## Abstract

A transient 1D mathematical model of whole-organ renal autoregulation in the rat is presented, examining the myogenic response on multiple levels of the renal vasculature. Morphological data derived from micro-CT imaging were employed to divide the vasculature via a Strahler ordering scheme. A previously published model of the myogenic response based on wall tension is expanded and adapted to fit the response of each level, corresponding to a distally dominant resistance distribution with the highest contributions localized to the afferent arterioles and interlobular arteries. The mathematical model was further developed to include the effects of in vivo viscosity variation and flow-induced dilation via endothelial nitric oxide production. Computer simulations of the autoregulatory response to pressure perturbations were examined and compared with experimental data. The model supports the hypothesis that change in circumferential wall tension is the catalyst for the myogenic response. The model provides a basis for examining the steady state and transient characteristics of the whole-organ renal myogenic response in the rat, as well as the modulatory influences of metabolic and hemodynamic factors.

- renal
- myogenic response

the myogenic response, a pressure-induced constriction or dilation of the vasculature first discovered 100 years ago by Bayliss, is now known to exist throughout the body (28). It is crucial in establishing basal arterial tone and maintaining a relatively constant blood supply to major organs, a phenomenon known as autoregulation (13). There are two key autoregulatory mechanisms in the kidney, the tubuloglomerular feedback (TGF) loop, which monitors NaCl delivery and filtration rate and adjusts afferent arteriolar resistance accordingly, and the myogenic response, which is the focus of current research and exists to varying degrees throughout the renal vasculature. Together, these two mechanisms account for ∼90% of renal autoregulation, with 50% attributed to the myogenic component (30, 31, 39). The estimation of this regulatory balance is based on an assumed linearity of the system and is subject to variations due to observed nonlinear interactions among mechanisms (41, 42, 62). Other possible mechanisms, with less impact and longer time courses, could be attributed to metabolic catalysts, sympathetic stimulation, or various other causes.

The myogenic response features predominantly in the small arteries and arterioles which make up the majority of peripheral resistance in the vasculature; this adjusts the vascular smooth muscle (VSM) tone, and hence diameter, to provide dynamic regulation of blood flow. The cellular mechanisms by which the myogenic response operates have been extensively studied (13, 23, 26, 50, 52). There is general agreement that an alteration in circumferential wall tension is the catalyst for change in VSM tone, mediated by an influx of extracellular calcium through voltage-gated ion channels and to a lesser extent by integrin recruitment of calcium from intracellular storage spaces such as the endoplasmic reticulum. The mechanism of membrane depolarization and the other signaling pathways and secondary messengers involved in changes in cytosolic calcium concentration are still under debate. The myogenic response is known to operate at 0.1–0.3 Hz, as confirmed by a multitude of direct experimental observations and frequency domain analyses in isolated resistance vessels ranging from 20 to 400 μm and in whole-organ studies (5, 11, 19, 26, 27, 30–32, 39, 45).

While there is evidence that it may be modulated in some part by TGF, the myogenic mechanism has been proven to exist independently by work done on the hydronephrotic kidney, a model known to be devoid of the TGF loop (11, 39). The functional role of the myogenic response is thought to be protection of the renal microvasculature from the damaging effects of high blood pressure, in addition to stabilizing fluctuations in renal blood flow (RBF), glomerular filtration rate (GFR), and filtrate delivery to the distal tubules. In 1980, Johnson (28) was the first to argue the possibility that myogenic constriction in proximal vessels of a vascular bed could limit the pressure elevations transmitted downstream to the smaller, distal vessels. Although the resistance of the renal vasculature is known to be predominantly distal in nature, it is not confined simply to the afferent arterioles, the feed vessels for the nephrons, as is evidenced by the large body of aforementioned work showing active myogenic constriction in vessels up to 400 μm in diameter. Therefore, the hypothesis presented here is that myogenic activation, via the maintenance of basal tone in larger arteries as well as active constriction in smaller arterioles, exerts its influence throughout the arterial structure of the kidney to facilitate autoregulation.

In this paper, a dynamic model of the myogenic response in the renal vasculature is presented. There are numerous mathematical and dynamic models of the myogenic response (8, 9, 38–40, 44), but this is the first that encompasses the entire renal vasculature. In vivo, the myogenic mechanism does not act in isolation, but rather interacts in a complex way with several other processes, such as TGF and cardiac and respiratory oscillations. However, to facilitate a clear modeling focus on the myogenic response, these confounding factors are, at this stage, neglected and a constant mean arterial pressure (MAP) is used as the relevant input variable. Only when the myogenic model has been fully investigated and validated can the effects of including these other factors be systematically addressed. The recently published model for isolated vessels by Carlson and Secomb (4) in 2005 uses circumferential wall tension as the variable determining the level of VSM tone, and this provides the basic framework for the expanded model presented in the current study. Other factors included in the system are the effects of flow-induced dilation via nitric oxide (NO) production, viscosity variation in the microvasculature, and the series coupling of different levels of the renal vasculature.

## METHODS

### Arterial Tree Representation

The rat renal vasculature is modeled as 12 compartments in series, where the first 11 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. (8, 9). The resistances of the efferent arterioles, 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. (43) employ information from microcomputer tomographic (CT) scans to divide the arterial structure via a Strahler ordering scheme (57), an algorithm originally developed for stream ordering and found to be highly applicable to vascular branching. 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. For further information regarding the reconstruction process and its robustness quantification via error analysis, the reader is referred elsewhere (43).

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 (51) is neglected at present. Additionally, it has been shown that there exist interactions and coupling phenomena among nephrons due to the TGF (41, 42), which have also been omitted to facilitate a clear focus on modeling and understanding the myogenic response.

Each compartment (*i* = 1, 2, …, 11) is assumed to represent *N*_{i} vessels in parallel, with average length *L*_{i} and diameter *d*_{i}, with the constant resistance of the venous and capillary system lumped in *compartment 12*. 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 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 1 represent the anatomic diameters (*d*_{0i}), defined as the passive value at a 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 anatomic diameters and are dependent on the hemodynamic properties of flow and pressure in the vessel. The Poiseuille relationship among pressure, flow, and resistance is applied throughout the renal vasculature, leading to the expression (1)

The subscript *i* refers to the vascular order, *R*_{i} denotes resistance, and η_{i} is 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 <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. (48, 49) in the rat mesentery vasculature observed this effect and produced an expression for in vivo viscosity variation (2)

Here, η_{i} is the viscosity, with units of centipoise, and *d*_{i} is 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 Pries et al. (49).

The resistance of each unit varies as a function of the diameter and the viscosity, which is diameter dependent itself. The local pressure P_{i} is taken to be the mean of the inlet (P_{in,i}) and outlet (P_{in,i+1}) pressures, where the outlet pressure of *compartment 12* is kept at a constant venous pressure (P_{ven}) 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 *Q*_{i} through each compartment is dependent on the pressure and resistance (3) (4) (5)

The representation of the preglomerular arterial tree corresponds to a distally dominant resistance distribution, with the highest active contributions localized to the arterioles, as shown in Fig. 1. 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 (10, 18, 56) and is not subject to variation. Given the inlet and outlet values, local pressures can be easily calculated from the condition that the flow in all orders must be the same.

### Vessel Model

Autoregulation is achieved by virtue of the capacity for change of the resistance vessels, with the strongest responses localized to small arterioles, but with a measurable contribution provided by the more proximal large arterioles and small arteries (7, 12, 18, 37). This occurs via dilation or constriction in response to alterations in blood pressure, which incite changes in circumferential wall tension. The active myogenic response closely follows an initial passive distension, as has been observed in a multitude of experimental studies (5, 11, 19, 26, 27, 30, 32, 39, 45, 58, 61, 63).

The vessel wall can therefore be treated as a nonlinear spring and contractile unit in parallel (4), representing the passive and active components of the myogenic response, respectively (see also Refs. 13, 39, 44, 50, 52). Passive tension, *T*_{pass}, is dependent on inherent properties of the vessel wall, primarily distensibility, while active tension only develops due to the activation of the VSM cells, and is represented as the maximal active tension, *T*_{act}^{max}, that can possibly be generated at a given circumference multiplied by φ_{MR}, the degree of VSM activation, also referred to as myogenic tone. This leads to an expression for total wall tension, *T*_{tot} (6)

Throughout the system, the subscript *i* refers to the particular Strahler order (*i* = 1, 2, …, 11) of the vascular tree, as all the parameters are unique for each level. As has been demonstrated in experimental length-tension studies, passive tension increases nonlinearly and quite rapidly at high levels of circumferential stretch (4, 13, 36–38), leading to the following exponential form (7) where *d*_{norm} is the diameter *d*_{i} normalized to the anatomic diameter *d*_{0i}, *C*_{Pi} is the passive tension at the anatomic diameter *d*_{0i}, and *C*′_{Pi} is a parameter determining the slope of the exponential curve. Total maximal tension is quantified in length-tension studies by bathing vessels in maximally activating physiological salt solutions (14). The maximally active component of tension is found by subtracting the passive component from the total and has been shown to follow a Gaussian-shaped curve (4), of the form (8) where *C*_{ai} is the peak magnitude of maximal active tension, *C*′_{ai} is the relative peak location, and *C*″_{ai} is the relative curve width. The Gaussian distribution signifies that active tension is at a maximum at a certain circumferential length and decreases symmetrically due to an increase or decrease in diameter. *Equation 8* assumes full VSM activation, and is directly related to the size of the vessel (via *C*_{ai}), as the wall thickness and hence the number and/or size of force-generating smooth muscle cells increase with increasing diameter. The dependence on the diameter *d*_{norm} (again normalized with respect to the anatomic diameter) dictates where along the Gaussian curve the maximally active tension value lies at that moment, while the contribution to the system is determined by the level of VSM tone, φ_{MRi}. *C*′_{ai} and *C*″_{ai} are dimensionless parameters that were found to not vary significantly over a wide range of vessel diameters (4), and therefore for our purposes were assumed to be the same for all *i*. Determination of all other parameters was dependent on the Strahler order to which they apply and is addressed in detail further on.

### Effect of NO

While blood pressure is the primary myogenic stimulus, flow-induced dilation modulated by shear stress and endothelial nitric oxide synthase (eNOS) production has been shown to attenuate increases in VSM tone (23, 29, 32, 36, 37, 46, 63). Furthermore, basal arterial tone is the result of a balance between myogenic activation and continuously synthesized NO released from the endothelium, thought to be particularly crucial in large-resistance vessels (15, 46, 60). A crucial stimulus of NO release is hemodynamic shear stress on the endothelium (29, 36, 37, 60), causing direct phosphorylation leading to mechanical eNOS activation, and more long-term production via eNOS gene transcription. Shear stress for each compartment per unit vessel was calculated using the following equation (2, 35) (9)

NO activates guanylate cyclase in VSM cells to synthesize cGMP, which results in several biological effects (17, 60), in this case the most relevant being a reduction in vascular tone via dilation. The activation factor, φ_{NO}, of this potent vasodilator has a sigmoidal dependence on shear stress (6), ranging from 0, inactive, to 1, fully active eNOS production (10)

This phenomenological expression for eNOS, and subsequently NO, activation cannot be directly physiologically quantified. However, it provides a useful tool for representing the effect of NO on the VSM cells. *C*_{NOi} and *C*′_{NOi} are parameters which determine the characteristics of the shear stress dependence for each vascular level. Smith et al. (55) used a similar approach to model the effects of NO on myogenic reactivity, via a sigmoidal relationship between concentration and percentage relaxation.

### VSM Activation

An analogous expression for the level of VSM activation, or myogenic tone, allows for the incorporation of NO-mediated dilatory effects into the system. Experimental data on renal vessels have shown the major effect of NO to be a decrease in sensitivity of the contractile mechanism (10, 46, 59). Therefore, while the steady-state myogenic activation factor for each vessel order, φ_{MRssi}, has a sigmoidal dependence on circumferential wall tension (4, 61), the slope of this curve, and hence the sensitivity of the dependence, is directly affected by the level of NO activation (11)

An example of the effect of eNOS activation on the relationship between φ_{MRss} and wall tension for vessel order *i* = 6 (*d*_{anat} = 88 μm) is shown in Fig. 2. As φ_{NO} ranges from virtually inactive (0.01) to partially active (0.5) to fully active (1), the rightward shift of the curve and decrease in slope represent a marked reduction in sensitivity of the contractile apparatus; stated a different way, a significantly greater increase in circumferential wall tension is required to elicit a corresponding change in VSM activation and vascular tone. The parameter α_{i} is also order specific, ranges from 0.2 to 1, and represents the relative influence of eNOS activation, a direct consequence of the varying responsiveness to flow throughout the arterial tree. A value of α = 1 allows for a potential halving in sensitivity of the contractile mechanism, as is the case in Fig. 2.

At a given arterial pressure, the diameter at each level of the renal vasculature adjusts to achieve a balance between actual tension, as determined by the law of Laplace, and the summation of the passive and active components (*Eq. 6*), leading to the following expression. (12)

This diameter response depends on the viscosity and inertia of the blood, the vessel wall, and the surrounding tissue, and is assumed to be instantaneous, as the time constant has been shown to be negligible compared with that involved in the development of active tension in the VSM cells (14). To model the dynamic behavior of the network, the actual VSM activation is treated as a time-dependent variable and satisfies the first-order dynamic equation (13)

The time constant *t*_{ai} determines the rate of change of VSM activation and varies throughout the vasculature. This system functions such that wall tension is controlled by a negative-feedback mechanism (28). An increase in tension due to a pressure jump elicits an increase in tone and a reduction in diameter, which in turn causes a decrease in wall tension by the law of Laplace. In addition, due to the pressure increase, flow increases, resulting in greater shear stress and eNOS activation, which modulates the diameter response. The reverse holds true in response to a decrease in MAP, effecting a delicate balance between myogenic and dilatory responses. A control diagram of the mathematical system is shown in Fig. 3. The inclusion of the TGF mechanism would affect the diameter response, the VSM activation, and the shear stress in the distal afferent arteriole.

### Parameter Determination

For the 11 Strahler orders that represent the renal arterial tree to the distal afferent arteriole, the parameters that determine the passive and active vessel responses to stimuli such as pressure, flow, wall tension, and shear stress are *C*_{pi}, *C*′_{pi}, *C*_{ai}, *C*′_{ai}, *C*″_{ai}, *C*_{ti}, *C*′_{ti}, *C*_{NOi}, *C*′_{NOi}, α_{i}, and *t*_{ai}. A literature review of the relevant experimental papers from the last 50 years provided pressure-response data on renal vessels ranging in diameter from 15 to 500 μm (5, 16, 19, 22, 26, 27, 29–32, 39, 45, 49, 50, 58, 64), which were used to estimate the corresponding parameters for each vascular level. When possible, preferential consideration was given to data obtained in vivo as opposed to in vitro, from rats rather than dogs, rabbits, or other animals, and from untreated vessels under free-flow conditions.

For each vascular order (*i* = 1, 2, …, 11) passive tension at *d*_{i} = *d*_{0i}, evaluated at MAP = 100 mmHg using Laplace's law, yields *C*_{pi}. This MAP was applied to the system at the level of the renal artery, and the subsequent pressure drops over each vascular order were calculated to yield the corresponding values for *C*_{pi}. The parameter in the exponential relationship between diameter and passive tension was estimated based on the work of Liao and Kuo (36, 38) and Cornelissen et al. (8, 9) in the coronary vasculature and used to estimate the parameter *C*′_{pi}. Their passive pressure-diameter relationship, shown below, was based on empirical data from arteries that fall into the range of *d*_{0} = 70–100 μm, encompassing intermediate and large arterioles and small arteries (14)

Tension was computed via Laplace's law and plotted against the ratio of the passive and anatomic diameters. They found that the distribution of data points fits an exponential curve with a value of 14.87 for the coefficient in the argument of the exponential (36, 38). Referring to *Eq. 7*, this value for *C*′_{pi} was applied to the Strahler orders within that range of diameters (*i* = 4, 5, 6, 7). This representation of the steepness of the passive response curve, which provides an indication of the distensibility of the vessel, has been shown to exhibit a negative trend in response to increasing diameter (4). For this reason, a slightly smaller value was assigned to the first three orders, representing the feeding renal and interlobar arteries, and, similarly, increasing values assigned to the smaller arterioles. These passive parameter values showed good agreement to pressure-diameter data obtained from wire myographic studies (3, 36–38).

The active parameters *C*_{ai}, *C*′_{ai}, *C*″_{ai}, *C*_{ti}, and *C*′_{ti} were determined for best fit to the available active pressure data as follows. The parameters for peak location, *C*′_{ai}, and range, *C*″_{ai}, of force generation were found not to exhibit significant dependence on the reference diameter and were thus taken to be the same for all vessel orders (4). However, *C*_{ai} exhibited significant variation over the orders due to the wide range of tension values throughout the vasculature. This peak value was defined as the active tension present when φ_{MR} = 1, *d*_{i} = *C*′_{ai}*d*_{0i}, and P = 180 mmHg, found by computing the total tension and subtracting the passive component. Various studies (11–13, 64) have shown that this pressure value approximates the upper bound of the autoregulatory range and therefore corresponds to the maximum tension possible. In contrast to the parameter determination for the passive behavior, this maximum pressure was applied uniformly at each Strahler order, rather than the renal artery, as this more accurately represents the experimental scenarios where the vessel is bathed in a maximally activating solution and forced to contract, providing active tension data.

The active data (5, 19, 22, 26, 27, 29, 64) were subsequently examined to determine the shape and position of the sigmoidal dependence between tension and VSM activation, providing values for *C*_{ti} and *C*′_{ti}. An iterative bisection method was used for each Strahler order to determine the sigmoidal relationships between φ_{MRss} and *T*_{tot} that, when incorporated into the mathematical system, minimized the error in vessel diameter responses between the model predictions and the experimental results (5, 19, 22, 26, 27, 29, 64). These active data also produced values for the time constant of VSM activation, *t*_{ai}, which shows a positive trend in response to increasing diameter. This means that the larger the artery, and hence the thicker the wall, the longer the delay in development of active tension. In addition, the parameter data from the model of Carlson and Secomb (4) provided guidelines and a more comprehensive overview of the varying smooth muscle response in vessels of differing sizes and species.

The same procedure was followed using the available data on flow-dependent dilation (2, 15, 24, 29, 32, 35–38, 46, 47, 63) to determine the relationship among flow, shear stress, and eNOS activation, φ_{NOi}. The parameters *C*_{NOi} and *C*′_{NOi} were estimated for the optimal fit to the data, by firstly determining the range of shear stress values experienced by each vessel, and that value which would correspond to φ_{NOi} = 0.5. Again, an iterative method was used to adjust *C*_{NOi} and *C*′_{NOi} to obtain the sigmoidal dependences between shear stress and eNOS activation for each vascular order that minimized the deviation between simulated and experimental diameter responses. The parameters were also evaluated in accordance with the observation that, for the majority of vascular levels, the magnitude of flow-induced dilation is usually highest at resting levels of vascular tone (46), highlighting the important contribution of this mechanism to the balance that maintains resting diameter. This influence of φ_{NOi} on the sensitivity of the contractile mechanism is governed by the gain α_{i}, which varies between 0 and 1 and is assigned based on the relative responsiveness to flow of arteries and arterioles of differing diameters (36, 37, 54).

### Numerical Methods

The system encompassed by *Eqs. 1*–*5* and *9*–*13* was solved using the math software package MATLAB. A 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 *Eq. 12* within an inner loop for a given level of VSM activation, φ_{MRi}. The time dynamics are incorporated via *Eqs. 11* and *13*, as the activation for each order adjusts over varying time scales to the steady-state value for a given circumferential tension, *T*_{toti}, and eNOS activation level, φ_{NOi}. 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 first based on achieving a balance between Laplacian and actual circumferential wall tension, and, second, on the flow in all compartments being equal, both with a tolerance of 10^{−6}. This convergence was typically achieved in <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 5 min per simulation. The MATLAB script is available upon request from the corresponding author.

## RESULTS

### Model Parameters

The values for the 11 parameters that govern the system, specific to each of the 11 Strahler orders, are shown in Table 2.

The passive tension at a MAP of 100 mmHg was calculated according to the law of Laplace, beginning in the renal artery. The baseline resistance of the branching structure of the preafferent arteriole renal vasculature (*i* = 1, 2, …, 9) was calculated using *Eq. 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 *C*_{p} and *C*′_{p} show acceptable agreement with passive data for various arterial sizes (3, 36–38). An example is shown in Fig. 4, where data were obtained from Bund et al. (3) on femoral arterioles with passive diameters equivalent to renal vascular order *i* = 4 (*d*_{0} = 172 μm).

An optimization procedure to determine best fit with active pressure data (5, 19, 22, 26, 27, 29, 64) provided values for *C*_{ai}, *C*_{ti}, and *C*′_{ti}. The peak magnitude of maximally active tension, *C*_{ai}, 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*′_{ai} and *C*″_{ai} were determined by Carlson and Secomb (4) to be the averages obtained from length-tension experiments performed on vessels from 50 to 300 μm in diameter, signifying that maximum active tension is reached at ∼91% of the passive anatomic diameter. The parameter *C*_{ti}, characterizing the dependence of vascular tone on circumferential wall tension, showed an inverse relationship with diameter, exhibiting a strong increase in smaller arterioles, while *C*′_{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 *C*_{NOi} values 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 φ_{NOi} is capable of changing over a larger range of shear stress values, as shown in Fig. 5. The parameter *C*_{NOi} 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.

The relationship between shear stress and φ_{NOi} was also determined by α_{i}, 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 α_{i} values, while those with diameters above and below this range have lower values. The parameter determining the time constant of VSM activation, *t*_{a}, 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 *C*_{ti}, *C*′_{ti}, *C*_{NOi}, and *C*′_{NOi} had the greatest impact on the system response.

### Active Response

Extensive comparison with experimental data is difficult, as this system mimics the functioning of the myogenic response in a fully intact kidney in vivo, without the influence of the TGF, a scenario that is challenging to reproduce in the laboratory. Most available renal data are obtained in vitro or under sedation, both factors that have a large influence on the dynamics and magnitude of the response. In addition, data are often unattainable throughout the vasculature of the kidney, due to the difficulties that arise when one attempts to measure the response of intermediate vessels that are embedded in the tissue and are not readily accessible. However, the limited data available show good agreement with the model results, especially concerning the qualitative trends of the myogenic response. To obtain results that lend themselves to comparison, the full model simulations were run and the diameter values were recorded for one particular Strahler order. In Fig. 6, both the dynamic and the steady-state behavior of a single vessel order in response to multiple pressure step increases are shown, and the steady-state values were compared with data from Takenaka et al. (58). This group examined the steady-state myogenic response of rat interlobular arteries in the hydronephrotic kidney, a preparation known to be devoid of the TGF. Following the method used by the experimentalists, the diameter values presented here were normalized with respect to the diameter at a local pressure of 80 mmHg. To prevent bias, these and all other data used for experimental comparison were not used in the parameter determination procedure.

The curves for the diameter response of every level in the series organization of the renal vasculature are shown in Fig. 7, *C* and *D*, as well as the myogenic activation, shown in Fig. 7*B*, in response to pressure step increases from 80 to 120 to 160 mmHg (*inset*, Fig. 7*A*). Also shown is the partial autoregulation of renal blood flow (Fig. 7*A*), which initially spikes due to the increase in pressure and is restabilized via the myogenic response of the renal vasculature.

Here, the larger vessels, while maintaining a certain level of tone, dilate in response to increases in pressure. The varying time scales are evident in the graph, as larger vessels require more time for activation than the smaller arterioles. The passive distensive response is taken to be instantaneous for all Strahler orders.

### Steady-State Whole-Organ Response

Another significant result of the model simulations is the overall renal autoregulation curve, demonstrating the ability of the kidney to stabilize renal blood flow under a wide range of mean arterial pressures. A fully functioning, healthy, intact kidney would produce a waveform similar to the idealized curve shown in Fig. 8*A*. However, one would expect less than perfect autoregulation from a system that solely exhibits a myogenic response, as shown in Fig. 8*B*. Pires et al. (45) performed experiments to determine spontaneous renal blood flow autoregulation curves in conscious sinoaortic baroreceptor-denervated rats, inhibiting the TGF with furosemide. This is known to cause near-complete suppression of the TGF response (30, 32, 45, 53, 55, 63).

The reported flow data for both the experimental and the model results have been normalized with respect to the flow value at a MAP of 70 mmHg to facilitate comparison. Both curves show a nonzero gradient within the autoregulatory range that is qualitatively similar. The percentage difference between model predictions and experimental data ranges from 0.38% at 100 mmHg, to 4.06% at 120 mmHg, with data for all other pressure values falling within this range of deviation. These differences fall well within the reported bounds of experimental error, and the simulated model results are therefore judged to be in close agreement with these experimental data, which are a representation of the autoregulatory mechanism resulting from a uniquely myogenic response.

### Inclusion of NO

The necessity of including the effects of NO is demonstrated by the diameter response of a single vessel order, *i* = 4, *d*_{0} = 172 μm, shown with and without the inclusion of shear stress-induced eNOS production and subsequent NO-mediated dilation, via *Eqs. 9*–*11* (Fig. 9). Pressure was increased in 20-mmHg increments from 40 to 100 mmHg. Holding the parameter φ_{NO} to 0 mimics the suppression of eNOS activation; in an experimental scenario, this is achieved with an inhibitor of NOS such as *N*^{G}-nitro-l-arginine methyl ester. Strahler *order 4* was chosen for illustrative purposes as it is within the range of vessels that exhibit the greatest responsiveness to changes in shear stress (36, 37, 38). The qualitative nature of the differing response between eNOS activation and suppression is representative of that for all orders, although the quantitative difference is smaller for other orders.

When NOS was inhibited, the baseline diameter of the vessel showed a marked decrease, indicating a significant contribution of NO to resting vascular tone. Additionally, in response to increases in MAP the artery exhibited sustained tone and constriction until the VSM cells were fully active and the vessel was forced to dilate in response to continued pressure increases. In contrast, the complete system, including the effects of NO, behaved as one would expect for this portion of the physiological range of blood pressure. The vessels undergo an initial passive dilation in response to the pressure step, followed by a new steady-state diameter value. Examining the steady-state values, the vessel dilated by 2.9% as MAP approached typical resting values and finally constricted by 4.1% as pressure continued to rise.

## DISCUSSION

Modeling the myogenic response throughout the kidney demonstrates the necessity of considering the entire vasculature when examining renal autoregulation. Although the capacity for resistance change, and therefore impact on blood flow stabilization, is greatest in the small vessels, it is not solely confined to the afferent arterioles as many models assume (39, 44, 50). The lower-order vessels with larger diameters are not capable of active myogenic constriction. However, they maintain a certain level of vascular tone which is required in a comprehensive renal model. Ultimately, this will allow extrapolation from normal baseline conditions to pathological scenarios ranging from hypertension to acute renal failure.

### Dynamic Response

The theoretical model for the myogenic response in an isolated vessel based on length-tension characteristics of vascular smooth muscle presented by Carlson and Secomb (4) in 2005 was designed to provide a foundation upon which to build a more comprehensive model of autoregulation, such as the current whole-organ renal model. The close fit of experimental data to our model supports the hypothesis that circumferential wall tension is the primary factor governing VSM activation, and expands upon this to account for modulation by NO and viscosity variation in the microvasculature. Furthermore, the results are consistent with the well-established proposal that flow, and subsequently shear stress, are important variables that cause eNOS activation and NO production, resulting in indirect dilation via a reduction in sensitivity of the contractile apparatus.

The passive behavior of the vessels is well represented by the mathematical model, as shown in Fig. 4. One potential reason for the discrepancy between the model prediction and the data for smaller diameter values is the experimental protocol, which simulated passive conditions using a Ca^{2+}-free solution. In this setting, the VSM activation was 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. It is therefore possible that the experimental values of passive tension at small diameters are slightly overestimated and represent a small active contribution as well. Confirming this hypothesis requires further experimental investigation.

The development of active tension in the VSM cells follows a different time course for each Strahler order. At normal physiological values, the lower orders are already significantly active and exist in a state of partial constriction (13, 25). In response to an increase in pressure, these larger vessels cannot change as drastically or as quickly as the smaller vessels, represented by the higher Strahler orders. Examining the diameter response to step increases in pressure throughout the vasculature reveals that the larger arteries are ultimately only capable of dilating, and thus decreasing resistance, in response to increased pressure, while the smaller arterioles exhibit active myogenic constriction, and are responsible for partially stabilizing renal blood flow. However, all the vessels exhibit varying degrees of myogenic reactivity via the maintenance of vascular tone, without which the diameter response would be a purely passive distension.

Model results for inhibition of NO in a single vessel presented in Fig. 9*B* demonstrate the exaggerated constrictor response of the vessel that is also seen experimentally (17, 32, 59). The response was examined and contrasted with one which includes NO over the lower part of the physiological pressure range (40 → 100 mmHg), as this is where NO concentration will vary most due to the large dependence of shear stress on diameter. The diameter changes most significantly over this segment of the pressure spectrum, producing the most variable shear stress and eNOS production.

The sigmoidal dependence between wall tension and VSM activation, and additionally between shear stress and eNOS activation, is at the heart of the behavior of the resistance vessels, and provides a physiologically based model of the autoregulatory response. The nature of the chemical cascade occurring in vascular smooth muscle is such that there exist lower and upper bounds of myogenic activation, between which there is a smooth transition whose characteristics depend on the physical attributes of the vessel and the availability of calcium, NO, and other ions. The present model shows that a sigmoidal variation in myogenic tone as a function of wall tension is consistent with the myogenic responses of a wide range of vessels observed experimentally. Similarly, fluid shear stress is known to be one of the most important stimuli with regard to NO production. The relationship between eNOS activation and shear stress follows the same sort of sigmoidal curve, as has been shown previously (1, 35). This negative feedback mechanism to induce dilation reduces shear stress levels, preventing additional energy expenditure and work for the cardiovascular system.

### Whole-Organ Autoregulation

The system exhibits partial autoregulation over a wide range of pressures (Fig. 9*B*), with a non-zero gradient of flow with respect to pressure in the autoregulatory range due to the omission of TGF from the model. This corresponds well to data from the experimental scenario that most closely simulates the condition of purely myogenic renal autoregulation (45). To prevent bias, this data set was not used to determine the parameters used in the mathematical system. The range of autoregulation was from 70 to 150 mmHg for the model results; the experimental data appear to follow the same trend, although this observation is limited by the lack of data for pressures >120 mmHg. The closeness of fit between experimental observations and model predictions suggests that NO may be one of the most important metabolites to consider in an examination of renal autoregulation. Another possibility is that the term φ_{NO}, taken here to be eNOS activation, actually encompasses the effects of prostaglandins and endothelium-derived hyperpolarizing factor as well. Both have been shown to be stimulated and released by changes in flow and shear stress (35), the key variables in *Eqs. 9* and *10*, and both influence renal autoregulation in the intact rat kidney in vitro (21). Renal NO production is also known to be severely affected in the early stages of diabetes mellitus and diabetic nephropathy (33, 34), as well as acute renal failure (20). Hence, a comprehensive model of autoregulation capable of representing diseased states requires this crucial component.

The autoregulatory range exhibited by the model appears to be shifted toward the lower end of that seen in perfect autoregulation. Additionally, the active higher order vessels exhibit damped oscillatory behavior (Fig. 8*D*) following the passive response, in contrast to the lower order arteries with larger diameters. This trend corresponds well with experimental observation (31, 32, 53, 63, 64). The dynamics of the response will also become more complex with the inclusion of oscillations induced by the cardiac and respiratory cycles and by the TGF. The existing oscillatory behavior, although small and highly damped, is due to interaction between flow-induced dilation, pressure-induced constriction, and (to a lesser extent) resistance change due to viscosity variation.

### Model Limitations

The assumption of Poiseuille flow has been widely accepted by the modeling community as appropriate and applicable to the renal vasculature, as are the use of MAP and nonpulsatile flow (13) for the sake of simplicity in mathematical models, although some groups have found systolic pressure to be an equally important stimulus (39). The series representation of the arterial structure assumes homogeneity of length, diameter, and wall thickness among vessels of the same order, an obvious simplification. However, this assumption is not only necessary for modeling purposes, it is particularly suitable for the kidney as this organ has no anastomoses between vessels. The renal vasculature is not a simple bifurcating tree, and it has been shown in dissection studies that 20–40 glomeruli originate from the same feed vessel. Nevertheless, the Strahler ordering scheme used by Nordsletten et al. (43) is a suitable description for the purposes of the model, as assumptions about the relationship between parent and daughter vessels of adjacent orders are unnecessary. The omission of nephron-nephron interaction, which adds complexity to the dynamics of the autoregulatory response (41, 42), is a related simplification. However, this effect is thought to arise from the synchronization of TGF oscillations and this model focuses solely on the myogenic response.

Blood pressure is easily controlled by the clinician or experimentalist and is therefore treated as an independent quantity. Due to the incomplete understanding of the cellular mechanics, this model and others (4, 8, 9, 12, 38, 40, 50) contain empirical elements to describe steady-state or reference properties, such as those that determine the shape of the sigmoidal dependence between circumferential wall tension and VSM activation or shear stress and eNOS activation. These parameters constitute modeling assumptions as activation factors that do not represent quantitative physiological entities, but rather qualitative effects.

The complexity of vascular regulation necessitates that the principal mechanisms be identified and modeled to a level that will facilitate a more holistic approach. Here, the focus lies on the pressure-induced myogenic response via VSM activation and the flow-induced dilation via eNOS activation. There are other crucial mechanisms, such as the renin-angiotensin system, baroreceptors, and the sympathetic nervous system, which form feedback loops to regulate blood pressure. The major autoregulatory mechanism unique to the kidney and absent from the current model is the TGF. The inclusion of this control loop via a system of partial differential equations that represent the macula densa cells measuring the NaCl concentration of the filtrate and sending feedback signals to the distal afferent arteriole would be expected to have a significant impact on the dynamics of the response, as well as temper the increase in shear stress. Additionally, the synchronization of the TGF oscillations among neighboring nephrons would add further complexities to the model. The current model could be extended to include these factors, and eventually applied to the human kidney as more accurate vascular imaging techniques and autoregulatory response data become available.

### Summary

The model results demonstrate good agreement with available experimental data and provide a valuable tool for examining the whole-organ myogenic response in the rat kidney. Pressure-induced myogenic activation functions in conjunction with changes in viscosity and flow-induced dilation to partially autoregulate renal blood flow. Differential myogenic and shear stress responsiveness throughout the renal circulation allows the mathematical model to accurately mimic the pressure- and flow-induced changes in the vasculature responsible for maintaining a relatively constant level of blood flow to the kidney.

## GRANTS

Financial support for N. Kleinstreuer is provided by the University of Canterbury Targeted Doctoral Scholarship.

## Acknowledgments

The authors thank Tim Secomb and Brian Carlson for previous work and discussion, David Nordsletten for micro-CT data and images of the rat renal vasculature, and two anonymous referees for comments that greatly improved the paper.

## Footnotes

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “

*advertisement*” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

- Copyright © 2008 the American Physiological Society