Mathematical modelling of oxygen transport in cardiac and skeletal muscle tissues

In this post Abdullah Al-Shammari describes the DPhil project he worked on at the WCMB during the last 4 years. This project was supervised by Dr Eamonn Gaffney in close collaboration with Prof Stuart Egginton, whose labs in Birmingham and Leeds kindly provided us with biological datasets and interpretation of our modelling results.

I am generally interested in exploring the effects of incorporating the spatial heterogeneities in skeletal and cardiac muscles on the transport of oxygen (O2), with the aim of developing indices and research tools that allow an objective assessment of tissue oxygenation. By ''heterogeneity'' here I mean basing my mathematical modelling on the micro-geometry extracted from images based on the histology of muscle tissue biopsies (Fig. 1A). This work has implications on muscle physiology and pathology [1-4], e.g. muscle fibre re-modelling, the spatial organisation of adaptive angiogenesis (capillary growth from pre-existing capillary bed), and muscle ischaemia. Moreover, it allows for in silico examination of the relative efficiency of changes in different components within the pathway of O2 transport to tissue [5], thus providing a framework for assessing experimental modulations in pathological situations.

Before delving into the details of my work, I will begin by emphasising the importance of O2 transport to muscles, among other tissues. The availability of energy within muscle cells (fibres) is essential for sustaining healthy functions, e.g. muscle contraction and heat production, where the preference is generally for aerobic (oxidative) respiration which generates large amounts of energy in the form ATP through oxidative phospohrylation in mitochondria. Hence, a continuous local supply of O2 to muscle fibres is necessary for matching their oxidative demand for energy. 

Figure 1. (A) typical tissue cross section of rat skeletal muscle (m. extensor digitorum longus) with capillary location identified by alkaline phosphatase staining. The dark structures are capillaries, the lighter objects are muscle fibres, and the lightest region is the interstitial space. The scale bar corresponds to 50 µm. (B) An expanded region of the original image on which Voronoi polygons are superimposed by dotted blue lines. A central Voronoi polygon is highlighted in dark gray and overlaps adjacent fibres, where the overlaps represent the fractional supply area of this Voronoi polygon to each fibre.

Such an aerobic respiration is contingent on a local capillary bed ensuring adequate O2 delivery within short diffusion distances of muscle fibres (Fig. 1B). In particular, the microvascular transport of oxygen to tissue depends on numerous factors including the distribution, permeability and tortuosity of capillaries; the location of mitochondria; variations in the capillary blood flow and haematocrit; the anatomical details of interstitial and cellular geometry; the levels of intracellular facilitated diffusion; the temperature and the interaction of haemoglobin and oxygen. How such factors orchestrate so as to dictate O2 functional capillary supply (FCS) to striated muscles, and whether this is particularly sensitive to one or more of these factors, are important questions that to date have far from a complete answer.

Consequently, our current understanding of experimental interventions to enhance FCS in diseased tissue, such as chronic ischaemia in skeletal and cardiac muscles, is far from complete [3, 6]. In particular, while treatment of such pathologies would certainly benefit from local enhancement of FCS of oxygen by inducing angiogenesis to match the local tissue demand, the classification of oxygen and metabolite supply that is based on conventional FCS measures can give conflicting results, thus leading to poor interpretations [6]. To this end, there has been a growing interest in improving the characterisation of FCS in effort to improve the interpretation of studies assessing pharmacological modulations of angiogenesis in response to skeletal or cardiac muscle pathologies (Fig. 1B).

Figure 2: A direct numerical exploration of the oxygen transport problem within tissue cross sections (A) can be pursued via image capture, overlaying a mesh which is faithful to the geometry captured from biopsies (B) and refined within regions of complex geometry (C). This allows a numerical solution of oxygen transport equations, which capture the biophysics of oxygen delivery while accounting for histological details. However, the complexity at the microvascular level limits the length scales which may be readily explored in this manner, especially for 3-D simulations, or for simulations within a large parameter space [7-9].

My work, in particular, seeks to isolate the important factors which govern oxygen supply and highlight factors whose influence is relatively weak. For example, I address the effect of detailed muscle micro-geometry such as incorporating (i) the distribution of capillaries [7], (ii) the distribution of muscle fibres, their sizes, and their different types [8], (iii) the detailed distribution of mitochondria within muscle fibres [11], and (iv) the facilitated diffusion by myoglobin [9, 11]. In addition, I seek to explain the relationship between muscle fibre size and mitochondria distribution, which influences O2 diffusivity and demand. This is better suited for finite elements models where a mathematical framework is used to simplify such investigations on complicated geometries and relatively large domain sizes, thus allowing their application to larger biological imaging datasets (Fig. 2).

My results highlight the relative importance of such heterogeneities in partitioning capillary O2 supply to muscle fibres, and emphasise that the extent of diffusion areas, rather than 1D diffusion distances, may better elucidate the link between capillary supply and tissue demand. In particular, the distributions of areas diffusively supplied by individual capillaries (capillary supply areas) provide a means to identify where deficient supply to tissue may occur (Fig. 3B), thereby naturally giving a methodological way to localise potential compensatory growth of new capillaries [8]. The possibility of such a localisation of angiogenesis is a direct result of modelling O2 supply in terms capillary supply areas. Indeed, everything else fixed, the larger the capillary supply area the lower the O2 levels [10]. This explains the usual pattern observed in skeletal muscle of all vertebrates, namely that capillary supply (O2 levels) scales with respect to fibre size [4]. Moreover, modifications to this basic growth process due to experimental modulations or pathologies can be conveniently explored within the framework of capillary supply areas [9].

Figure 3: Investigation of metabolic and anatomic heterogeneities. (A) A tissue with heterogeneous distribution of fibre sizes and types (red shading = oxidative fibers; blue shading = glycolytic fibres; red polygons = Voronoi polygons; black = trapping regions). (B) Distribution of capillary supply areas (Voronoi polygons & trapping regions). (C) Area histograms of oxygen tension for the same tissue. (D) A heat map visualising the spatial distribution of oxygen established by diffusion from capillaries and consumption by tissue (mitochondria).

Also, of key interest is using this framework to investigate the appropriateness of common oxygen supply indices (e.g. Voronoi polygons; Figs. 1B, 3A-B) that are aimed at specifying estimates for functional capillary supply, especially in the presence of the aforementioned heterogeneities (see [6] for a detailed list of indices). While such polygons are typically assumed to represent the actual supply area of individual capillaries in physiological studies, their use has yet to be fully validated by mathematical models of oxygen transport.

I explored this question by calculating O2 supply regions (trapping regions) from the oxygen flux lines established by diffusion from capillaries; this is an intensive computational alternative to Voronoi polygons (Fig. 4). My results [8-9] indicated that methods based on Voronoi polygons may be useful for assessing capillary supply in homogeneous and heterogeneous muscle tissues (Fig. 3A-B), but their use may become problematic in the presence of extensive capillary rarefaction (localised low capillary density), substantial spatial differences in O2 extraction capacities, and where extensive non-uniformity in capillary O2 content occurs. In such cases, trapping regions provide a more general representation of capillary supply areas. In addition, trapping regions can account for additional influences of heterogeneity that are absent in the consideration of Voronoi polygons (e.g. differences in local metabolism or muscle fibre size).

Figure 4: Computation of the biophysical modelling predictions of oxygen capillary supply areas (trapping regions) in 2D muscle tissue. Trapping regions (black lines) characterise capillary supply by delimiting the oxygen flux lines (red lines) diffusing from capillaries (red disks).

Nonetheless, I also observed that the correlation between Voronoi polygons and trapping regions can provide insight into the control of local tissue remodelling in striated muscles [8]. For example, in aerobic muscle, this correlation suggests a sensitive local control of angiogenesis on the length scale of fibre diameter. In particular, in mixed muscles, the effect of capillary loss becomes insignificant when localised to glycolytic fibres, suggesting that such rarefactions may be mitigated by reducing fibre O2 consumption.

The aforementioned models and computational framework have been coded, optimised, packaged, and made available in Matlab for easy use in Prof Egginton's experimental laboratory at the University of Leeds.

References

1. Hudlická O, Brown M, Egginton S. Angiogenesis in skeletal and cardiac muscle. Physiological Reviews, 1992; 72(2):369-417.
2. Badr I, Brown M, Egginton S, Hudlická O, Milkiewicz M, Verhaeg J. Differences in local environment determine the site of physiological angiogenesis in rat skeletal muscle. Experimental Physiology, 2003; 88(5):565-568. 
3. Deveci D, Egginton S. Muscle ischaemia in rats may be relieved by overload-induced angiogenesis. Experimental Physiology, 2002; 87(4):479-488.  
4. Wüst RCI, Gibbings SL, Degens H, Fiber capillary supply related to fiber size and oxidative capacity in human and rat skeletal muscle. In: Liss, P and Hansell, P and Bruley, D F and Harrison, D K, editor. Oxygen Transport to Tissue XXX. vol. 645 of Advances in Experimental Medicine and Biology; 2009, pp. 75-80.
5. Hauton D, Winter J, Al-Shammari AA, Gaffney EA, Evans RD, Egginton S, Changes to both metabolism and performance accompany acute reductions in functional capillary supply. Biochimica et Biophysica Acta (BBA)-General Subjects, 2015, 1850(4): 681-690. DOI: 10.1016/j.bbagen.2014.12.014
6. Egginton S, Morphometric analysis of tissue capillary supply. In: Boutilier, RG (ed) Vertebrate Gas Exchange from Environment to Cell.  Advances in Comparative and Environmental Physiology, 1990: 6, 73-141. 
7. Egginton S, Gaffney EA, Tissue capillary supply – it’s quality not quantity that counts! Experimental Physiology, 2010; 95(10): 971-979. 
8. Al-Shammari AA, Gaffney EA, Egginton S, Re-evaluating the use of Voronoi tessellations in the assessment of oxygen supply from capillaries in muscle. Bulletin of Mathematical Biology, 2012; 74(9): 2204-223. DOI: 10.1007/s11538-012-9753-x
9. Al-Shammari AA, Gaffney EA, Egginton S, Modelling capillary oxygen supply capacity in mixed muscles: Capillary domains revisited. Journal of Theoretical Biology, 2014; 356:47-61. DOI: 10.1016/j.jtbi.2014.04.016.
10. Al-Shammari AA, Gaffney EA, Egginton S, Modelling oxygen capillary supply to striated muscle tissues. Advances in Applied Mathematics. Springer International Publishing, 2014. 13-21. DOI: 10.1007/978-3-319-06923-4_2
11. Al-Shammari AA, Gaffney EA, Egginton S, The effect of heterogeneity in mitochondrial spatial distribution on oxygen transport in mixed muscles, (to appear).