Fig 1.
Top panel: Main steps of the recursive bifurcating algorithm to create an ABT. Starting with a vessel of a certain diameter, (a) determine the vessel length in accordance with its diameter; (b) estimate diameters and lengths of daughter vessels; (c) apply (b) to the first daughter vessel and continue recursively until Dstop is reached; (d) apply (b) to the last “unprocessed” daughter vessel; (e) apply (b), (c), and (d) until the whole tree is built. Bottom panel: Additional segmentation for kidney specific structure KSABT. (A) Determine initial diameter and length of the vessel; (B) Grow afferent arterioles (instead of daughter vessels); this is accompanied by the diameter reduction of the initial vessel in accordance with Murray’s law; (C) after the entire length of the vessel is processed, estimate the diameter of daughter vessels based on the new (reduced) diameter of the current vessel. Except segmentation, other steps of creating KSABT are the same as of ABT. Detailed description of the algorithm can be found in Materials and Methods.
Fig 2.
Distributions based on the data published by Nordsletten et al. [15] (red) and their approximations (blue).
Left panel: Daughter vessel diameter as a function of the parent vessel diameter. Right panel: Vessel length as a function of the vessel diameter.
Fig 3.
Left panel: Measurements of the distance between neighboring AAs based on the data from cleared renal tissue (blue) and the exponential approximation (red). Experimental data are normalized to the area under the distribution. Right panel: Reconstructed image shows a stack of renal vessels. Afferent arterioles branch from large vessels that continue their branching (glomeruli are depicted in red color). The dimensions of the stack are 850 x 850 x 211 μm. Axis labels show the original coordinates in μm, but note that the tissue shrinks ≈ 20% in each direction during the optical clearing procedure (see Material and Methods). A correction factor for tissue shrinkage was included into the measurements in the left panel.
Fig 4.
A fragment of the vascular structure.
pn, dn1, dn2 and j denote the parent node, the first daughter node, the second daughter node, and the current node, respectively. Each vessel has its own hemodynamic resistance.
Fig 5.
Equivalent electrical circuit of a vessel segment (light gray) located between two branch points.
Branch point conductance is modeled as triangle-shaped coupling Gg1–Gg2–Gg3. A vessel segment consists of several units (dark gray) whose length is equal to the length of an endothelial cell with the conductance Gc and the capacitance Cc. Units are coupled via gap junctions with conductance Gg. Details are given in Materials and Methods.
Fig 6.
Sketch of the main components of the nephron.
Note particularly how the terminal part of the loop of Henle passes within cellular distances of the afferent arteriole, allowing the TGF mechanism to control the incoming blood flow in response to the ionic composition of the fluid leaving the loop of Henle.
Fig 7.
Autoregulation dynamics of the model.
Relative response of afferent the arteriole radius r (left panel) and flow in the efferent arteriole Feff (right panel) on increasing arterial pressure Pa. Well pronounced regulation is observed within the range of 10–13 kPa.
Fig 8.
Simulated vascular structure based on the data published by Nordsletten et al [15] with afferent arterioles only at the top of bifurcating tree (ABT).
Top panel: 3D visualizations of the vascular trees with Dstop = 22μm and Dinitial = 200μm and 70μm (left and right panels, respectively). Bottom panel: Vessel length as a function of the vessel diameter (left) and daughter vessel diameter as a function of the parent vessel diameter (right) for simulations (blue) and data described in Ref [15] (red). Note that the difference between simulated and experimental results for the daughter vessel diameter is related to the fact that we use Murray’s law to calculate the diameter of the second daughter vessel.
Fig 9.
Simulated vascular structure with exponential distribution of afferent arterioles along the tree (KSABT).
Top panel: 3D visualizations of vascular trees with Dstop = 22 μm and Dinitial = 200 μm and 70 μm (left and right panels, respectively). Middle panel: Vessel length as a function of the vessel diameter (left) and daughter vessel diameter as a function of the parent vessel diameter (afferent arterioles excluded from calculations)(right) for simulation (blue) and data described in Ref [15] (red). Bottom panel: Probability distribution of the distances between two neighboring afferent arterioles obtained from simulations (blue) and the approximation to the experimental data (red).
Table 1.
Example of statistical data on afferent arterioles in ABT and KSABT.
Simulations were done at the same parameters Dstop = 22 μm and Dinitial = 530 μm. RA denotes renal artery, AA denotes afferent arteriole.
Fig 10.
Effect of electrotonic interactions.
Left panel: Color coded diagram of the strength of electrical signal propagation along afferent arteriole to the second daughter vessel. Simulations are performed for the vacular network without nephrons (test current was applied to the end of the first daughter vessel and measured at the end of the second daughter vessel). Right panel: Tubular pressure Pt variations of two neighboring nephrons with ≈0.05 (top) and ≈0.1 strength of electrical interaction (bottom).
Fig 11.
Simulation results for ABT (left) and KSABT (right) for Dinitial = 40 μm and Dstop = 22 μm.
Top panel: 3D visualization of simulated structures. Middle panel: Possible tubular pressure values for each nephron as function of the feeding pressure. For the ABT structure, the nephrons are inactive for the whole range of the root feeding pressure, while for the KSABT structure all nephrons are active within a wide range of the root feeding pressure. Bottom panel: Dynamics of tubular pressure Pt variations in all nephrons for pressure in the feeding vessel equal to 10 kPa (∼ 75 mmHg). This pressure level is insufficient for nephrons in the ABT structure to oscillate while for the KSABT structure the nephrons show oscillatory dynamics. Negative values of tubular pressure for the ABT structure are the consequence of insufficient pressure in the afferent arteriole to balance venous and filtration pressure and flow. Tubular pressure shown on the middle and bottom panels is color coded for each nephron according to the color markers on the top panel.
Fig 12.
Comparison of net flow in efferent arterioles and glomerular filtration rate for ABT (left) and KSABT (right) structures shown in Fig 11.
Top panel: Blood flow in efferent arterioles as a function of root pressure. Bottom panel: Glomerular filtration rate as a function of root pressure. Solid black curves (with left axes) show net blood flow and net glomerular filtration rate. Gray curves (with right axes) represent the same characteristics for individual nephro-vascular unit. Dashed black lines highlight the region of the strongest autoregulation.
Fig 13.
The image shows the full kidney stack after reconstruction.
The dimensions of the stack are 850 x 850 x 1018 μm. The axis labels represent the original coordinates in μm, but note that the reconstructed data has shrunk about 20% in all dimensions compared to living tissue.
Fig 14.
Functional diagram of the algorithm.
Black arrows show common steps to create both structures. Green arrows are the main steps of recursively bifurcating algorithm for creation of ABT. Red arrows are the main steps for recursively bifurcating algorithm with additional segmentation for KSABT. Dinitial and Dstop are parameters; Gddp, Gvlvd, Gaad, and Eaad are the distributions obtained from experimental data; Dp, Ddv1, Ddv2 are the diameter of current segment of the parent vessel, and the first and second daughter vessel diameters; L is the length of the current vessel; X is the position of AA origin along the vessel; S is a shift of AA position that accounts for the distance to the neighboring AA located on the other (e.g. parent) vessel; Daff denotes the diameter of the afferent arteriole.
Fig 15.
Schematic presentation of electrical coupling.
A cross-section of a three vessel segment at the branch point with N endothelial cells and gap junction conductances Gg.
Table 2.
Table of constants and parameters.