The exercise is to write and test two Scheme procedures that carry out routine computations relating to human physiology.
Long before birth, the metabolism of a human fetus requires a plentiful supply of oxygen. Fetuses do not breathe; they obtain oxygen from their mothers. In the body of the mother oxygen is carried in the bloodstream, attached to the hemoglobin in the mother's red blood cells. But the blood of the mother and the fetus are normally kept completely separate by the placenta -- the mother's red blood cells are not exchanged with the fetus's. How does the oxygen they contain manage to reach the fetus?
The answer is that although oxygen is bound to hemoglobin while it is carried around the body, at each point there is a continual process of exchange between the hemoglobin and the surrounding environment, in the form of attachment and release of oxygen molecules -- an equilibrium between hemoglobin and unbound oxygen on one hand, and ``oxyhemoglobin'' (hemoglobin with oxygen bound to it) on the other. So when blood containing oxyhemoglobin reaches the placenta, some of the oxygen that it carries is released. (At the same time, some of the unbound oxygen molecules at the placenta are bound -- the reaction can go in either direction -- but since there is more oxygen in the oxyhemoglobin, available to be released, than at the placenta, available to be bound, the net effect is that oxygen is released from the oxyhemoglobin. Also, since carbon dioxide can also attach itself to hemoglobin, the nature of the equilibrium that is reached is affected by the amount of carbon dioxide in the vicinity. However, this fact doesn't invalidate the computation that you'll be working on.)
Once oxygen molecules are released from their attachment to hemoglobin, they tend to migrate through the placenta (which is a membrane that shuts out anything as large as a red blood cell but can be permeated by oxygen molecules). Again, such migrations can occur in either direction, but the partial pressure of oxygen is higher on the mother's side of the placenta than on the fetus's, so more oxygen migrates from the mother to the fetus than in the opposite direction. Once the unbound oxygen molecules are on the fetus's side of the placenta, they can be picked up by the hemoglobin in the fetus's red blood cells and carried (again in the form of oxyhemoglobin) through the fetus's bloodstream to other parts of its body.
For a long time, there was a question about whether enough oxygen could reach the fetus by simple diffusion across the placenta, or whether there are other ways in which the mother's body might push or carry oxygen to the fetus. To resolve this question, physiologists compared the answers to two other quantitative questions: (1) What is the maximum amount of oxygen that the fetus consumes (just before birth, say), in a given amount of time? (2) How much oxygen could diffuse through the placenta in that same amount of time?
To answer the first question, one can take blood samples from umbilical arteries and veins and determine the concentration of oxygen molecules in each, as measured in milliliters of oxygen per hundred milliliters of blood. One can also measure the rate at which blood flows through the umbilical arteries and veins, in milliliters per minute. The fetal oxygen consumption rate is the result of multiplying the rate of flow by the difference between the concentration of oxygen in the fetus's venous blood and its concentration in the fetus's arterial blood.
Write a Scheme procedure that computes the fetal oxygen consumption rate, given the measured rate of flow and the two measured concentrations of oxygen. Be sure to describe the units in which the fetal oxygen consumption rate is measured and to use units of measurement consistently throughout your computation.
The maximum rate of oxygen diffusion through the placenta is directly proportional to the surface area of the placenta, inversely proportional to its thickness, and directly proportional to the difference between the partial pressure of oxygen in the mother's bloodstream and its partial pressure in the fetus's bloodstream. (The ``partial pressure'' of oxygen, within a mixture of gases such as air, is the pressure exerted by the mixture multiplied by the ratio of the amount of oxygen present to the total amount of gas present. Like other pressures, it is measured in millimeters of mercury -- `mm Hg' for short -- or in ``atmospheres.'') The constant of proportionality depends on the physical properties of the placenta; the observed value of the constant is 3.09 x 10-8 square centimeters per minute-mm Hg. To compute the rate of oxygen diffusion, subtract the partial pressure of oxygen on the fetus's side from its partial pressure on the mother's side, multiply by the surface area of the placenta, divide by the thickness of the placenta, and finally multiply by the constant of proportionality.
Write a Scheme procedure that computes the maximum rate of oxygen diffusion, given the two partial pressures of oxygen and the surface area and thickness of the placenta. Again, be careful about units! (Note: one milliliter, considered as a measure of capacity, occupies one cubic centimeter; in this exercise the two measures are equivalent.)
If the maximum rate of oxygen diffusion is substantially greater than the fetal oxygen consumption rate, then simple diffusion can explain how the fetus gets enough oxygen; otherwise, we need to keep looking for some other mechanism that boosts the oxygen supply.
Use your Scheme procedures and the following empirical observations to determine whether the simple diffusion mechanism is an adequate explanation:
Concentration of O2 in umbilical vein: 13.5 ml O2 per 100 ml blood
Concentration of O2 in umbilical artery: 4.5 ml O2 per 100 ml blood
Rate of flow of blood in umbilical veins and arteries: 250 ml per minute
Partial pressure of O2, maternal side: 36.5 mm Hg
Partial pressure of O2, fetal side: 21.5 mm Hg
Thickness of placenta: 3.5 x 10-4 cm
Surface area of placenta: 12 m2
What is the ratio between the fetal oxygen consumption rate and the maximum rate of oxygen diffusion through the placenta? (In other words, what fraction of the placenta's capacity for oxygen diffusion is being used, if simple diffusion is the only mechanism for the transfer of oxygen?)
Here's what you need to turn in for this exercise:
The two procedure definitions that you write for parts A and B of the exercise. It would be natural to write these two definitions in a file (with a name ending in .ss), supplying an opening comment in which you give your name, the date, the occasion for which the program was written, and any background information that a reader who hasn't already seen this statement of the exercise might need in order to understand what the procedures are for. Each of the procedure definitions should also be preceded by a comment that explains what the procedure does and (unless this is completely obvious) how it does it.
Any other definitions that you wish to make available. For instance, it might be a good idea to give the constant of proportionality a name.
A transcript or log file in which you load in the file containing the procedure definitions and then apply your procedures to the empirical observations presented in part C. A web page (``Using the submit program'') describes how to construct such a log file.
Another web page (``Mailing files'') describes how to submit your Scheme source code files and log files by electronic mail. I will accept printed copies instead, but I prefer e-mail.
The physiology used in this exercise is derived, with some simplifications, from sections 6.3 and 6.4 of An introduction to the mathematics of biology, by Edward K. Yeargers, Ronald W. Shonkwiler, and James V. Herod (Boston: Birkhäuser, 1996), pp. 180-186. Any errors are almost certainly mine. The actual data values are taken from the same source but are derived from three research papers cited in the bibliography for chapter 6 of that book:
H. Bartels, W. Moll, and J. Metcalfe, ``Physiology of gas exchange in the human placenta,'' American journal of obstetrics and gynecology 84 (1962), 1714-1730.
J. Metcalfe, H. Bartels, and W. Moll, ``Gas exchange in the pregnant uterus,'' Physiological review 47 (1967), 782-838.
A. Costa, M. L. Costantino, and R. Fumero, ``Oxygen exchange mechanisms in the human placenta: mathematical modelling and simulation,'' Journal of biomedical engineering 14 (1992), 85-389.
This document is available on the World Wide Web as
http://www.math.grin.edu/courses/Scheme/spring-1998/exercise-1.html
created January 28, 1998
last revised June 21, 1998