During my time at Texas A&M (1997-2001), students in biomedical engineering had to take three courses in calculus and one in differential equations, two courses in physics, at least two electrical engineering courses, at least two mechanical engineering courses, and two courses in physiology designed for biomedical engineers, all before being admitted to "upper classes" (one also needed to have at least a 3.3 GPA) and taking the BMEN courses for our major.
To divide biomedical engineering into three headings: 1) modeling; 2) artificial devices; 3) signal processing/equipment, that were covered in various courses in my last two years as an undergrad. (Texas A&M's College of Engineering is very peculiar in that any engineering degree would take someone 5 years to complete if they never exceeded the 18 hour per semester limit - I got around this through alot of AP credits when entering and summer school, but it is quite odd to have one's scheduling book advise 21 hours of engineering classes per semester for one's final four semesters - one would have no time to do ANYTHING but homework.)
a. Blood flow in the body is pulsatile non-Newtonian flow, and blood can have variable viscosity. The blood vessels in your body experience time-dependent forces, including shear stress. Indeed, growth factors in the epithelial cells of your blood vessels are only released when the shear stress from the movement of blood through the vessel is applied, and when the stress is outside normal limits is when all sorts of nasty things begin to happen to your blood vessels (there is also shear rate, separate from shear stress, and the relation between the two depends on the viscosity, which itself is largely dependent on the clotting factor fibrinogen). Abnormalities in blood vessels (such as from artherosclerosis) then form secondary flow streamlines in the vessels, and biomedical engineers who work on these issues would calculate the patterns, forces, axial velocities, and shears and strains that exist. The results are applicable to creating artificial tissue that could duplicate the functions of blood vessels, but it's also just a good ol' extension of the idea that math is the language that underlies the universe.
b. Clearly enough, your nervous system is a huge electrical circuit. So are parts of your muscles, and even transfer in your kidneys can be modeled as en electrical circuit.
c. Fluid transfer in your body, it is assumed, operates the same way as fluid transfer in any other system. One can model the body and different systems as a number of different compartments that interact according to the permabilities/diffusive properties of each membrane. Your lungs work this way, your lymph system works this way, oxygen moving from the hemoglobin of your red blood cells to another cell operates this way.
For example, a wonderful derivation (I'm not going to type out the equation):
We ignore the particulate nature of blood as well as the mass transfer resistance of the red blood cell. The blood is assumed to be in plug flow with an average velocity represented by V. Also note that the hemoglobin is carried along by the red blood cell at the average blood velocity (V). R HBO represents the volumetric production rate of oxygenated hemoglobin. After dividing by 2πr∆r∆z, and taking the limit as ∆z→0, we obtain the following differential equation that describes the mass balance for ozygenated hemoglobin within the blood flowing through the capillary....(From Fournier's Basic Transport Phenomena in Biomedical Engineering)I only included it because I once had to spend an entire semester doing such derivations, and it was alot of fun. Math is great! Even reading that book again made my heart flutter. If the above paragraph does not do the same for you, biomedical engineering and engineering in general are probably not for you.
2) Artificial devices. Tied to modeling, it does no good to build a prosthetic leg if one doesn't know the forces, stresses, and strains that the device may experience. After modeling the act of walking, in terms of forces, angles, rates, etc., one can design a device that can perform the task of walking, but at the same time does not alter the forces experienced by the other bones in the body. Those forces are important for proper bone growth, and using too strong of a material for a prosthetic weight-bearing device, or using a material that vibrates could affect the other bones of your body. We encounter similar problems with building artificial hearts and lungs: constructing these organs on nothing but mechanical engineering principles can be done easily enough, but one must also consider the particular additional features of these organs that contribute to proper function in the body.
Several of the students in the department had internships at NASA. Of course, it is of great interest to NASA to know the conditions under which the human body functions here on earth (specifically, with gravity), to devise ways to monitor the health of astronauts and counteract the effects of weightlessness, including perhaps devising special exercise equipment.
3) And finally, designing all that fancy equipment one sees in a hospital is now primarily the job of biomedical engineers - EEG and ECG recorders, heart rate monitors, ventilators, etc. A lot of signal processing is going on in those machines, and I hope to never again have to do Fourier and Laplace transforms by hand (that was a wicked professor). My design project for the second semester of my senior year was to write a program that would allow someone to input a night of EEG recordings, and could output the time spent in each stage of sleep with each major wave event noted. Such is the work of biomedical engineers who act as the interface between the performance of the body and processing that performance into language that can be understood by clinicians.
So that is the 'brief' introduction to biomedical engineering that I will provide here. It was a very fun major that incorporated lots of soldering in an advanced clinical engineering class, the shop class I never took (in constructing a plastic injection molding machine that could make skin buttons for insulin injection ports), and learning FDA device regulations. But mostly, it was doing lots and lots of math that I love, and applying it to the human body. Fluid dynamics, pressure, strain, YEAH! So don't consider it unless you enjoy calculus (triple integral-type calculus) and differential equations, and prefacing every solution with a list of assumptions, as all good engineers do.