Laura Causey

The City College of New York, New York, New York

Beth E. Lewandowski

Glenn Research Center, Cleveland, Ohio

Sheldon Weinbaum

The City College of New York, New York, New York

A Mathematical Model of Oxygen Transport in

Skeletal Muscle During Hindlimb Unloading

NASA/TM—2014-216631

March 2014
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Laura Causey

The City College of New York, New York, New York

Beth E. Lewandowski

Glenn Research Center, Cleveland, Ohio

Sheldon Weinbaum

The City College of New York, New York, New York

A Mathematical Model of Oxygen Transport in

Skeletal Muscle During Hindlimb Unloading

NASA/TM—2014-216631

March 2014

National Aeronautics and

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Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2014-216631 1 

A Mathematical Model of Oxygen Transport in 

Skeletal Muscle During Hindlimb Unloading 

Laura Causey 

The City College of New York 

New York, New York 10031 

Beth E. Lewandowski 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Sheldon Weinbaum 

The City College of New York 

New York, New York 10031 

Abstract 

During hindlimb unloading (HU) dramatic fluid shifts occur within minutes of the suspension, 

leading to a less precise matching of blood flow to O2 demands of skeletal muscle. Vascular resistance 

directs blood away from certain muscles, such as the soleus (SOL). The muscle volume gradually reduces 

in these muscles so that eventually the relative blood flow returns to normal. It is generally believed that 

muscle volume change is not due to O2 depletion, but a consequence of disuse. However, the volume of 

the unloaded rat muscle declines over the course of weeks, whereas the redistribution of blood flow 

occurs immediately. Using a Krogh Cylinder Model, the distribution of O2 was predicted in two skeletal 

muscles: SOL and gastrocnemius (GAS). Effects of the muscle blood flow, volume, capillary density, and 

O2 uptake, are included to calculate the pO2 at rest and after 10 min and 15 days of unloading. The model 

predicts that 32 percent of the SOL muscle tissue has a pO2 1.25 mmHg within 10 min, whereas the 

GAS maintains normal O2 levels, and that equilibrium is reached only as the SOL muscle cells 

degenerate. The results provide evidence that there is an inadequate O2 supply to the mitochondria in the 

SOL muscle after 10 min HU. 

Introduction 

In 1992 McDonald et al. wrote two papers describing how blood flow is redirected upon 15 days of 

hindlimb unloading (HU) in the rat (Refs. 1 and 2). Hindlimb unloading is of interest since it mimics 

aspects of microgravity by removing weightbearing loads from the hindlimbs and produces a cephalic 

fluid shift, similar to that which occurs in astronauts in a microgravity environment (Refs. 3 and 4). Both 

papers evaluated the possibility that the changes in perfusion to the soleus (SOL) and gastrocnemius 

(GAS) muscle during HU contributed to their degeneration. Hindlimb unloading reduces the release of 

vasodilatory metabolites, such as nitric oxide, and decreases the responsiveness of the sympathetic 

nervous system. These actions change the blood flow path (Refs. 1 and 2). The first study compared the 

blood flow and perfusion rate (blood flow relative to the muscle weight) to the SOL in standing, 10 min 

HU (the control group), and 15 day HU rats (Ref. 2). The group found that the relative blood flow to the 

SOL in the 15 day HU rats was higher than what was considered to be the control group (10 vs. 8 mL 

(min  100 g)–1, respectively). The second study examined these parameters in 15 day HU and cagecontrolled rats and concluded that the relative flow to the 15 day HU rats was about the same as the 

control (12 vs. 20  (min  100 g–1, respectively)) (Ref. 1) even though the relative flow was reduced by 
NASA/TM—2014-216631 2 

40 percent. The studies mutually concluded that the fatigability of the SOL was not a result of reduced 

muscle blood flow. 

Upon a closer look, this deduction is problematic. The studies neglected to compare the O2 uptake of 

the SOL to the O2 delivery to determine if the muscle was adequately oxygenated. Furthermore, one study 

assumed that the rats that underwent 10 min of HU would be adequate controls and that in these animals 

the SOL was sufficiently perfused and oxygenated (Ref. 2). An analysis by Bloomfield et al., 2006 based 

on data collected by Colleran et al., 2000 reviewed how the perfusion rate to the SOL changes after 

10 min, 7 days, and 28 days HU in a rat (Refs. 5 and 6). The study found that the vast majority (80 percent) 

of the perfusion rate changes when moving from a standing to a HU position within the first 10 min. 

In this analysis, the data predominately from the aforementioned studies was reexamined to consider 

the possibility that hypoxia in the SOL may cause adverse effects, such as degeneration. Due to limited 

information regarding how the O2 uptake in the rat SOL changes upon HU and exercise, the scope was 

narrowed to predict only the percentage of the SOL that becomes hypoxic if this muscle maintains normal 

resting metabolism as it would on earth. This is contrary to that which actually occurs during spaceflight 

since slow twitch SOL muscles transition to fast-twitch muscles. For example, after 5 weeks of hindlimb 

suspension, the maximum rate of O2 uptake is decreased by 19 percent in the SOL muscle (Ref. 7). 

Further, there is evidence in the SOL of a cat that suggests that as muscle perfusion is diminished there is 

a downregulation in O2 uptake (Ref. 8). Despite this, the model provides evidence that within the first 

week of HU the SOL cannot maintain a normal resting metabolism without facing the consequences of 

partial tissue hypoxia. It therefore seems plausible that an O2 deficit may have adverse results on this 

muscle during HU. 

Methods 

The extent of hypoxia was first estimated by treating the tissue and blood as well-mixed regions. 

Table 1 lists the flowrate, Q, perfusion rate, Q/MT, and volume, VT, of the SOL and GAS at rest and after 

10 min and 15 days of HU, as recorded by the two studies by McDonald, et al., 1992 (Refs. 1 and 2). The 

volume was determined by dividing Q into Q/MT and assuming that the density of muscle is 1.056 g  cm–3. 

Arterial blood is normally saturated with O2 so that the concentration of O2 is CaO2 = 0.2 mLO2/ 

mLblood. To estimate the O2 supply to each muscle Q/MT data from the two McDonald, et al., 1992 studies 

were multiplied by this value, so that O2 Supply = 0.2 mLO2/mLblood  Q/MT. The O2 uptake of SOL and 

GAS are 2.2 mLO2/(100 g  min) and 0.8 mLO2/(100 g  min), respectively (Refs. 9 and 10). The O2

supply for each muscle was plotted at rest and after 10 min and 15 days of HU and compared with the O2

uptake of the respective muscles. The results of this calculation are plotted in Figure 2. 

The calculations using a spatially lumped region depend on the fact that the radial gradient of pO2 is 

quite small, which was checked using the Krogh model in which the tissue surrounding each capillary is 

treated as a cylinder. The validity of the assumptions made for this spatially lumped region model are 

analyzed in the discussion. 

TABLE 1.—FLOW AND TISSUE VOLUME FROM MCDONALD, ET AL., 1992 STUDIES

Resting 10 min HU 15 days HU Ref.

Qa Q/MT

b VT 

cm3

Qa Q/MT

b VT 

cm3

Qa Q/MT

b VT 

cm3

SOL ---- ---- ---- 1.6 8.0 0.19 0.8 10.0 0.08 (Ref. 2) 

3.7 20.0 0.18 ---- --- ---- 1.3 12.0 0.10 (Ref. 1) 

GAS ---- ---- ---- 13.3 7.1 1.79 13.6 9.2 1.40 (Ref. 2) 

26.1 12.4 1.99 ---- --- ---- 23.6 14.9 1.50 (Ref. 1) a

Q is in units of mL  (100  min)–1

b

Q/MT is in units of mL  (100 g  min)–1
NASA/TM—2014-216631 3 

Figure 1.—Krogh cylinder cross-section with dimensions labeled.

The cross section parallel to the axial direction of a Krogh cylinder is shown in Figure 1, with 

dimensions labeled. To find the amount of oxygen in a given physical system a mass balance was 

performed. In Equation (1) the change in oxygen concentration with respect to time is equal to the 

gradient of the mass flux of the oxygen minus the rate of oxygen uptake by the muscle fibers, written as, 

	C / 	t    n  R . (1) 

The mass flux of the system is a sum of the diffusive flux, according to Fick’s Law of Diffusion, and the 

convective flux where D is the diffusivity and v0 is the volume average velocity, given by, 

n  DC 
 Cv0 . (2) 

Using Equations (1) and (2) the change in oxygen concentration per unit time is 

	C /	t     DC 
 Cv0  R . (3) 

Equation (3) was used to determine the mass balance of oxygen in the capillaries and tissue. A detailed 

description of the Krogh cylinder calculations can be found in Reference 41. Calculations are briefly 

described for this Krogh Cylinder problem. 

Tissue 

In the tissue a number of assumptions were made to simplify the problem: (1) steady state system, 

(2) negligible convection effects, and (3) diffusion in the axial direction is neglected. Henry’s Law was 

used to express the concentration of oxygen in terms of partial pressure (pO2 = C  H). The boundary 

conditions were set for this equation. At the interface between the capillary and tissue (r = rc) the partial 

pressure of oxygen is the same (pO2 = pO2

T

) and the oxygen flux is continuous (D  2 = DT x 

O2

T

). At the Krogh tissue cylinder diameter (r = rT) the flux is zero (O2

T

 = 0) since there is a 

plane of symmetry between cylinders. When the level of O2 in the muscle tissue reaches a state in which 

pO2

T

 = 0 it will not continue to uptake O2 since there is no O2 to be taken and it is impossible to have a 

negative oxygen distribution. To account for this, rT (z) was replaced with ra (z) (the radius at which 

tissue becomes anoxic) in cases in which the tissue became anoxic at a given radius along the length of 

the tissue cylinder. After applying the aforementioned assumptions and boundary conditions Equation (3) 

becomes 
NASA/TM—2014-216631 4 

      
  c T c T T T

pO r,z pO z RH / 4D r r 2r z ln r/r

2 2 2

2  2    , (4) 

for rC r rT ,where pO2(z) is the partial pressure of oxygen in the capillary, H is Henry’s Constant, and 

T denotes the properties of the tissue. 

Capillary 

Equation (3) was used to solve for the concentration of oxygen in the blood by assuming: (1) steady 

state, (2) no convection in the radial direction, and (3) axial diffusion is negligible when compared with 

axial convection. The mass balance for concentration of hemoglobin was found using Equation (3) as 

well, replacing C with C’, with the apostrophe denoting that it is the concentration of hemoglobin. Some 

key assumptions were made: (1) steady state, (2) no diffusive transport of hemoglobin, and (3) negligible 

change in the concentration of hemoglobin in the radial direction. It was assumed that there was adequate 

conductive mixing of O2 in the plasma. The mass balance of the oxygen was combined with that of the 

hemoglobin by recognizing that the rate of disappearance of oxygenated hemoglobin must equal the rate 

of appearance of dissolved oxygen. Henry’s Law was used to represent the equation in terms of the partial 

pressure of oxygen. Equation (5) is the result where m is the change in hemoglobin concentration with 

respect to the oxygen concentration, as given by, 

	pO / 	z  D /   v   1
 m 
  1/r  	 / 	r r  	pO / 	r 2 0 2 . (5) 

The boundary condition was applied which states that at the capillary radius the flux of the oxygen is 

continuous and the radial average of the oxygen within the capillary was taken since the bulk of the mass 

transfer resistance is in the tissue and not the capillary (Ref. 11), leading to 

    cr T c T d pO / dz 2D / v 1 m r dpO / dr | 2  0 
 2 . (6) 

The term dpO2

T

/dr was found by taking the derivative of Equation (4) with respect to r and evaluating it 

at rc. An additional boundary condition was applied since at the capillary entrance (z = 0) pO2in is 

known. Using these methods, the partial pressure of oxygen within the capillary for 0 z L is 

p  z p RH    v   m 
  r z r z T c T

in

O O / 1 / 1 2 2

2  2  0 
  . (7) 

Oxygen-Hemoglobin Dissociation Curve and Myoglobin Effects 

The effects of the O2 carrying proteins, hemoglobin and myoglobin, were considered. Oxygen 

dissociates from hemoglobin as O2 is lowered, enhancing its availability. The Hill Equation 

(Equation (B7) in the Appendix) was used to calculate the dependence of the slope of the O2-hemoglobin 

dissociation curve (m) on the pO2 along the length of the capillary. We used the average m value for the 

range of pO2 considered (0 to 71 mmHg normally and 0 to 64 mmHg in environmental hypoxic 

conditions, as will be discussed shortly) along the length of the capillary. Myoglobin enhances O2

transport at low O2 concentrations. At critical O2 concentration myoglobin-assisted transport is greatest at 

which point the diffusivity is enhanced by a factor of 1.06 in skeletal muscle (Ref. 12). If any portion of 

the SOL or GAS reached levels 1.25 mmHg the diffusion coefficient was multiplied by this factor to 

account for the modest myoglobin effect. 
NASA/TM—2014-216631 5 

Parameters 

The perfusion rate equals the flowrate of blood per unit volume of the total muscle, Q/VT. Therefore, 

the velocity of the blood flowing through each capillary, v0, is 

  2 2

0v  Q  r

T  L/ VT  rC . (8) 

The Q, VT, and rT were variable, whereas other parameters in this equation remained constant. 

The tissue cylinder radius, rT, was calculated by assuming that the capillaries within the skeletal 

muscle are evenly spaced. The calculation for rT is 

r nc T 1/ 2  , (9) 

where nc is capillary density. The capillary density in the SOL and GAS muscles of the rat is 836 and 

628 capillaries/mm2

, respectively (Refs. 13 and 14). The results of this calculation are listed in Table 2. 

Literature values for the capillary density of rats under HU conditions were used for both muscles to 

estimate the tissue cylinder radius after 15 days HU. The capillary density in the SOL muscles of the rat 

after HU for 9 days is 994 capillaries/mm2

 (Ref. 13). It was assumed that this value would not change 

significantly after 6 more days. We inserted this value into Equation (9). After 10 days of HU in the GAS 

of a mouse the capillary to fiber ratio decreases by 19.5 percent (Ref. 21). However, after 15 days the 

volume of the GAS muscle changes from 1.99 to 1.50 cm3

. Equation (10) was used to estimate the rT in 

the GAS muscle of the hindlimb unloaded rat after 15 days and is given by 

   1.50/1.99 /628/mm 1 0.195  2 2

r

T    . (10) 

The values for the muscle tissue cylinder radii are listed in Table 2. 

TABLE 2.—MODEL PARAMETERS 

Parameter Value Source 

DT 1.65 to 3.0410–5 cm2

/s (Refs. 15 and 16) 

N 2.64 (Ref. 17) 

pO50 26.2 (Ref. 17) 

Csat’ 8800 M (Ref. 18) 

H, HTb 0.74 mmHg/M (Ref. 18) 

pO2,in (normal) 71.4 mmHg (Ref. 19) 

pO2,in 

(HYP) 

64.4 mmHg a 

L 500 m (Ref. 20) 

rc 2.5 m (Ref. 20) 

Rsol 17.04 M/s a 

Rgas 6.20 M/s a

rT,sol (normal) 20 m a

rT,gas (normal) 23 m a

rT,sol

(15 days HU) 

18 m a

rT,gas 

(15 days HU) 

22 m a

a

Calculated 

b

Assumed that H = HT
NASA/TM—2014-216631 6 

Groebe and Thews (1990) suggested that extracellular and intracellular pO2 diffusivities are different. 

They referred to the space containing plasma, capillary endothelium, and interstitial space as the 

“perierythrotic regions” and claimed that the diffusivity of this region, 1.6510–5 cm2

/s (Ref. 15), has 

higher diffusion properties than that within the myocyte, which was assumed to be comparable to water 

(3.0410–5 cm2

/s) (Ref. 16). The Krogh Cylinder Model was used to compare the percent volume of the 

tissue that becomes hypoxic using these two diffusion coefficients. 

The amount of O2 extracted from the blood in resting SOL and GAS muscles is 2.2/20.0 mLO2/mLbl

(equivalent to 4840 M) and 0.8/12.4 mLO2/mLblood (equivalent to 2839 M), respectively. The rate of O2

uptake of the resting SOL muscle tissue may be estimated as 

s L s

Rsol

M 17.04

1000mL

1L

L 

1000cm mol

cm

1.056g 4840 100g 60

20mL

bl

bl

tiss bl

3

3

bl        

  . (11) 

The rate of O2 uptake of the GAS muscle tissue was similarly found to be 6.20 M/s. The model 

parameters are listed in Table 2. The parameters pO50, Csat’

, and N were used to calculate the slope of the 

O2-hemoglobin dissociation curve (m) in Equation (B7), as discussed in Appendix B. 

Simulated Hypoxic Environmental Conditions 

Astronauts perform a pre-breathe protocol to remove nitrogen from the body before a spacewalk. To 

minimize the pre-breathe time crewmembers may live in an atmosphere that has a low partial pressure of 

nitrogen, which can be achieved at lower ambient pressures. To analyze this effect in our model, we 

considered a case in which the atmospheric pressure is lowered from 14.7 to 8 psi and the O2

concentration is raised from 21 to 32 percent, causing the cabin partial pressure of O2 to be reduced. 

The normal partial pressure of O2 in air is: 760 mmHg  0.21 = 159 mmHg. Under these normal 

conditions, the partial pressure of O2 that enters the capillary bed of the muscle is ~71.4 mmHg (Ref. 19) 

since there is a longitudinal gradient of pO2 in precapillary microvessels (Ref. 22). If we lower the cabin 

pressure to 414 mmHg and increase the O2 concentration to 32 percent, the partial pressure of O2 is 

132 mmHg. The arterial blood partial pressure of O2 was measured by Fisher, et al., 1992 in a rat after it 

was exposed to various degrees of hypoxia. Data excerpts from the aforementioned study are listed in 

Table 3. To estimate the arterial pO2 for an inspired partial pressure of O2 of 132 mmHg (the equivalent 

of 17 percent O2 at a cabin pressure of 760 mmHg) a curve was fitted to the data from Table 3. We found 

the arterial pO2 to be approximately 64.4 mmHg. We used this value as an input for our model. 

Mathematical Model 

MATLAB (V. R2007b, MathWorks, Natick, MA) software was used to perform the calculations. The 

pO2 at the capillary entrance was assigned and it was first assumed that there were no hypoxic areas. The 

pO2 in the capillary and tissue was then calculated. If pO2

T 0 at any point along the capillary rT (z) was 

reduced to a value ra (z) at that location and the model was iterated once again. 

TABLE 3.—ARTERIAL PO2 AT DIFFERENT 

O2 CONCENTRATIONS 

(Data from Fisher, et al., 1992) 

10% 21% 30% 

Arterial pO2, mmHg 55.1 71.4 91.4 
NASA/TM—2014-216631 7 

Results 

The first analysis of data that we made treated the tissue and blood as well-mixed regions. In a resting 

state both the SOL and GAS O2 supplies exceed the demand by 1.8 mL/(min  100 g) and 1.7 mL/(min 

100 g), respectively, assuming that the radial gradient of pO2

T

 is negligible. After 10 min of HU the flow 

in the SOL and GAS is reduced from 20.0 and 12.4 mL/(min  100 g) to 8.0 and 7.1 mL/(min  100 g) 

and the O2 supply becomes 1.6 and 1.4 mL/(min  100 g), respectively, as shown in Figure 2 (Refs. 1 and 

2). In the case of the SOL this means that portions of the muscle may become hypoxic, assuming normal 

muscle metabolism is maintained, whereas the GAS is adequately perfused. After 15 days HU the SOL 

and GAS experience a 44 to 58 percent and 22 to 25 percent volume decrease. The O2 deficit is 

ameliorated by the reduced muscle volume. 

The distribution of O2 within a cross-section of the Krogh tissue cylinder in the SOL and GAS 

muscles is represented by Figure 3. The arrow located at the bottom left-hand corner indicates the 

location at which the “simulated blood” enters the Krogh cylinder cross-section at z = 0. At this point 

normally pO2,in = 71.4 mmHg. The space in which the blood is flowing is from r = 0 to 2.5 m and 

z = 0 to 500 m. The endomysial and perimysial tissue resides in the space from r = 2.5 m to rT. The 

partial pressure of O2 was color coded to indicate the pO2

T

. The partial pressure of O2 at which 

mitochondrial metabolism becomes inhibited in the muscle is ~1.25 mmHg (Ref. 23). The black region 

represents the portion of the tissue reduced below this level. 

Figure 2.—Comparison of O2 supply and demand. SOL 1, GAS 1 data from 

Reference 1; SOL 2, GAS 2 data from Reference 2. The O2 demand in the 

SOL and GAS was experimentally determined by References 9 and 10.
NASA/TM—2014-216631 8 

Figure 3.—Distribution of O2 in the SOL (A to C) and GAS (D to F) muscles. The left-most plots (A and D) depict the 

resting tissue. The center plots (B and E) show the tissue after the blood flow is redistributed after 10 min HU. The 

right-most plots depict the tissue when the blood flow and volume of the muscle has been reduced after 15 days 

HU. The mean pO2

T for each plot was (A) 50 mmHg, (B) 25 mmHg, (C) 37 mmHg, (D) 59 mmHg, (E) 50 mmHg, 

and (F) 61 mmHg. The percent volume of the tissue that is hypoxic in (B) is 32 percent. (DT

 = 3.0410–5 cm2/s). 

Note that (C) and (F) were generated using data from Reference 1. 

Normal Conditions 

The model predicts that the average pO2

T

 in the SOL and GAS muscles during normal conditions is 

50 and 59 mmHg, respectively. Experimentally, the average SOL and GAS intramuscular pO2

T

 is 

~51 mmHg in the rat (Ref. 24) and ~48 mmHg in the canine (Ref. 25), respectively. We estimate that 

the pO2

T

 in the SOL muscle ranges from 30 to 71 mmHg and the GAS muscle has a pO2

T

 from 47 to 

71 mmHg. During normal resting conditions the SOL muscle is perfused by blood over 1.5 times that of 

the amount of blood perfused to the GAS muscle per tissue weight (Ref. 1). The capillary density of the 

SOL and GAS muscles are nearly the same. Therefore, the reason the level of pO2

T

 is less in SOL muscle 

as compared with the GAS is almost entirely attributed to the differences in O2 consumption of the two 

muscles. To achieve our results, the value of tissue diffusivity, DT

, was first assumed to be that of water, 

3.0410–5 cm2

/s (Ref. 16). This is the best possible scenario since the DT is equal to 1.6510–5 cm2/s at the 

wall of the capillary (Ref. 15). The influence of DT

 on the O2 distribution was further analyzed, as 

described shortly. 
NASA/TM—2014-216631 9 

10 min—Most of the changes in blood flow that arise in the SOL and GAS muscles during HU occur 

within a timeframe of 10 min of weightlessness (Ref. 5). When the blood flow changes due to HU the O2

distribution is reduced, as shown in Figure 3(B) and (E). The percent volume of the SOL in which pO2

T

1.25 mmHg is 32 percent. Our model predicts that the average pO2

T

 in the SOL and GAS muscles is 

25 and 50 mmHg, respectively. The pO2

T

 in the SOL muscle ranges from 0 to 71 mmHg and the GAS 

muscle has a pO2

T

 from 29 to 71 mmHg; still within a normal range. 

15 days—Within weeks of decreased blood flow, the muscle volume decreases in both muscles. 

Using our model, we found that the average pO2

T

 in the SOL and GAS muscles when the blood flow and 

muscle volume were changed to the values provided by Reference 1 and the capillary density was 

changed to the HU values listed in Table 2 was 37 and 61 mmHg, respectively. The pO2

T

 in the SOL and 

GAS muscles ranged from 3 to 71 and 51 to 71 mmHg, respectively. 

Similarly, the mean pO2

T

 predicted using values from Reference 2 was 31 mmHg for the SOL and 

55 mmHg for the GAS. In the same study the pO2

T

 in the SOL and GAS muscles ranged from 0 to 71 and 

39 to 71 mmHg, respectively. Figure 3(C) and (F) demonstrate that the pO2

T

 returns to higher 

oxygenation levels after the volume of the muscle is reduced as compared with Figure 3(B) and (E). 

According to Reference 1 this O2 distribution is only achieved if the SOL and GAS experience a 44 and 

25 vol% decrease, respectively. 

Simulated Hypoxic Environmental Conditions 

When the partial pressure of O2 is decreased to 132 mmHg, the slightly hypoxic environment reduces 

the O2 distribution so that the range of pO2

T

 is 26 to 64 and 42 to 64 mmHg in the SOL and GAS, 

respectively. 

10 min—The exposure to a hypoxic environment in addition to the changes in the distribution of 

blood flow causes further O2 depletion in the tissue. In this scenario, the SOL and GAS muscle pO2

T

ranges from 0 to 64 and 26 to 64 mmHg, respectively. The average pO2

T

 for the SOL and GAS after these 

changes is 22 and 45 mmHg, respectively. 

15 days—In the same environment the O2 distribution was predicted after the blood flow and volume 

were changed and the muscle reaches a state of equilibrium using the data from the two studies listed in 

Table 1. When the perfusion rate of Reference 1 was used in the model the pO2

T

 in the SOL and GAS 

muscles ranged from 2 to 64 and 46 to 64 mmHg, respectively. The average pO2

T

 in the SOL and GAS 

was predicted to be 33 and 55 mmHg, respectively. Using data from Reference 2, the pO2

T

 in the SOL 

and GAS muscles ranged from 0 to 64 and 35 to 64 mmHg and the average pO2

T

 was predicted to be 

28 and 49 mmHg, respectively. Table 4 shows the estimated percentage of the tissue that has an O2 deficit 

under these different circumstances for DT

 = 3.0410–5 cm2/s. 

TABLE 4.—PERCENTAGE OF HYPOXIC TISSUE 

IN DIFFERENT CONDITIONS IN THE SOL 

% Hypoxic tissue 

DT = 3.0410–5

cm2/s 

% Hypoxic tissue 

DT = 1.6510–5

cm2/s 

Restingb 0.0 0.0 

Restingb HYP 0.0 0.0 

10 min HUa 31.7 32.5 

10 min HUa HYP 33.2 34.0 

15 daysa 14.6 15.4 

15 daysa HYP 15.8 16.7 

15 daysb 0.0 0.0 

15 daysb HYP 0.0 0.6 a

Mcdonald, et al., 1992 (Ref. 2) 

b

Mcdonald, et al., 1992 (Ref. 1) 
NASA/TM—2014-216631 10 

The Krogh Cylinder was evaluated for each circumstance in the SOL using two different diffusion 

coefficients. After 10 min HU in an ambient environment there was 31.7 and 32.5 percent hypoxic tissue 

when DT

 = 1.6510–5 and 3.0410–5 cm2/s, respectfully. This information suggests that the percentage of 

hypoxic tissue is mostly dependent on the tissue perfusion with the tissue diffusivity being a less 

important factor. 

Discussion 

In order for long duration human space exploration, such as from Earth to Mars, to become a reality, 

the impact of the missions on the health of the astronauts needs to be understood and mitigated. Since 

limited astronaut spaceflight data exists and is difficult to obtain, other ground-based methods have been 

established to simulate spaceflight conditions, such as HU. 

The conventional belief is that during spaceflight muscle fibers atrophy and, due to a smaller fiber 

diameter, the distance between the O2 in blood residing in the capillaries and the mitochondria decreases, 

improving O2 transport (Ref. 26). However, the blood is redistributed within the first 10 min of HU 

(Ref. 5), while the capillary density and muscle volume changes are observed after at least one and two 

weeks have passed, respectively (Refs. 2 and 1). There are two reasons that the muscle decreases in 

volume: short-term redistribution of fluids in the interstitium and long-term muscle degeneration 

(Ref. 27). Dramatic fluid shifts occur within minutes of arriving in microgravity. During this time the 

hydrostatic pressure in the vascular system of the legs decreases and fluid from the interstitium is 

reabsorbed into the vascular system, as described by Starling’s Principle (Ref. 5). However, this action 

accounts for only 4 percent of the total SOL muscle loss whereas the remaining 96 percent of the total 

muscle volume lost occurs due to the downregulation of protein synthesis or muscle atrophy, which is 

measured over the course of weeks in rats (Ref. 28). Therefore, the distance between the capillary and the 

muscle fiber does not decrease immediately. In the past the conclusion was made that since the volume of 

the muscle changes, the capillaries are able to fully oxygenate the adjacent muscle fibers since the 

distance between them is reduced (Ref. 26). However, as we show, the oxygen is depleted from the 

muscle after the blood flow is redistributed within the first 10 min. By analyzing the O2 distribution using 

a mathematical model that simulates the fluid shift, the O2 distribution was predicted at different 

timeframes. The results support the idea that the O2 equilibrium is reached only as the muscle fibers 

degenerate. 

The redistribution of blood flow in the SOL and GAS muscle upon HU is most likely acutely due to a 

reduced release of vasodilatory metabolites, such as nitric oxide (Ref. 2). The chronic decreased 

responsiveness of the sympathetic nervous system is responsible for the inadequate regulation of blood 

flow in the long-term (Refs. 1 and 2). In the rat, muscle composed primarily of fast-twitch fibers has the 

potential for greater control of vascular tone by the sympathetic nervous system than that composed of 

slow-twitch fibers. Thus, a reduction in sympathetic mediated vascular tone would be expected to 

increase blood flow to muscle composed of fast-twitch fibers (Ref. 2). 

In skeletal muscle cells (also referred to as fibers), the mitochondria require O2 to undergo 

mitochondrial oxidative phosphorylation, the main source of ATP for a muscle fiber. The partial pressure 

of O2 at which mitochondrial metabolism becomes inhibited in skeletal muscle is ~1.25 mmHg (Ref. 23). 

When O2 is limited ATP generation begins to diminish but the cellular demands remain constant, leading 

to an energy deficit. This can be temporarily ameliorated by anaerobic ATP production. Certain muscle 

types are more successful than others in this regard. Type I (slow-twitch) fibers have many mitochondria, 

a high concentration of myoglobin, and a low glycolytic capacity so that they principally respire 

aerobically. Type II (fast-twitch) fibers rely mostly on anaerobic respiration (Ref. 26). Type IIA fibers 

have a high glycolytic capacity and a moderate oxidative activity whereas type IIB have a high glycolytic 

capacity and a low oxidative capacity. The SOL muscle contains approximately 91 percent type I fibers 

and 9 percent type IIA fibers whereas the GAS consists of 18 percent type I fibers, 30 percent type IIA 

fibers and 52 percent type IIB fibers (Ref. 29). Therefore, the SOL is more dependent on O2 than the GAS 

to generate ATP (Ref. 30). If ATP production does not meet the energetic demands the cell may survive 
NASA/TM—2014-216631 11 

only if it is able to initiate mechanisms that preserve its ionic integrity. Since protein synthesis is a major 

ATP sink it is a target for downregulation (Ref. 31). In microgravity muscle degeneration can be 

attributed to protein loss primarily due to a decline in protein synthesis (Refs. 32 and 33). 

The validity of using a Krogh cylinder to estimate O2 transport in a muscle has been debated. Ellis 

et al., 1983 questioned the Krogh cylinder approach to O2 diffusion given the substantial tortuosity and 

branching of the skeletal muscle capillary bed. A mathematical model for the transport of O2 in a 

contracted muscle becomes complicated since the capillaries become tortuous as the muscle shortens 

(Ref. 34). This finding limits the mathematical model described herein so that it is only valid for muscles 

in the resting and not the contracted state, which is the case for hindlimb unloaded muscles. Further, 

emphasis has been placed on events within the capillary and across the RBC-subsarcolemmal boundary as 

elucidated by Federspiel and Popel, 1986 and Groebe and Thews, 1990, which is not considered in the 

original Krogh Cylinder Model. Despite elegant calculations that predict the limitation of O2 transport in 

the plasma space (Ref. 35), recent studies with crocetin, a compound that enhances O2 transport in plasma 

failed to improve O2 conductance (Ref. 11). In a review article, Wagner, 2000 suggested that the reason 

for this is that there is adequate conductive mixing in the microcirculation. Richardson et al., 1995 

provided experimental evidence that there is a steep O2 gradient between the blood and myocyte and 

revealed that O2 pressure in the blood is ~10 times higher than in the cytoplasm during exercise in 

humans. He supported the idea that the path between the red cell to the sarcolemma plays an important 

role in determining maximal muscle O2 uptake (Ref. 36). Groebe and Thews (1990) suggested that 

extracellular and intracellular pO2 diffusivities are different. They referred to the space containing plasma, 

capillary endothelium, and interstitial space as the “perierythrotic regions” and claimed that the 

diffusivity of this region has different diffusion properties than that within a myocyte (Ref. 37). For this 

reason, the chosen mathematical model was evaluated using a diffusivity value of the perierythrotic 

region (DT

 = 1.6510–5 cm2/s), which has a high resistance as described by Groebe and Thews, 1990, and 

the diffusivity of the muscle fibers (DT

 = 3.0410–5 cm2/s). However, the O2 distribution in the model was 

found to be attributed mostly to the perfusion rate with the capillary density and diffusivity having less 

influence. 

In the present model, we assumed that the O2 uptake remains constant regardless of the fiber type 

transition that may occur in the muscle. This is contrary to that which actually occurs during spaceflight 

since slow twitch SOL muscles transition to fast-twitch muscles. After 5 weeks HU, the maximum rate of 

O2 uptake in the SOL is decreased by 19 percent (Ref. 7). However, our purpose was to focus on the role 

of the circulatory system in a microgravity environment, though the inclusion of the changes in the fibertype transition may be pursued in future work. 

In the mathematical model described herein there was no region of the GAS that exhibited a 

pO2

T

1.25 mmHg. However, the rat GAS muscle is reduced by 23 percent after 15 days of HU with 

4 percent of this loss accounted for by the degeneration of the white GAS (Ref. 1). Hepple et al., 2000 

showed a 45 percent reduction of GAS muscle fiber cross-sectional area when the absolute blood flow 

was reduced 30 percent in the canine model as a result of immobilization; however, the study did not 

investigate the changes in blood distribution within the muscle or the percentage of the type I fibers and 

whether they contributed to most of the degeneration (Ref. 38). The loss of GAS muscle could be caused 

by other factors in addition to O2 depletion such as stress and diet. Portions of the red GAS may have 

become deoxygenated due to the disproportional distribution of blood flow. The depletion of O2 may be 

part of the reason, but perhaps not the sole cause of the muscle degeneration. 

The downregulation of protein synthesis observed in the hindlimb unloaded SOL muscle may be a 

consequence of an inadequate O2 supply to the mitochondria which limits ATP production and, in turn, 

protein synthesis. In the rat SOL muscle the lack of O2 inhibits protein synthesis and increases the rate of 

protein degradation (Ref. 39). In microgravity muscle degeneration can be attributed to protein loss 

primarily due to a decline in protein synthesis (Refs. 32 and 33). Given that the SOL muscle is composed 

of predominately slow-twitch fibers which rely on aerobic respiration and that the SOL muscle has less 
NASA/TM—2014-216631 12 

blood when it is unloaded, it is possible that this is the reason that the SOL exhibits more degeneration 

than the GAS. This occurs in humans as well. In humans 6 months of weightlessness has been shown to 

lead to a loss of fiber mass, force, and power with the hierarchy of the effects being SOL type I  SOL 

type II  GAS type I  GAS type II (Ref. 40). 

Conclusions 

The model described herein predicts that the short-term simulated HU causes the rat SOL muscle to 

become hypoxic whereas the GAS maintains a normal tissue O2 distribution. The depletion of O2 may be, 

at least in part, an instigator of the degeneration of skeletal muscles. A reduced cabin partial pressure of 

O2 may cause further muscle degeneration. 
NASA/TM—2014-216631 13 

Appendix A.—Acronyms 

C oxygen concentration 

C’ hemoglobin concentration 

Csat' saturated concentration of oxygen bound to hemoglobin 

D diffusivity in the capillary 

DT

 diffusivity in the tissue 

H Henrys Law constant in blood 

HT Henrys Law constant in tissue

L capillary length 

m slope of the oxygen-hemoglobin dissociation curve

MT mass of the tissue 

n oxygen flux 

N Hill Equation coefficient 

nc capillarity 

pO2 partial pressure of oxygen 

pO2

T

 partial pressure of oxygen in the tissue

pO50 partial pressure of oxygen at which 50 percent of the O2 binding sites are filled

Q volumetric flowrate of blood 

ra radius at which the tissue becomes anoxic 

rC capillary radius 

rT tissue cylinder radius 

r radius in cylindrical coordinates 

R rate per unit volume of oxygen uptake 

t time 

v0 volume average velocity 

VT volume of the tissue 

Z distance in z direction 

NASA/TM—2014-216631 15 

Appendix B.—Derivation of Equations 

After expanding Equation (3) the result is 

  R

z

C C

r

rC

r r

v

z

C C

r r

C

r

r r

C t D 



 





	

	




	

	




	

	 

















	

	




	

	  




 





	

	

	

	 	 	  1 1 1 1 / 0

2

2

2

2

2 . (B1) 

We will refer back to Equation (B1). The Krogh model consists of (1) O2 transport through blood and (2) 

reaction in tissues. 

Tissue 

Henry’s law was used and Equation (B1) was simplified using the assumptions listed in the methods 

section so that 

T

T T

D

RH

dr



r

dr

d

r 













 2 1 . (B2) 

The properties of the tissue are denoted with a T. The boundary conditions were set for this equation. At 

the interface between the capillary and tissue the pO2 is the same and the O2 flux is continuous. 

BC1: c r  r , T 2  2 , r

 D

r

 D

T

T

	

	  		 2 2

At the Krogh tissue cylinder diameter the flux is zero since there is a plane of symmetry between the 

cylinders. 

BC2: T r  r , 	2 /	r  0 T

Equation (B2) was integrated twice and the boundary conditions BC1 and BC2 were applied. 

Equation (4) is the result. 

Capillary 

Equation (B1) was simplified using the assumptions listed in the methods section so that 

R

r

C

r

r r

D

z

C

v   



 



 



 





	

	

	

	  		 1

0 . (B3) 

Hemoglobin—Equation (B1) was simplified using the assumptions listed in the methods section where 

C’ denotes the hemoglobin concentration so that 

R	

z

C

v  	

	 ' 0 . (B4) 
NASA/TM—2014-216631 16 

The slope of the oxygen hemoglobin curve is 

C

C

m

	

	  '

. (B5) 

Combining Equations (B3) to (B5), recognizing R = R	, and applying Henry’s Law we have 

  



 



 



 





	

	

	

	


  	

	

r

p r

v m r r

D

z

p 2

0

2 1 O

1

O . (B6) 

The slope of the oxygen-hemoglobin curve can be expressed as a function of the Hill Equation 

coefficient, N, the partial pressure of O2 at which 50 percent of the O2 binding sites are filled, pO50, 

Henry’s Constant, H, the partial pressure of O2 in the blood, pO2, and the saturated concentration of O2

bound to hemoglobin, Csat' so that 


 2

50 50 2

1 O2 O / O O N N

sat

N N m  Np p HC p 
 p  . (B7) 

A derivation of Equation (B7) may be found on page 242 in Fournier, R L. Basic transport phenomena in 

biomedical engineering. New York: Taylor & Francis, 2007 (Ref. 41). 

The radial average of the capillary pO2 was taken since the bulk of the oxygen mass transfer 

resistance is not within the capillary, but in the tissue. The radially averaged pO2 level, pO2, is the sum 

of many thin-walled annuluses with radii from 0 to rc, each multiplied times their respective pO2, divided 

by the entire area. Therefore, Equation (B6) becomes 

  r r

r

 r

v m r r r

D

r r

z



r

c cr

c

r

c

   	 

 





	

	

	

	


 

  	  	

	

   2 1 1

1

2 1 2

0 2

0 0

2

2 . (B8) 

The left-hand size of this equation resembles the radially averaged pO2 level equation. The equation 

becomes 

  rc

c dr

dp

v m r

D

p dz

d | O

1

2 O 2

0

2


  . (B9) 

After applying BC1 Equation (B9) becomes 

  rc

T

c

T

dr

dp

v m r

D

p dz

d | O

1

2 O 2

0

2


  . (B10) 

By differentiating Equation (4) dpO2

T

/dr|rc can be solved. Equation (B10) becomes 

  













  
   1

1

O 2

2

0

2

c

T

T

r

r

v m

RH

p dz

d . (B11) 

Equation (B11) was integrated to find the axial change in the capillary oxygen partial pressure, 

Equation (7). 
NASA/TM—2014-216631 17 

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