**5. Present objective**

In this work we model the time delay of release of ATP as supporting work shows by Wan et al. [11] for shear-induced ATP release from red blood cells. A release rate which is a function of time and introduces a delay mechanism is introduced to show how the concentration of ATP is thus affected.

RBC-ATP Theory of Regulation for Tissue Oxygenation-ATP Concentration Model 161

*<sup>d</sup> Qc H S x q dx* (1)

[ 3].

0 1, ( ) *j j q Mr r* 

0 1 *R Sx R RSx* () 1 () (2)

 

(4)

By conservation of mass, the decline in oxygen flux must equal the rate of oxygen consumption, giving the following equation for the change in oxygen saturation, S(x), with

( ) *o D*

where Q is volume flow rate in an individual vessel, *<sup>o</sup> c* is the carrying capacity of RBCs at

The release rate of ATP from an RBC, R[S(x)], is defined by a decreasing linear function of oxyhemoglobin saturation based on experimental data. ATP release from human erythrocytes in response to normoxia and hypoxia was observed in in vitro

In general,the rate of change in plasma ATP concentration, C(x,t), is given by the difference

2 2 [ (1 ) ] [(1 ) ( )] [ ( , )] 2 ( , ) *R H C H QC x R H R S x t k RC x t T D Td t x*

where *HT* is tube hematocrit, R is radius of vessel and *<sup>d</sup> k* is a concentration rate

We assume that there is no convection in the vessel and there is no x-dependence. Equation

<sup>2</sup> ( ( )) 1 1 *T d T T*

In this chapter a model is developed which predicts tissue ATP concentrations as a variation

The ATP concentration within plasma as a variation of time due to changes in oxygen tension at the tissue surface is related to the release and degradation of ATP, by the

> ( ) *ATP dC RtC dt*

of time and depth into the tissue due to changing oxygen tensions.

*dC <sup>H</sup> <sup>k</sup> RSt <sup>C</sup> dt H R H*

(3)

A linear fit of experimental values defines the ATP release function of saturation :

100% saturation. *HD* is the discharge hematocrit, and 2 2

between the rates of ATP release and degradation:

(3) simplifies to the following equation:

**6. Method of solution** 

distance, x, along each arteriole:

experiments.[3].

constant.([3])

following equation:

**Figure 4.** [11]

#### **6. Method of solution**

160 Blood Cell – An Overview of Studies in Hematology

**Figure 4.** [11]

By conservation of mass, the decline in oxygen flux must equal the rate of oxygen consumption, giving the following equation for the change in oxygen saturation, S(x), with distance, x, along each arteriole:

$$\frac{d}{d\mathfrak{x}}\Big[\mathcal{Q}c\_oH\_DS(\mathfrak{x})\Big] = -q\tag{1}$$

where Q is volume flow rate in an individual vessel, *<sup>o</sup> c* is the carrying capacity of RBCs at 100% saturation. *HD* is the discharge hematocrit, and 2 2 0 1, ( ) *j j q Mr r* [ 3].

The release rate of ATP from an RBC, R[S(x)], is defined by a decreasing linear function of oxyhemoglobin saturation based on experimental data. ATP release from human erythrocytes in response to normoxia and hypoxia was observed in in vitro experiments.[3].

A linear fit of experimental values defines the ATP release function of saturation :

$$R\left[S(\mathbf{x})\right] = R\_0\left[1 - R\_1S(\mathbf{x})\right] \tag{2}$$

In general,the rate of change in plasma ATP concentration, C(x,t), is given by the difference between the rates of ATP release and degradation:

$$\frac{\partial}{\partial t}[\pi \mathbf{R}^2 \cdot (1 - H\_T)\mathbf{C}] + \frac{\partial}{\partial \mathbf{x}}[(1 - H\_D)\mathbf{Q}\mathbf{C}(\mathbf{x})] = \pi \mathbf{R}^2 H\_T \mathbf{R} [\mathbf{S}(\mathbf{x}, t)] - 2k\_d \pi \mathbf{R} \mathbf{C}(\mathbf{x}, t) \tag{3}$$

where *HT* is tube hematocrit, R is radius of vessel and *<sup>d</sup> k* is a concentration rate constant.([3])

We assume that there is no convection in the vessel and there is no x-dependence. Equation (3) simplifies to the following equation:

$$\frac{dC}{dt} = \frac{H\_T}{1 - H\_T} R(S(t)) - \frac{2}{R} \frac{k\_d}{1 - H\_T} C$$

In this chapter a model is developed which predicts tissue ATP concentrations as a variation of time and depth into the tissue due to changing oxygen tensions.

The ATP concentration within plasma as a variation of time due to changes in oxygen tension at the tissue surface is related to the release and degradation of ATP, by the following equation:

$$\frac{d\mathcal{C}}{dt} = \eta \cdot \mathcal{R}\_{ATP}(t) - \delta \mathcal{C} \tag{4}$$

where is some constant of degradation. <sup>2</sup> 1 *d T k R H* [Related to the RBC fraction], R is the release of ATP from the RBC and C is ATP concentration.

RBC-ATP Theory of Regulation for Tissue Oxygenation-ATP Concentration Model 163

**Figure 6.**

rate of release".[11]

**Author details** 

Terry E. Moschandreou

*London Ontario, Canada* 

**8. Summary and conclusion** 

to compression and deform thus resulting in ATP release.

to ATP release due to a decrease in saturation.[3]

*Department of Medical Biophysics, University of Western Ontario,* 

In this work we have outlined the importance of ATP as a signaling molecule in the microcirculation and have discussed the biochemical aspects of ATP and ADP as well as introduce a model for ATP release as used in microfluidic devices where RBC's are subject

It is shown by Wan et al. [11 ] that even though the RBCs deform significantly in short constrictions (duration of increased stress <3 ms), no measurable ATP is released. This critical timescale is in proportion with a characteristic membrane relaxation time determined from observations of the cell deformation by using high-speed video[11]. "The results suggest a model wherein the retraction of the spectrin-actin cytoskeleton network triggers the mechano-sensitive ATP release and a shear-dependent membrane viscositycontrols the

It is noteworthy to see that these results for shear-induced ATP release can also be extended

In this model the ATP concentration maximizes at some constant value, depending on the oxygen saturation. This model can be used to predict the plasma ATP concentration based on different oxygen saturations.

#### **7. Discussion**

From Equation (4), with varying degradation constant, related to the RBC fraction, and release rate , ( ) ( 3) *R t Ht ATP* where H is the Heaviside function as a function of t, we show results in Figure 5 for concentration of ATP, ( *M* ), versus time (ms). The value in the shifted Heaviside function corresponds to the experimental results of Wan et al [11] for time t< 3 ms. This is the time interval where there is no ATP release and can be confirmed in Figure 4 where there is no ATP release before and throughout the stenosis of the microfluidic device. In fact there is ATP release in the low shear expansion on the right of the microfluidic device. This can be seen in the increase in concentration of ATP released by RBC as shown in Figure 4. The parameter η represents the rate of increase of ATP release and for large η, the concentration of ATP increases rather steeply, wheras for smaller η the concentration of ATP increases less steeply as can be seen in Figure 6 for varying values of degradation constant. Also our model is consistent with the experimental results of Wan et al.[11] in that the greatest increase in ATP occurs approximately 29 ms after the onset of increased shear stress.

(See Figure 6 for 0.1 at time t=30ms.) It is noteworthy to see that these results for shearinduced ATP release can also be extended to ATP release due to a decrease in saturation.

**Figure 5.**

**Figure 6.**

162 Blood Cell – An Overview of Studies in Hematology

on different oxygen saturations.

results in Figure 5 for concentration of ATP, (

is some constant of degradation. <sup>2</sup>

the release of ATP from the RBC and C is ATP concentration.

1 *d T*

In this model the ATP concentration maximizes at some constant value, depending on the oxygen saturation. This model can be used to predict the plasma ATP concentration based

From Equation (4), with varying degradation constant, related to the RBC fraction, and release rate , ( ) ( 3) *R t Ht ATP* where H is the Heaviside function as a function of t, we show

Heaviside function corresponds to the experimental results of Wan et al [11] for time t< 3 ms. This is the time interval where there is no ATP release and can be confirmed in Figure 4 where there is no ATP release before and throughout the stenosis of the microfluidic device. In fact there is ATP release in the low shear expansion on the right of the microfluidic device. This can be seen in the increase in concentration of ATP released by RBC as shown in Figure 4. The parameter η represents the rate of increase of ATP release and for large η, the concentration of ATP increases rather steeply, wheras for smaller η the concentration of ATP increases less steeply as can be seen in Figure 6 for varying values of degradation constant. Also our model is consistent with the experimental results of Wan et al.[11] in that the greatest increase in ATP occurs approximately 29 ms after the onset of increased shear stress.

induced ATP release can also be extended to ATP release due to a decrease in saturation.

0.1 at time t=30ms.) It is noteworthy to see that these results for shear-

*k R H*

[Related to the RBC fraction], R is

*M* ), versus time (ms). The value in the shifted

where

**7. Discussion** 

(See Figure 6 for

**Figure 5.**

#### **8. Summary and conclusion**

In this work we have outlined the importance of ATP as a signaling molecule in the microcirculation and have discussed the biochemical aspects of ATP and ADP as well as introduce a model for ATP release as used in microfluidic devices where RBC's are subject to compression and deform thus resulting in ATP release.

It is shown by Wan et al. [11 ] that even though the RBCs deform significantly in short constrictions (duration of increased stress <3 ms), no measurable ATP is released. This critical timescale is in proportion with a characteristic membrane relaxation time determined from observations of the cell deformation by using high-speed video[11]. "The results suggest a model wherein the retraction of the spectrin-actin cytoskeleton network triggers the mechano-sensitive ATP release and a shear-dependent membrane viscositycontrols the rate of release".[11]

It is noteworthy to see that these results for shear-induced ATP release can also be extended to ATP release due to a decrease in saturation.[3]

#### **Author details**

Terry E. Moschandreou *Department of Medical Biophysics, University of Western Ontario, London Ontario, Canada* 
