Biomedical Engineering Reference
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which the property is generated inside the system. The
sum of these two terms is the rate of change of
the property, inside the system, with respect to time.
The advantage of the rate form of the law is that the laws
of physics generally make it easy to find the rates at
which things are happening. The disadvantage of the rate
form is that it generates differential equations.
Although the accounting principle can be applied for
any extensive property, it is most useful when the trans-
port and generation/consumption terms have physical
significance. The most useful applications of this princi-
ple occur when something is known a priori about the
generation/consumption term.
For conserved extensive properties the equations that
apply the accounting principle are significantly simpler.
In the accumulation form, the equations become:
For the derivation of the Nernst equation, the system
being considered is a cell membrane surrounded by ex-
tracellular fluid, with ion flow across the membrane.
Assume that one of K þ ,Na þ ,orCl ions flow across
themembrane, with the positive direction taken to be from
the outside, across the membrane, to inside the cell. The
rate form above leads to the following three relationships:
1. Fick's Law is a model that relates the flow of ions due
to diffusion and the ion concentration gradient across
the cell membrane. It is a law that expresses the
influence of chemical force on the conservation of
mass in the system. The left-hand side of the
conservation law is the flow rate of mass, that is, the
mass flux in the system, which is the flow due to
diffusion. The right-hand side is the sum of the rates
of change of mass through the system inlets and
outlets, that is, the flow through the ion channels.
The rate form of the principle of conservation of
mass leads to:
P final
inside
P initial
inside
¼ðP i P o Þ:
J diffusion ¼D dI
2.1a.3.3 Using balance equations
dx
The mathematical formulation of problems that bio-
medical engineers solve is based on one or more of the
conservation laws from chemistry and physics. The
solution to a problem becomes obvious when one has the
right formulation. The approach used herein is that the
solution to a problem can be formulated using one or
more of the forms of the following conservation laws.
The model formulation and problem-solving frame-
work presented in Section 2.1a.3.1 shows that there are
two forms for each of the conservation laws: the accu-
mulation (or sum) form and the rate form. The former is
used in solutions of steady-state or finite-time problems,
whereas the latter is used in solving problems with
transient behavior.
that is, the flow of ions due to diffusion is equal to
the ion concentration gradient across the membrane
scaled by the diffusivity constant D. That is, there
is ion flow in response to an ion concentration gra-
dient. The negative sign indicates that the ion flow is
in the opposite direction of the gradient.
2. Ohm's Law is a model for the influence of electrical
force on ion flow across the cell membrane. It is
still based on conservation of mass, but the driving
force on the right-hand side is mass flow due to an
electric field induced by other charged particles. An
electric field E is applied, creating a rate of change
of potential, dv/dx. The right-hand side is the ion
flow due to this potential, and leads to:
Example 2.1a.1 How conservation laws lead to the
Nernst equation.
J drift ¼ m 1 dv
Show how Fick's law, Ohm's law and the Einstein re-
lationship can be derived from the conservation laws in
Section 2.1a.3.1. Show how these three conservation
models lead to the Nernst equation. This problem is
derived from section 3.4 of Enderle et al. (2000) .
The rate form of the principle of conservation of
mass is:
dx
in which J drift is the ion flux due to the electric
field, m is the mobility, Z is the ion valence, [ I ]is
the ion concentration, and v is the voltage.
3. The Einstein relationship is a form of conservation of
momentum that expresses a relationship between
diffusion and ion mobility. The electric field induces
a force on the ions that causes a flow that is balanced
by osmotic pressure. The conservation law,
¼ X
inlets
_ m i X
outlets
dm sys
dt
_ m e
meaning that the overall time rate of change of mass in
the system is equal to the difference between the sum of
the rates of change of mass into the system and the sum
of the rates of change of mass out of the system.
¼ X F ext þ X
inlets
_ P i X
outlets
dP sys
dt
_ P o
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