Page 67 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

44 BIOMECHANICS OF THE HUMAN BODY

T t is the weighted average tissue temperature defined by Eq. (2.9) and is given by
ρω T − T V = ∫ ∫∫∫ ρ ω T − T dV
c
(
)
c
(
)
a0 t body a t body (2.11)
body volume
where in Eq. (2.11), T can be determined by solving the Pennes bioheat equation.
t
During clinical applications, external heating or cooling of the blood can be implemented to
manipulate the body temperature. In the study of Zhu et al. (2009), the blood in the human body is
represented as a lumped system. It is assumed that a typical value of the blood volume of body, V ,
b
is approximately 5 L. External heating or cooling approaches can be implemented via an intravas-
cular catheter or intravenous fluid infusion. A mathematical expression of the energy absorbed or
removed per unit time is determined by the temperature change of the blood, and is written as
bb b[
a ] Δ
ρ cV T t + Δ t) − T t) / t ≈ ρ cV dT a (2.12)
(
(
bb b
a
dt
where T (t) is the blood temperature at time, t, and T (t +Δt) is at time t +Δt. In the mathematical
a a
model, we propose that energy change in blood is due to the energy added or removed by external
heating or cooling (Q ), and heat loss to the body tissue in the systemic circulation (Q ).
ext blood-tissue
Therefore, the governing equation for the blood temperature can be written as
dT
T t −
c ω
Tt
ρ cV a = Q (, ) Q blood-tissue ( t = Q (, ) − ρ b b V body (T − T t ) (2.13)
)
bb b
ext
t
a
a
ext
a
dt
where Q ext can be a function of time and the blood temperature due to thermal interaction between
blood and the external cooling approach, T , , and ω can be a function of time. Equation (2.13)
T
a t
cannot be solved alone since T t is determined by solving the Pennes bioheat equation. One needs
to solve Eqs. (2.8) and (2.13) simultaneously.
One application of blood cooling involves pumping coolant into the inner tube of a catheter
inserted into the femoral vein and advanced to the veno-vera. Once the coolant reaches the catheter,
it flows back from the outer layer of the catheter and out of the cooling device. This cooling device
has been used in clinical trials in recent years as an effective approach to decrease the temperature
of the body for stroke or head injury patients. Based on previous research of this device, the cooling
capacity of the device is around −100 W [Q in Eq. (2.13)].
ext
Figure 2.4 gives the maximum tissue temperature, the minimum tissue temperature at the skin
surface, the volumetric-average body temperature (T ), and the weighted-average body temperature
avg
(T t ). The difference between the volumetric-average body temperature and the weighted-average-
body temperature is due to their different definitions. All tissue temperatures decrease almost lin-
early with time and after 20 minutes, the cooling results in approximately 0.3 to 0.5°C tissue
temperature drop. The cooling rate of the skin temperature is smaller (0.2°C/20 min). As shown
in Fig. 2.5, the initial cooling rate of the blood temperature in the detailed model is very high (~0.14°C/min),
and then it decreases gradually until it is stabilized after approximately 20 minutes. On the other
hand, cooling the entire body (the volumetric average body temperature) starts slowly and gradually
catches up. It may be due to the inertia of the body mass in responding to the cooling of the blood.
Figure 2.5 also illustrates that after the initial cooling rate variation, the stabilized cooling rates of
all temperatures approach each other and they are approximately 0.019°C/min or 1.15°C/h. The sim-
ulated results demonstrate the feasibility of inducing mild body hypothermia (34°C) within 3 hours
using the cooling approach.
The developed model in Zhu et al. (2009) using the Pennes perfusion term and lumped system
of the blood is simple to use in comparison with these previous whole body models while providing
meaningful and accurate theoretical estimates. It also requires less computational resources and time.
Although the model was developed for applications involving blood cooling or rewarming, the
detailed geometry can also be used to accurately predict the body temperature changes during exercise.