3.1 Models That
Assume a Constant
Tissue Size
3.1.1 Fick’s Law
of Diffusion
Fick’s law of diffusion is commonly used to model mass transfer in
tissues. Fick’s first law describes the diffusive flux of a component
i ( j i ) in terms of its concentration gradient:
j i ¼ ÀD i ∇c i
ð2Þ
where D i represents the diffusion coefficient of component i in a
designated media, and c i represents the concentration of component i. If we assume only diffusive flux, a mass balance in the
absence of a chemical reaction results in Fick’s second law:
∂c i
∂t
¼ D i ∇
2
c i
ð3Þ
Fick’s law as defined in Eq. 3 above is fairly straightforward to
implement, and the analytical solutions for the common simplifications of homogenous, isotropic media in one dimension are prevalent and can be found in references such as the classic discussion of
diffusion by Crank [31]. However, this modeling approach is limited since it assumes that the size of the tissue remains constant, it
neglects the effects of cell membrane transport on interstitial diffusion, and it is based on the assumption that the solution is ideal and
dilute.
To make mass transfer predictions using Fick’s law, it is necessary to know the value of the diffusion coefficient D i . Diffusion in
porous materials such as tissues is typically described using an
effective diffusion coefficient, which takes into account the fact
that only a fraction of the tissue volume is available for diffusion,
and that the diffusing species typically must follow a tortuous path
as it makes its way around solid obstacles in the porous network.
The effective diffusion coefficient is typically expressed in terms of
the free solution diffusion coefficient as follows:
D eff ¼ D
ε
λ
ð4Þ
where D eff is the effective diffusion coefficient, D is the diffusion
coefficient in free solution, ε is the void fraction of the tissue, and λ
is the tortuosity. Effective diffusion coefficients have been measured
for various CPAs in various tissue types, resulting in a ratio D eff /D
of about 0.3 [9].
There are several examples in the literature of the use of Fick’s
law to design methods for CPA equilibration in tissues. For
instance, a series of papers were recently published characterizing
CPA diffusion in articular cartilage and describing the use of Fick’s
law for designing methods for delivery of a mixture of CPAs into
the cartilage [17, 32, 33]. Diffusion predictions were used to
design a multistep method to reach a desired minimal CPA concentration throughout the cartilage sample in a minimal amount of
time [32]. Han and colleagues [11] used Fick’s law to design a
method for delivery of CPA into whole ovaries that was predicted
Tissue Transport Modeling
181
Assume a Constant
Tissue Size
3.1.1 Fick’s Law
of Diffusion
Fick’s law of diffusion is commonly used to model mass transfer in
tissues. Fick’s first law describes the diffusive flux of a component
i ( j i ) in terms of its concentration gradient:
j i ¼ ÀD i ∇c i
ð2Þ
where D i represents the diffusion coefficient of component i in a
designated media, and c i represents the concentration of component i. If we assume only diffusive flux, a mass balance in the
absence of a chemical reaction results in Fick’s second law:
∂c i
∂t
¼ D i ∇
2
c i
ð3Þ
Fick’s law as defined in Eq. 3 above is fairly straightforward to
implement, and the analytical solutions for the common simplifications of homogenous, isotropic media in one dimension are prevalent and can be found in references such as the classic discussion of
diffusion by Crank [31]. However, this modeling approach is limited since it assumes that the size of the tissue remains constant, it
neglects the effects of cell membrane transport on interstitial diffusion, and it is based on the assumption that the solution is ideal and
dilute.
To make mass transfer predictions using Fick’s law, it is necessary to know the value of the diffusion coefficient D i . Diffusion in
porous materials such as tissues is typically described using an
effective diffusion coefficient, which takes into account the fact
that only a fraction of the tissue volume is available for diffusion,
and that the diffusing species typically must follow a tortuous path
as it makes its way around solid obstacles in the porous network.
The effective diffusion coefficient is typically expressed in terms of
the free solution diffusion coefficient as follows:
D eff ¼ D
ε
λ
ð4Þ
where D eff is the effective diffusion coefficient, D is the diffusion
coefficient in free solution, ε is the void fraction of the tissue, and λ
is the tortuosity. Effective diffusion coefficients have been measured
for various CPAs in various tissue types, resulting in a ratio D eff /D
of about 0.3 [9].
There are several examples in the literature of the use of Fick’s
law to design methods for CPA equilibration in tissues. For
instance, a series of papers were recently published characterizing
CPA diffusion in articular cartilage and describing the use of Fick’s
law for designing methods for delivery of a mixture of CPAs into
the cartilage [17, 32, 33]. Diffusion predictions were used to
design a multistep method to reach a desired minimal CPA concentration throughout the cartilage sample in a minimal amount of
time [32]. Han and colleagues [11] used Fick’s law to design a
method for delivery of CPA into whole ovaries that was predicted
Tissue Transport Modeling
181