Biomedical Engineering Reference
In-Depth Information
the case where they are not negligible. Interstitial flow is slow through a particular
tissue because it is 8 liters per day or 5
.
55 cubic centimeters per minute for the en-
tire human body (Levick,
1995
;Cowin,
2011
). The exact velocity ranges of this flow
are unknown but measurements in limb tissue have suggested they are on the order
of 0
.
1 to 2 microns per second (0.03 to 0.57 feet per day) (Levick,
1995
;Cowin,
2011
). Deformation-driven interstitial flows, such as those that occur in bone tissue,
are greater, on the order of tens of microns per second. No tissue has a mass al-
ways composed of the same proteins and fluids; they are always changing, however
slowly. Thus tissues form open systems.
For the growth of soft tissue it is reasonable to assume that the blood supply to
the tissue will deliver the proteins and the supply of energy. If these proteins carried
by the blood are not employed or deployed by the liver in their first pass through the
tissue in need they will likely be transported across the blood vessel walls to pass
into the interstitial fluid of another tissue. The interstitial fluid will then pass through
the tissue to a collecting lymph node and then pass into the lymphatic system. The
lymphatic system collects the lymph from all the tissues, concentrates the proteins
and passes them back into the circulatory system at the left subclavian vein before
it enters the heart. The tissue building proteins are then recirculated again and again
before they find deployment in a tissue or are passed out of the body (Levick,
1995
).
The coupling of these related transport problems to growth problems is not difficult
due to very slow transport velocities involved.
It is assumed that the stress tensor associated with each constituent is symmetric
and that there are no action-at-a-distance couples, as there would be, for example, if
the material contained electric dipoles and was subjected to an electrical field. The
restrictions associated with each of these assumptions may be removed; they are
imposed to restrict this presentation to an economical path for the development of a
tissue-appropriate model for normal physiological growth phenomena.
18.4.1 Open and Closed System Models at the Constituent and
Mixture Levels
Open systems permit the transport of mass, momentum and energy across their
boundaries, closed systems do not. Bertalanffy (
1950
) pointed out that 'From the
physical point of view the characteristic state of the living organism is that of an
open system'. Thus an open system model is desired to model growth. In traditional
mixture theory (Truesdell and Toupin,
1960
), each constituent is considered to be
an open system, but the entire mixture is considered to be a closed system. In the
Cowin and Cardoso (
2012
) development, the mixture is also considered to be an
open system as a mechanism by which growth may be modeled. This means that
the statements of the balance principles for the mixture may have supply terms as
well as the statements of the balance principles for each constituent.
An alternative attractive approach to achieving an open system by allowing sup-
ply terms in the balance equations for the mixture is to, instead, allow the mixture to
