Page 29 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

6 BIOMEDICAL SYSTEMS ANALYSIS

normal individual is usually 15 mg% (mg% = milligrams of the substance per 100 mL of blood), the
BUN in uremic patients could reach 50 mg%. The purpose of the dialysis is to bring the BUN level
closer to the normal. In the artificial kidney, blood flows on one side of the dialyzer membrane and
dialysate fluid flows on the other side. Mass transfer across the dialyzer membrane occurs by diffu-
sion due to concentration difference across the membrane. Dialysate fluid consists of a makeup solu-
tion consisting of saline, ions, and the essential nutrients so as to maintain zero concentration
difference for these essential materials across the membrane. However, during the dialysis, some
hormones also diffuse out of the dialyzer membrane along with the urea molecule. Too rapid dialy-
sis often leads to depression in the individual due to the rapid loss of hormones. On the other hand,
too slow dialysis may lead to unreasonable time required at the hospital. Simple modeling can be
used to calculate the treatment protocols of mass coming into the body from the dialyzer, plus the
metabolic production rate. When the patient is not on dialysis, the concentration of urea would
increase linearly if the metabolic production rate is constant or will increase exponentially if the
metabolic production rate is a linear function of the concentration (first order reaction). When the
patient is on dialysis, the concentration would decrease exponentially. This way, the treatment pro-
tocol can be prescribed after simulating different on and off times (e.g., turn on the dialyzer for
4 hours every 3 days) to bring the BUN under control. In the chapter on artificial kidney devices, a
simple one compartmental model is used to compute the patient protocol.
Compartmental models are often used in the analysis of thermal interactions. Simon and Reddy
(1994) formulated a mathematical model of the infant-incubator dynamics. Neonates who are born
preterm often do not have the maturity for thermal regulation and do not have enough metabolic heat
production. Moreover, these infants have a large surface area to volume ratio. Since these preterm
babies cannot regulate heat, they are often kept in an incubator until they reach thermal maturity. The
incubator is usually a forced convection heating system with hot air flowing over the infant.
Incubators are usually designed to provide a choice of air control or the skin control. In air control,
the temperature probe is placed in the incubator air space and the incubator air temperature is controlled.
In the skin control operation, the temperature sensor is placed on the skin and infant’s skin temper-
ature is controlled. Simon et al. (1994) used a five compartmental model (Fig. 1.2) to compare the
adequacy of air control and skin control on the core temperature of the infant. They considered the
infant’s core, infant’s skin, incubator air, mattress, and the incubator wall to be four separate well-
mixed compartments.
The rate of change of energy in each compartment is equal to the net heat transfer via con-
duction, convection, radiation, evaporation, and the sensible heat loss. There is a convective heat
loss from the infant’s core to the skin via the blood flow to the skin. There is also conductive heat
transfer from the core to the skin. The infant is breathing incubator air, drawing in dry cold air
at the incubator air temperature and exhaling humidified hot air at body temperature. There is
heat transfer associated with heating the air from incubator air temperature to the infant’s body
(core) temperature. In addition, there is a convective heat transfer from the incubator air to the
skin. This heat transfer is forced convection when the hot air is blowing into the incubator space
and free convection when the heater manifolds are closed. Moreover, there is an evaporative heat
loss from the skin to the incubator air. This is enhanced in premature infants as their skin may
not be mature. Also, there is a conductive heat transfer from the back surface of the skin to the
mattress. Also, exposed skin may radiate to the incubator wall. The incubator air is receiving hot
air (convective heat transfer) from the hot air blower when the blower is in the on position. There
is convective heat transfer from the incubator air to the incubator wall and to the mattress. In
addition, there is metabolic heat production in the core. The energy balance for each compart-
ment can be expressed as
mCp(dT/dt) =ΣQ − ΣQ + G (1.4)
in out
where m is the mass of the compartment, T is the temperature, t is the time, Q is the heat transfer
rate, and G is the metabolic heat production rate. G is nonzero for the core and zero for all other com-
partments. G is low in low-birth-weight and significantly premature babies. Simon et al. (1992)
investigated infant-incubator dynamics in normal, low birth weight, and different degrees of prema-
turity under skin and air control. Recently, Reddy et al. (2008) used the lumped compartmental