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TECHNICAL PAPERS: Fluids/Heat/Transport

# Readdressing the Issue of Thermally Significant Blood Vessels Using a Countercurrent Vessel Network

[+] Author and Article Information
Devashish Shrivastava

Department of Mechanical Engineering,  University of Utah, Salt Lake City, UT 84102

Robert B. Roemer

Department of Mechanical Engineering,  University of Utah, Salt Lake City, UT 84102bob.roemer@utah.edu

J Biomech Eng 128(2), 210-216 (Sep 19, 2005) (7 pages) doi:10.1115/1.2165693 History: Received October 29, 2004; Revised September 19, 2005

## Abstract

A physiologically realistic arterio-venous countercurrent vessel network model consisting of ten branching vessel generations, where the diameter of each generation of vessels is smaller than the previous ones, has been created and used to determine the thermal significance of different vessel generations by investigating their ability to exchange thermal energy with the tissue. The temperature distribution in the 3D network (8178 vessels; diameters from 10 to $1000μm$) is obtained by solving the conduction equation in the tissue and the convective energy equation with a specified Nusselt number in the vessels. The sensitivity of the exchange of energy between the vessels and the tissue to changes in the network parameters is studied for two cases; a high temperature thermal therapy case when tissue is heated by a uniformly distributed source term and the network cools the tissue, and a hypothermia related case, when tissue is cooled from the surface and the blood heats the tissue. Results show that first, the relative roles of vessels of different diameters are strongly determined by the inlet temperatures to those vessels (e.g., as affected by changing mass flow rates), and the surrounding tissue temperature, but not by their diameter. Second, changes in the following do not significantly affect the heat transfer rates between tissue and vessels; (a) the ratio of arterial to venous vessel diameter, (b) the diameter reduction coefficient (the ratio of diameters of successive vessel generations), and (c) the Nusselt number. Third, both arteries and veins play significant roles in the exchange of energy between tissue and vessels, with arteries playing a more significant role. These results suggest that the determination of which diameter vessels are thermally important should be performed on a case-by-case, problem dependent basis. And, that in the development of site-specific vessel network models, reasonable predictions of the relative roles of different vessel diameters can be obtained by using any physiologically realistic values of Nusselt number and the diameter reduction coefficient.

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## Figures

Figure 3

Absolute thermal energy exchanged between the different vessel levels and the tissue is presented as a percentage of the total energy exchanged by all vessel levels (ΔQL0‐L9) for RD=0.8 and 0.6. For the case of hyperthermia ΔQL0‐L9=8.51W(RD=0.8) and 8.18W(RD=0.6). For the case of hypothermia (ΔQL0‐L9)=4.73W(RD=0.8) and 4.60W(RD=0.6).

Figure 4

Absolute thermal energy exchanged between the different vessel levels and the tissue is presented as a percentage of the total energy exchanged by all vessel levels (ΔQL0‐L9) for Mar,in=23.16×10−4kg∕s. This value of mass flow rate gives the average perfusion value of 5kg∕m3∕s. (ΔQL0‐L9)=11W (hyperthermia) and 12.73W (hypothermia).

Figure 5

Absolute thermal energy exchanged between the different vessel levels and the tissue is presented as a percentage of the total energy exchanged by all vessel levels (ΔQL0‐L9) for Tar,in=42°C. (ΔQL0‐L9)=7.81W (hyperthermia) and 9.49W (hypothermia).

Figure 6

Absolute thermal energy exchanged between the different vessel levels and the tissue is presented as a percentage of the total energy exchanged by all vessel levels (ΔQL0‐L9) for NTU=4(48∕11). (ΔQL0‐L9)=9.64W (hyperthermia) and 5.24W (hypothermia).

Figure 7

Distribution of the mixed mean temperature of the arterial and venous blood along two representative paths is shown from level 0 to 9 to illustrate the complex behavior of the vessel-tissue heat transfer rate. One of the two paths goes towards the center of the tissue and the other goes towards the periphery. The paths start from the center of the first and second level 0 branching points and end at the level 9 terminal vessels.

Figure 8

Distribution of the wall temperature of level 0 vessels along z direction

Figure 9

Net thermal energy exchanged between the different vessel levels and the tissue is presented as a percentage of the total energy exchanged by all vessel levels (ΔQL0‐L9) for reference values. ΔQL0‐L9=8.51W (hyperthermia) and 4.73W (hypothermia).

Figure 1

Schematic of the partial arterial vessel network. All ten vessel levels (level 0 to level 9) are shown. The venous network is parallel to the arterial network.

Figure 2

Absolute thermal energy exchanged between the different vessel levels and the tissue is presented as a percentage of the total energy exchanged by all vessel levels (ΔQL0‐L9) for Rar,L0=500 and 125μm. For the case of hyperthermia ΔQL0‐L9=8.51W(Rar,L0=500μm) and 8.16W(Rar,L0=125μm). For the case of hypothermia ΔQL0‐L9=4.73W(Rar,L0=500μm) and 4.61W(Rar,L0=125μm).

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