2 Tissue Modeling Considerations
2.1 Mass Transfer
in the Extracellular
Space
Preparation of a tissue sample for cryopreservation requires effective delivery of CPAs to all regions of the tissue. This involves
transport of the CPA through the tissue from the surface to the
center. Compared to isolated cells, the amount of time required to
deliver CPA into tissue is often much longer, particularly for large
tissues. The time scale for diffusion of CPA into a tissue sample can
be estimated as:
t diff ¼ L
2
=D eff
ð1Þ
where L is the diffusion length and D eff is the effective diffusivity of
the CPA in the tissue. The effective diffusivity for various CPAs has
been estimated in a variety of tissue types, resulting in values that
range from approximately 10
À6 to 10
À5 cm
2 /s at room temperature [9–17]. Diffusion length can vary depending on the dimensions of the tissue of interest. A pancreatic islet has a radius of about
50 μm, which results in a CPA diffusion time scale of between 3 and
30 s. Larger tissues require much longer for CPA diffusion. For
example, articular cartilage with a half-thickness of about 1 mm
would have a diffusion time scale of between 20 min and 3 h.
For comparison, it is useful to consider the typical time scale of
the osmotic response of a cell after exposure to a CPA solution.
Exposure to CPA causes a shrink-swell response due to initial water
efflux followed by influx of both water and CPA. The duration of
this shrink-swell response varies depending on the cell type and
CPA. For example, human RBCs undergo a shrink-swell response
after exposure to glycerol with a time scale on the order of 10 s at
room temperature [18]. Oocytes, on the other hand, are much
larger and hence respond more slowly after exposure to CPA,
exhibiting a shrink-swell response with a time scale of about
10 min [19]. A typical mammalian cell exhibits an intermediate
time scale of about 1 min.
A comparison of time scales reveals that transport from the
surface to the center of the tissue is often the rate limiting step for
delivery of CPA into tissues. Therefore, it is essential to consider
interstitial transport for effectively modeling CPA transport in
tissues.
2.2 Tissue Size
Changes Due to CPA
Exposure
Mass transfer in tissues has been studied for decades and is typically
modeled using Fick’s law of diffusion, assuming that the tissue is
comprised of a rigid porous matrix and does not change size. This is
reasonable in many cases relevant to physiology and medicine.
However, the concentrated solutions used for cryopreservation
can create substantial osmotic gradients that cause the size of the
tissue to change. For example, exposure of pancreatic islets to
2 molal dimethyl sulfoxide (DMSO) causes the islet to initially
shrink to about 70% of its original volume, followed by swelling
Tissue Transport Modeling
175
2.1 Mass Transfer
in the Extracellular
Space
Preparation of a tissue sample for cryopreservation requires effective delivery of CPAs to all regions of the tissue. This involves
transport of the CPA through the tissue from the surface to the
center. Compared to isolated cells, the amount of time required to
deliver CPA into tissue is often much longer, particularly for large
tissues. The time scale for diffusion of CPA into a tissue sample can
be estimated as:
t diff ¼ L
2
=D eff
ð1Þ
where L is the diffusion length and D eff is the effective diffusivity of
the CPA in the tissue. The effective diffusivity for various CPAs has
been estimated in a variety of tissue types, resulting in values that
range from approximately 10
À6 to 10
À5 cm
2 /s at room temperature [9–17]. Diffusion length can vary depending on the dimensions of the tissue of interest. A pancreatic islet has a radius of about
50 μm, which results in a CPA diffusion time scale of between 3 and
30 s. Larger tissues require much longer for CPA diffusion. For
example, articular cartilage with a half-thickness of about 1 mm
would have a diffusion time scale of between 20 min and 3 h.
For comparison, it is useful to consider the typical time scale of
the osmotic response of a cell after exposure to a CPA solution.
Exposure to CPA causes a shrink-swell response due to initial water
efflux followed by influx of both water and CPA. The duration of
this shrink-swell response varies depending on the cell type and
CPA. For example, human RBCs undergo a shrink-swell response
after exposure to glycerol with a time scale on the order of 10 s at
room temperature [18]. Oocytes, on the other hand, are much
larger and hence respond more slowly after exposure to CPA,
exhibiting a shrink-swell response with a time scale of about
10 min [19]. A typical mammalian cell exhibits an intermediate
time scale of about 1 min.
A comparison of time scales reveals that transport from the
surface to the center of the tissue is often the rate limiting step for
delivery of CPA into tissues. Therefore, it is essential to consider
interstitial transport for effectively modeling CPA transport in
tissues.
2.2 Tissue Size
Changes Due to CPA
Exposure
Mass transfer in tissues has been studied for decades and is typically
modeled using Fick’s law of diffusion, assuming that the tissue is
comprised of a rigid porous matrix and does not change size. This is
reasonable in many cases relevant to physiology and medicine.
However, the concentrated solutions used for cryopreservation
can create substantial osmotic gradients that cause the size of the
tissue to change. For example, exposure of pancreatic islets to
2 molal dimethyl sulfoxide (DMSO) causes the islet to initially
shrink to about 70% of its original volume, followed by swelling
Tissue Transport Modeling
175