Fluid and protein exchange in microvascular networks: Importance of modelling heterogeneity in geometrical and biophysical properties

Abstract

Integration of functional results obtained across scales, from chemical signalling to the whole organism is a daunting task requiring the marriage of experimental data with mathematical modelling. In this paper a novel coupled computational fluid dynamics model is developed incorporating fluid and protein transport using measurements in an in vivo frog (Rana pipiens) mesenteric microvascular network. The influences of network architecture and exchange are explored systematically under the common assumptions of structurally and functionally identical microvessels (Homogeneous Scenario) or microvessels classified by position in flow (Class Uniform Scenario), which are compared to realistic microvascular network components (Heterogeneous Scenario). The model incorporates ten quantities that vary within a microvessel; pressure boundary conditions are calibrated against experimental measurements. The Homogeneous Scenario standard model showed that assuming a single ‘typical’ capillary hides the influence of vessels arranged into a network architecture, where capillary hydrostatic pressures (p_T) are reduced resulting in both a nonuniform distribution of blood flow and reduced volume flow rate (J_f,T). In the Class Uniform Scenario p_T was further attenuated to produce ≈ 60% reduction in J_f,T. Finally, the Heterogeneous Scenario, incorporating measures of individual vessel surface area, demonstrates additional lowering of p_T from inlet values favoring a > 70% reduction of J_f,T in the face of a ≈ 120% increase in protein movement into the tissues relative to the Homogeneous Scenario. Beyond the impacts of network architecture, an unanticipated finding was the influence of a blind-end microvessel on model convergence, indicating a profound influence of the largely unexplored dynamics of vascular remodelling on tissue perfusion.

1. Introduction

Optimal organ function requires the matching of blood flow to metabolic demand. As organ function is dynamic, metabolic demand becomes variable necessitating that both blood flow through, and exchange properties of, the vessels within the organ possess the capacity to adapt. In turn, vascular delivery depends on vessel architecture which varies with organ function, spatial distribution, and dimensions as well as the distribution of driving forces [14, 48]. At present, data interpretation is hindered by the lack of appropriate mathematical models that can account for results derived at the cellular, single vessel segment, and whole organ levels.

Most of the circulation relies on blood flow through branching microvascular network structures as are found in skeletal muscle and skin. In the majority, models of mass transfer from microvascular networks focus on gas exchange, particularly molecular oxygen (O₂) [12, 36, 37, 50, 38]. Beside the fact that metabolizing cells require O₂ delivery and CO₂ removal, this focus is not surprising as experimental study of this process has been relatively straight forward. Given that most O₂ is associated with hemoglobin within the red blood cells, red cell flux, along with vascular dimensions can be obtained with minimal disturbance in a selection of living tissues. From these measures the fraction of red cells in a vessel, the haematocrit (HCT), is calculated. Even so, it was a comprehensive study of hemodynamic parameters with the aim of identifying functionally germane interrelationships determining O₂ delivery by Pries et al. [39] that demonstrated the need to appreciate network structures. Measured parameters (vessel segment length, generation, diameter, HCT, red cell velocity), and those derived from these measures, displayed widely varying distributions and sets of strong correlations. They further demonstrated that assuming scaling from the “typical vessel” to a network of vessels could lead to erroneous conclusions. With respect to O₂ transport, for example, red cell transit time calculated using ‘average’ values was 60% slower than actually observed.

Blood, a complex fluid consisting of red blood cells, white cells, and platelets suspended in plasma, transports more than the respiratory gases. Everything else that moves between the vascular space and the tissue space, from water to substrates like glucose, to vasoactive peptides and hormones, to proteins, to leukocytes, is carried in the plasma spaces (1-HCT) between the red cells carrying O₂. Mathematical modelling of exchange of most everything else carried in blood also relies on the characteristics of a “typical” exchange vessel. In the case of O₂ the permeability of the red cell, vessel walls and extracellular matrices are fairly constant. By contrast the permeabilities to all of the other materials carried in plasma varies by orders of magnitude, governed by biophysical parameters (size, fluid and lipid solubility, pH, charge, etc.), the composition and structure of the vessel walls which, in turn, vary by position in the network as well as by the organ through which they traverse [40]. Further, while the classical view is that the permeability of the barrier to a particular solute is a constant, as given by Fick’s First Law, it has been shown that in the intact circulation permeability can be a controlled variable [48] with the ability to change up or down from basal levels in both health and in disease. Results from experiments conducted at the different scales (cell to single vessel to whole organ) are not easily reconciled using current approaches and differences are widely attributed to the differences of methods employed.

In this paper we focus on modelling plasma/tissue exchange of fluid and protein in a network of capillary microvessels of known architecture to test the hypothesis that network behavior can be modelled using the characteristics of a “typical” exchange vessel. This approach, as pointed out by Pries et al. in [39], was used successfully for the Krogh cylinder of diffusive exchange and Fick’s analysis of capillary pressures. What we determined, though, is that the ‘typical’ vessel approach fails to provide understanding of the basal exchange characteristics of the network, first with respect to the distribution of hydrostatic pressures that are the primary determinant of volume mass transfer or second the distribution of protein mass transfer within the network. Understanding of these parameters are required for the further analysis of exchange when the biophysical properties governing fluid and mass transfer change in a time- and spatial-dependent manner.

The paper is organized as follows. Section 2 presents the microvascular network of a frog mesentery that is analysed in this work, along with the experimental and image processing methods that are utilised to extract quantitative information. Section 3 presents the mathematical model that is used to study vascular fluid and protein transport and exchange. The influences of network architecture and exchange are explored systematically in Section 4 under the common assumptions of structurally and functionally identical microvessels (Homogeneous Scenario) or microvessels classified by position in flow (Class Uniform Scenario), which are compared to realistic microvascular network components (Heterogeneous Scenario). Section 5 addresses the challenge of setting appropriate boundary conditions for the simulations. Simulation results are presented in Section 6 and discussed in Section 7. Finally, conclusions and future perspectives are summarised in Section 8.

2. Observed microvascular network of a frog mesentery

2.1. Ethical Approval

In this work, we consider a representative, experimentally observed capillary microvascular network of the mesentery of a frog (Rana pipiens, 6.5–7 cm body length, male, obtained from J.M. Hazen, VT), sketched in Fig. 1.

Figure 1: Mesenteric microvascular network of a real frog mesentery with numbered vessel segments and fluid flow directions shown as observed during a live animal experiment with dynamic flows.

(a) Capillary network with each vessel segment labeled, and black arrow(s) indicating flow direction(s) based on predominant physical observations over one to two minutes. Eleven pressure measurements taken in top-to-bottom time order along vessels, 1, 2, 12, 31, 13, 5, 11, 26, 21, 32, and 22, are also reported. (b) Vessels are filled in red, green or blue color based on their primary classification as arterial (AC), true (TC) or venular (VC) capillaries, respectively (see reference flow directions in Fig. 4). Annotations were digitized from the original lab notebook sketch.

All experiments were carried out in accordance with the National Research Council’s “Guide for the Care and Human Use of Laboratory Animals” under Protocol #8983 approved by the Office of Laboratory Medicine at the University of Missouri (MU ACUC). The frogs used in this, and other studies, were housed in a climate-controlled room (15°C ± 2°C) with a 12:12 light/dark cycle in chambers containing bricks immersed in fresh water allowing the animals access to full immersion or a porous dry surface.

2.2. Animal model

Frogs have been the traditional animal model for the observation of blood flow in heart, tongue, web, and mesentery [24] and measurement of hydrostatic pressures in an intact microvascular network [25, 8, 30], since the circulation at room temperature remains functional when the animals are insensate following cerebral pith in the absence of vascular depressing anaesthetics. At the termination of the experiment, the frog was euthanized by exsanguination following removal of the heart. Furthermore, since the blood pressures generated by the 3-chambered heart are low and their erythrocytes retain their nuclei, the calibre of the capillaries are large compared to mammalian vessels (10–20 μm versus 4–8 μm). These dimensions facilitate placement of perfusion pipettes for the measurement of pressures with minimal tip resistance.

Briefly, following cerebral pith the frog was placed in the supine position, the abdominal cavity was exposed via a right lateral incision, and an intestinal loop was floated out over a quartz pillar (1 cm diameter, Haerus-Amersil, NJ). The mesenteric tissue was suffused with air-equilibrated Frog Ringer solution at pH 7.4 and 15±2 °C. All perfusate solutions contained 1 mg/ml (0.3 cmH₂O oncotic pressure) bovine serum albumin (BSA; Sigma, USA; A4378, Lot No. 125F-9350) [9].

The intact circulation in a pie-shaped section of the mesenteric circulation was observed during transillumination on a Leitz Diavert microscope (x75–125) following imaging of a calibration micrometer. The microscope stage with frog was moved by an X-Y drive to capture the entire area for later analysis and manual construction of the intact network (see Fig. 1). Following the scan, hydrostatic pressure was measured in selected vessels following placement of a single-barreled glass micropipette (15 μm tip) containing Frog Ringer’s solution and 1–3 % (v/v) human red blood cells as flow markers. Details of the solution compositions, fabrication of the perfusion pipettes, and preparation of the microtools can be found in [9, 16]. Measurements were made in the lower pressure vessels proceeding to the higher-pressure segments to minimise disruption of the circulation following pipette removal. The pressure recorded in the vessel segment was the pressure in the perfusion system where the marker red cells were still at the pipette tip (balance pressure). Under these conditions of constant suffusion, the tissue was considered to be in equilibrium with atmospheric pressure.

2.3. Processing of experimental data

Off-line analysis of the microscopy video recording included manual construction of the mesenteric microvascular network with measurements of vessel segment length and radius as well as the flow patterns observed in free flow; the latter are annotated with arrows in Fig. 1(a) and each capillary vessel segment type is shown in Fig. 1(b). The classification scheme defined by Chambers and Zweifach [8] and summarised in Fig. 2, was used to identify capillary vessel type (i.e., arteriolar, true, or venular capillary). Briefly, capillaries with divergent flow at junctions upstream and downstream from the cannulation site were labeled arteriolar capillaries (AC) (see Fig. 2a). Those with divergent flow upstream and convergent flow downstream were labeled true capillaries (TC) (see Fig. 2b and 2c). Finally, venular capillaries (VC) had convergent flow both upstream and downstream from the cannulation site (see Fig. 2d). Venular capillaries empty into larger diameter vessels with scant smooth muscle coverage known as venules. In some vessels, dynamic changes in flow direction were observed causing the AC/TC/VC classification to change accordingly. Vessel type classifications are shown in Fig. 1(b) and included in Table 1, with the convention that in the case of dynamic changes, they are ordered by frequency of occurrence. One vessel (segment 7) had very weak or no flow and is labeled as TC. The relevance of vessel 7 will be discussed in Section 7.

Figure 2: Capillary vessel segment type classification depending on flow direction, which can change dynamically in capillaries.

(a) Arterial capillary (AC), where the flow diverges at both ends. (b), (c) True capillary (TC), where the flow diverges at one end and converges at the other end. (d) Venular capillary (VC), where the flow converges at both ends.

Table 1:

Summary of linearised geometry of the vessel network in Fig. 1 with vessel segment numerical labels, class type, radius and length.

Vessel Seg#	Vessel Class Type	Radius [μm]	Length [μm]	Vessel Seg#	Vessel Class Type	Radius [μm]	Length [μm]
1	AC	10.0	534.7	17	TC	10.3	325.2
2	TC/AC	10.0	442.6	18	TC	12.5	363.8
3	TC	8.3	140.2	19	TC	11.5	205.0
4	AC/TC	12.3	177.0	20	VC	16.5	150.2
5	VC/TC	9.0	222.8	21	VC	21.0	246.1
6	TC	20.5	95.4	22	VC	15.0	149.3
7	TC *	10.5	233.0	23	VC/TC	10.5	154.8
8	VC	16.5	319.8	24	TC	11.0	109.8
9	VC	10.5	94.6	25	TC/VC	12.5	368.1
10	VC	37.5	321.2	26	VC	22.5	429.2
11	VC	31.0	340.0	27	AC	12.5	93.0
12	AC	8.2	222.7	28	AC/TC	14.0	45.0
13	TC	11.0	460.6	29	AC	14.0	138.2
14	AC	14.0	210.9	30	TC	14.5	272.3
15	TC	15.0	106.4	31	TC	16.5	869.5
16	TC/AC	10.0	482.4	32	VC	15.0	467.3

A single vessel can be categorized into múltiple classes as flow direction changes dynamically. Predominant class types based on experimental observations are reported in boldface. Abbreviations: AC: Arterial capillary; TC: true capillary; VC: venular capillary.

^*Vessel 7 is a blind-end segment with no red cell flow.

2.4. Estimation of tissue perfusion areas

The microscopy videos collected in situ can be mosaiced manually or semi-automatically [1] for extracting the microvascular network architecture [6, 7] and red blood cell flow velocities in vessel segments [32, 34, 10]. In this paper, the vessel segmentation traces and flows shown in Fig. 1 were manually analysed to provide gold standard architecture networks for the simulation models and can be used as training data for automated computer vision methods [21, 11, 22]. Once the microvascular network graph is constructed and capillary vessel segment types identified, we utilized a Generalized Voronoi Diagram (GVD) to estimate tissue perfusion regions. The GVD is an extension of the Voronoi diagram for sized objects with extent and complex shape like curved vessels [31]. In the GVD computation we used a threshold of 0.275 and minimum hole size of 1000 [31].

We use the vessel classifications shown in Fig. 1(b) and Table 1, to detach the microvascular network into connected vessels of the same class as shown in Fig. 3. For example, vessels {13, 15, 16, 17, 18, 19, 24, 25, 30, 31} form a connected component set of ten TC which we label TCSet-3. The two other true capillary connected vessels are TCSet-1={6, 7} and TCSet-2={2, 3}. All vessels, at the boundary of neighboring connected components, were detached at their bifurcation points manually, using straight cuts without smoothing the contours. The angle and width (i.e. pixel thickness) of the cut can significantly affect the shape of the detached vessel and we adopted the rule that the flow in the continuous segment should be as laminar as possible, to guide the shape of the digital cuts and minimise spurs in the GVD. Each cut was a 2 – 5 pixel wide gap to enable the GVD to find accurate boundary regions between connected vessels. Once the microvascular network graph has been decomposed into connected components based on vessel classification, then the GVD algorithm is applied to the set of detached vessels. In the case of the frog mesentery network there were nine connected vessel components (Fig. 1(b)), which gives rise to the nine corresponding colored GVD regions (Fig. 3(b)); the light gray contours identify the borders between GVD boundaries as (Fig. 3(a)). The GVD boundaries are the set of points that are equidistant to two or more vessels and sometimes have high curvature regions due to the approximate discrete geometry computation of the GVD.

Figure 3: Geometric estimate of tissue perfusion regions using the Generalized Voronoi Diagram (GVD).

(a) Estimated GVD boundaries using connected segments of the same vessel type and boundaries between capillary segments of different types. The light interior lines indicate the Voronoi region boundary between adjacent vessel segments. (b) Islands of perfusion areas colored by their associated capillary vessel type: red for AC, green for TC and blue for VC perfused tissue.

Once the GVD boundaries are identified, adjacent regions of the same type are merged and the border regions are resampled with an equal number of pixels of both colors (boundary types) to reduce bias in the digital estimation of regions using discrete geometry. In this case two regions {TCSet-2, TCSet-3} were merged to create a large TC (green) perfusion area. We refer to the GVD with colored regions after merging by vessel classification as the modified GVD (mGVD), shown in Fig. 3(b). The mGVD is used to accurately compute the tissue perfusion regions by digitally estimating the areas of each connected component. Table 2 shows the pixel areas classified by arterial, true and venular capillary type converted to square microns using the sampling resolution (0.785 μm/pixel). We also computed the area occupied by the vessel in each mGVD region as shown in Table 2. The GVD computation applied to the network in Fig. 1, shows 44.5% of the tissue is perfused by true capillaries, 37.0% by venular capillaries and 18.5% by arterial capillaries (Fig. 3). This analysis reveals a large surface area for exchange, with small vessels that may shunt and redirect blood flow to better meet tissue metabolic demand.

Table 2:

GVD-based perfusion tissue areas by capillary vessel segment type classification.

Vessel Class Type	Vessel Segment Areas		Tissue Region Areas
	(μm2)	(%)	(μm2)	(%)
Arterial Capillary (AC, Red)	31,070	13.5%	251,149	18.5%
True Capillary (TC, Green)	93,608	40.5%	603,359	44.5%
Venular Capillary (VC, Blue)	106,309	46.0%	500,675	37.0%
Total	230,987	100.0%	1,355,184	100.0%

Tissue region area includes both vessels and tissue. Areas are reported in square microns, with calculations based on the scale 0.785 μm/pixel, and as a percent with respect to the total area.

3. Mathematical model of fluid and protein transport in a microvascular network

The main goal of this section is to formulate a mathematical description of fluid and protein transport throughout the capillary microvascular network, which simultaneously accounts for phenomena occurring along the vessel longitudinal axes and across their permeable walls into the surrounding tissue. For computational fluid dynamics modelling, the frog microvascular network reported in Fig. 1, is geometrically approximated using the linearised network morphology shown as a schematized graph representation in Fig. 4. The main assumption underlying the simplified network geometry is that all vessels in the network can be approximated using straight rigid cylinders with constant circular cross-section, whose radius and length are as reported in Table 1. The geometrical network maintains the same number of vessels (32 in this network) and the same connectivity among vessels as in the real frog mesentery microvasculature. The schematic vessel network graph in Fig. 4 shows numerical labels for each vessel (ranging from 1 to 32), which correspond to the original annotation sketch in Fig. 1. Numerical labels for nodes in the vessel network graph are shown with internal nodes (circled black numbers) and boundary nodes (double circled black numbers) distinguished.

Figure 4: Directed graph schematic representation of the linearised capillary network structure for the computational microvascular blood flow model corresponding to the real network in Fig. 1.

Vessels are assumed to be cylinders of circular cross-section and are identified with the same numerical labels from 1 to 32 as in Fig. 1. The arrows indicate the positive convention for the reference flow direction in each vessel based on predominant blood flow observed over one to two minutes. Network node labels are shown, ranging from 1 to 28, and identified as outer boundary or inner nodes using double- and single-lined circles, respectively

In order to introduce a mathematical description of the problem, let us denote by $\mathcal{T}$ and $\mathcal{N}$ the sets of N_vessels cylindrical vessel segments, and N_nodes nodes constituting the geometrical vessel network. In the case of the real network in Fig. 1 modelled using the linearised network graph in Fig. 4, we have N_vessels = 32 and N_nodes = 28. Furthermore, $\mathcal{N}$ can be written as the union of the set ${\mathcal{N}}_{int}$ of internal nodes (denoted with the circled numerical labels) and the set ${\mathcal{N}}_b$ of boundary nodes (denoted with the double circled numerical labels), so that $\mathcal{N}={\mathcal{N}}_{int}\cup {\mathcal{N}}_b$. In the case of the directed graph microvasculature network model in Fig. 4, we have:

$${\mathcal{N}}_{int}=(2,4,5,6,8,9,13,14,15,16,18,19,20,23,24,25,27)$$ (1a)

$${\mathcal{N}}_b=(1,3,7,10,11,12,17,2126,28)$$ (1b)

where the nodes are listed in simple ascending numerical order.

For each vessel, an arrow indicates the positive orientation of the reference axis along its centerline. For example, the reference orientation for vessel 18 goes from node 16 to node 8. This reference orientation is extremely helpful when analysing the results of the model simulations, which include fluid and mass flows through the network. For example, if the model predicts a negative flow along vessel 18, this means that the flow goes in the opposite direction with respect to the reference direction, namely from node 8 to node 16. Vessel 7, classified as TC, was actually a blind-end vessel with no red cell flow observable during the experiment. As discussed in Sections 6 and 7, this vessel presented a number of interesting properties.

The mathematical model of fluid and protein transport is detailed in the next sections. We begin with the model for a single vessel in the network (Section 3.1) which, combined with suitable boundary and junction conditions (Section 3.2), leads to the full network transport model, whose solution require an iterative numerical method (3.3).

3. 1. Single Vessel model

Let us consider a single vessel in $\mathcal{T}$ and refer to it by T. Each vessel T is assumed to be a cylinder of length L_T and constant radius R_T; thus, we can introduce a local system of cylindrical coordinates in which r, s and θ are the radial, axial and angular coordinates, with r ∈ [0, R_T), s ∈ (0, L_T) and θ ∈ [0,2𝜋), as shown in Fig. 5. The convention for the positive orientation of the s-axis in each vessel of the network is consistent with the arrows (directed graph) shown in Fig. 4.

Figure 5: Schematic representation of one vessel T in the network $\mathcal{T}$.

The vessel is assumed to be a cylinder of length L_T and constant circular cross-section of radius R_T. A local system of cylindrical coordinates is introduced, where r, s and θ denote the radial, axial and angular coordinates, respectively. The model includes fluid (blue shading) and proteins (yellow dots), which are transported along the vessel through the network and are exchanged with the tissue across the permeable vessel walls (orange shading).

Each vessel hosts the flow of a complex medium made of fluid, solutes and cells. In this work, we will focus on two main components of this complex medium, namely fluid (f) and proteins (p), which we will model as a dilute solution (see [42, Section 8.1]). This dilute solution flows throughout the network and exchanges fluid and proteins with the tissue across the permeable vessel walls, as schematized in Fig. 5. As stated in the introduction, the focus of this model is on the movement of water-soluble materials carried in the plasma phase of blood within a network. Therefore, consideration of red cells and O₂ flux will be neglected at this time. Obviously, in the future, a more comprehensive vascular network model will include these components.

In this work, we will focus on stationary conditions in the presence of axial symmetry and we will assume that hydrostatic pressures p_T (cmH₂O⁻¹) and protein concentrations C_T (gcm⁻³) are constant over each vessel cross-section. Protein concentration C_T and oncotic pressure Π_T (cmH₂O) are assumed to be related via the phenomenological relationship (see [15]):

$${\Pi }_T\left(C_T\right)=a_1{C}_T+a_2{C}_T^2+a_3{C}_T^3$$ (2a)

where a₁, a₂ and a₃ are given constants. Under these assumptions, the transport of fluid and proteins along each vessel in the network can be described by the following differential equations:

$$\frac{d{q}_{f,T}(s)}{ds}+2\pi R_T{L}_{p,T}\left[\left(p_T(s)-{\tilde{p}}_T(s)\right)-{\sigma }_T\left({\Pi }_T(s)-{\tilde{\Pi }}_T(s)\right)\right]=0$$ (2b)

$$\frac{d{q}_{p,T}(s)}{ds}+2\pi R_T\frac{1-{\sigma }_T}{{\sigma }_T}{P}_{d,T}\left(C_T(s)-{\tilde{C}}_T(s)\right)=0$$ (2c)

holding for s ∈ (0, L_T). The volumetric rate of fluid flow q_f,T(s) (cm³ s⁻¹ ) and the mass rate of protein flow q_p,T(s) (g s⁻¹) at each point s ∈ (0, L_T) along the axial direction of vessel T are defined as follows:

$$q_{f,T}(s)=-\frac{\pi R_T^4}{2\mu (\gamma +2)}\frac{d{p}_T(s)}{ds}$$ (2d)

$$q_{p,T}=C_T(s)q_{f,T}(s)-\pi R_T^{2}D\frac{d{C}_T(s)}{ds}$$ (2e)

Model parameters include the hydraulic conductivity L_p,T of the vessel wall (cm s⁻¹ cmH₂O⁻¹), the solute permeability 𝑃_d,T of the vessel wall (cm s⁻¹), the reflection coefficient 𝜎_T (dimensionless) satisfying the constraint 0 < 𝜎_T ≤ 1, the fluid dynamic viscosity μ (cmH₂O s), the diffusion coefficient D (cm² s⁻¹) of proteins in the fluid and the steepness parameter 𝛾 > 0 characterizing the velocity profile on the vessel cross-section, with 𝛾 = 2 corresponding to the classic Hagen-Poiseuille profile.

From the mathematical viewpoint, the model described by Eqs. (2a)–(2e) is a first-order system of nonlinearly coupled differential equations whose solution consists of finding the four dependent variables q_f,T = q_f,T(s), p_T = p_T(s), q_p,T = q_p,T(s) and C_T = C_T(s) in each vessel $T\in \mathcal{T}$ for s ∈ (0, L_T), provided suitable expressions are given for the data in the tissue ${\tilde{p}}_T={\tilde{p}}_T(s)$, ${\tilde{\Pi }}_t={\tilde{\Pi }}_T(s)$ and ${\tilde{C}}_T={\tilde{C}}_T(s)$, and suitable conditions are provided at the end nodes located at s = 0 and s = L_T, which may belong to the set of boundary nodes ${\mathcal{N}}_b$ or internal nodes ${\mathcal{N}}_{int}$, as listed in Eqs. (1a) – (1b) and discussed in Section 3.2. We refer the interested reader to [48], for a summary of the fundamental laws of physics and physiology on which Eqs (2a) – (2e) are build and to [42, Chapter 15] for the details on how to reduce transport laws in a three-dimensional vessel to one-dimensional differential equations along the vessel axis.

For each vessel T, we define additional quantities of physiological interest that will be used to illustrate the results in Section 6. The average axial volumetric flow rate Q_f,T (cm³ s⁻¹) is defined as:

$$Q_{f,T}:=\frac{1}{L_T}{\int }_0^{L_T}{q}_{f,T}(s)ds.$$ (3)

The sign of Q_f,T is positive or negative depending on whether or not the fluid flow along the vessel axis has the same direction as the arrow annotated in Fig. 4. The specific lateral volumetric flow rate of fluid j_f,T(s) (cm² s⁻¹) is defined as:

$$j_{f,T}(s):=2\pi R_T{L}_p\Delta p_{net,T}(s)$$ (4)

for s ∈ (0, L_T) and it depends on the net pressure 𝛥p_net,T(s) given by:

$$\Delta p_{net,T}(s):=\left(p_T(s)-{\tilde{p}}_T(s)\right)-{\sigma }_T\left({\Pi }_T(s)-{\tilde{\Pi }}_T(s)\right).$$ (5)

We remark that, by combining (4) with (2b), the specific lateral volumetric flow rate of fluid j_f,T can also be written as:

$$j_{f,T}(s)=-\frac{d{q}_{f,T}(s)}{ds}.$$ (6)

The net pressure difference 𝛥p_net,T(s) across the wall of vessel T is the driving force for fluid in Starling’s Law for filtration. In the physiology literature when 𝛥p_net,T(s) = 0, the Starling balance point is achieved, and no fluid moves across the vessel wall. If the vessel walls are impermeable to water, then j_f,T(s) = 0 for every s ∈ (0, L_T) and, as a consequence of Eq. (2b), the axial volumetric flow rate q_f,T is constant along the vessel. Conversely, if the vessel walls are leaky, then j_f,T(s) is positive or negative depending on whether fluid is moving from the vessel into the tissue or vice versa (see Fig. 6(a)). The integral of j_f,T(s) over the length of the vessel provides the total lateral volumetric flow rate of fluid J_f,T (cm³ s⁻¹), namely:

$$J_{f,T}:={\int }_0^{L_T}{j}_{f,T}(s)ds.$$ (7)

We emphasise that J_f,T represents the fluid volume exchanged through the whole lateral surface of the vessel T per unit time. Thus, J_f,T is positive (resp. negative) if the fluid volume exiting the vessel into the tissue per unit time is higher (resp. lower) than the fluid volume entering the vessel from the tissue; interestingly, J_f,T = 0 not only characterises impermeable vessels, but also leaky vessels where the fluid volume leaving the vessel is equal to that entering the vessel.

Figure 6: Schematic illustration of the variables of interest relating to microvasculature transport with lateral flows in the basal physiological state.

(a) Fluid filtration and reabsorption. (b) Protein transport. See Table 3 for definition of variables.

Let us now consider the protein variables. We define the diffusive axial mass flow rate q_p,T,diff(s) (g s⁻¹) as:

$$q_{p,T,diff}(s)=-\pi R_T^{2}D\frac{d{C}_T}{ds}(s)$$ (8)

for s ∈ (0, L_T). This variable is very useful to specify the boundary conditions, as discussed in Section 3.2. In addition, we define the specific lateral mass flow rate j_p,T(s) (cm⁻¹ g s⁻¹) as:

$$j_{p,T}(s):=2\pi R_T\frac{1-{\sigma }_T}{{\sigma }_T}{P}_{d,T}\left(C_T(s)-{\tilde{C}}_T(s)\right)$$ (9)

for ∈ (0, L_T). If the vessel walls are impermeable to proteins (as is the case in specialized vessels in the brain and testes, see for example [48], then j_p,T(s) = 0 for every s ∈ (0, L_T) and, as a consequence of Eq. (2c), the axial mass flow rate q_p,T is constant along the vessel. Conversely, if the vessel walls are leaky to proteins (as occurs in the pathophysiological state of vessel injury from sepsis to trauma), then j_p,T(s) is positive or negative depending on whether proteins are moving from the vessel into the tissue or vice versa (see Fig. 6(b)). The integral of j_p,T(s) over the length of the vessel provides the total lateral mass flow rate of proteins J_p,T (g s⁻¹), namely:

$$J_{p,T}:={\int }_0^{L_T}{j}_{p,T}(s)ds.$$ (10)

We emphasise that J_p,T represents the mass of proteins exchanged through the whole lateral surface of the capillary vessel T per unit time. Thus, J_p,T is positive (resp. negative) if the mass of proteins exiting the vessel into the tissue per unit time is higher (resp. lower) than the mass of proteins entering the vessel from the tissue; interestingly, J_p,T = 0 not only characterises impermeable vessels, but also leaky vessels where there is a balance between the mass of proteins leaving and entering the vessel.

For ease of reference, the notation and parameters characterising the reduced-order model for the coupled fluid-protein transport in each vessel T and in the whole network are listed in Table 3 together with a description of the physical meaning of the symbols and its units. An illustration of the key variables of interest are shown in Fig. 6.

Table 3:

Summary of model variables and parameters.

Description	Symbols	Unit
Quantities that vary within a vessel
Axial coordinate along the length of vessel T	s	[cm]
Hydrostatic pressure in vessel T	pT (s)	[cmH2O]
Protein concentration in vessel T	CT (s)	[g cm−3]
Oncotic pressure in vessel T	ΠT (s)	[cmH2O]
Net pressure difference across the wall of vessel T	Δpnet,T (s)	[cmH2O]
Axial volumetric flow rate in vessel T	qf,T (s)	[cm3 s−1]
Specific lateral volumetric flow rate in vessel T	jf,T (s)	[cm2 s−1]
Axial mass flow rate in vessel T	qp,T (s)	[g s−1]
Diffusive axial mass flow rate in vessel T	qp,T,diff (s)	[g s−1]
Specific lateral mass flow rate in vessel T	jp,T (s)	[cm−1 g s−1]
Quantities that are constant for each vessel
Radius of vessel T	RT	[cm]
Length of vessel T	LT	[cm]
Hydraulic conductivity of the wall of vessel T	Lp,T	[cm s−1 cmH2O−1]
Protein reflection coefficient of the wall of vessel T	σT	[-]
Solute permeability of the wall of vessel T	Pd,T	[cm s−1]
Average axial volumetric flow rate in vessel T	Qf,T	[cm3 s−1]
Lateral volumetric flow rate in vessel T	Jf,T	[cm3 s−1]
Average axial mass flow rate in vessel T	Qp,T	[g s−1]
Lateral mass flow rate in vessel T	Jp,T	[g s−1]
Quantities that characterise the whole network
Total lateral volumetric flow rate	Jf,tot	[cm3 s−1]
Total lateral mass flow rate	Jp,tot	[g s−1]

The table distinguishes among quantities that vary within a vessel, are constant for each vessel but vary from vessel to vessel or characterise the whole network. For quantities that vary within a vessel, their dependence on the axial coordinate s is explicitly indicated.

3.2. Boundary and junction conditions for the vessel network

Boundary conditions must be prescribed for both fluid and proteins at each network node in the set ${\mathcal{N}}_b$. For the specific mesenteric capillary network under consideration in Fig. 1 and modelled as the directed graph in Fig. 4, the value of the hydrostatic pressure is assigned at all nodes in ${\mathcal{N}}_b$ except for node 7, where a condition of zero volumetric axial flow rate is imposed. This choice is justified by the fact that node 7 was experimentally observed to be a dead-end for vessel segment 7. For what concerns the proteins, concentration values are assigned at the boundary nodes 1, 3, 11, 12, 21, and 26; zero diffusive flux is imposed at the remaining boundary nodes, where we set q_p,T,diff(s) = 0, with s = 0 or s = L_T depending on which of the vessel ends constitutes a network boundary. We remark that the boundary values for the hydrostatic pressure are not directly available from the experiments and, therefore, require a delicate calibration procedure, which is illustrated in Section 5.

Junction conditions must be prescribed at each network node in the set ${\mathcal{N}}_{int}$, with the goal of ensuring that pressures and concentrations are continuous, and mass is not lost at the junctions. To this end, for every node in ${\mathcal{N}}_{int}$, we identify the inlet/outlet surfaces of the connecting vessels and we impose that: (i) the sum of the axial volumetric flow rates q_f,T entering or leaving the junction through each of the connecting vessels equals to zero; (ii) the sum of the axial mass flow rates q_p,T entering or leaving the junction through each of the connecting vessels equals to zero; (iii) the values of the hydrostatic pressure p_T at the connecting end of each vessel are equal; (iv) the values of the protein concentration C_T at the connecting end of each vessel are equal.

3.3. Microvascular network: Complete coupled model

The complete coupled model for the microvascular network consists in finding:

$$q_{f,T}=q_{f,T}(s),p_T=p_T(s),q_{p,T}=q_{p,T}(s),C_T=C_T(s).$$ (11)

for s ∈ (0, L_T) and for all $T\in \mathcal{T}$ as solution of problem (2) equipped with relationship between Π_T and C_T given in Eq. (2a), the boundary and junction conditions described in Section 3.2 and suitable data for hydrostatic and oncotic pressures in the tissue. Once the variables in Eq. (11) are obtained, oncotic pressures and lateral fluxes in each vessel can be calculated via Eqs. (2a), (4), (7), (9) and (10). To quantitatively evaluate the total volume of fluid exchanged between vessels and tissue throughout the whole network, we define the total lateral volumetric flow rate J_f,t𝑜t (cm³ s⁻¹) and the total lateral mass flow rate J_p,t𝑜t (g s⁻¹) as:

$$J_{f,tot}:=\sum _{T=1}^{N_{vessels}}{J}_{f,T},J_{p,tot}:=\sum _{T=1}^{N_{vessels}}{J}_{p,T}.$$ (12)

The complete coupled model is solved using a fixed-point iteration whose basic goal is to decouple the equations for the fluid and for the proteins, in such a way that the solution of the nonlinear system is reduced to the successive solution of two model sub-blocks (fluid and protein blocks) in an iterative fashion until convergence is achieved. We note that referring to the list of symbols, parameters, descriptions and units in Table 3 may facilitate following the iterative algorithm below. In essence, each step k, k ≥ 0, of the iterative procedure consists of:

Step 1: Fluid block

1.1 Set $C_T(s)=C_T^{(k)}(s)$ for s ∈ (0, L_T) and $T\in \mathcal{T}$ in Eq. (2a);

1.2 Solve the fluid Eqs. (2b) and (2d), supplied by the boundary and interface conditions illustrated in Section 3.2, to determine the functions $p_T(s)=p_T^{(k+1)}(s)$ and $q_{f,T}(s)=q_{f,T}^{(k+1)}(s)$ for s ∈ (0, L_T) and $T\in \mathcal{T}$.

Step 2: Protein block

2.1 Set $q_{f,T}(s)=q_{f,T}^{(k+1)}(s)$ for s ∈ (0, L_T) and $T\in \mathcal{T}$ in Eq. (2e);

2.2 Solve the protein Eqs. (2c) and (2e), supplied by the boundary and interface conditions illustrated in Section 3.2, to determine the functions $C_T(s)=C_T^{(k+1)}(s)$ and $q_{p,T}(s)=q_{p,T}^{(k+1)}(s)$ for s ∈ (0, L_T) and $T\in \mathcal{T}$.

Step 3: Convergence checks

3.1 Compute the estimates of the iteration errors:

$$er{r}_p^{(k)}:=\underset{T\in \mathcal{T}}{\max }\frac{{\Vert p_T^{(k+1)}-p_T^{(k+1)}\Vert }_{\infty }}{{\Vert p_T^{(k+1)}\Vert }_{\infty }}er{r}_C^{(k)}:=\underset{T\in \mathcal{T}}{\max }\frac{{\Vert C_T^{(k+1)}-C_T^{(k+1)}\Vert }_{\infty }}{{\Vert C_T^{(k+1)}\Vert }_{\infty }}$$ (13)

where $\Vert \varphi {\Vert }_{\infty }:=\underset{s\in \left(0,L_T\right)}{\max }|\varphi (s)|$ for any continuous function $\varphi :\left(0,L_T\right)\to ℝ$ with $T\in \mathcal{T}$.

3.2 Set $er{r}^{(k)}:=\max \left\{er{r}_p^{(k)},er{r}_C^{(k)}\right\}$.

3.3 If err^(k)< toll, then the fixed-point iteration has reached convergence and we can proceed to Step 4, otherwise the procedure is repeated starting from Step 1.

Step 4: Post-processing

4.1 Solve for lateral fluxes via Eqs. (4), (7), (9) and (10).

The fixed-point algorithm requires the choice of a specific value for the tolerance toll and an initial guess for the distribution of hydrostatic pressures and protein concentrations in the whole network, corresponding to k = 0. The results reported in Section 6 have been obtained by setting toll = 10⁻⁶, $p_T^{(0)}(s)=16.11$ cmH₂O and $C_T^{(0)}(s)=0.025$ gcm⁻³ for s ∈ (0,L_T ) and $T\in \mathcal{T}$.

Steps 1 and 2 require the solution of diffusion-advection-reaction problems on each vessel $T\in \mathcal{T}$, for which we adopted a displacement-based form of the Galerkin finite element approximation with piecewise linear finite elements that are continuous over the network $\mathcal{T}$. For a detailed description of the method and of its theoretical properties, we refer to Sacco et al. [42, Chapter 23]. In the case of the protein block, the influence of the advective field $q_{f,T}^{(k+1)}$ is significant and makes the diffusion-advection-reaction model problem (2e) strongly advection-dominated. This requires the adoption of a stabilisation technique to prevent the numerical solution from being affected by spurious unphysical oscillations which might pollute the computed protein concentration so seriously to violate the property of positivity. In the numerical algorithm implemented in this article we have used the exponentially-fitted stabilisation proposed by Scharfetter and Gummel for the simulation of the drift-diffusion equations in semiconductor devices [49].

4. Simulation scenarios: Homogeneous, Class Uniform and Heterogeneous vessel properties

The modelling framework presented in the previous sections is used to estimate how and to what extent the transport and exchange of fluid and protein is impacted by assuming homogeneous or heterogeneous distributions of geometrical vessel properties (radius R_T and length L_T ) and biophysical vessel properties (hydraulic permeability L_p,T and solute permeability P_d,T ) within the network. To this end, we consider three simulation scenarios, referred to as Homogeneous Scenario (see Section 4.1), Class Uniform Scenario (see Section 4.2) and Heterogeneous Scenario (see Section 4.3).

The values of the fluid dynamic viscosity μ, the steepness parameter γ, the protein reflection coefficient σ_T, the diffusion constant D, the parameters a₁, a₂ and a₃ for the relationship between oncotic pressure and concentration are assumed to be the same for all the three simulation scenarios and are reported in Table 4. We remark that the reflection coefficient of vessel 7, denoted as σ₇, is assumed to be one order of magnitude smaller than in the rest of the network. During the experimental observations, vessel 7 showed to be ‘leaky’ but did not exhibit an axial fluid flow strong enough to drive red blood cells into the vessel, a behavior typically observed in vessel segments undergoing growth (angiogenesis) or remodelling [48, 14].

Table 4:

Summary of model parameter values that are common to all simulation scenarios.

Parameter	Unit	Value	Reference
μ	[cmH2O s]	1.1 × 10−5	[2], page 33
γ	[-]	2	[42]
σT, for T ≠ 7	[-]	0.8	[48]
σ 7	[-]	0.08	This work
D	[cm2 s−1	61 × 10−8	[51]
a 1	[cmH2O (g cm−3)−1]	380.8	[15]
a 2	[cmH2O (g cm−3)−2]	2448.0	[15]
a 3	[cmH2O (g cm−3)−3]	1.632 × 104	[15]
${\tilde{p}}_T$, for all $T\in \mathcal{T}$	[cmH2O]	0	[9]
${\tilde{\Pi }}_T$, for all $T\in \mathcal{T}$	[cmH2O]	0	[9]
${\tilde{C}}_T$, for all $T\in \mathcal{T}$	[g cm−3]	0	[9]

The reflection coefficient of vessel 7 has been adapted to account for its leaky nature.

The values of the hydrostatic pressure ${\tilde{p}}_T$, the oncotic pressure ${\tilde{\Pi }}_T$ and the protein concentration ${\tilde{C}}_T$ in the tissue outside of each vessel T are assumed to be equal to zero, as reported in Table 4, mirroring the specific experimental conditions illustrated in Section 2.

4.1. Homogeneous Scenario

In the Homogeneous Scenario, all vessels in the network are assumed to be identical with respect to their geometrical and biophysical properties, so that we can write

$$R_T=\overline{R},L_T=\overline{L},L_{p,T}={\overline{L}}_p,P_{d,T}={\overline{P}}_d,\text{for}\text{all}T\in \mathcal{T}$$ (14)

where $\overline{R}$, $\overline{L}$, ${\overline{L}}_p$ and ${\overline{P}}_d$ represent network averages estimated as:

$$\begin{gathered} \overline{R}=\frac{1}{N_{vessels}}\sum _{T=1}^{N_{vessels}}{R}_T,\overline{L}=\frac{1}{N_{vessels}}\sum _{T=1}^{N_{vessels}}{L}_T, \\ {\overline{L}}_p=\frac{1}{3}\left({\overline{L}}_{p,AC}+{\overline{L}}_{p,TC}+{\overline{L}}_{p,VC}\right),{\overline{P}}_d=\frac{1}{3}\left({\overline{P}}_{d,AC}+{\overline{P}}_{d,TC}+{\overline{P}}_{d,VC}\right). \end{gathered}$$ (15)

The values of R, L, L_p and P_d used in this work are reported in Table 5.

Table 5:

Summary of geometrical and biophysical parameters for the class Homogeneous Scenario.

Parameter	Unit	Value	Reference
$\overline{R}$	[μm]	14.5	Eq. (22) and Table 1
$\overline{L}$	[μm]	274.7	Eq. (22) and Table 1
${\overline{L}}_p$	[cm s−1 cmH2O−1]	3.17 × 10−7	Eq. (23) and Table 6
${\overline{P}}_d$	[cm s−1]	3.31 × 10−7	Eq. (23) and Table 6

All vessels in the network are assumed to be identical, characterised by the same radius R, length L, hydraulic conductivity L_p and solute permeability P_d.

4.2. Class Uniform Scenario

In the Class Uniform Scenario, the experimental flow-based categorization as AC, TC and VC, see Table 1, is utilised to prescribe identical geometrical and biophysical properties only to vessels within the same vessel type category. Thus, subdividing the set of vessels as $\mathcal{T}={\mathcal{T}}_{AC}\cup {\mathcal{T}}_{TC}\cup {\mathcal{T}}_{VC}$, we can write:

$$R_T={\overline{R}}_i,L_T={\overline{L}}_i,L_{p,T}={\overline{L}}_{p,i},P_{d,T}={\overline{P}}_{d,i},\text{for}T\in {\mathcal{T}}_i,\text{with}i=AC,TC,VC$$ (16)

where ${\overline{R}}_i$, ${\overline{L}}_i$, ${\overline{L}}_{p,i}$ and ${\overline{P}}_{d,i}$ represent class averages. Here, ${\overline{R}}_i$ and ${\overline{L}}_i$ are computed as:

$${\overline{R}}_i=\frac{1}{N_{vessels,i}}\sum _{T\in {\mathcal{T}}_i}{R}_T,{\overline{L}}_i=\frac{1}{N_{vessels,i}}\sum _{T\in {\mathcal{T}}_i}{L}_T,\text{with}i=AC,TC,VC$$ (17)

where N_vessels,i, for i = AC, TC, VC, are the total number of vessel segments in each of the three classes ${\mathcal{T}}_{AC}$, ${\mathcal{T}}_{TC}$ and ${\mathcal{T}}_{VC}$. The values of ${\overline{L}}_{p,i}$ and ${\overline{P}}_{d,i}$ are adopted directly from [19]. The values of ${\overline{R}}_i$, ${\overline{L}}_i$, ${\overline{L}}_{p,i}$ and ${\overline{P}}_{d,i}$ for i = AC, TC, VC used in this work are reported in Table 6.

Table 6:

Summary of geometrical and biophysical parameters for the Class Uniform Scenario.

Parameter	Unit	Value	Reference
Class Uniform – Vessel Radius
${\overline{R}}_{AC}$	[μm]	12.1	Eq. (17) and Table 1
${\overline{R}}_{TC}$	[μm]	12.4	Eq. (17) and Table 1
${\overline{R}}_{VC}$	[μm]	18.6	Eq. (17) and Table 1
Class Uniform – Vessel Length
${\overline{L}}_{AC}$	[μm]	203.1	Eq. (17) and Table 1
${\overline{L}}_{TC}$	[μm]	319.6	Eq. (17) and Table 1
${\overline{L}}_{VC}$	[μm]	263.2	Eq. (17) and Table 1
Class Uniform – Hydraulic Conductivity
${\overline{L}}_{p,AC}$	[cm s−1 cmH2O−1]	1.5 × 10−7	[19]
${\overline{L}}_{p,TC}$	[cm s−1 cmH2O−1]	3.0 × 10−7	[19]
${\overline{L}}_{p,VC}$	[cm s−1 cmH2O−1]	5.0 × 10−7	[19]
Class Uniform – Solute Permeability
${\overline{P}}_{d,AC}$	[cm s−1]	1.7 × 10−7	[19]
${\overline{P}}_{d,TC}$	[cm s−1]	2.9 × 10−7	[19]
${\overline{P}}_{d,VC}$	[cm s−1]	4.8 × 10−7	[19]

All vessels within the same class (i.e., AC, TC, VC) are assumed to be identical and characterised by the same radius (i.e. ${\overline{R}}_{AC}$, ${\overline{R}}_{TC}$, ${\overline{R}}_{VC}$), length (i.e. ${\overline{L}}_{AC}$, ${\overline{L}}_{TC}$, ${\overline{L}}_{VC}$),, hydraulic conductivity (i.e. ${\overline{L}}_{p,AC}$, ${\overline{L}}_{p,TC}$, ${\overline{L}}_{p,VC}$), and solute permeability(i.e. ${\overline{P}}_{d,AC}$, ${\overline{P}}_{d,TC}$, ${\overline{P}}_{d,VC}$),.

4.3. Heterogeneous Scenario

In the Heterogeneous Scenario, each vessel in the network is assumed to be geometrically different and characterised by its own values of radius R_T and length L_T, as listed in Table 1. Since the values of hydraulic conductivity and solute permeability are not available for each single vessel in the network, their values are considered identical for vessels within the same class, so that $L_{p,T}={\overline{L}}_{p,i}$, $P_{d,T}={\overline{P}}_{d,i}$, for $T\in {\mathcal{T}}_i$ with i = AC, TC, VC, with the same values listed in Table 6.

5. Calibration of boundary pressure conditions

The calibration of the boundary conditions is a very delicate issue. Overall, the flow through the network is driven by a hydrostatic pressure gradient determined by the pressure values prescribed at the boundary nodes Nb, marked as double lined circles in Fig. 7(b), and yet a direct measurement of these boundary pressures is not available. Pressure measurements, however, were made in 11 out of the 32 vessels in the network. Thus, we will leverage the available pressure data to calibrate the boundary values for the pressures needed for the model. To this end, we divide the set of hydrostatic pressures measured experimentally in two subsets: a training set and a validation set. The training set consists of all the pressures that are measured in a vessel for which one end constitutes a boundary node for the whole network, denoted in black in Fig. 7(b). The validation set consists of all the pressures that are measured in vessels for which both ends are internal nodes, denoted in dark gray in Fig. 7(b).

Figure 7:

Calibration of boundary values for the hydrostatic pressure in the microvascular network: (a) experimental annotations; (b) mathematical network graph representation distinguishing boundary nodes (double lined circles) and internal nodes (single lined circles). Experimental pressure measurements are available in selected vessels at the boundary of the network (black) or in its interior (dark gray). Vessel segments without a pressure measurement are shown in light gray.

We consider the heterogeneous simulation scenario, since it is the most realistic among the three scenarios considered here (see Section 4), and we tune the pressure values prescribed at the nodes in ${\mathcal{N}}_b$ till the predicted pressures along the boundary vessels are sufficiently close to the experimentally measured pressure values in the training set (black vessels, Fig. 7(b)). Next, we verify that the predicted pressures pertaining to the internal vessels in the validation set (dark gray vessels, Fig. 7(b)) are in good agreement with the measured values. The results of the training and validation procedures are summarised in Table 7.

Table 7:

Comparison between measured and predicted pressure values at selected vessels for which experimental values are available.

Training Set [cmH2O]			Validation Set [cmH2O]
Vessel Number	Measured Pressure	Predicted Pressure	Vessel Number	Measured Pressure	Predicted Pressure
1	24.0	23.8	5	12.2	13.9
2	22.0	21.6	13	16.7	16.1
11	13.8	8.3	21	8.0	7.2
12	19.2	19.2	26	10.8	11.1
22	4.5	4.4	31	12.6	13.2
32	7.6	7.7

Vessels in the training sets are those for which one end is located at the network boundary, whereas vessels in the validation set are internal to the network.

For all vessels in the validation set, measured and estimated pressures differ by less than 1 cmH₂O, which is within experimental error. Vessel 11 in the training set (Table 7) is the single vessel exhibiting a difference between the measured and predicted pressures of greater than 1cmH₂O. In fact, this discrepancy could well have reflected an artifact resulting from the measuring pipette laying close to the wall of this 62 μm diameter vessel. The perfusion pipettes used to make the measures of balance pressure are generally on the order of 8–10 μm in diameter and the venular vessels can actually be elliptical instead of cylindrical in cross section making it possible to have the pipette placed outside of the center stream. In any case, the predicted value of 8.3 cmH₂O is more consistent with the pressure values more often obtained in these vessel segments (see the results for vessels 22 and 32, for example that are of similar dimensions and locations in the vessel network).

Finally, the calibrated boundary pressure values are listed in Table 8, along with all other boundary conditions that will be used henceforth in all model simulations. The first column in the table indicates the boundary node in the set ${\mathcal{N}}_b$ where boundary conditions for fluid (second column, Table 8) and proteins (third column, Table 8) are prescribed. The notation in Table 8 can be easily interpreted by referring to the summary of model variables reported in Table 3. We recall that p_T (s), C_T (s), q_f,T (s), and q_p,T,diff (s) indicate the hydrostatic pressure, the concentration, the axial volumetric flow rate and the diffusive axial mass flow rate along the length of vessel T, with s = 0 and s = L_T indicating the vessel endpoints according to the oriented graph of Fig. 4. For example, the boundary conditions at node 3 has to be imposed at the end s = 0 of vessel 2, whereas the boundary condition at node 17 has to be imposed at the end s = L₂₃ of vessel 23.

Table 8:

Summary of the conditions imposed at the boundary nodes of the network for fluid and proteins.

${\mathcal{N}}_b$	Fluid	Proteins
1	p1(0) = 28.5 cmH2O	C1(0) = 0.025 g cm−3
3	p2(0) = 23.5 cmH2O	C2(0) = 0.025 g cm−3
7	qf,7(L7) = 0 cm3 s−1	qp,7,diff (L7) = 0 g s−1
10	p10(L10) = 2 cmH2O	qp,10,diff (L10) = 0 g s−1
11	p11(0) = 15 cmH2O	C11(0)= 0.025 g cm−3
12	p12(0) = 26 cmH2O	C12(0) = 0.025 g cm−3
17	p23(L23) = 10 cmH2O	qp,23,diff (L23) = 0 g s−1
21	p27(0) = 25 cmH2O	C27(0) = 0.025 g cm−3
26	p32(0) = 8 cmH2O	C32(0) = 0.025 g cm−3
28	p22(L22) = 2 cmH2O	qp,22,diff (L22) = 0 g s−1

The conditions are common to all simulation scenarios.

6. Coupled model simulation results

The three frameworks considered in this paper are a Homogeneous Scenario, a Class Uniform Scenario and a Heterogeneous Scenario, based on biophysical vessel properties, as described in Section 4. The results obtained for each scenario are summarised and compared in Figs. 8 to 11 and Table 9. Specifically, Fig. 8 visualizes the distributions of hydrostatic pressures p_T (first row) and oncotic pressures Π_T (second row) along each vessel in the network. Fig. 9 and 10 plot the values of volumetric flow rate of fluid J_f,T and the mass flow rate of protein J_p,T for each vessel T in the network, respectively. Fig. 11 provides colormaps of the net pressure difference Δp_net,T (first row), the specific lateral volumetric flow rate j_f,T (second row) and the specific lateral mass flow rate j_p,T that describe the exchange of fluid and protein across the wall of each vessel in the intact network. The Starling balance points are reported in all panels of Fig. 11 as short ticks, as they mark the locations at which the vessels are predicted to switch from filtrating to absorbing fluid. The total lateral volumetric and mass flow rates J_f,tot and J_p,tot represent the overall fluid and protein exchange between the microvascular network and the tissue and are provided in Table 9. The values of these quantities are also provided in the case where vessel 7 is excluded from the calculation, given its peculiar nature of being a leaky, dead-end vessel. In the next three sections, the results are presented by scenario.

Figure 8: Visualisation of hydrostatic and oncotic pressures in cmH₂O for the Homogeneous, Class Uniform and Heterogeneous Scenarios.

The colormaps represent p_T (top row) and Π_T (bottom row), with axial flow directions reported as arrows along the vessels. Flow directions are reported in red or black color depending on whether or not they differ from the reference direction reported in Fig. 4.

Figure 11: Visualisation of transmural exchange of fluid and proteins for the Homogeneous, Class Uniform and Heterogeneous Scenarios.

The colormaps represent the net pressure difference Δp_net,T (top row, units: cmH₂O), the specific lateral volumetric flow rate j_f,T (center row, units: cm² s⁻¹) and the specific lateral mass flow rate j_p,T (bottom row, units: cm⁻¹ g s⁻¹). Starling balance points are marked with transversal ticks (top and bottom rows). Flow directions are reported with arrows in red or black color depending on whether or not they differ from the reference direction reported in Fig. 4.

Table 9:

Total volume flow rate of fluid (J_f,tot) and mass flow rate of proteins (J_p,tot) for the Homogeneous, Class Uniform and Heterogeneous Scenarios.

Quantity	Units	Homogeneous	Class uniform	Heterogeneous
Jf,tot	[cm3 s−1]	21.0 × 10−9	9.5 × 10−9	5.9 × 10−9
Jp,tot	[g s−1]	54.0 × 10−12	49.1 × 10−12	31.1 × 10−12
Jf,tot – Jf,7	[cm3 s−1]	19.4 × 10−9	8.1 × 10−9	5.4 × 10−9
Jp,tot – Jp,7	[g s−1]	15.2 × 10−12	17.2 × 10−12	18.3 × 10−12
J f,7 /Jf,tot	[%]	8	14	9
J p,7 /Jp,tot	[%]	72	65	41

The flow rates without terminal Vessel 7, which is very leaky, are shown in rows three and four.

Figure 9: Lateral volumetric flow rates of fluid J_f,T for each vessel T in the network belonging to the AC category (a), TC category (b) and VC category (c).

Results are compared for the Homogeneous Scenario (gray), Class Uniform Scenario (blue) and Heterogeneous Scenario (red). Underneath each graph, the cumulative value of volumetric flow rates for all vessels in the AC, TC and VC categories is reported for the three scenarios utilising the same color-coding.

Figure 10: Lateral volumetric flow rates of fluid J_f,T for each vessel T in the capillary network belonging to the AC Lateral mass flow rates of protein J_p,T for each vessel T in the network belonging to the AC category (a), TC category (b) and VC category (c).

Results are compared for the Homogeneous Scenario (gray), Class Uniform Scenario (blue) and Heterogeneous Scenario (red). Underneath each graph, the cumulative value of lateral mass rates for all vessels in the AC, TC and VC categories is reported for the three scenarios utilising the same color-coding.

6.1. Homogeneous Scenario

The Homogeneous Scenario is the approach taught and used most commonly in the physiology and medical literature when considering fluid and protein fluxes at the level of whole organs or systemic vasculature. In this scenario, all vessels in the network are assumed to have the same radius, length, hydraulic conductivity, and permeability to proteins, with the rationale that the mean values of these parameters primarily influence the transmural movements of fluid and protein. For the ability to compare results in the other two scenarios, in Figs. 9–10 the exchange microvessels are subclassified as in Table 2 albeit they were modelled as identical elements. Our results show that, when the vessels are assumed to possess identical anatomical and exchange characteristics, the hydrostatic pressure gradients throughout the network illustrated in Fig. 8 become the sole determinant of fluid flow through the network and volume flow across the walls of the individual vessels. Consequently, hydrostatic pressures, set by the inflow values specified in Table 7, are highest at the points of inflow and drop with successive branching to become lowest at the outflow (see top-left plot in Fig. 8). Interestingly, the branching architecture of the network induces pressures not only to fall from inflow (top) to outflow (bottom) but also from left to right. These patterns of hydrostatic pressures are reflected in the calculated values of volume flux across the vessel walls (center-left plot in Fig. 11) wherein volume flow rates j_f,T are highest at the top and left dropping to the bottom and right making the left upper quadrant ”leakier” than the right lower quadrant. It should also be noted (see left plots in Fig. 8 and Fig. 11) that the direction of flow in vessel 15 changes from the reference flow direction shown in Fig. 7, as indicated by the red arrow. Only two Starling points are present in the network, as marked by the ticks in vessels 10 and 21. This means that fluid reabsorption is occurring in only 4 of the 32 vessels in the network, namely vessels 10, 21, 22 and 32. The outcome of the Homogeneous Scenario also results in a uniform flux of protein across the vessel walls (bottom-left plot in Fig. 11) as the transmural concentration gradient, the only driving force, is effectively a constant with Π_T being practically the same in every vessel segment (see bottom-left plot in Fig. 8). In the Homogeneous Scenario, 35.9%, 48.3% and 15.8% of the volume flow occurs across the AC, TC and VC elements of the network, respectively. With respect to the lateral protein mass flows, 22.6%, 41.9% and 35.5% occurs across the AC, TC and VC elements. Not surprisingly, the percentages for protein mass flows reflects the frequencies of the 3 vessels classes: 21.8% AC, 43.8% TC and 34.4% VC (Table 2).

6.2. Class Uniform Scenario

In the Class Uniform Scenario, two experimentally determined components in living tissue are considered. First, from the observations of blood cells flowing within the vessels of the network a classification scheme first used by Chambers and Zweifach in 1944 was applied [8]. This approach permitted the exchange microvessels to be subdivided by their position in flow within the network. Further, the average anatomical measurements of vessel segment radius and length by vessel class were applied. Second, from a decade of measurements of L_p and/or P_d in single microvessel segments classified as AC, TC, or VC, average values of the exchange parameters were applied [19].

Inclusion of these data also accounted for the observation of the gradients in permeability properties from the high pressure to the low pressure ends of networks ( [19], Table 6). Inclusion of these anatomical features, as well as the subclassification of the vessel types by flow and inclusion of their average permeability properties, resulted in changes of individual vessel and network volume and protein flow compared to the Homogeneous Scenario. With respect to the distribution of hydrostatic pressures, given no changes in inlet pressures, inclusion of the vessel segment surface areas and radii and class-specific L_p, hydrostatic pressures in the top and right portions of the network were reduced (see top-center plot in Fig. 8). The impact of the reduction in net driving pressures, especially in vessels of the AC classification, is illustrated in Fig. 8. In this scenario, the bulk of the volume flow lateral exchange occurs in the TC segments (65.2% and compared to 48.5% in the Homogeneous Scenario). Once again, it is notable that, in this scenario, the changes in hydrostatic pressure result in an inversion of flow direction compared to the reference state in vessel 15. In addition, the number of vessels with pressures at or below the Starling balance point increased from 4 to 6 (see vessels 10, 20, 21, 22, 31 and 32 in the center plots in Fig. 11). In this scenario vessels with the predominantly VC classification experience fluid reabsorption of −0.21×10⁻⁹ cm³s⁻¹. Overall, the reduction in driving force for fluids resulted in a 55% reduction of network volume flux rate from 19.4 to 8.1 ·10⁻⁹ cm³s⁻¹, as reported in Table 9. The AC and VC vessel segments experienced the largest reduction in volume flux compared to the Homogeneous Scenario with 25% and 57% reductions, respectively (Fig. 9). With respect to the protein flux, j_p,T is no longer distributed uniformly across the network reflecting inclusion of the vessel subtype specific permeability characteristics (see bottom-center plot Fig. 11). Of particular interest solute flux increased by 13% compared to the Homogeneous Scenario, in contrast to the behavior observed for volume flux (Table 9). As illustrated in Fig. 10, the increase reflects the permeability to protein and relative frequency of VC classification. Further, inspection of Table 9 and Fig. 10 reveals that compared to the predictions of the Homogeneous Scenario, J_p,tot was reduced to 34% in the AC segments and relatively unchanged at 93% in the TC segments. The largest changes were found in the VC segments that now exuded 189% more protein than those vessels in the Homogeneous Scenario. Even with these significant changes in protein flux distribution, the increase from 15.2 to 17.2 ·10⁻¹² gs⁻¹ was insufficient to result in a change in Π_T across the network (see bottom-center plot in Fig. 8).

6.3. Heterogeneous Scenario

The third scenario augmented the Class Uniform Scenario by including measured radii and lengths thus providing the vessel-specific anatomical characteristics to the network. As seen with p_T and Π_T (see right plots in Fig. 8), this additional refinement resulted in a further reduction in the hydrostatic pressures from the inlet pressures compared to the Class Uniform Scenario. Once again there were changes in flow directions compared to the reference set. In the Heterogeneous Scenario the flow in vessel 15 remained as originally set, while flow in vessel 3 changed direction. Further, the Starling balance points were shifted further upstream to include 2 additional vessels (9, 10, 11, 20, 21, 22, 31, 32) resulting in a greater fluid flow amongst the vessels of the VC classification. The influence of the changes in driving pressure on the magnitude and direction of fluid movement in each of the vessels in the network is clearly illustrated in Fig. 9. This shift in the Starling balance points not only reduced the total fluid flux rate to 29% of that predicted in the Homogeneous Scenario but resulted in a net fluid flux into several vessels of the VC classification (fluid reabsorption, −1.38 ·10⁻⁹ cm³s⁻¹) that is −77% of the fluid filtration predicted by in the Homogeneous case and 6.6 times greater than that experienced by VC in the Class Uniform Scenario. Driving forces for fluid flux reflected the gradients in hydrostatic pressure across the network as the distribution of oncotic pressure throughout the network was negligible. Overall, the reduction in driving force for fluids resulted in a 27.9% reduction of network volume flux rate from 19.4 ·10⁻⁹ cm³s⁻¹ to 5.4 ·10⁻⁹ cm³s⁻¹. The AC and VC vessel segments experienced further reductions in volume flux compared to the Homogeneous Scenario to 21% and −29% reductions, respectively. Although the Heterogeneous Scenario does not result in a change of oncotic pressure distribution within the network, changes in the distribution and magnitude of protein mass flow rates were observed (Table 9, Fig. 10). Unlike fluid flux, total protein flux was found to increase, relative to the Homogeneous Scenario. The distributions across the networks are similar to that for the Class Uniform Scenario with the largest difference in the VC, wherein those vessels had a flux rate of 111 gs⁻¹ representing a 206% increase compared to the VC in the Homogeneous Scenario. Overall, protein flux was 182 gs⁻¹, 120% of that predicted for the Homogeneous Scenario.

7. Discussion

The model analyses in this paper demonstrates the profound influences of network architecture and network-specific measures of permeability coefficients on the distribution of fluid and protein and their exchange between vessels and tissue. Failure to consider network structure not only hides changes in the distribution of flow within the network but also the locations and magnitudes of basal lateral fluid and protein flux into and out of the surrounding tissue. Further, inclusion of data reflecting position-dependent permeability properties reveals details on how and where movements of fluid and proteins segregate to different portions of the network.

It is imperative that understanding of these fluxes under basal conditions be appreciated. Discerning the mechanisms whereby solute flux is matched to meet metabolic demand is not possible in the absence of this foundation information. In turn, the basal mechanisms provide the starting points necessary to comprehend and then treat failure of transport under either acute or chronic conditions. The movement of fluid and proteins, as well as other materials, from respiratory gases to cells, occurs across the walls of interconnected microvessels coursing through tissues of varying metabolic demand. In the case of oxygen flux, the mathematical models have become increasingly more sophisticated to account for conditions of changing demand, such as exercise, in tissues of increasingly more complex design, such as skeletal muscle [36, 27, 35, 40]. By comparison, the models currently available to study fluid and water-soluble solutes are less sophisticated, reflecting the difficulty of studying these processes in a living tissue.

In this work we set out three scenarios for fluid and protein flux using data collected from an in vivo network of exchange vessels in the relatively simple planar tissue of the frog mesentery. The first, the Homogenous Scenario, treated all the exchange vessels as equivalent with respect to surface area for exchange and permeability properties. The second, the Class Uniform Scenario, classified the vessels into three groups defined by the predominant flow patterns at the time of observation. Further, the anatomical and permeability properties for L_p and P_d assigned to each group reflected the average behavior of these groups. Finally, while the third, the Heterogeneous Scenario, used the same permeability properties as the class uniform scenario, each vessel in the network had its unique radius and length allowing calculation of a vessel-specific surface area for exchange.

The major findings from the analyses to this point are as follows. First and foremost, the architectural arrangement and dimensions of the network of interconnected vessels can only be ignored at one’s peril. This is because the number and dimensions of the capillaries are the determinants of the hydrostatic pressures within each segment controlling the distribution of flow, as well as the transmural movement of fluid. Further, small changes in the hydrostatic pressure gradients within the smallest anastomosing vessels can rapidly and reversibly alter flow direction in the adjoining capillaries. With the additional network-dependent architectural and biophysical parameters determining exchange, both fluid and protein, the capillary pressures were further reduced from the boundary inlet values, which also reduced the net movement of fluid into the tissue and produced selected regions that favoured fluid reabsorption in the face of protein efflux.

7.1. Oncotic pressures and protein concentrations

Among the variables central to the regulation of both transmural fluid and protein movement is the oncotic pressure. Recalling that oncotic pressure (Π) is a function of protein concentration (see Eq. (2a)), in each of the 3 scenarios it was found that, given the basal values of σ and P_d, the amount of protein moving across the wall was orders of magnitude smaller than the concentration of protein within the vascular space. As a consequence, protein flux had a negligible influence on the distribution of oncotic pressure within the network under nonpathological conditions and did not differ amongst the three scenarios, as shown in the bottom row of Fig. 8. This outcome is not expected under conditions where the reflection coefficient σ is closer to 0 and/or P_d is increased, as would occur in inflammation, sepsis, or vascular remodelling (such as vessel 7). In these conditions, protein flux into the tissue could indeed be of sufficient magnitude to induce a gradient in oncotic pressure within the microvascular network. Under those conditions, both fluid and protein can accumulate in the tissue (edema). In the extreme pathology resulting from loss of barrier function, blood flow through the network can actually cease due to insufficient plasma in the vascular space to carry the formed elements of blood. The pathophysiology of these cases represents the remarkable dynamic range of barrier properties.

7.2. The special case of vessel 7

As illustrated in Fig. 1, vessel 7 was a blind-ended structure of the sort observed during network remodelling. Red cells flowing upstream of such segments bypass blind-end vessels; addition of a dye to the vasculature demonstrates flow into this segment and leak into the tissue, commensurate with a sufficient hydrostatic pressure gradient for plasma flow without cell flow, a low value of σ and a high P_d. When vessel 7 was assigned the same value for σ as for the other vessels, the entire model ‘blew up’ for all the 3 scenarios. In other words, the protein concentration in the network grew so much that the numerical algorithm failed to converge to a solution. Only when this blind-end segment was allowed to be ‘leaky’ was convergence restored. Inspection of Table 9 documents that vessel 7 alone, depending on the scenario, would have accounted for 8% to 14% of the total lateral fluid flow rate and 41% to 72% of the total lateral mass flow rate from the network. Therefore, while the results for vessel 7 are included in the body of the paper, this vessel was excluded from the subsequent analyses of this section.

It is to be noted that future studies need to consider the dynamics of microvascular network remodelling as the majority of in vitro models from which the cell biology of vessel angiogenesis is being determined, while now including a 3-D gel matrix to simulate the interstitial environment, do not include flow in the growing tubes nor flow through the interstitial matrix as would occur in vivo.

7.3. What we have learnt from the Homogenous Scenario

An even more simplified version of the Homogeneous Scenario, using a single pressure drop has been widely applied in multiple whole organ studies of fluid and protein flux with a reasonable matching of the experimental data to model predictions. In part the reliance on the Homogeneous model reflects the difficulty of obtaining independent measures of exchange surface aera and permeability to solute. Unlike whole organ models which use an estimate of capillary hydrostatic pressure, the Homogeneous Scenario as modelled in this work includes 11 measures of hydrostatic pressures at multiple points within the network, as shown in Fig. 1. In turn, these measures from vessels located at a network boundary were used as a training set to calculate the pressures at the boundaries of the network, which were then used to generate a prediction set compared to pressures in vessels within the network, as summarised in Table 7. In an intact organ what is usually obtained is an estimate of the average pressure in the exchange vasculature from measures of inlet and outlet pressure [23].

When fluid movement is assessed within an organ, the quantity measured is net organ weight at the start and throughout the experiment reflecting the amount of fluid in the vascular and tissue compartments, respectively. To measure solute movement within an organ a known amount of a tracer substance, most often the specific solute or a surrogate solute (similar size, for example but no biological activity), with a radio- or dye label is injected into the arterial supply and concentrations in blood samples from the venous outflow are measured over time. In both cases a vasodilator is present to relax the arterial smooth muscle thereby providing a maximal and constant surface area for exchange. In the case of fluid, what is measured is the Capillary Filtration Coefficient (CFC) which is the hydraulic conductivity per unit surface area (L_p S); for solutes the lumped parameter is the Permeability-Surface area product (PS) [5]. Inherent in both sets of measures is the assumption of homogeneity as CFC is really the sum of the L_p and S of a number of vessels, just as PS is the sum of the P_d and S of a number of vessels. In mathematical terms, these concepts can be written as follows:

$$\text{CFC}=\sum _{T=1}^{N_{vessels}}{L}_{p,T}{S}_T\text{and}\text{PS}=\sum _{T=1}^{N_{vessels}}{P}_{d,T}{S}_T.$$ (18)

Both approaches also assume that the vasodilator is without influence on either vascular permeability to fluid (L_p) and/or solute (P_d) [33]. This central assumption has been shown experimentally to be invalid [17, 18, 41, 43, 52]. The analysis in this paper demonstrates that application of the Homogeneous Scenario to a ‘real network’ removes the influence of the changing hydrostatic driving force for lateral volume flow simply because of the distribution of blood flow into the daughter vessels. The variation of lateral volume flow as a consequence of the drops in hydrostatic pressures results in 90% of the vessels experiencing, net filtration of fluid out of the circulation. Fluid movement into the circulation is predicted to occur in only 4 vessels (10, 21, 22 and 32, Fig. 9(c)). The immediate impact of applying the homogenous scenario, even to describe normal microvascular exchange, can be appreciated when looking at the predictions of the lateral protein mass flow. In this case, a uniform lateral flow of proteins is predicted to occur given the negligible influence of the hydrostatic pressure gradients (Figs. 8, 10 and 11). This uniform lateral flow of proteins across all elements of the microvascular network masks the well documented leakiness to solutes of the venular elements [30, 14].

7.4. What we have learnt from the Class Uniform Scenario

The major contribution of the Class Uniform Scenario is the inclusion of vessel class average values for radius, length, L_p and P_d. This results in a more varied distribution in hydrostatic pressures and lateral volume flows favoring the majority of fluid filtration occurring from the TC or the “middle” of the network. In addition, while there is net filtration of fluid into the tissue (84% of the vessels), 5 of the VC (vessels 10, 21–23, 32, Fig. 9) are predicted to be sites of reabsorption of fluid back into the circulation. The net lateral volume flow from the network results in a 58% reduction of fluid filtration compared to the Homogeneous Scenario.

With respect to lateral protein flux, again given the relatively small influence of hydrostatic pressure gradients on solute flux under our basal boundary conditions, the major observation is a gradient in lateral protein mass flow reflecting the inclusion of class-specific permeabilities to protein. The higher permeability to protein of the VC elements, compared to the lower permeability of the AC vessels, resulted in a net 13% increase in protein flux compared to the predictions of the Class Uniform Scenario (Fig. 10). Although fluid is predicted to move from the tissue to blood spaces in VC vessels 10, 21–23 and 32 (Fig. 9), a net movement of protein into the tissue is predicted by the Class Uniform Scenario.

7.5. What we have learnt from the Heterogeneous Scenario

The inclusion of the anatomical parameters of vessel segment length and radius in the Heterogenous Scenario permitted consideration of the surface area for exchange in addition to average class-specific permeability properties. In turn, a profound effect on the movement of fluid across the network was observed. As with the Class Uniform Scenario, the Heterogeneous Scenario preserved the reduction in hydrostatic pressures compared to the Homogenous Scenario. Inspection of the colormaps in Fig. 11 row 2 demonstrate that inclusion of the anatomical surface areas resulted in further reduction of fluid filtration in the shorter, smaller diameter AC vessels. Of interest, inspection of Fig. 10(a) shows that the lateral mass flow rate from the AC segments in the Heterogenous Scenario is very similar to that predicted by the Class Uniform Scenario. Half of the total filtration in both the Class Uniform and Heterogeneous Scenarios, respectively, occurred in vessels classified as TC. Of greater interest, inclusion of the architectural variables resulted in movement of the Starling balance point upstream, which provides a larger area for reabsorption of fluid into the circulation accounting for the further reduction in network fluid in the Heterogeneous Scenario. It is notable that inclusion of the anatomical variables that influence the surface area available for exchange had little impact on protein flux from the AC and TC segments, as shown in Fig. 11 row 3. The anatomical considerations had the largest influence on protein flux from the venular elements. First, it is in the VC that the behavior of fluid (Fig. 9) and protein (Fig. 10) flows are contrasted. Whereas the VC, particularly vessels 10, 21, and 32, where fluid reabsorption is predicted to occur (Fig. 9(c)), protein movement is increased (Fig. 10(c)) compared to the other scenarios and continues to move from the vascular to the tissue space. Only in vessel 22 is lateral protein movement (Fig. 10(c)) expected to be somewhat diminished in the face of fluid reabsorption (Fig. 9(c)).

7.6. Comparison of the movement of volume and solute in the venous portion of the network

As a consequence of including the distribution of hydrostatic pressures within the network in both the Class Uniform and Heterogenous Scenarios, the largest changes in amount and direction of fluid movement were observed in the venous capillaries. The influences of the 3 scenarios for the spatial distribution of the volumetric flow rate for 4 venous capillaries, 9, 20, 21 and 31, are illustrated in Fig. 12. The textbook behavior of the Modified Starling relationship is observed in only a limited number of conditions [14]. Overall, inclusion of additional architectural and barrier properties results in a 2- to 4-fold reduction in volume flow rate. In contrast to volume flow rate, under these basal conditions the sensitivity of solute flow rate, while differing with scenario, was of lesser magnitude. Fig. 13 provides an illustration of the scenario-dependent influences on volume (Fig. 13(a)) and protein flow rate (Fig. 13(b)) in a single venous capillary (vessel 10). First under the Heterogenous Scenario, fluid is filtered over the first 20% of the segment; under the Class Uniform Scenario filtration occurs over the first 10% of the vessel while in the Heterogeneous Scenario significant fluid movement into the vessel occurs over the entire length. In all cases, (Fig. 13(b)) the change in direction of fluid movement is without influence on protein flow rate. In fact, although we observe greater movement of fluid into the vascular space going from the Homogenous to Class Uniform to Heterogenous Scenarios, solute movement from the vascular space increases by 2- and 3-fold, respectively compared to the Homogenous Scenario. Interestingly, this outcome of protein movement occurring in a direction opposite to that of fluid movement is notable as there are data from both endothelial monolayer culture and in vivo microvessels consistent with pathways for transmural protein flux separate from pathways that conduct fluid [20, 16].

Figure 12: Spatial distribution of the volumetric flow rate of fluid per unit area through the lateral boundary of vessels 9, 20, 21, and 31.

Homogeneous Scenario: grey lines. Class Uniform Scenario: blue lines. Heterogeneous Scenario: red lines.

Figure 13: Fluid and protein exchange across the wall of vessel 10.

(a) Lateral volumetric flow rate, (b) Lateral mass flow rate. Homogenenous Scenario: grey lines. Uniform Scenario: blue lines. Heterogeneous Scenario: red lines.

7.7. Considerations regarding the present experimental model

The major contribution to the understanding of fluid and protein from a network of interconnected exchange vessels are the experimentally derived data in living tissue. At present the ability to observe blood flow and identify vessel dimensions and orientations is best achieved in a subset of relatively thin tissues. The exchange vessels (terminal arterioles, capillaries, and venules) are small requiring use of microscopes with limited depth and field of view, respectively. The presence of a microscope lens in the field further limits access for the microtools used measure hydrostatic pressure, fluid and solute flux from which the permeability coefficients are calculated. Finally, the microvessel barrier can be compromised by the placements of pipettes into the vessel lumen. Therefore, when hydrostatic pressures are ascertained, measures are made in the low-pressure, down-stream vessels prior to the higher-pressure vessels. Whether the presence of the perfusion pipettes alters the characteristics of the network is not known although repeated measures within a vessel under basal conditions remains fairly constant [9].

To calculate L_p, measures of fluid flux tend to be made in long (> 500μm), relatively straight vessels to segregate the movement of marker cells from fluid displacement on occlusion [9]. Measures of P_d can be made on shorter vessel segments as vessel occlusion does not occur. On the other hand, a subset of vessel geometries upstream of the measurement site can preclude the rapid and sustained filling of the vessel required to set and maintain a constant known solute concentration in the vessel lumen [9]. These issues, for example, limit where and how many quantitative measures can be performed on a given network and the ability to provide a more comprehensive model.

As discussed in Section 7.2, the growth of new or remodelling of existing vessels presents a challenge given the increase in permeability to fluid and solute. This process is likely a necessary component in vascular remodelling as it permits high tissue concentrations and gradients of growth factors, degradative enzymes, immune cells, and substrates in addition to altering fluid flux in the interstitial matrix. The results obtained in this paper provide a blueprint for the next level of experimental design, especially with regard to the pressure measurements at the boundaries.

7.8. Mesentery as an organ system

Historically the mesentery has been employed as a model tissue for the study of microvascular transport of everything from molecular oxygen to migrating leucocytes [3, 8, 9, 16, 25, 28, 29, 30, 43, 45, 52]. The tissue is thin, well vascularised, similar between species from amphibians to mammals, relatively easy to exteriorise for intravital observations and experimental manipulation and is robust allowing for measurements to be made over minutes to hours [9]. In studies of the actions of vascular endothelial growth factor (VEGF) the mesenteric preparation was used over a period of days [4, 3].

The ability to develop and validate models of the microvascular networks of higher structural and functional complexity in health and disease will benefit from the knowledge gained using the mesenteric model given its thin planar structure. Until recently the mesentery was considered a tissue with little function and relatively low metabolic demand. We took advantage of the planar distribution of a network of interconnected microvessels in this tissue to model the lateral volume and protein mass flow given a set of in vivo anatomical and physical measurements when blood flow could also be monitored. In fact, this seemingly ‘simple’ tissue has additional aspects that will further the utility of constructing a rigorous model of fluxes beyond the sophisticated considerations of red cell flux and oxygen delivery [40]. There are two structures in the spaces between and among the microvessels: lymphatics and nerves. These two components require additional skills to observe in living tissue; consequently, their functions have been understudied. Recently the gut-brain axis has been recognized as a major contributor to a variety of functions and pathologies ranging from mood, irritable bowel syndrome, to Parkinson’s disease, among others. In addition, the mesenteric space contains the immune cells that migrate between structures and participate in tissue surveillance, an especially important function given the position of the mesentery between the gut and the systemic circulation. We have data demonstrating that several neuroactive agents such as norepinephrine, acetyl choline, serotonin, and CGRP can reversibly change (increase and/or decrease) both L_p and P_d [17]. Further these changes in permeability properties on exposure to these agents, as well as the vasoactive hormones such as insulin [43] and atrial, brain, and C-type natriuretic peptide (ANP, BNP and CNP, respectively) depend on position within the network [52, 28]. Several of these same vasoactive compounds influence lymphatic permeability and contractility in the mesenteric preparation which will in turn influence the magnitude and direction of the gradients within the tissue [45, 47, 44]. Finally, microvascular network properties and structures are also dynamic with both their architecture and permeability properties changing in response to whole body status, such as volume expansion [29] and, in the case of the frog, season [13].

7.9. Limitations of the current mathematical model

The current mathematical model is static, meaning that it does not attempt to capture the observed dynamics of the flow. As observed on inspection of Table 1, as a consequence of changes in flow direction in 2 small anastomosing capillaries, 3 and 24, classification of 7 adjoining vessels, or 22% of the vessel segments, could change. The fact that some vessels placed strategically in a particular location (in this case representing 22% of the segments in the network), may be more prone to change dynamically the flow direction because they favor a particular function including but not limited to gas transport, heat dissipation, particular solute delivery or immune cell delivery. Furthermore, the values of water conductivity and solute permeability are not constant, nor do they exhibit a normal distribution [18, 41]. In this regard, even the more sophisticated versions of our model that adopt different L_p and P_d values for AC, TC and VC vessels, still fall short from accounting for the real distribution of these values along the network. In all scenarios, vessels are assumed to be straight cylinders, with varying radius and length whereas in real life they have varying degrees of tortuosity and may vary in diameter along their length. We used a classification of vessels based on flow patterns in attempt to capture an aspect of the dynamic behaviors observed in vivo. Other classification methods are possible and used widely, such as those based on branching order or diameter [40]. Even within the same classification method, differences in implementation exist. For example: defining generational branch orders down from the terminal arterioles or branch up from venules. Each classification method has pros and cons (and is often motivated by methodological and/or anatomical constraints). Here we chose AC/TC/VC because we are mostly interested in the flow and related transport and exchange of proteins. Redistribution of the vessels and permeability properties on the basis of other classification schemes may yield alternative insights into network behavior. The other component that was not included was anything relating to the duration of flow direction experienced by each segment indicative of changing hydrostatic pressure gradients that obviously determine volume mass flux. In addition, these changes in hydrostatic pressure will alter the shear stress experienced along the vessel walls and any shear-dependent modulation of exchange or endothelial cell signaling. The models, as constructed, are amenable to include the dynamics once new experiments are carried out to define these variables with greater precision.

Finally, because the tissue preparation on which these experimental measurements were made was exteriorised and suffused with fluid, the tissue hydrostatic and oncotic pressures were set to zero. Of course, future work will need to account for non-zero values as these parameters influence not only how far and how quickly fluid and protein move in the interstitial space but also the functioning of the lymphatics [46, 45, 44].

8. Conclusions and perspectives

The main finding was that there is no “typical” capillary and that vascular architecture, reflecting with greater fidelity the organization of the exchange microvessel segments into a network, is an essential determinant of the magnitude and distribution of hydrostatic pressures. In turn the hydrostatic pressures along the vessel walls control the sites, direction, and amount of fluid movement while the gradients in pressure determine axial flow in each segment. These architectural features are appreciated with respect to the more widely studied field of oxygen dynamics [12, 36, 37, 38, 40]. Interestingly, the added details on heterogeneous distributions of values for geometrical parameters (vessel radius, vessel length) and biophysical parameters (L_p, P_d) had a lesser influence on the distribution of hydrostatic pressure, oncotic pressure and axial flow through the network under basal conditions. However, they do lead to notable differences on the amount of fluid exchanged between vessels and tissue as a consequence of including the surface area for exchange. Further, inclusion of a rudimentary distribution of P_d amongst vessel types lead to site-specific areas for fluid and protein efflux raising the question as to whether the networks are designed to support the functions of structures in the tissue yet to be included in the present models, such as terminal and collecting lymphatics and nerves. This is of paramount importance to clarify the roles of the lymphatic systems in its ability to move excess filtration and act in immune surveillance. Although the present focus was on fluid and protein, we know the intact microvasculature includes red cell flux and oxygen delivery, two components could be included in an extended version of the model presented in Section 3.

Key Points:

• Microvascular network architecture defines coupling of fluid and protein exchange.

• Network arrangements markedly reduce capillary hydrostatic pressures and resting fluid movement while increasing the capacity for change

• The presence of vascular remodelling or angiogenesis puts constraints of network behaviour

• The sites of fluid and protein exchange can be segregated to different portions of the network

• Although there is a net filtration of fluid from a network of exchange vessels, there are specific areas where fluid moves into the circulation (reabsorption) and while protein is moving into tissue the amount is insufficient under basal conditions to result in changes in oncotic pressure.