California State Polytechnic University, Pomona
Chemical & Materials Engineering Department
CHE 313: Transport Modeling
CASE STUDY
Oxygen Transport from a Perfluorocarbon Blood Substitute
May 25, 2012
by
Souza, Betty E.
This case study is given under the Honor System and by signing here
I have agreed that the work submitted is my work alone and that I
neither sought nor received help from others.
Presentation
20%
Model Development
25%
Solution
25%
Discussion
20%
Completeness and Neatness
10%
February 6, 2016
Dr. T.K. Nguyen
Chemical engineering Department
California State Polytechnic University, Pomona
3801 West Temple Avenue
Pomona, CA 91768
Dr. Nguyen:
I have researched the development and use of artificial blood products for
the ChE 313 case study. I have developed and solved a model for oxygen
transport in the capillaries and tissues for a perfluorocarbon emulsion used as a
blood substitute. As the shortage of blood is a pressing concern in this country
and world wide, this seems to be an important area of study. The development
of a substitute for blood is difficult because of the human body’s tendency to
reject or destroy any foreign materials. This has been a problem in developing a
hemoglobin based blood substitute. Perfluorocarbons (PFCs) are organic
compounds similar to Teflon. They are inert and therefore not rejected or
attacked by the body’s immune system. PFCs are good oxygen carriers and
several PFC blood substitutes are being developed and tested.
Following the oxygen transport model developed for natural blood in the Fournier
text, I have developed and solved a model for the transport of oxygen in the
capillaries and tissue for the case when PFC is used as a blood substitute.
Figure 1 illustrates the capillary and tissue system of interest. The model for
PFC is much the same as that for natural blood with the exception of the
parameter m. This difference is outlined in the model development. Figure 2
shows a comparison of the oxygen profile for natural blood and PFC with
identical inlet oxygen concentrations. The profiles are very similar. As expected,
the profile for PFC is steeper than that for natural blood. This is reasonable
because PFC will release its oxygen more readily into the plasma than will
hemoglobin, making more oxygen available for transport to the tissues, and
depleting the oxygen more quickly.
The amount of oxygen absorbed by PFCs is directly proportional to the oxygen
available for transfer in the lungs. To illustrate the effects of this phenomenon, I
have graphed (figure 3) the resulting oxygen profiles for the case when the
entering PFC PO2 is about half and one and a half times the nominal value
assumed for arterial blood PO2. This graph shows that anoxia is an issue at inlet
concentrations less than the nominal value.
Given that blood transfusions do not replace all of a patients blood, I have also
calculated the oxygen profiles for a 1:1 mixture of natural blood and PFC at
various inlet oxygen concentrations. The results of this analysis can be found in
figures 4, 5, and 6.
Analysis of the results of this model for oxygen transport from PFC shows that if
initial oxygen concentration of the blood stream can be kept slightly above that
for natural blood, PFC will be an effective in carrying oxygen to the various
tissues in the body.
Sincerely,
Betty Souza
Background
Each year, in the United States alone, about 12 million units (6 million liters) of
blood is transfused, and worldwide demand is estimated to increase by 7.5
million liters per year. It is predicted that by the year 2030 the United States will
be experiencing a shortage of 2 million liters per year. This shortage is due to
low overall donation rates – only about 5% of Americans donate blood --, a short
shelf life for donated blood, and difficulty finding perfectly matched blood (Nucci).
The search is on for an artificial blood product that can make up this shortfall as
well as solve some other problems encountered when transfusing blood. Ideally
a blood substitute would be acceptable to patients of all blood types, nontoxic,
have a long shelf life with minimal storage restrictions, stay in the blood stream
long enough for the patient to rebuild their own blood, and be a good oxygen
carrier (Nucci).
Natural blood transports oxygen by way of hemoglobin molecules within the red
blood cells (RBC). Each milliliter of human blood contains about 5 million red
blood cells, each of which contain about 250 million molecules of hemoglobin.
Each molecule of hemoglobin can transport a maximum of 4 molecules of
oxygen from the lungs to various parts of the body. Each RBC also contains 2,3diphosphoglycerate (2,3-DPG). 2,3-DPG is a compound that makes it possible
for hemoglobin to release its oxygen throughout the body and it also prevents
auto-oxidation and the breakdown of the hemoglobin. In addition to transporting
oxygen from the lungs, hemoglobin can bind with carbon dioxide and transport it
back to the lungs to be exhaled (Nucci).
More than half a dozen biotechnology companies have developed products
designed to replicate the role of hemoglobin in the blood. These products
generally fall into two classes; synthetic oxygen carriers and hemoglobin based
blood substitutes. Hemoglobin based substitutes utilize human hemoglobin from
outdated donated blood or hemoglobin from animals. Hemoglobin cannot be
transfused without the protection of the RBC, because naked hemoglobin breaks
down rapidly and is toxic to the kidneys. Proposed solutions include the
encapsulation of hemoglobin in liposomes, attaching polyethylene glycol to the
molecule to protect it in a shell of water, and polymerizing the molecule to
strengthen its bonds (Lawton). So far, none of the hemoglobin-based products is
commercially available. Clinical trials of Baxter Healthcare’s HemAssist blood
substitute were halted in April 1998 when death rates due to an increase in blood
pressure exceeded projections (AP).
Perfluorocarbons (PFCs), synthetic organic compounds similar in composition to
Teflon, are the most effective class of synthetic oxygen carriers (Lawton). PFCs
will not dissolve in plasma but can be emulsified with various agents that allow
them to be dispersed as tiny particles in the blood. They deliver gas passively
and absorb oxygen preferentially over RBCs. They also release oxygen more
quickly into the blood plasma since they are not housed inside a cell membrane
(Nucci). Unlike hemoglobin, the amount of oxygen PFCs can carry is directly
proportional to the oxygen available to them in the lungs. The drawback to
PFCs, as stated by Dr. Winslow of UC San Diego, is that their “oxygen carrying
capacity is so low that patients have to breathe pure oxygen” to make them
effective (Lawton). While several PFC based products have made it to clinical
trials, storage and effectiveness issues have kept them from experiencing much
success.
The following model will attempt to predict the average oxygen profile in the
capillary and at the tissue cylinder radius when a PFC based blood substitute is
used as compared to natural blood. I will follow the model development for
oxygen transport from natural blood in the Krogh tissue cylinder model presented
in the Fournier reference. In this model, oxygen is released from the hemoglobin
into the plasma, and only the oxygen in the plasma can diffuse into the
surrounding tissue. Only minor modifications to this model will need to be made
to simulate the oxygen transport by PFCs. It has been noted that blood
transfusions are required when more than 40% of the total volume is lost (Nucci).
I will therefore model the oxygen profile for the cases of 100% natural blood,
100% PFC, and a 1:1 ratio of natural blood to PFC.
Oxygen Transport Model Development
To develop the model for oxygen transport from the capillary to the tissues, we
will use a shell balance and the Krogh tissue cylinder model. The Krogh tissue
cylinder model is a simplified model of the tissue surrounding the capillary. It
assumes a cylindrical layer of tissue surrounding each capillary, that is fed from
only that capillary (Fournier, 44). The capillary is assumed to be cylindrical and
of constant radius. Three shell balances will be completed to describe the
transport of oxygen bound to hemoglobin, oxygen dissolved in the plasma, and
oxygen in the tissue. These balances will then be combined and simplified to
complete the model for oxygen transport.
Shell balance on hemoglobin bound oxygen in the capillary:
Variables:
C’ = concentration of oxygen in capillary that is bound to hemoglobin.
V = Average velocity of blood in the capillary
RHb0 = Production rate of oxygenated hemoglobin
r = distance in radial direction
z = distance in direction of flow

2rrzC '  2rrVC ' z  2rrVC '
t
z  z
 RHbO 2rrz
The first two terms on the right side of this equation represent the convective
transport of oxygen. No diffusive transport terms are include in this balance
since the oxygen is bound to hemoglobin (which is enclosed in the red blood
cells and travels with the flow of blood) and is not free to diffuse.
Dividing by 2rrz and taking the limit as z  0 gives:
C '
C '
 V
 R HbO
t
z
Shell balance on dissolved oxygen in the capillary:
Variables:
C = concentration of oxygen in dissolved in capillary
V = Average velocity of blood in the capillary
D = diffusivity of oxygen in bloodstream
rc = radius of capillary
Roxygen = Production rate of oxygen in the bloodstream
r = distance in radial direction
z = distance in direction of flow

2rrzC    2rrVC z  2rrVC
t
C

  2rr D
z

z  z
z
   2rz D C
r

  2rr D
C
z
z  z
r
C

   2rz
r

r  r




  Roxygen 2rrz

The first bracketed term is the convective mass transport of the dissolved
oxygen. The second bracketed term is the diffusive mass transport in the radial
direction, and the third bracketed term is the diffusive mass transport in the axial
direction.
Dividing by 2rrz and taking the limit as z  0 and r  0 gives:
 1   C   2 C 
C
C
 V
 D
r
  2   Roxygen
t
r
 r r  r  z 
We can now simplify and combine these two balances by noting the following:
C ' C '  C 
C


m
t
C  t 
t
C ' C '  C 
C


m
z
C  z 
z
where m = dC’/dC
RHbO = - Roxygen
by Henry’s law, PO2=HC
where PO2 =partial pressure of O2
and H=Henry’s constant
This gives us a combined balance on oxygen dissolved in the plasma and bound
to hemoglobin as follows:
1  m
 1   PO 2   2 PO 2 
PO 2
P
 V 1  m O 2  D 
r

2 
t
z
 r r  r  z 
(Equation A)
Shell balance on dissolved oxygen in the tissue:
Variables:
CT = concentration of dissolved oxygen in tissue
DT = diffusivity of oxygen in tissue
rT = radius of tissue cylinder
metabolic = Volumetric oxygen consumption rate in tissue
r = distance in radial direction
z = distance in direction of flow
T



C T
T C


2rrzC T   2rz D T


2

r

z
D
r
r  r 
t
r
r


T

C T
T C


  2rr D T


2

r

r
D
z
z
z



z  z

  metabolic 2rrz

The first bracketed term is the diffusive mass transport in the radial direction, and
the second bracketed term is the diffusive mass transport in the axial direction.
Dividing by 2rrz and taking the limit as z  0 and r  0 , and again noting
that PTO2=HCT where PTO2 is the partial pressure of oxygen in the tissue:

 POT2
POT2
T 1 
r
D 

t
 r r  r
  2 POT2 
T

 z 2   metabolicH


(Equation B)
Equations A & B can be further simplified with the following
Additional Assumptions:
1) steady state conditions – neglect all time derivatives
2) axial diffusion of O2 in bloodstream negligible compared to convective
transfer
3) since radial dimensions of tissue region much less than axial dimensions,
axial diffusion in the tissue can be ignored
4) negligible concentration gradients in the radial direction within the capillary,
and radially average the capillary oxygen levels – this is illustrated below
resulting equations:
   POT2 
  rmetabolicH T / D T
  r

 r  r 
 1   PO 2 
P
V 1  m O 2  D 
r

z
 r r  r 
(Equation B2)
(Equation A2)
To find the average radial PO2, we integrate equation A2 over the r direction
rc
2  1  mV
0
2 1  mV
rc
PO 2
1    P
rdr  2D    r O 2
z
r  r  r
0
rc
PO 2
d
P
r
dr

2

Dr
O
2
c
dz 0
r

 rdr

rc
rc
Recognizing that the average PO2 can be defined as PO 2 rc2  2  PO 2 rdr , the
0
above equation becomes
1  m 
d PO 2

dz
2 D dPO 2
rcV dr
rc
Further assuming that the capillary wall provides negligible resistance to mass
transfer, so that the oxygen flux at rc is continuous,
1  m 
d PO 2

dz
2 D T dPO 2
rcV dr
rc
(Equation A3)
Analytical solutions for oxygen profiles in capillary and tissue regions:
Tissue:
   POT2
  r
 r  r

  rmetabolicH T / D T


Boundary conditions:
(Equation B2)
@ r = rc
POT2  PO 2
@r = rT
dPOT2
0
dr
Integrating twice gives:
r 2
HT   r
1  
P r , z   PO 2 z   c metabolic
4D T
  rc
T
O2



2
 r 2
HT  r 
  T metabolic
ln  
2D T

 rc 
(Equation B3)
Capillary:
1  m 
d PO 2
dz
T
2 D T dPO 2

rcV dr
rc
(Equation A3)
To solve for this equation for the oxygen profile in the capillary PO2 z  , we need
dPOT2
rc . We can differentiate equation B3 with respect to r to
dr
T
metabolicH T rc  rT2 
T dPO 2
 2  1 .
obtain:
D
rc  
r

dr
2
 c

Substituting this back into equation A3 and integrating gives:
to evaluate D T
PO 2 z   PO 2
metabolicH T

in
1  mV
 r
 T
 rc
2


  1 z


(Equation A4)
Equations A4 and B3 can be used to determine the oxygen profile in the capillary
or in the tissue region. These equations can be applied to natural blood or an
artificial oxygen carrier by correctly evaluating the parameter m.
POn2
C'

For blood, recognizing that Y 
,
C ' sat P50n  POn2
n 1
m
PO 2
dC '
dY
dY
 C ' sat
 HC ' sat
 nP50n HC ' sat
dC
dC
dPO 2
P50n  POn2
For the artificial oxygen carrier PFC,

m
dC '

dC
d

2
PO 2
T
H PFC  H
P
H PFC
d OT2
H
Tissue
Membrane
O2
O2
O2
O2
O2
O2
O2
O2
O2
O2
O2
O2
Capillary
O2
O2
O2
O2
O2
O2
O2
O2
O2
O2
Flow
O2
O2
Tissue
Hemoglobin or artificial oxygen carrier
Plasma
This figure illustrates hemoglobin or an artificial oxygen carrier flowing in the plasma.
Oxygen is released from the oxygen carrier to the plasma. Oxygen dissolved in the
plasma is then transported across the capillary wall to the tissues.
Figure 1
Results and Discussion
Several assumptions were made in the development of the oxygen
transport equations that will cause the values in figures 2 through 4 to be
overstated. Radial diffusion in the capillary has been partially neglected by
averaging the radial concentration. Resistance to diffusion through the capillary
wall has been neglected. Since oxygen is such a small molecule, neglecting this
resistance should not cause a significant error. The values for the physical
properties used to obtain solutions for this model were values given for the
similar model in the Fournier reference. The tissue perfusion rate of 0.7
ml/cm3*min is the given value for the blood perfusion rate to the heart. The
assumed exit PO2 level of 31mmHg has been verified by the calculations for
PO2(z) at the capillary exit (30.9mmHg).
All oxygen values given for the tissue region are at the outside radius of
the tissue cylinder. For the given physical values, Figure 2 shows that for natural
blood, no anoxic regions exist in the capillary or tissue. For PFC at PO2in =
95mmHg, the tissue becomes anoxic at about z=. 088cm. The partial pressure
of oxygen at the entrance to the capillary was also evaluated at 45mmHg and
145mmHg to simulate the possibility of the patient receiving less than or more
than the ideal amount of oxygen during transfusion. Figure 3 illustrates that at
45mmHg, The tissue becomes anoxic after the PFC has traveled only about 30%
of the way down the capillary. The fluid in the capillary will become anoxic at
about 0.055cm. This illustrates the need to keep the patient well oxygenated
while using this product.
The oxygen profiles were also evaluated for the case of a 1:1 mixture of
PFCs and natural blood. I evaluated these mixtures with PO2in for blood constant
at 95mmHg, and allowed the PO2in for PFC to vary from 45mmHg to 145mmHg. I
have insufficient data to determine the relationship between blood and PFC P O2
levels. For the case of PO2in=45mmHg, it does seem reasonable that since PFC
can give up its oxygen so easily, there would be a lower concentration in the
blood due to the PFC as compared to natural blood. The fact that PFC absorbs
oxygen in the lungs in preference to the red blood cells gives credence to the
possibility that the concentration of O2 in the blood due to PFC could exceed that
due to natural blood. Either case would be contingent on the concentration of O 2
available in the lungs. Figures 4,5, and 6 compare the oxygen profiles for the
combination of blood and PFC to natural blood at PO2in=95mmHg. Figure 4 shows
that for both components having a PO2in=95mmHg the tissue just becomes anoxic
at the corner of the cylinder where z=L. Figure 5 shows no anoxic region, but
there is the possibility of tissue damage from over oxygenation. Figure 6 shows
significant anoxic regions in the tissue at PFC PO2in=45mmHg.
120.0
Partial Pressure Oxygen (mmHg)
100.0
Bloodcapillary
blood-tissue
80.0
60.0
40.0
20.0
0.0
0.000
0.020
0.040
0.060
0.080
0.100
0.120
-20.0
Distance (cm)
Figure 2. Partial Pressure of Oxygen vs. distance along capillary (PO2in = 95 mmHg)
Partial Pressure of Oxygen (mmHg)
200.0
95mmHg-capillary
95mmHg-Tissue
145mmHg-capillary
145mmHg-tissue
45mmHg-capillary
45mmHg-Tissue
150.0
100.0
50.0
0.0
0.000
0.020
0.040
0.060
0.080
-50.0
-100.0
Distance (cm)
Figure 3. Oxygen Profile for PFC
0.100
0.120
Partial Pressure O2 (mmHg)
100.0
blood- capillary
80.0
blood - tissue
60.0
40.0
20.0
0.0
0.000
0.020
0.040
0.060
0.080
0.100
0.120
-20.0
Distance (cm)
Figure 4. Partial Pressure of Oxygen vs. distance along capillary (Blood and PFC 1:1,
PFC PO2=95 mmHg)
140.0
Partial Pressure O2 (mmHg)
120.0
blood - capillary
blood - tissue
100.0
80.0
60.0
40.0
20.0
0.0
0.000
0.020
0.040
0.060
Distance (cm)
0.080
0.100
Figure 5. Partial Pressure of Oxygen vs. distance along capillary( Blood and PFC 1:1,
PFC PO2=145 mmHg)
0.120
120.0
100.0
blood - capillary
Partial Pressure O2 (mmHg)
80.0
blood - tissue
60.0
40.0
20.0
0.0
0.000
0.020
0.040
0.060
0.080
0.100
0.120
-20.0
-40.0
Distance (cm)
Figure 6. Partial Pressure of Oxygen vs. distance along capillary (Blood and PFC 1:1,
PFC PO2=45 mmHg)
References
AP – Associated Press Release, Houston, 10 April 1998
Fournier, Ronald L. Basic Transport Phenomena in Biomedical Engineering.
Philadelphia, PA: Taylor & Francis, 1995
Lawton, Graham. “Can Substitutes Solve the Blood Crisis?” Chemistry and
Industry. 04 January, 1999: 9
Nucci, Mary L. and Abraham Abuchowski. “The Search for Blood Substitutes.”
Scientific American. February 1998:72-77.