A mathematical model was used to investigate the filter properties of the thick ascending limb (TAL), that is, the response of TAL luminal NaCl concentration to oscillations in tubular fluid flow. For the special case of no transtubular NaCl backleak and for spatially homogeneous transport parameters, the model predicts that NaCl concentration in intratubular fluid at each location along the TAL depends only on the fluid transit time up the TAL to that location. This exact mathematical result has four important consequences: 1) when a sinusoidal component is added to steady-state TAL flow, the NaCl concentration at the macula densa (MD) undergoes oscillations that are bounded by a range interval envelope with magnitude that decreases as a function of oscillatory frequency; 2) the frequency response within the range envelope exhibits nodes at those frequencies where the oscillatory flow has a transit time to the MD that equals the steady-state fluid transit time (this nodal structure arises from the establishment of standing waves in luminal concentration, relative to the steady-state concentration profile, along the length of the TAL); 3) for any dynamically changing but positive TAL flow rate, the luminal TAL NaCl concentration profile along the TAL decreases monotonically as a function of TAL length; and 4) sinusoidal oscillations in TAL flow, except at nodal frequencies, result in nonsinusoidal oscillations in NaCl concentration at the MD. Numerical calculations that include NaCl backleak exhibit solutions with these same four properties. For parameters in the physiological range, the first few nodes in the frequency response curve are separated by antinodes of significant amplitude, and the nodes arise at frequencies well below the frequency of respiration in rat. Therefore, the nodal structure and nonsinusoidal oscillations should be detectable in experiments, and they may influence the dynamic behavior of the tubuloglomerular feedback system.
Steady-state response of constant coefficient discrete-time differential systems