Arterial Tree Representation

The rat renal vasculature is modeled as twelve compartments in series, where the first eleven represent the renal artery through to the distal afferent arteriole, each with unique myogenic and flow-dependent properties. The compartment model of the arterial tree is similar in structure to that used for the coronary circulation by Cornelissen et al [15, 16]. The resistances of the venules and capillaries are assumed to be constant and are lumped together in the final compartment. The recently published morphological data of Nordsletten et al [79] employ information from micro-computer tomography (CT) scans to divide the arterial structure via a Strahler ordering scheme [104], an algorithm originally developed for stream ordering and

found to be highly applicable to vascular branching. An example of Strahler ordering is shown in Figure 6.2; this method is particularly suited for the kidney due to the complete lack of anastomotic connections between vessels.

Figure 6.2: Strahler ordering, reprinted from [79]

Initially a 20 µm voxel resolution image was used to segment the arterial and venous trees, identifying vessels down to 30 µm in radius, followed by the division of a renal subtree using a higher resolution 4 µm voxel image. An iterative scheme was developed to integrate the two bodies of information and map the entire topology of the renal vascular tree. A color-coded representation of the rat renal vasculature is shown in Figure 6.3, divided by Strahler order. For further information regarding the reconstruction process and its robustness quantification via error analysis, the reader is referred to [79].

The relevant radial, length, and connectivity data are shown in Table 1 with order 1 designated as the renal artery and orders 10 and 11 the proximal and distal afferent arteriole respectively. Because this model does not include the TGF system, the influence of the TGF on the distal afferent arteriole [96] is neglected at present. Additionally, it has been shown that there exist interactions and coupling phenomena among nephrons due to the TGF [74, 75], which have also been omitted to facilitate a clear focus on modeling and an understanding of the myogenic response.

Figure 6.3: Rat renal vasculature color-coded by Strahler order, reprinted from [79]

Each compartment of the vasculature (i = 1, 2, ..., 11) is assumed to represent Ni vessels in parallel, with

average length Liand diameter di, with the constant resistance of the venous and capillary system lumped in

the eleventh compartment. Assumptions about bifurcational structure of the vascular tree are not necessary in the sense that the number of vessels of order i+1 that originate from a given single vessel of order i is unspecified; the important parameter is simply the total number of vessels in each order. The efferent arteriole is also assumed to be part of the postglomerular vasculature in the most distal compartment. The diameter values in Table 6.1 represent the anatomical diameters (d0i), defined as the passive value at a mean

arterial pressure (MAP) of 100 mmHg. Because of the existence of myogenic tone, and the ability of the smaller vessels to constrict, the typical diameter values in vivo will be less than the anatomical diameters, and

Table 6.1: Renal arterial tree measurements [79].

Order Anatomical Diameter Std Dev Vessel Length Std Dev No. of Elements Std Dev

µm µm mm mm 1 432.20 4.74 0.185 − 1 − 2 384.84 17.79 1.440 0.647 3 − 3 279.66 20.11 8.975 1.331 6 − 4 172.30 24.06 2.516 2.053 24 1 5 107.74 12.51 1.031 0.674 90 6 6 88.46 9.81 0.511 0.000 247 23 7 78.58 1.08 1.001 0.216 578 71 8 59.74 0.35 0.656 0.286 1,245 198 9 40.12 6.90 0.404 0.390 4,373 664 10 27.80 3.80 0.423 0.283 13,070 2,293 11 20.16 0.14 0.312 0.285 29,566 5,965

are dependent on the hemodynamic properties of flow and pressure in the vessel. The Poiseuille relationship between pressure, flow, and resistance is applied throughout the renal vasculature, leading to the expression:

Ri =

128ηiLi

Niπd4i

(6.1)

The subscript i refers to the vascular order, and ηiis the viscosity of the perfusate, which varies throughout

the vasculature and exhibits diameter dependence. In accordance with the Fahraeus-Lindqvist effect, effective viscosity decreases with decreasing vessel diameter to a certain point. However, in the microvasculature, consisting of vessel diameters below 30 µm, the effective viscosity begins to increase again, providing a profound impact on the magnitude of resistance changes due to constriction. The work of Pries et al. [90, 91] in the rat mesentery vasculature observed this effect and produced an expression for in vivo viscosity variation, shown in Eqn 6.2.

ηi= " 1 +6e−0.085di_{+ 2.2 − 2.44e}−0.06d0.645i  d_i di− 1.1 2# _d i di− 1.1 2 (6.2)

Here ηiis the viscosity, with units of cP, and diis the diameter in micrometers. This equation assumes an

invariant systemic hematocrit of 45% and was developed using an approach combining novel experimental methods for the measurement of hematocrit and flow velocity in vessel segments of large microvascular networks and theoretical blood flow simulations through these networks based on experimentally determined architecture. For further details the reader is referred to [91]. The resistance of each unit varies as a function of the diameter and the viscosity, which is diameter-dependent itself. The local pressure Piwithin each vessel

of the twelfth compartment is kept at the constant venous pressure (Pven) of 5 mmHg. It is assumed that

the local pressure is steady (with the input pressure to Strahler order 1 being determined by MAP), with pressure variations arising from the cardiac cycle being ignored. The volumetric flow rate Qi through each

compartment is dependent on the pressure and resistance. We thus have the following,

Pi = Pin,i+Pin,i+1 2 for i = 1, 2, ..., 11 (6.3) Qi = Pin,i−Pin,i+1 Ri for i = 1, 2, ..., 11 (6.4) Q11 = Pin,11−Pven R11 . (6.5)

The representation of the pre-glomerular arterial tree corresponds to a distally dominant resistance distribution, with the highest active contributions localized to the arterioles, as shown in Figure 6.4. The resistance of the most distal compartment, representing the capillaries and veins, is determined by applying a normal glomerular pressure of 60 mmHg in the rat at a typical MAP of 120 mmHg [30], and is not subject to variation. Given the inlet and outlet values, local pressures for each Strahler order can be easily calculated from the condition that the flow in all orders must be the same.