The first stage of urine formation is the production of a filtrate of the blood plasma. The glomerulus, the primary filter, is a network of up to 50 parallel branching and anastomosing (rejoining) capillaries covered by epithelial cells and encased by Bowman's capsule. Blood enters the glomerulus by way of the afferent arteriole and leaves through the efferent arteriole. Pressure of the blood in the glomerulus causes the fluid to filter into Bowman's capsule, carrying with the filtrate all the dissolved substances of small molecular weight. The glomerular membrane is almost completely impermeable
to all plasma proteins, the smallest of which is albumin (molecular weight 69,000). As a result, the glomerular filtrate is identical to plasma except that it contains no significant amount of protein.
Figure 20.3 Schematic diagram of the glomerular filtration.
of renal plasma flow that becomes glomerular filtrate and is typically about 20 percent. Over 99 percent of the filtrate is reabsorbed in the tubules, with the remaining small portion passing into the urine.
There are three pressures that affect the rate of glomerular filtration. These are the pressure inside the glomerular capillaries that promote filtration, the pressure inside Bowman's capsule that opposes filtration, and the colloidal osmotic pressure (cf. Chapter 2, Section 2.7) of the plasma proteins inside the capillaries that opposes filtration.
A mathematical model of the glomerular filter can be described simply as follows. We assume that the glomerular capillaries comprise a one-dimensional tube with flow q1 and that the surrounding Bowman's capsule is also effectively a one-dimensional tube with flow q2 (Fig. 20.3). Since the flow across the glomerular capillaries is proportional to the pressure difference across the capillary wall, the rate of change of the flow in the capillary is d1 = Kf (P2 - Pi + nc), (20.1)
where P1 and P2 are the hydrostatic fluid pressures in tubes 1 and 2, respectively, nc is the osmotic pressure of suspended proteins and formed elements of blood, and Kf is the capillary filtration rate. The osmotic pressure of the suspended proteins is given by nc = RTc, (20.2)
where c, the concentration expressed in moles per liter, is a function of x, since the suspension becomes more concentrated as it moves through the glomerulus. Since the large proteins bypass the filter, we have the conservation equation
where ci is the input concentration and Qi is the input flux. It follows that
where ni = RTci is the input osmotic pressure. Since the hydrostatic pressure drop in the glomerulus is small compared to the pressure drop in the efferent and afferent arterioles, we take P1 and P2 to be constants.
Equation (20.1) along with (20.4) gives a first-order differential equation for q1, which is easily solved. Setting q1(L) = Qe we find that
Qi \ 1 _ a / ' aQi where Qe is the efflux through the efferent arterioles, L is the length of the filter, and a = n/(P1 _ P2).
Finally, we assume that the pressures and flow rates are controlled by the input and output arterioles, via
and that the flow out of the glomerulus into the proximal tubule is governed by
where Pa, Pe, and Pd are the afferent arteriole, efferent arteriole, and descending tubule pressures, respectively, and Ra, Re, and Rd are the resistances of the afferent and efferent arterioles and proximal tubule, respectively. Typical values are P1 = 60,P2 = 18,Pa = 100, Pe = 18,Pd = 14 _ 18, n = 25 mm Hg, with Qi = 650, Qd = Qi _ Qe = 125 ml/min.
The flow rates and pressures vary as functions of the arterial pressure. To understand something of this variation, in Fig. 20.4 is shown the renal blood flow rate Qi and the glomerular filtration flow rate as functions of the arterial pressure. It is no surprise that both of these are increasing functions of arterial pressure Pa.
The strategy for numerically computing this curve is as follows: with resistances Ra and Re and pressures Pe, Pd, and ni specified and fixed at typical levels, we pick a value for glomerular filtrate Qd = Qi _ Qe. For this value, we solve (20.5) (using a simple bisection algorithm) to find both Qi and Qe. From these, the corresponding pressures Pa,P1, and P2 are determined from (20.6) and (20.7), and plotted.
For this model, the filtration rate varies substantially as a function of arterial pressure. However, in reality (according to data shown in Fig. 20.5), the glomerular filtration rate remains relatively constant even when the arterial pressure varies between 75 to 160 mm Hg, indicating that there is some autoregulation of the flow rate.
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This ebook provides an introductory explanation of the workings of the human body, with an effort to draw connections between the body systems and explain their interdependencies. A framework for the book is homeostasis and how the body maintains balance within each system. This is intended as a first introduction to physiology for a college-level course.