiber level, each cobweb type has a different characteristic fiber size, oxygen consumption rate (Thousand _c), oxygen diffusivity (D_O2 ), myoglobin concentration (C_Mb ), and number of surrounding capillaries. We constructed a set of tissue geometries; each geometry has a dissimilar combination of parameters set to be uniform and parameters fix to be fiber type-specific. This allows u.s.a. to capture the dissimilar effects of each parameter on spatial heterogeneity in the musculus. For each geometry, the oxygen send model is used to simulate the spatial profiles of oxygen under exercise conditions of varying intensities. Our simulation results suggest a marked increase of spatial heterogeneity of oxygen due to cobweb size distribution in a mixed musculus. The divergence in oxygen consumption rates of different fiber types also causes higher oxygen heterogeneity compared to control cases with uniform fiber properties. In contrast, incorporating variation in other fiber type-specific properties, such every bit myoglobin concentration, causes little modify in spatial tissue oxygenation profiles. To the best of our knowledge, this is the first computational model used to examine the importance of cobweb type and size in tissue oxygen transport.

Methods

We adapted our multiscale computational model of oxygen transport [18], [23], [24] to study the effects of fiber type and fiber size on the oxygen distribution and oxygen gradients in working muscle. Two new features were introduced into the model. Outset, we incorporated six variables representing experimentally-measured physiological properties of specific fiber types in skeletal musculus. These variables are cobweb type composition, fiber size, D_O2 , Thousand _c, C_Mb and capillary distribution. Second, the modified model describes the oxygen transport process using coupled fractional differential equations in three regions: inside the muscle fibers, in the interstitial space, and within the vascular infinite. Previously, fibers and interstitial infinite had not been treated separately.

The model consists of three components: the muscle geometry module, the microvascular claret flow module, and the oxygen transport module. We volition describe each module, with the focus on the new model features in this study.

Skeletal Muscle Geometry Module

We constructed a set of skeletal muscle tissue geometries. These constructions were done computationally, using a like algorithm to that applied in previous studies [23]–[28], with modifications described beneath. The muscle in our model is the rat extensor digitorum longus (EDL) that has been used extensively in experimental studies of activity-induced angiogenesis. To assess the effects of cobweb type, cobweb size and also fiber-type dependent microvascular structures on oxygen distribution profiles, we specifically introduced four types of skeletal musculus geometry. Each geometry (designated G1–G4) incorporates an experimentally-observed number (or per centum) of each fiber type. However they differ as follows: G1, uniform cobweb size and compatible capillary distribution; G2, uniform fiber size and cobweb-type-specific capillary distribution; G3, non-uniform cobweb size and uniform capillary distribution; and G4, non-uniform fiber size and fiber-type-specific capillary distribution.

Computational generation of these 3D muscle geometries (Figures i and two) is implemented as a two-step method, with fiber geometry constructed first followed by microvascular network insertion. The outline of the design algorithm is as follows: (i) In the rectangular cuboid tissue volume, muscle fibers are represented every bit correct circular cylinders; (2) Microvessels, specifically capillaries, are placed betwixt the muscle fibers; the vessels are divided into short segments and each segment is represented as a right circular cylinder; tortuosity and anastomoses are introduced to mimic physiological vascular networks by varying the orientation of adjoining segments; (three) Pre-capillary arterioles and postal service-capillary venules are placed in a staggered pattern to connect with the microvessel network; the capillaries originate from an arteriole and terminate at a venule approximately 350 µm away.

Figure 1. Musculus geometries with uniform-size fibers.

A) An example of the 3D structure of skeletal musculus with uniform-size fibers and microvasculature. B) 3D structure of the microvasculature. C) 2D cantankerous department of skeletal muscle with uniform-size fibers and uniform capillary distribution (G1). D) 2D cross department of skeletal musculus with uniform size cobweb and cobweb blazon dependent capillary distribution (G2). Fiber type I is shown in blue, type IIa in green and type IIb in yellow. Microvessels are shown in red.

https://doi.org/10.1371/journal.pone.0044375.g001

Figure 2. Muscle geometries with not-compatible-size fibers.

A) An example of the 3D structure of skeletal muscle including non-uniform-size fibers and microvasculature. B) 3D structure of the microvasculature. C) second cross section of skeletal musculus with non-uniform size fibers and compatible capillary distribution (G3). D) 2d cross section of skeletal muscle with not-uniform size fibers and fiber-type-dependent capillary distribution (G4). Fiber type I is shown in blue, type IIa in light-green and type IIb in xanthous. Microvessels are shown in red.

https://doi.org/x.1371/journal.pone.0044375.g002

For cobweb geometry with uniform-size fibers (G1 and G2), we place the fibers on a regular filigree [23], [25]–[28]. To avert edge effects, a number of fibers are bisected by the edges, but appear on the other side in a periodic fashion. The simulations apply periodic boundary atmospheric condition for oxygen improvidence. We then use a randomizing algorithm, combined with experimentally-observed rules, to assign fiber types to fibers. The fiber type composition is dependent on the muscle blazon, creature species, and individual differences. The proportion of fiber type I in the EDL ranges from 2 to 9%, cobweb blazon IIa from 11 to 27%, and fiber blazon IIb from 61 to 82% [29]. For this model, nosotros only consider fiber blazon I, fiber blazon IIa and cobweb blazon IIb due to incomplete information most cobweb type IId/10 (for example, values for the myoglobin concentration and oxygen diffusion coefficient in this fiber type are not available). In our constructed geometries, we consider 48 fibers in the tissue book (358×233×800 µm), of which iv fibers are type I (viii.3%), 11 are type IIa (22.9%) and 33 are type IIb (68.viii%). 2d, based on histochemical staining reported in the literature [29], there are two rules for fiber type distribution in rat EDL musculus: (i) type I fibers are not typically found next to each other; (ii) type IIa fibers are typically found close to blazon I fibers.

For our tissue geometries with fiber-type-specific fiber sizes (G3 and G4), nosotros employ a hybrid simulated-annealing algorithm [30] to construct the cobweb geometry. The optimization function is designed to find the best fit of several experimentally-observed physiological parameters. These parameters are: interstitial fluid book fraction 6%–20% [31]–[33]; cobweb type proportion two–9%, eleven–27%, 61–82% for fiber type I, IIa and IIb respectively [29]); the radii for cobweb type I, type IIa, type IIb are thirteen, 16, and 23 µm respectively, based on reported fiber areas [34].

For the microvascular network in the tissue, capillary placement is also governed by a stochastic algorithm, augmented by several rules including: (1) the average capillary-to-fiber ratio (C∶F) is 1.1 (i.eastward., the total number of capillaries is virtually 52). The capillary density is 623/mm², which is close to 650 (recalibrated value, number per cobweb area including interstitial space, reported in [35]); (2) the radii of capillary, venule and artery segments are set equally uniform in the vascular network (iii, 6, 6 µm); and (3) the number of capillaries around a fiber (NCAF) is either specific (G2, G4) or non-specific (G1, G3) to fiber type, depending on the constructed geometry (i.e. compatible or heterogeneous fiber-blazon-specific microvascular distribution. For the uniform vascular network geometries (G1, G3), the average NCAF values for 3 fiber blazon groups are the same (approximately 3.24) and we likewise seek to minimize the variability of NCAF between private fibers in the geometry (Fig. 1 and two). For cobweb-type-specific vascular geometries (G2, G4), we follow experimental observations in rat skeletal musculus that muscle fibers with a high oxidative potential are generally associated with a denser capillary network [36], [37], although not all studies agree with this finding [38]. In this blazon of fiber-type-specific geometry (Fig. ane and 2), fiber type I has the largest number of surrounding capillaries, followed by type IIa so blazon IIb. NCAF values for each fiber varies from one to 6. In addition, we too designed the geometry with a meaning caste of heterogeneity in each fiber type group.

Flow Module

Nosotros use an in vivo hemorheological model [39] to calculate the claret flow rate (Q) and discharge hematocrit (H_D ) in each of the blood vessel segments in the muscle, which incude last arterioles, capillaries and collecting venules. The vascular network is digitized as a fix of nodes (vascular bifurcations) and a set up of vessel segments linking those nodes together. The governing equations for volumetric claret flow charge per unit and ruby blood cell menstruation rate at the j^thursday node are: where Q_ij and H_{D ij} are the volumetric flow charge per unit and discharge hematocrit in vessel segment ij, a cylindrical segment whose ends are the ith and jth nodes of the network. Using the governing equations combined with empirical equations describing ruddy blood prison cell-plasma separation at vascular bifurcations [39], we obtained a set of nonlinear algebraic equations for all N segments and the equations were solved for pressure and hematocrit, from which menses in each segment is calculated.

Oxygen Module

Nosotros employ a modified version of our previously published oxygen transport model to compute oxygen distribution in the tissue. In previous studies we causeless that muscle fibers and the interstitial space were a unmarried tissue phase [xviii], [23]. Here the model consists of three partial differential equations, governing: intravascular oxygen transport; oxygen diffusion in interstitial space; and oxygen diffusion and consumption inside fibers.

Local oxygen tension in the fibers, P_f , is governed past complimentary oxygen diffusion, myoglobin-facilitated diffusion, and oxygen consumption by myofibers: where D_O2,f and D_Mb are the diffusivities of oxygen (O_two) and myoglobin (Mb) inside the cobweb; Southward_Mb is the oxygen-myoglobin saturation; α_f is the oxygen solubility in the fiber; is the binding chapters of myoglobin for oxygen; M _c is the Michaelis-Menten-blazon maximal oxygen consumption rate; P_crit is the critical P_f at which oxygen consumption rate equals 50% of M_c . S_Mb is defined equally P_f/(P_f +P _50,Mb ), assuming local bounden equilibrium between oxygen and myoglobin, where P_50,Mb is the P_f corresponding to fifty% myoglobin saturation with oxygen. It should be noted that the myoglobin-facilitated improvidence conception is simplified in that information technology does non consider diffusion of myoglobin equally a divide molecular species and assumes that myoglobin and oxygen are locally in chemical equilibrium; yet, this assumption is justified a posteriori by the minor effect of myoglobin under steady-state weather condition.

Local oxygen tension in the interstitial space, P_i , is governed only by costless oxygen diffusion (we assume negligible consumption in this space): Oxygen transport within the blood vessels, P_b , is governed by hemoglobin bounden and blood convection: Here is the oxygen-hemoglobin saturation in claret vessel; α_RBC and α_pl are oxygen solubility in cherry blood jail cell and plasma, respectively; P_b is the oxygen tension in blood plasma; ν_b is the mean blood velocity (ν_b  =Q/(πR ²)); H_T is the vessel (tube) hematocrit calculated from blood flow model; is binding chapters of hemoglobin with oxygen; ξ is the altitude along a vessel'south longitudinal axis; J_wall is the capillary wall flux. is defined every bit assuming binding equilibrium betwixt oxygen and hemoglobin, where P_50,Hb is the P_b at 50% hemoglobin oxygen saturation.

Continuity of oxygen flux at the interface between blood vessels and interstitial space, and between muscle fiber and interstitial space, yields: where n is the unit of measurement normal vector, and yard ₀ is the mass transfer coefficient. This system of nonlinear partial differential equations was solved using the finite deviation method, with a filigree size of i micron as described in [23].

Model parameters and simulation platform

Parameters used in the model are listed in Table 1 (fiber-type specific parameters) and Tabular array 2 (non-fiber-type-specific parameters). Most of the model parameters were taken from experimental data for rat EDL; some were based on measurements in other muscle types; theoretical estimates were used when parameters are unavailable from literature. Nosotros utilise the multiscale modeling platform developed previously [24] to combine the three modules described above into an integrated model. This platform allows integration of different modules written in dissimilar programming languages and using different modeling methodologies. While the period module remained unchanged from the previous version, the geometry and oxygen transport modules were updated based on the changes described to a higher place. The simulation experiments were run on an IBM cluster and each simulation was run on 1 node with viii cores and 32 Gbyte retentivity. The oxygen transport module was implemented with OPENMP back up and can be parallelized. JDK (Coffee Development Kit) 1.six.xvi (Oracle, Redwood Shores, CA) is used as Coffee compiler, and Intel Fortran/C++ compiler suite xi.one (Intel, Santa Clara, CA) is used as Fortran/C++ compiler.

Results

Nosotros computationally evaluated the contribution of fiber type limerick and fiber type-specific parameters to spatial heterogeneity of tissue oxygenation. These parameters include: cobweb blazon-dependent oxygen consumption (M _c), oxygen diffusivity in fiber (D_O2 ), myoglobin concentration (C_Mb ), cobweb size, and number of capillaries around a fiber (NCAF). To understand the effects of each factor separately as well as their combined furnishings, we ran a series of computational experiments to examine the heterogeneity of oxygen distribution in skeletal musculus during exercise in different muscle geometries (G1–G4) and with scenarios varying a number of the other factors under consideration (S1–S5). The annotations for each simulation are summarized in Table three. For instance, G3S1 refers to G3 geometry (non-compatible fiber size and uniform capillary distribution) with scenario S1 (compatible K_c , D_O2 , and C_Mb ). The respective microvascular and fiber structures are shown in Figs. 1C (G1), 1D (G2), 2C (G3) and 2nd (G4), representing the aforementioned location in networks (z = 400 µm). The iii-dimensional geometry of G1 is shown in Fig. 1A,B and that of G3 in Fig. 2A,B. The number of capillaries around a fiber (NCAF) for each of the simulation geometries is shown in Table S1. In the following sections, we report the mean and the coefficient of variation (CV) of the tissue oxygen profile equally a measure out of the heterogeneity in the tissue for each simulation.

Information technology is worth noting that microvascular hemodynamics is a well-known important determinant of tissue oxygenation and information technology is not the focus of this study. Our blood flow simulations advise that these vascular networks in two geometries (G1 and G2) share similar distribution patterns of blood flow velocity and hematocrit, and have similar total blood volume flow rates (run across Table S2 and Fig. S1). In addition, vascular networks G3 and G4 share like microvascular blood flow and red blood cell distribution (Table S2 and Fig. S2).

Capillary distribution is an important determinant of oxygen distribution

For the G1 and G2 geometries (compatible fiber size; Fig. 1) we start computed the steady-state tissue oxygen distribution without consideration for fiber type composition, not varying O₂ consumption rates, O_ii diffusivities, and myoglobin concentrations in unlike fiber types. In other words, compatible fibers with the aforementioned parameters: book-averaged M_c (3.34, six.68, ten.02×10^−four mlO₂ ml⁻¹ south⁻¹ for low, moderate, high intensity exercise), D_O2 (1.72×10⁻⁵ mlO₂ ml⁻¹ s⁻¹), and C_Mb (5.vii×10^−three mlO₂ ml⁻¹ s⁻¹) were used for every fiber in the simulation. In this case, with calorie-free intensity exercise, an increase in the heterogeneity of microvascular structure (G2 compared to G1), and therefore in the heterogeneity of oxygen supply, is predicted to lead to a slight increase in the heterogeneity of O_ii (CV_G1S1: 0.09, CV_G2S1: 0.10) and a slight lowering of mean tissue PO₂ level (P_G1S1: 28.9 mmHg, P_G2S1: 27.6 mmHg) (Fig. 3 and Tabular array four). In exercise of moderate or high intensity, the increase in oxygen consumption levels causes fifty-fifty greater heterogeneity. When capillaries are non-uniformly distributed around the fibers (Fig. 3, solid lines), the variation further increases; hateful PO₂ levels decrease from 17.4 to fifteen.one mmHg (moderate intensity exercise), and from 8.84 to half-dozen.82 mmHg (loftier intensity exercise). These results indicate that capillary distribution in muscle tissue affects non-uniform oxygen distribution and heterogeneity.

Figure three. PO₂ histograms for muscle geometries with compatible-size fibers nether exercising conditions.

Fiber PO_ii probability distribution profiles in muscle tissue under A) light intensity exercise conditions, book-averaged Grand _c = 3.34×x⁻⁴ mlO₂ ml⁻¹ due south⁻¹; B) moderate intensity practise conditions, book-averaged Thou_c  = 6.68×x⁻⁴ mlO₂ ml⁻¹ s⁻¹ ; and C) loftier intensity exercise conditions, volume-averaged M _c = 1.02×10⁻³ mlO₂ ml⁻ⁱ s⁻ⁱ, for simulation cases G1S1, G1S2, G2S1, G2S2. Dotted lines are simulation results for G1 geometry, while solid lines are results for G2 geometry. Black lines are the simulation scenarios S1 with uniform fiber-blazon properties (Yard _c, D_O2 , C_Mb ) and red lines are the simulation scenarios (S2 using fiber-type-specific parameters (except size) for different fiber types.

https://doi.org/x.1371/periodical.pone.0044375.g003

Fiber blazon composition does not contribute significantly to oxygen distribution

We then computed steady-state tissue oxygen distribution in geometries G1 and G2 when fiber type composition and varying Chiliad _c, D_O2 , and C _Mb in different fiber types (fiber type-specific parameters shown in Table one) are considered; in other words, all fiber-type-specific parameters except for size. The simulation results with fiber type limerick considered (scenario S2) were compared to uniform properties (scenario S1) (Tabular array iv). Fig. iii shows the histogram of muscle fiber PO₂ in two muscle geometries (G1 and G2). Table 4 summarizes the central characteristics from all the simulation results of PO₂ distribution under different intensities of practice in G1 and G2.

At all levels of exercise, the difference in oxygen distribution between the heterogeneous fiber-type specific parameters (S2) and the uniform fiber parameters (S1, control cases) is minimal (Fig. 3, red vs. black lines). This includes the mean, range and variance of oxygen levels and the portion of tissue that is hypoxic, i.e. with PO₂<1 or ii% (Table iv). For case, compared to the control cases with uniform cobweb backdrop (i.eastward. G1S1, G2S1), the effect of fiber type composition on oxygen spatial heterogeneity and mean PO_ii level is pocket-sized when muscles are stimulated with low-cal or moderate exercise intensity (due east.g., for light exercise, CV for the geometry with uniform capillary distribution, CV_G1S2/S1: ∼0.09, CV for the geometry with fiber-type-specific capillary distribution, CV_G2S2/S1: ∼0.ten, P_G1S2/S1: ∼28.9 mmHg, P_G2S2/S1: ∼27.vi mmHg). Under high intensity exercise conditions, oxygen spatial heterogeneity changes slightly compared to command cases (i.eastward. G1S1, G2S1) (CV_G1S2/S1: 0.56 vs. 0.59; CV_G2S2/S1 : 0.65 vs. 0.68) and mean PO₂ remains at the same levels (PO_{ii G1S2/S1}: 8.8 vs. 8.5 mmHg, high;PO_{two G2S2/S1}: 6.8 vs. 6.6 mmHg, high). The oxygen distribution is much more dependent on capillary distribution (Fig. 3, solid vs. dotted lines) than on heterogeneity of fiber backdrop (Fig. 3, red vs. black lines).

Non-compatible fiber size significantly enhances the heterogeneity in oxygen distribution

To investigate the effect of fiber size on oxygen distribution in the EDL, we constructed geometries G3 and G4 (Fig. 2C,D), with non-uniform size fibers within the same dimension cuboid (358×233×800 µm³) and the same total fiber volume (79%) as G1 and G2 (Fig. 1C,1D). We kickoff compared the tissue PO₂ profile in geometry G3S1 with heterogeneous cobweb sizes to the contour in geometry G1S1 with homogeneous fiber sizes. Both of them use compatible fiber type backdrop (scenario S1, i.e., Yard _c, D_O2 , C_Mb ) and have uniform capillary distribution. Our oxygen send simulation results suggest that cobweb size distribution in muscle geometry plays an of import office in determining tissue oxygen contour and spatial heterogeneity. Figure iv shows that PO₂ in heterogeneous fiber size geometry is much more broadly distributed compared to the control instance (i.e., G1S1) with uniform size cobweb under all practise atmospheric condition (G3S1 vs. G1S1, black dashed lines in Fig. 4 vs. Fig. 3; 3D graphical representation shown in Fig. 5A,B). Heterogeneous fiber size distribution shifts tissue PO₂ to lower values (mean PO₂ from 28.9 to 26.5 mmHg, light exercise; 17.1 vs. xiii.6 mmHg, moderate; eight.84 vs. 6.96 mmHg, high), and increases its spatial heterogeneity (0. 09 vs. 0.xiv, light; 0.26 vs. 0.42, moderate; 0.56 vs. 0.71, loftier). Under high intensity exercise weather condition, the proportion of hypoxic tissue is much larger than in control case (18% vs. 7%).

Effigy 4. PO_ii histograms for muscle geometries with non-compatible-size fibers nether exercise conditions.

PO₂ probability distribution profiles for all points within fibers in musculus tissues under A) low-cal intensity exercise. B) moderate intensity exercise. C) high intensity exercise, for simulation cases G3S1, G3S2, G4S1, G4S2. Dotted lines are simulation results for G3 geometry, while solid lines are results for G4 geometry. Black lines are the simulation scenarios S1 and red lines are the simulation scenarios S2.

https://doi.org/ten.1371/periodical.pone.0044375.g004

We further examined the furnishings of fiber-type-specific properties on oxygen distribution in the not-uniform size cobweb geometries (geometries G3 and G4, results in Figure four and Table five). Conclusions similar to those described above can be drawn from these data. First, oxygen distribution is sensitive to local capillary distribution around the fibers. Notwithstanding, their effects are different from previous example with uniform size cobweb distribution (Fig. 4, solid vs. dashed lines). With fiber-type-dependent capillary distribution (smaller fibers take more than adjacent capillaries), the coefficients of variation of the tissue PO₂ profiles nether light practise weather are slightly decreased compared to the tissue with uniform capillary distribution (CV: 0.14 vs. 0.12, light; 0.42 vs. 0.42, moderate), and their mean PO₂ levels shift to slightly lower values (PO_two: 26.47 vs. 26.09 mmHg, light; 13.56 vs. 12.54 mmHg). Second, with the consideration of all fiber type properties (scenario S2) and cobweb size, tissue PO_ii profiles do not change significantly from tissue PO₂ profiles computed using the uniform fiber properties. This conclusion is based on the comparisons of tissues PO₂ in geometries with uniform/non-uniform capillary distribution in Fig. 4 and three-dimensional graphical representation in Fig. 5B,C. There are only slight differences between black and ruby lines (including dashed and solid cases) at all exercise intensities, with 1 exception: under high intensity exercise and with cobweb-type-specific capillary distribution, heterogeneous fiber type composition increases spatial heterogeneity (CV: 0.81 vs. 1.00) and reduces its tissue PO₂ level (PO₂: 5.93 vs. 3.22 mmHg).

We further studied the distribution of average PO₂ for each fiber from three fiber blazon groups under various cases (Fig. 6). This shows that under light intensity exercise, the difference in average cobweb PO₂ level among the three fiber type groups is depression. With the increase of exercise intensity (to moderate and high intensity), fibers of blazon I have college local oxygenation levels than types IIa and IIb. With fiber-blazon specific capillary distribution (G4 vs. G3), the oxygenation levels increase for oxidative fibers blazon I and type IIa, but subtract for type IIb. Therefore, at high intensity exercise weather, the divergence of PO₂ levels between fiber blazon IIb and fiber type I increased significantly, suggesting that local oxygen supply matches local need well for oxidative fiber only not glycolytic cobweb in exercise. Greater heterogeneity of average PO₂ level in each fiber type was also observed for higher do intensities and with capillary heterogeneity.

Figure six. Distribution of average PO₂ levels of individual fibers in skeletal musculus under exercise weather.

Boilerplate (across all fibers of each blazon) of the mean PO_ii levels of individual fibers nether A) light intensity exercise; B) moderate intensity do; and C) high intensity exercise, for simulation cases G3S1, G3S2, G4S1 and G4S2. Blazon I is shown in blue, type IIa in green, type IIb in yellow.

https://doi.org/10.1371/journal.pone.0044375.g006

Fiber-type-specific 1000 _c and D_O2 , just not C_Mb , cause greater heterogeneity in oxygen distribution

We side by side investigated the contribution of the heterogeneity of each fiber-type-dependent variable by itself (Thousand _c, D_O2 , and C_Mb ) to tissue PO₂ spatial heterogeneity. To do this, we performed simulations for the non-uniform cobweb size geometry with alterations of the cistron to exist considered (scenarios S3–S5; see Table 3 for a detailed description). Control cases were imitation with compatible cobweb backdrop for comparison. Figures seven, eight, 9 and Table 6 show the simulation results for the effects of M _c, D_O2 , C_Mb heterogeneity in different cobweb types. The results indicate that the heterogeneity of M _c in unlike cobweb types (Fig. 7, black vs. reddish lines) volition shift mean PO_two levels to lower values (east.chiliad. G3S1 vs. G3S3, 26.5 vs. 25.4 mmHg, light; thirteen.6 vs. 12.3 mmHg, moderate; 6.9 vs. 5.9 mmHg, high; Tabular array 5), and increase its spatial heterogeneity at all exercise conditions (eastward.thousand. G3S1 vs. G3S3, 0.10 vs. 0.xvi, light; 0.29 vs. 0.48, moderate; 0.65 vs. 0.80, high; Tabular array 5). Fig. 7 shows that the tissue oxygen levels are distributed in a much broader range. Incorporation of heterogeneity of oxygen diffusivity in different fiber types (Fig. 8, black vs. cherry lines) too affects tissue oxygen profiles. The PO_ii spatial heterogeneity increased significantly for both G3 (dashed lines) and G4 (solid lines) geometries. However, incorporation of heterogeneity of oxygen diffusivity in different fiber types causes a small-scale increment in the mean PO₂ level (e.k., G3S1vs G3S4, 26.5 vs. 27.5 mmHg, light; thirteen.vi vs. xv.04 mmHg, moderate; 6.9 vs. 7.six mmHg, loftier; Table 5). Lastly, comparison of simulation results for G3 and G4 geometries with incorporation of myoglobin concentration variation in different fiber types (Fig. 9, black vs. red lines) to control cases indicates that the contribution of C _Mb heterogeneity to PO₂ spatial heterogeneity is small (Fig. 9 and Table 6). Further investigation of the sensitivity of PO₂ contour to myoglobin-facilitated diffusion by using a much smaller D_Mb (2×10⁻¹⁴ cm²/s, i.e. 7 orders of magnitude lower) shows that the tissue PO₂ contour with no myoglobin-facilitated diffusion does not deviate significantly from control cases at all exercise atmospheric condition (Fig. S3).

Figure 7. Furnishings of variation of oxygen consumption rate in dissimilar fiber types on PO₂ histograms.

Fiber PO_two probability distribution profiles in muscle tissues nether A) lite intensity practise; B) moderate intensity exercise; and C) high intensity exercise, for simulation cases G3S1, G3S3, G4S1 and G4S3. Dotted lines are simulation results from G3 geometry, solid lines are results from G4 geometry. Black lines are the simulation scenarios S1 and ruddy lines are the simulation scenarios (S3) when Yard _c is fiber-type-dependent while others utilize compatible values for different cobweb types.

https://doi.org/10.1371/periodical.pone.0044375.g007

Figure 8. Effects of variation of oxygen diffusivity in unlike cobweb types on PO₂ histograms.

Fiber PO₂ probability distribution profiles in muscle tissues under A) light intensity exercise; B) moderate intensity exercise; and C) high intensity exercise, for simulation cases G3S1, G3S4, G4S1 and G4S4. Dotted lines are simulation results for G3 geometry, solid lines are results for G4 geometry. Blackness lines are the simulation scenarios S1 and red lines are simulations scenarios S4 when D_O2 is cobweb-type-dependent while others use uniform values for dissimilar fiber types.

https://doi.org/x.1371/journal.pone.0044375.g008

Effigy 9. Furnishings of variation of myoglobin concentration in unlike fiber types on PO₂ histograms.

Fiber PO₂ probability distribution profiles in musculus tissues under A) light intensity exercise; B) moderate intensity exercise; and C) high intensity practice, for simulation cases of G3S1, G3S5, G4S1 and G4S5. Dotted lines are simulation results for G3 geometry with compatible capillary distribution, solid lines are results for G4 geometry with fiber type-dependent distribution. Black lines are simulation scenarios S1 and carmine lines are simulations scenarios S5 when C_Mb is fiber-blazon-dependent while others use uniform values for different fiber types.

https://doi.org/10.1371/periodical.pone.0044375.g009

Discussion and Conclusions

Insufficient oxygen supply from microvascular networks nether conditions of increased metabolic demand may lead to new capillary growth from pre-existing microvasculature during physiological weather (e.g., exercise). Spatial heterogeneity is nowadays in the angiogenic response within exercising muscles. The study of inherent geometric and cellular heterogeneities at the tissue level is critical to a fundamental understanding of organ function and may provide answers for drug resistance and suggestions for efficient therapeutic strategy. The sources of tissue heterogeneity can be owing to complex microvascular structure and geometry (such as irregular vessel structure, vessel length and diameter distributions), and correspondingly heterogeneous hemodynamic variables such as blood flow charge per unit, shear stress, and hematocrit [40], [41]. These factors have been shown to play an of import role in determining heterogeneous oxygen perfusion in tissue using both experimental and theoretical approaches [18], [40], [41]. The upshot of the heterogeneity can be hard to predict without a computational model; for example, inclusion of interstitial space as a separate not-oxygen-consuming volume decreased oxygen heterogeneity from 0.12 to 0.09 in light practise. However, the heterogeneity in local capillary supply is often correlated with the difference of oxygen consumption chapters in different types of muscle fibers to maintain tissue homeostasis. In this study, we applied a multiscale oxygen send model to quantitatively evaluate the impact of incorporation of fiber type-specific properties including fiber size on the oxygen distribution. Using rat EDL skeletal muscle as an example, we simulated the distribution of oxygen partial force per unit area under three exercise conditions (light, moderate, and high intensities) and studied the contributions of cobweb type-dependent backdrop and cobweb size to tissue oxygenation profiles.

Our results show that combined effects of fiber type backdrop other than size do not contribute significantly to tissue oxygen spatial heterogeneity and practise not significantly change oxygen profiles compared to the control cases with compatible fiber distribution. By itself, variation of oxygen consumption ratio across the three cobweb types has a significant consequence (Fig. seven). But that heterogeneity tin exist compensated by fiber blazon-specific oxygen diffusivity based on experimentally-measured physiological parameters (Fig. 8); the heterogeneity of myoglobin concentration in different fiber types does not significantly affect PO₂ profiles (Fig. 9). These iii effects are thus collectively modest. Even so, cobweb size heterogeneity increases oxygen spatial heterogeneity significantly under all exercise atmospheric condition. These data are consistent with experimental observations that oxygen supply is closely correlated with fiber size [42]. In addition, it was shown in rat skeletal muscle that the degree of hypoxia-induced angiogenesis tin be due to a cobweb size consequence [43]. Our results also advise a small contribution of myoglobin-facilitated diffusion, and consequently differences of myoglobin concentration in dissimilar fiber types crusade small deviations in oxygen distribution. Its small-scale contribution at steady land is consistent with other studies [23], [44], [45], although it might play an of import office in regulating oxygen PO₂ at non-equilibrium states and high-intensity exercise conditions [46], [47].

Our model has several limitations. Start, current heterogeneous capillary distribution used in the geometry construction is based on the findings that capillary density is primary correlated with cobweb oxidative chapters. Other studies have shown that capillary distribution is too a function of fiber size, musculus position (superficial or deep region), muscle type and creature species [48]. These factors can be considered in the future. Second, oxygen distribution in tissue is dependent on the balance of oxygen consumption, oxygen supply and oxygen transport. Thus the parameters related to these variables are critical for tissue PO_ii profiles. Well-nigh of these parameters used in our model are from rat EDL skeletal muscle and thus the conclusions we draw hither may only hold true for the EDL. These hypotheses tin be tested in other muscle types when muscle-specific parameters are available. Third, in these simulations, we hold blood menses constant (mean capillary velocity at approximately 1 mm/sec, [49]) in order to investigate and dissect the effect of the heterogeneity of cobweb type. Quaternary, depending on the level of exercise, different muscle fibers are likely recruited. While nosotros do not consider these physiological effects in the current study, the model makes it possible to comprise such regulatory changes in the future. 5th, our oxygen transport model is limited to simulation of endurance practice (primarily aerobic metabolism), but not resistance exercise. Sixth, in our model, exercise is simulated as an increase of oxygen consumption, without taking account the physical contractions and consequent fluid displacement and compartment deformation. Seventh, nosotros do not include remodeling in this study. Yet, clearly oxygen perfusion is a dynamic process that is regulated by microvascular network geometry, which itself is regulated by oxygen-dependent vascular remodeling. Models that include such a feedback organisation have been proposed [50], and our current model can exist extended to include remodeling. Finally, specific vascular networks may accept specific heterogeneous structure that causes claret flow to be imbalanced. Our tissue geometry models are somewhat idealized and bodily distribution of hypoxia may be sensitive to the verbal placement of vessels – for example, a fiber with half dozen rather than five capillaries nearby may skew the local distribution. Notwithstanding, we hypothesize that significant outliers would be removed past the physiological processes of angiogenesis and vessel regression. Nosotros generated two additional sets of tissue geometries and ran all the simulations for these to determine whether the results could be generalized. The conclusions presented here concord also for those additional geometries (see Additional Supplemental Material S2).

In summary, we conclude that differences in fiber size and capillary arrangement (but not other fiber type properties) contribute significantly to oxygen tension profiles in EDL skeletal muscle. Our extended heterogeneous-fiber oxygen send model tin exist used to understand local oxygen distribution in other muscle types when cobweb-type-specific parameters become bachelor.

Supporting Information

Figure S1.

Distribution of claret flow rate and hematocrit distribution of the two computed geometries G1 and G2. Histograms of velocity (A, C) and hematocrit (B,D) distributions for the vascular network G1 and G2. G1 refers to the geometry with uniform fiber size and uniform capillary distribution; G2, uniform fiber size and fiber type-dependent capillary distribution.

https://doi.org/x.1371/periodical.pone.0044375.s003

(PNG)

Figure S2.

Distribution of blood flow rate and hematocrit distribution of the two computed geometries G3 and G4. Histograms of velocity (A, C) and hematocrit (B,D) distributions for the vascular networks G3 and G4. G3 refers to the geometry with non-uniform cobweb size and uniform capillary distribution; G4, non-uniform cobweb size and fiber blazon-dependent capillary distribution.

https://doi.org/ten.1371/journal.pone.0044375.s004

(PNG)

Figure S3.

Effects of fiber-type specific myoglobin diffusivity on oxygen distribution in geometries of non-uniform-size fibers. Fiber PO2 nether A) light intensity exercise, volume-averaged yard_c  = 3.64×10^−four mlO2 ml⁻¹ due south⁻¹; B) moderate intensity exercise, volume-averaged m_c  = 6.68×10⁻⁴ mlO2 ml⁻¹ southward⁻¹; and C) loftier intensity do, volume-averaged g_c  = 1.02×10⁻³ mlO2 ml⁻ⁱ southward^−one in musculus tissue of G3S1,G3S6,G4S1,G3S6. Dashed lines are simulations for G3 geometry with uniform capillary considered; solid lines are results from G4 geometry with cobweb type-dependent capillary distribution. Black lines are the simulation cases (S1) with uniform cobweb-type properties (thousand_c , D_O2 ,C_Mb ) and red lines are simulations when D_Mb uses low value 3×x⁻¹⁴ instead of three×10⁻⁷ (S6).

https://doi.org/10.1371/journal.pone.0044375.s005

(PNG)

Acknowledgments

The authors thank Dr. Corban Rivera and Alex Liu for their contribution to musculus geometry construction.

Author Contributions

Conceived and designed the experiments: GL FMG ASP. Performed the experiments: GL. Analyzed the information: GL FMG ASP. Contributed reagents/materials/analysis tools: GL FMG ASP. Wrote the paper: GL FMG ASP.

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Increased Oxygen Supply Results In,

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