Biology is defined as the science of life or living matter in all its forms and phenomena, especially concerning the origin, growth, reproduction, structure, and function (Webster’s 149). To study the wide array of topics in biology it is quite necessary to use mathematics. This common use of mathematics is seen in the many subfields of biology including physiology, cardiology, genetics, and hemodynamics just to name a few. Because biology is such a broad field, there will be a look at how mathematics is specifically involved in renal physiology or kidney function.

Before beginning a discussion of mathematics in the physiology of the kidney it is important to discuss the general anatomy and its basic function. This will make it easier to understand the use of mathematics in this study. The kidneys are paired organs that lie in the back of the abdominal cavity, one on each side of the vertebral column, slightly above the waistline. Each kidney is supplied by a renal artery and a renal vein, which enter and leave the kidney at the indentation that gives the kidney its "bean-like" shape (See Diagram below). (Purves 1020).

Each kidney is composed of about 1 million microscopic units referred to as nephrons. The functional unit is the smallest unit within an organ capable of performing all of the organ’s functions. Every nephron consists of a vascular component and a tubular component, both of which are intimately related structurally and functionally. The major part of the vascular component is referred to as the glomerulus (a ball-like structure of capillaries). The glomerulus serves to filter the blood which passes through. This filtered blood then passes through the tubular component of the nephrons (proximal tubule, Loop of Henle, and distal tubule) where it becomes modified by various transport processes that eventually convert it into urine. (Sherwood 470-471)

Let us consider how much water is taken in and eliminated from the human body on a daily basis when considering an average individual of 150 pounds.

2550 ml/day

Output: Insensible loss (skin and lung) 900 ml/day

Sweat 50 ml/day

In Feces 100 ml/day

+ Urine 1500 ml/day

2550 ml/day

It is apparent from looking at these two values, that the amount of water taken into the body and that which is eliminated is equal. So, the equation: Input = Output can be used. Another thing to consider here is the amount of fluid that makes up the human body. Sixty percent of body weight is made up of water. So, in looking at our average human being who is 150 lbs or 70 kg, we can calculate the total volume of fluids in the body.

Calculation: 70kg x 0.6 = 42 L

This 42L of fluid can be looked at more closely in order to determine the amounts found extracellularly and intracellularly. Without calculation, 33% (14L) is found outside the cells (extracellular) and 67% (28L) is found to be within the cells (intracellular). (Purves 1022)

Now it is time to look at glomerular filtration and how mathematics is involved in this part of renal physiology. There is a significantly high pressure in the glomerular capillaries when compared to capillaries throughout the body – it is 55mmHg in the glomerulus versus 15-35mmHg found elsewhere. High pressures such as this favor filtration. The following is Starling’s Law for filtration:

m = fluid movement

K = filtration coefficient (varies in different organs)

π = osmotic forces

P_t = Pressure in Bowman’s capsule = 15 mmHg

P_c = Pressure in glomerular capillary = 55 mmHg

π_c = Osmotic pressure in capillaries due to proteins = 30 mmHg

π_t = Osmotic Pressure in Bowman’s capsule = 0 mmHg (no proteins)

Therefore, the fluid movement through the glomerular capillaries is:

Calculation:

m = (55-15) – (30 – 0) = + 10 net filtration pressure

This is a net filtration that favors filtration, forcing large volumes of fluid from the blood through the highly permeable glomerular membrane. From knowing this value, the plasma volume that is filtered by the kidneys can be determined. Scientists know that 180 L/day (45 gallons, 450 lbs) of fluid is filtered by the kidney. There is an equation for the calculation of this GFR. A study was done using inulin, a sugar that is neither absorbed nor secreted. The inulin was infused into the blood at a specific rate and concentration. The individual’s urine was collected over a time and analyzed. This enabled scientists to study the filtration rate in the kidney.

GFR = Glomerular Filtration Rate

Pl_inulin = Concentration of inulin in the plasma = 1 mg/ml

U_inulin = Concentration of inulin in the urine = 125 mg/ml

V_urine = Volume of urine = 1 ml/min

GFR = 1ml/min x 125mg/ml

1mg/ml

GFR = 125 ml/min

125ml/min x 60min/hr x 24hr/day = 180 L/day

The total volume of plasma filtered by other capillaries can also be calculated given that the human body contains approximately 5 L of blood.

This means, that the kidney filters the entire plasma volume 60 times/day. This is found simply through division:

Calculation:

(180L/day)/(3L) = 60 times/day

The urine output, however, is not so large. The range is anywhere from 1.5 L to 2.0 L per day. (Sherwood 478-480)

It is important to know that the blood flow through the glomerulus is highly regulated. The regulation is primarily due to the changing of the radius of the afferent arteriole leading into the glomerulus. This follows from Poiseuille’s Law:

GFR = Blood flow

D P = Change in Pressure

p = Osmotic Pressure

r = Radius of blood vessel considering

l = Length of the blood vessel considering

h = Viscosity of the fluid

R = Resistance

It is quite obvious from this equation, that the radius of the blood vessel is very important in the regulation of the blood flow into the nephron. By having the radius ( r) raised to the fourth power it makes the quantity very large, thus affecting the rate of flow tremendously. For example, if there is an increase in arterial pressure, with all other factors remaining constant, the glomerular filtration rate would increase due to the direct relationship. Similarly, if there is a fall in the arterial pressure, there will be a subsequent fall in the glomerular filtration rate. The mechanism by which this works is rather interesting. When the arterial pressure becomes increased, it causes the response of constriction on the afferent arteriole. As a result, the blood is able to be filtrated by the nephron efficiently. In contrast, when the pressure is reduced, the afferent arteriole dilates, thus creating "a pool" of blood. This is the reason why the glomerular filtration decreases. (Sherwood 478- 482)

Mathematics is important in renal physiology in another way – in tubular flow through the rest of the nephron. The important thing to consider here, is not so much direct calculation, but rather concentrations and moving of substances in and out of the tubule. The filtered fluid from the glomerular filtration contains nutrients, electrolytes, and other substances that the body cannot afford to lose in the urine. It is important that the necessary materials that are filtered are reabsorbed and returned to the blood. (Sherwood 482)

Tubular reabsorption is a highly selective process. All constituents except plasma proteins are at the same concentration in the glomerular filtrate as the plasma. In general terms, the tubules have a high reabsorptive capacity for substances needed by the body and little or no reabsorptive capacity for substances of little or no value. That which is of little or no value is referred to as wastes and are eliminated in the urine. As these "wastes" remain in the tubule while useful substances are being absorbed, the tubular fluid becomes highly concentrated. (Sherwood 482)

There have been many studies that calculated the percentages of the substances that get reabsorbed by the body. The tubules typically reabsorb 99% of the filtered water, 100% of the sugar, 100% if the amino acids, and 99.5% of the filtered salt. The flow of salt and water flows through the tubule in specific ways in order to concentrate the wastes that need to be removed from the urine. In the proximal tubule, sodium flows out due to the concentration gradient being higher inside the tubule. Water, as a rule always follows the sodium here because of the difference in osmotic potential created by the transport of solutes. When entering the descending loop of Henle, only water moves out of the loop and into the interstitial fluid which is now more concentrated. Once the concentrations of the interstitial fluid and the descending loop are equilibrated, the water flow ceases. At the bottom of the loop of Henle the fluid is found to be most concentrated. As the fluid climbs the ascending part of the loop of Henle, there is a transport of sodium chloride, but this time water does not follow, meaning that the urine is becoming more concentrated. In the distal tubule, water again follows sodium for the same reasons as before, and the fluid passes into the collecting tubule. This fluid becomes more concentrated as it proceeds through the collecting ducts because more water is reabsorbed in the presence of antidiuretic hormone and is released as urine.

It is clear that mathematics is significant in renal physiology. There is a great deal seen in calculating rates of filtration, the amount of blood being filtered, concentrations affecting flows of ions, and much more. This type of mathematical significance is seen throughout the world of biology and in all of the natural sciences. So in reality, math and science work hand in hand as if it were a symbiotic relationship. Without mathematics, scientists would not be able to calculate and determine many of things that are necessary to understanding life.

Purves,William K. Life: The Science of Biology. Sinauer Associates, Inc.: 1995.

Sherwood, Lauralee. Human Physiology: From Cells to Systems. New York:

Wadsworth Publishing Company: 1997.

Webster’s Encyclopedic Unabridged Dictionary of the English Language. New York:

Portland House: 1989.