Re-evaluating the Use of Voronoi Tessellations in the Assessment of Oxygen Supply from Capillaries in Muscle

Abstract

The ability to characterise capillary supply plays a key role in developing effective therapeutic interventions for numerous pathological conditions, such as capillary loss in skeletal or cardiac muscle. However, quantifying capillary supply is fraught with difficulties. Averaged measures such as capillary density or mean inter-capillary distance cannot account for the local geometry of the underlying capillary distribution, and thus can only highlight a tissue wide, global hypoxia. Detailed tissue geometry, such as muscle fibre size, has been incorporated into indices of capillary supply by considering the distribution of Voronoi tessellations generated from capillary locations in a plane perpendicular to muscle fibre orientation, implicitly assuming that each Voronoi polygon represents the area of supply of its enclosed capillary. Using a modelling framework to assess the capillary supply capacity under maximal sustainable conditions in muscle, we theoretically demonstrate that Voronoi tessellations often provide an accurate representation of the regions supplied by each capillary. However, we highlight that this use of Voronoi tessellations is inappropriate and inaccurate in the presence of extensive capillary rarefaction and pathological variations in oxygen tension of different capillaries. In such cases, oxygen flux trapping regions are developed to provide a more general representation of the capillary supply regions, in particular incorporating the additional influences of heterogeneity that are absent in the consideration of Voronoi tessellations.

Appendices

Appendix A: Mathematical Details

1.1 A.1 Uniqueness of Solution

Consider the partial differential equation for the oxygen partial pressure, p:

(13)

(14)

where C is the unit disk, C _i is the ith circular capillary, and n _i is the normal inward to the ith capillary and outward to C. For every i∈{1,…,N _c }, the boundary condition on ∂C _i is n _i ⋅∇p=κ(η _i −p). When the oxygen tension is higher inside capillaries (p<η _i ) the flux is then outward to the capillary (−∂ _r p>0), so that κ is positive. The solution is unique, as we now demonstrate.

Let $q:=p-\tilde{p}$ for two possibly different solutions $p, \tilde {p}$ which satisfy Eq. (13). Thus, ∇² q=0 with boundary conditions n _i ⋅∇q=−κq on capillaries and n _i ⋅∇q=0 on the unit circle. By Green’s identity, we employ these boundary conditions to get

and, therefore, ∫ _C (∇q)²  dA=0, which implies q= is constant on C. Consequently, $\oint_{\partial C_{i}} q^{2} \,ds = 0$ which implies q=0 on all ∂C _i . Thus, $p-\tilde{p}=q=0$ on C and the solution is unique.

1.2 A.2 Trapping Regions

Recall that the streamlines associated with the oxygen partial pressure, p, are the integral paths of the dynamical system

(15)

and that the trapping region associated with a capillary is the area of the muscle reached by streamlines emerging from the capillary boundary. Below we assume that all capillaries act as oxygen sources with no regions on their boundaries where the oxygen flux is from the tissue into the blood compartment, unless explicitly stated otherwise. This is valid for all examples considered in the main text except non-uniform capillary PO₂. In addition, note that the trapping regions considered below will tessellate the domain, requiring that oxygen supply is sufficient to prevent anoxia; the latter is highly pathological and the required parameter regimes are not considered in this study.

1.3 A.3 Area of Trapping Regions

The trapping region D _i ⊆D of a capillary with centre x _i , with outward pointing normal n _i is defined to be the union of the smallest domain satisfying

(16)

Using by A _i to denote the area of the ith trapping region, D _i , we integrate Eq. (13) over this region to obtain

(17)

where ds is the anti-clockwise parameterisation element around the capillary and the trapping region. Note that, for a fixed capillary radius ϵ, this further simplifies to

(18)

When any region of a capillary boundary acts as a sink, with an oxygen flux from the tissue into the blood compartment, the above equation for trapping region areas is not valid. This only occurs in our study for the case of non-uniform capillary PO₂, and then we observe that all such cases have all streamlines at the capillary boundary pointing away from the tissue and into the capillary interior. In this instance, the trapping region is of zero area, allowing the evaluation of numerous statistics in the main text even in the case of non-uniform capillary PO₂.

1.4 A.4 General Properties of the Trapping Regions

When ∇p=0, we have a stationary point in the phase plane diagram associated with Eq. (15). Given the above is a gradient dynamical system, the Jacobian matrix must be symmetric. In particular, a non-degenerate stationary point of this system is restricted to a dynamical system sink, source, or saddle (Guckenheimer and Holmes 1997) and neither periodic streamlines (limit cycles) nor homoclinic connections can exist in a gradient dynamical system (Guckenheimer and Holmes 1997; Hirsch and Smale 1974).

Furthermore, no source exists other than possibly on capillary boundaries: at an interior source point, both p _xx and p _yy are negative or zero, giving the contradiction 0≥∇² p=μ>0. Similarly, maxima may not occur on the domain boundary (the unit circle). In polar coordinates, a maximum on the boundary satisfies p _θ =0 and p _θθ ≤0. The boundary condition is p _r =0 which, by Taylor expansion about r=1, gives p _rr ≤0. Then, as above, we have the contradiction 0≥∇² p=μ>0. In addition, when no region of a capillary boundary acts as a sink, with instead the oxygen flux always from the capillary into the tissue, or vice-versa, no point on a capillary is a stationary point as p _r ≠0.

Hence, when streamlines emanate from capillary boundaries, the fact there are no sources in the tissue implies, from the Poincaré–Bendixson theorem (Hirsch and Smale 1974) that a streamline within a trapping region can only approach a closed loop that is made up by hetero-clinic connections of dynamical system stationary points. In principle, these stationary points may be degenerate, and thus non-hyperbolic, but this is not observed in practice (and the appearance of non-hyperbolic stationary points would require mathematical precision in parameter values, with a concomitant sensitivity in model behaviour). Hence, we have that the trapping region boundaries are connections between the allowed hyperbolic stationary points, i.e. saddles to saddles or saddles to sinks, noting that sink-sink connections are inconsistent with Eq. (15). Furthermore, Wang and Bassingthwaighte (2001) have shown that only saddle-minimum hetero-clinic connections are possible on the boundary of a trapping region. This latter property, combined with stationary point hyperbolicity (and that stationary points are finite in number), as observed in practice, implies that Eq. (15) is a Morse–Smale gradient field (Guckenheimer and Holmes 1997), and thus the dynamical system is structurally stable (Palis and Smale 1970).

The above properties of gradient dynamical systems are used below for a numerical algorithm that calculates the trapping region boundary given that all capillaries act as oxygen sources with no regions on their boundaries where the oxygen flux is from the tissue into the blood compartment.

Appendix B: Numerical Computations

2.1 B.1 Finite Element Method for the Oxygen Partial Pressure

We solve Eqs. (13)–(14) to determine the oxygen partial pressure. This is accomplished by using the finite element method (see, for example, Reddy 1993). The mesh used was generated by the PDE toolbox provided by Matlab (The MathWorks Inc., Natick, MA). Adaptive meshing was used to improve the accuracy of the computed solution around capillaries to resolve areas of rapid changes in oxygen partial pressure gradients.

2.2 B.2 Streamlines

Streamlines were computed by numerically solving Eq. (15) via Heun’s method. As frequent numerical evaluations of ∇p are required by this numerical solver, linear interpolation is called upon to provide approximate values within elements of the finite element solution.

2.3 B.3 Trapping Regions

As above, in discussing trapping region properties, we assume that the flux of oxygen on capillary boundaries always points into the tissue compartment. We have that each trapping region is delimited by a collection of heteroclinic connections of type saddle-minimum. Such orbits of Eq. (15) are estimated numerically as follows:

• (1) The unstable manifolds emerging from each saddle point are determined within machine level tolerances exploiting the local Hartman–Grobman theorem for hyperbolic equilibrium points of dynamical systems (Guckenheimer and Holmes 1997).

• (2) Integration proceeds until ending at a dynamical system sink, thus giving the saddle-sink connection.

• (3) All these connections are determined and the resulting tessellation of the domain gives the trapping regions, each of which is identified with the capillary it encloses.

Appendix C: Dependence on Parameters

In the non-dimensionalised model, there are N _c +3 parameters: κ, μ, ϵ and η _i for i=1,…,N _c . The non-dimensional metabolic demand, μ, determines how much O₂ is absorbed by the tissue, and the non-dimensional mass transfer coefficient or capillary permeability, κ, is a measure of the O₂ flux leaving the capillary. The areas of the trapping regions are given by Eq. (18) and so are ostensibly dependent on κ, μ, the non-dimensional capillary radius ϵ, the ith capillary non-dimensionalised partial pressure, η _i , and the solution p. Below, we consider uniform p _cap so that all the η _i are equal and denoted by η. For non-uniform capillary supply, there are no analogous simplifications in the parameter dependencies of the trapping regions; however, studying such complications is not our focus.

3.1 C.1 Streamlines and Trapping Regions Are Independent of the Non-dimensional Capillary Partial Pressure and Oxygen Uptake

We consider a re-scaling of the equations to show that trapping region areas and shapes are independent of μ, η, and a re-scaling of p.

Fixing κ, η and ϵ, we re-scale μ↦ξμ. Then let $\tilde{p}$ denote the solution of Eqs. (13)–(14) with parameters κ, η _i =η, ϵ, and (ξμ). By uniqueness, the solution $\tilde{p}$ must satisfy $\tilde {p}=\xi p - \xi\eta+ \eta$. This clearly satisfies Eq. (13) for the given parameter values, while direct differentiation gives $\nabla\tilde{p}=\xi\nabla p$. Thus, all no-flux boundary conditions and stationary points are unchanged because $\nabla p = 0 \Longrightarrow\nabla\tilde{p} = 0$ provided ξ≠0.

In addition, the streamlines and trapping regions, are unchanged as the streamline parameter may be rescaled to give

Integrating Eq. (13) over the trapping region gives

where the ξ cancel to give the same expression as Eq. (17). Thus, the modelling predictions do not depend on changes in μ, while holding other parameters fixed.

Instead, fix κ, ϵ, and μ, and re-scale η↦ζη. Then $\tilde{p}= p - \eta+ \zeta\eta$ satisfies Eqs. (13)–(14) with the rescaled value of the nondimensional capillary partial pressure, ζη. By analogous reasoning to the above, we have that the streamlines and trapping regions are unchanged.

3.2 C.2 Varying the Non-dimensional Capillary Permeability, κ, and the Capillary Density ρ

Analogous to Table 3 in the main text, Table 4 above presents statistical measures for the geometry of rat EDL muscle (see Fig. 4(f)), but with the non-dimensional capillary permeability, κ, reduced by factors of 2 and 4, reducing oxygen supply into the tissue. Note that there is not an extensive difference on comparison with the unperturbed case, highlighting an insensitivity to κ, though the agreement between the Voronoi polygons and trapping regions slightly reduces. This trend holds given the permeability and oxygen supply is sufficient that oxygen levels are sufficient for maximal uptake; analogous comments hold for variations in the dimensional capillary density ρ.

Table 4 Statistics determined from areas of Voronoi polygons and trapping regions for the geometry of rat EDL muscle cross section as in plots 5(f) within the main text. Results for unperturbed and perturbed parameters are presented. We consider reducing the non-dimensional capillary permeability, κ, by factors of 2 and 4, and increasing the dimensional capillary density, ρ, by factors of 4 and 16. For a description of the above statistics, the reader is referred to the main text

Note that the non-dimensional capillary radius, ϵ, depends on the dimensional value and the scaling used to non-dimensionalise. The latter length scale is $L=\sqrt{4N_{c}/\pi\rho}$, as discussed in Sect. 3.2 of the main text, where N _c is the number of capillaries in the non-dimensional unit disc and ρ is the dimensional capillary density. Thus, assuming the dimensional capillary radius does not vary between capillaries in the cross-section, varying ϵ is equivalent to varying ρ. Thus, the above observations of insensitivity also apply and more generally we have that the trapping region statistics used in the main text are robust to variations in the model parameters, at least in the case of uniform capillary PO₂.