Dynamics of Nephron-Vascular Network

Abstract

The paper presents a modeling study of the spatial dynamics of a nephro-vascular network consisting of individual nephrons connected via a tree-like vascular branching structure. We focus on the effects of nonlinear mechanisms that are responsible for the formation of synchronous patterns in order to learn about processes not directly amenable to experimentation. We demonstrate that: (i) the nearest nephrons are synchronized in-phase due to a vascular propagated electrical coupling, (ii) the next few branching levels display a formation of phase-shifted patterns due to hemodynamic coupling and mode elimination, and (iii) distantly located areas show asynchronous behavior or, if all nephrons and branches are perfectly identical, an infinitely long transient behavior. These results contribute to the understanding of mechanisms responsible for the highly dynamic and limited synchronization observed among groups of nephrons despite of the fairly strong interaction between the individual units.

Appendix A

Over the years, significant efforts have been made to develop mathematical models that can account for the experimentally observed behavior of nephrons. While early papers (Jensen et al. 1986; Holstein-Rathlou and Marsh 1990; Holstein-Rathlou and Leyssac 1987; Deen et al. 1984) have delivered a reasonably simple model (Barfred et al. 1996), recent studies (Laugesen et al. 2010; Layton et al. 2000, 2009) have been focused on detailed modeling of nephron autoregulation. In the present study, we use a simplified model (Barfred et al. 1996) that captures that main dynamical features of nephron autoregulation.

1.1 A.1 Pressure Variations

The pressure P _t in the proximal tubule of nephron changes in response to differences between the in and outgoing fluid flows:

$$ \dot{P_t}= \bigl[ F_{f}(P_t, P_a,r)-F_\mathrm{reab}-F_{H} \bigr]/C_\mathrm{tub}. $$

(15)

Here, F _f is the glomerular filtration rate, F _reab represents the reabsorption that takes place in the proximal tubule, F _H is the flow of fluid into the loop of Henle, and C _tub is the elastic compliance of the tubule. The Henle flow,

$$ F_{H}=\frac{P_t-P_d}{R_{H}}, $$

(16)

is determined by the difference between the proximal P _t and the distal P _d tubular pressures and by the flow resistance R _H . It gives a good approximation to the experimentally determined pressure-flow relation (Jensen et al. 1986). In this model, the reabsorption F _reab in the proximal tubule and the flow resistance R _H are treated as constants. These processes are accounted for in considerable detail in Holstein-Rathlou and Marsh (1990).

The glomerular filtration rate is given by

$$ F_{f}= ( 1-H_a ) \biggl( 1-\frac{C_a}{C_e} \biggr) \frac{P_a-P_g}{R_a} $$

(17)

as discussed in Holstein-Rathlou and Leyssac (1987). Here, the afferent hematocrit H _a represents the volume fraction of blood cells in blood at the entrance to the glomerular. C _a and C _e are the protein concentrations of the afferent and efferent blood plasma, respectively. R _a is the flow resistance of the afferent arteriole. (P _a −P _g )/R _a determines the incoming blood flow. Multiplied by (1−H _a ) this gives the plasma flow. The factor (1−C _a /C _e ) relates the filtration rate to the change in protein concentration for the plasma in the vessel (Deen et al. 1984). To define C _e , the cubic equation based on experimentally established relation between C _e and glomerular osmotic pressure (Deen et al. 1984) is used. This equation is solved during each time step of numeric simulation (Barfred et al. 1996).

1.2 A.2 Tubuloglomerular Feedback

The effect of changes in the loop of Henle flow can be described by a sigmoidal relation between the muscular activation ψ of the afferent arteriole and the delayed flow through the loop of Henle:

$$ \varPsi=\varPsi_{\max}-\frac{\varPsi_{\max}-\varPsi_{\min}}{1+\exp(\alpha(3X_3/TF_{H_0}-S))}. $$

(18)

Here, Ψ _max and Ψ _min are the upper and lower activation limits of TGF mechanism. α referred to as TGF-feedback is the maximum slope and S the inflection point of the S-shaped curve that characterizes the TGF relation. 3X ₃/T is a delayed flow into the loop of Henle and $F_{H_{0}}$ is a normalized value of the flow through the loop of Henle.

1.3 A.3 Afferent Arteriole

The afferent arteriole is modeled as a damped second-order system

$$ \ddot{r}+ d \dot{r}=\frac{P_\mathrm{av}-P_\mathrm{eq}}{\omega}, $$

(19)

where r is the radius of afferent arteriolar normalized to its resting value. P _av and P _eq are average and equilibrium values of the vascular pressure in the arteriole, respectively. d is a damping coefficient, and ω is a measure of the mass relative to the elastic compliance of the arteriolar wall. The pressure P _av refers to the fact that this pressure is evaluated as the average hydrostatic pressure along the length of the afferent arteriole:

$$ P_\mathrm{av}=\frac{1}{2}\biggl(P_a-(P_a-P_g) \beta\frac{R_{a0}}{R_a}+P_g\biggr) $$

(20)

with the glomerular pressure given by

$$ P_g=P_v+R_e\biggl(\frac{P_a -P_g}{R_a}-F_{f} \biggr). $$

(21)

Equations (20) and (21) express simple linear relations between flow rates and pressure drops in the arteriolar system. P _v is the venous pressure. R _a and R _e are the flow resistances of the afferent and the efferent arterioles, R _a0 is a resting value of R _a . β is the fraction of the total afferent arteriolar length that responds to the TGF signal.

Table 1 Table of constant parameters

Table 2 Table of variable parameters

Analytical calculation of equilibrium pressure P _eq delivers the expression with two integrals that can be solved only numerically. Instead, the following approximation was suggested in Barfred et al. (1996):

$$ P_\mathrm{eq}=1.6(r-1)+ 0.006\exp{10(r-0.8)}+\varPsi\biggl( \frac{4.7}{1+\exp{13(0.4-r)}} +7.2(r+0.9)\biggr). $$

(22)

Here, P _eq is the hydrostatic pressure at which the afferent arteriole is at equilibrium for a given muscular activation Ψ and a given radius r. The first two terms represent the elastic strain properties of arteriole, whereas other terms that are proportional to Ψ represent the active (myogenic) response. Expression (22) gives good approximation within the range of physiologically relevant values of P _eq and r.

1.4 A.4 List of Parameters

The values of dimension parameters were adopted from Jensen et al. (1986) where their origin and physiologically relevant range have been discussed. Dimensionless parameters were adjusted to obtain physiologically relevant behavior.