AMBITO DE APLICACION

The system encompassed by Equations 6.1−6.5 and 6.9−6.13 was solved using the math software package MATLAB. A user-written first-order temporal finite differencing scheme with step size of 0.01 s was used, combined with inner iterations to fulfill convergence criteria. For each Strahler order at every time step, the instantaneous diameter response was evaluated using Eqn 6.12 within an inner while loop for a given level of VSM activation, φM Ri. The time dynamics are incorporated via Eqns 6.11 and 6.13, as the activation

for each order adjusts over varying time scales to the steady-state value for a given circumferential tension, Ttoti, and eNOS activation level, φN Oi. Finally, in the outermost loop, the resistance values are adjusted

based on the changes in diameter and subsequent compartmental pressures are calculated, as well as the overall renal blood flow.

To determine the steady-state response of the system, the procedure described above was allowed to reach a steady-state renal blood flow value for a wide range of arterial pressures. Convergence criteria were firstly based on achieving a balance between Laplacian and actual circumferential wall tension, and secondly on the flow in all compartments being equal, both with a tolerance of 10−6. This convergence was typically achieved in less than 10, 000 iterations for each pressure input. The numerical simulations were performed on a desktop PC with a 2.2 GHz AMD Athlon 64 processor, and were completed in under 2 minutes per simulation. The MATLAB script is contained in Appendix F.

Chapter 7

Renal Myogenic Model Results

The mathematical model of the myogenic mechanism in the rat kidney was developed to include order- specific parameters that mimic the response of each level of the arterial tree. The system was subjected to computer simulations of pressure perturbations in the form of step increases. The dynamic response is examined, as well as the steady state flow-pressure relationship over the physiological pressure range.

7.1

Model Parameters

The values for the eleven parameters that govern the system, specific to each of the eleven Strahler orders, are shown in Table 7.1.

Table 7.1: Model parameters for rat renal vasculature

Order Anatomical Diameter Cp Cp0 Ca Ca0 Ca00 Ct Ct0 CN O CN O0 g ta

µm N/m N/m m/N m2_{/N s} 1 432.20 2.88 10 3.55 0.91 0.374 1.06 1.33 0.95 16.20 0.2 45 2 384.84 2.53 10 3.14 0.91 0.374 1.95 2.16 0.87 5.48 0.45 40 3 279.66 1.77 10 2.34 0.91 0.374 5.40 3.49 0.71 6.25 0.7 35 4 172.30 1.02 14.87 1.61 0.91 0.374 15.74 4.72 0.43 4.90 1.0 30 5 107.74 0.62 14.87 1.02 0.91 0.374 26.94 4.71 0.34 4.87 1.0 25 6 88.46 0.49 14.87 0.83 0.91 0.374 31.87 4.78 0.47 3.89 1.0 20 7 78.58 0.43 14.87 0.74 0.91 0.374 32.93 4.68 0.72 3.26 0.9 15 8 59.74 0.33 18.5 0.59 0.91 0.374 35.28 4.03 0.81 3.21 0.75 10 9 40.12 0.21 21 0.40 0.91 0.374 38.15 3.34 0.89 4.05 0.6 5 10 27.80 0.15 25 0.29 0.91 0.374 54.56 2.64 0.92 6.41 0.5 2 11 20.16 0.10 28 0.21 0.91 0.374 67.92 2.91 0.93 7.57 0.4 1.8

in the renal artery. The baseline resistance of the branching structure of the pre-afferent arteriole renal vasculature (i = 1, 2, ..., 9) was calculated using Equation 6.1 and the reported anatomical diameters for each order. At a MAP of 100 mmHg, the first nine Strahler orders account for a pressure drop of 19.1 mmHg, corresponding to an inlet pressure of 80.9 mmHg at the proximal afferent arteriole in this passive state. The parameters Cp and Cp0 show acceptable agreement with passive data for various arterial sizes [7, 65, 66, 70].

An example is shown in Fig 7.1, where data were obtained from Bund et al [7] on femoral arterioles with passive diameters equivalent to renal vascular order i = 4 (d0=172 µm). The minor discrepancies between

the model predictions and the data for smaller diameter values is likely due to the experimental protocol, where passive conditions were simulated using a Ca2+-free solution. Here, the VSM activation is taken to be zero, representing solely the passive component of wall tension. However, although the extracellular calcium has been abolished, the contribution to VSM activation from intracellular calcium sources is neglected. Therefore, the experimental values of passive tension at small diameters are slightly overestimated and most likely represent a small active contribution as well.

Figure 7.1: Passive tension data [7] and model results, i = 4 (d0=172 µm)

An optimization procedure to determine best fit with active pressure data [10, 31, 37, 48, 52, 54, 121] provided values for Cai, Cti, and C

0

ti. The peak magnitude of maximally active tension, Cai, was found

to exhibit a strong positive correlation with increasing diameter, due to the increased wall thickness and therefore increased number of force-generating smooth muscle cells. The values of C_a0

i and C

00

ai were deter-

mined by Carlson and Secomb [9] to be the averages obtained from length-tension experiments performed on vessels from 50 µm to 300 µm in diameter, signifying that maximum active tension is reached at around 91% of the passive anatomical diameter. The parameter Cti, characterizing the dependence of vascular tone

on circumferential wall tension, showed an inverse relationship with diameter, exhibiting a strong increase in smaller arterioles, while C0

ti did not exhibit significant diameter dependence.

The vessels experiencing the most variable shear stresses were also the orders known to be most responsive to flow, resulting in smaller CN Oivalues for the small arteries and large arterioles. The sigmoidal dependence

on shear stress for each of the intermediate Strahler orders had a more gradual slope, and φN Oi is capable of

changing over a larger range of shear stress values, as shown in Figure 7.2. The parameter C_{N O}0

i determines

the midpoint of the curve, and did not show significant correlation with diameter, depending instead on the typical shear stress values for each level.

Figure 7.2: NO activation: dependence on shear stress for vascular orders: i = 2 (d0=385 µm), i = 5 (d0=108

µm) and i = 9 (d0=40 µm).

The relationship between shear stress and φN Oi is also determined by gi which varies from 1, allowing

for a halving in sensitivity of the contractile apparatus in the case of full eNOS activation, to 0.2, which only permits up to a 16.7% decrease in sensitivity. The large arterioles most sensitive to changes in flow have higher gi values, while those with diameters above and below this range have lower values. The parameter

determining the time constant of VSM activation, ta, has a positive correlation with diameter, also due to

the increase in wall thickness and variation in size and number of smooth muscle cells. The fastest responses are seen in the higher Strahler orders corresponding to the interlobular arteries and afferent arterioles, contributing to the distally dominant nature of the autoregulatory response.

Parameter sensitivity analysis was performed by varying each of the parameters individually within a reasonable range and observing the change in outcome of the simulation. Those that could not be directly calculated, and were therefore estimated based on the available data, were subject to the most variability and

exhibited the highest sensitivity. Specifically, changes in Cti, C

0

ti, CN Oi, and C

0

N Oi had the greatest impact

on the system response.