Biorheology Presentation O-268 S271 A PROBABILISTIC APPROACH TO TWO-PHASE FLOW IN MICROVESSELS Stephen Payne, Chang Sub Park Department of Engineering Science, University of Oxford, UK Introduction Most models used in physiological fluid dynamics make use of a purely deterministic approach, based on physical equations. However, it has been shown in hydraulic flows that a probabilistic approach can provide a greater insight into the flow behaviour [Chiu, 1987] and this is explored here. Methods The system information entropy is maximised, subject to the physical constraints imposed on the system (the probability function used must integrate to 1 and the averaged value is the mean flow velocity). This gives [Chiu, 1987]: u F e M u u M 1 1 ln 1 max , (1) dependent upon M, termed the ‘entropy parameter’ by Chiu. The cumulative distribution function is set to that for two-phase flow [Sharan, 2001]: 1 1 0 1 2 2 2 2 u F , (2) using the terminology given in [Sharan, 2001]. The flow rate can be calculated by integration of the flow velocity over the cross-sectional area, substituting equation 2 into equation 1. The result is analytical, but not quoted here for reasons of space. The apparent total flow viscosity and tube haematocrit can be calculated, using mass balance. There are three experimental relationships from the literature (total haematocrit, relative viscosity and core to plasma viscosity ratio), dependent upon tube haematocrit and diameter. These are dependent upon the flow model used, which in turn depends upon three model parameters, including M and . A non-linear fitting algorithm was used to calculate the values of model parameters, since the equations are complex and non-linear. Results Figures 1 and 2 show the variation in M and plasma layer thickness fraction 1- with tube diameter and haematocrit. Neither M nor 1- are strongly dependent upon the haematocrit, and both show strong dependence upon tube diameter, both tending towards 0 for large diameters, as expected. The plasma layer thickness fraction shows a much better agreement with the data of [Reinke, 1987] (data not shown here), being significantly lower than the model of [Sharan, 2001]. Figure 1: Flow parameter M. Figure 2: Plasma layer thickness fraction. Discussion The use of a probabilistic approach to two-phase flow, which is widely used to model blood flow in microvessels, shows a greater preliminary agreement with experimental data than a purely deterministic approach. There is still significant uncertainty over the physical interpretation of the parameter M, in particular the fact that the values obtained here are negative. However, the promising results obtained here indicate that this is a productive avenue for future exploration. References Chiu, J Hydraul Eng, 113:583-600, 1987. Reinke et al, Am J Physiol, 253:H540-547, 1987. Sharan and Popel, Biorheology, 38:415-428, 2001. 16th ESB Congress, Oral Presentations, Wednesday 9 July 2008 Journal of Biomechanics 41(S1)

A PROBABILISTIC APPROACH TO TWO-PHASE FLOW IN MICROVESSELS

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Biorheology Presentation O-268 S271

A PROBABILISTIC APPROACH TO TWO-PHASE FLOW IN MICROVESSELS

Stephen Payne, Chang Sub Park

Department of Engineering Science, University of Oxford, UK

Introduction Most models used in physiological fluid dynamics make use of a purely deterministic approach, based on physical equations. However, it has been shown in hydraulic flows that a probabilistic approach can provide a greater insight into the flow behaviour [Chiu, 1987] and this is explored here. Methods The system information entropy is maximised, subject to the physical constraints imposed on the system (the probability function used must integrate to 1 and the averaged value is the mean flow velocity). This gives [Chiu, 1987]:

uFeMu

u M 11ln1

max

, (1)

dependent upon M, termed the ‘entropy parameter’ by Chiu. The cumulative distribution function is set to that for two-phase flow [Sharan, 2001]:

1101

222

uF , (2)

using the terminology given in [Sharan, 2001]. The flow rate can be calculated by integration of the flow velocity over the cross-sectional area, substituting equation 2 into equation 1. The result is analytical, but not quoted here for reasons of space. The apparent total flow viscosity and tube haematocrit can be calculated, using mass balance. There are three experimental relationships from the literature (total haematocrit, relative viscosity and core to plasma viscosity ratio), dependent upon tube haematocrit and diameter. These are dependent upon the flow model used, which in turn depends upon three model parameters, including M and . A non-linear fitting algorithm was used to calculate the values of model parameters, since the equations are complex and non-linear. Results Figures 1 and 2 show the variation in M and plasma layer thickness fraction 1- with tube diameter and haematocrit. Neither M nor 1- are strongly dependent upon the haematocrit, and both show strong dependence upon tube diameter, both tending towards 0 for large diameters, as expected. The plasma layer thickness fraction shows a much better agreement with the data of [Reinke, 1987]

(data not shown here), being significantly lower than the model of [Sharan, 2001].

Figure 1: Flow parameter M.

Figure 2: Plasma layer thickness fraction. Discussion The use of a probabilistic approach to two-phase flow, which is widely used to model blood flow in microvessels, shows a greater preliminary agreement with experimental data than a purely deterministic approach. There is still significant uncertainty over the physical interpretation of the parameter M, in particular the fact that the values obtained here are negative. However, the promising results obtained here indicate that this is a productive avenue for future exploration. References Chiu, J Hydraul Eng, 113:583-600, 1987. Reinke et al, Am J Physiol, 253:H540-547, 1987. Sharan and Popel, Biorheology, 38:415-428, 2001.

16th ESB Congress, Oral Presentations, Wednesday 9 July 2008 Journal of Biomechanics 41(S1)