Steady laminar heat transfer in a circular tube with prescribed wall heat flux

Thermal entrance problem for blood flow inside an axisymmetric tube: The classical Graetz problem extended for Quemada’s bio-rheological fluid with axial conduction

The heat-conducting nature of blood is critical in the human circulatory system and features also in important thermal regulation and blood processing systems in biomedicine. Motivated by these applications, in the present investigation, the classical Graetz problem in heat transfer is extended to the case of a bio-rheological fluid model. The Quemada bio-rheological fluid model is selected since it has been shown to be accurate in mimicking physiological flows (blood) at different shear rates and hematocrits. The steady two-dimensional energy equation without viscous dissipation in stationary regime is tackled via a separation of variables approach for the isothermal wall temperature case. Following the introduction of transformation variables, the ensuing dimensionless boundary value problem is solved numerically via MATLAB based algorithm known as bvp5c (a finite difference code that implements the four-stage Lobatto IIIa collocation formula). Numerical validation is also presented against two analytical approaches namely, series solutions and Kummer function techniques. Axial conduction in terms of Péclet number is also considered. Typical values of Reynolds number and Prandtl number are used to categorize the vascular regions. The graphical representation of mean temperature, temperature gradient, and Nusselt numbers along with detail discussions are presented for the effects of Quemada non-Newtonian parameters and Péclet number. The current analysis may also have potential applications for the development of microfluidic and biofluidic devices particularly which are used in the diagnosis of diseases in addition to blood oxygenation technologies.

Theoretical and Experimental Investigation of Alginate Microtube Extrusion for Cell Culture Applications

A novel cell culture technology, consisting of hollow alginate tubes, OD ~550 μm, ID ~450μm containing a cell suspension, provides stress-free conditions. Cells reach confluency in approximately ten days with cell densities of 0.5 - 1 billion cells per mL. Tubes are manufactured in a tri-axial needle extruder with three concentric flows. The cell suspension flows in the inner needle (N1), the alginate solution flows in the annulus between N1 and the second needle (N2) and a CaCl ₂ solution is the sheath fluid between the second and third needle (N3). Beyond the tip of N2, the sheath solution is in contact with the alginate and Ca ²⁺ diffuses into the alginate solution and crosslinks it to form an alginate microtube around the core fluid. The cross-linked layer moves radially inwards like a front, starting at the sheath/annulus interface and ends at the annulus/core interface. A mathematical model is used to find the minimum length z _C of direct contact between the CaCl ₂ solution and the alginate solution to complete the cross-linking. Experimental results support the theoretical findings that stable tubes can only be manufactured if the contact length exceeds z _C . Experiments also show that the extruder configuration N3>N2 is best for alginate tube manufacture.

In vitro intermittent hypoxia: challenges for creating hypoxia in cell culture

Intermittent hypoxia has been implicated in morbidities associated with sleep apnea, and may be a novel cellular signal for inflammation [J. Appl. Physiol. 90 (2001) 1986]. Standard cell culture has two major limitations for studying the effects of steady-state P(O(2)) and intermittent hypoxia. First, convective mixing in the culture media can be variable, making precise control of cellular P(O(2)) difficult. Second, diffusion of oxygen through the culture media slows changes in cellular P(O(2)) after rapid changes in the gas phase P(O(2)). Our estimates of diffusional transients for standard cell culture suggest significant restrictions in the ability to cycle P(O(2)) at frequencies relevant to intermittent hypoxia. We present a novel system for forced convection cell culture with adherent cells inside capillary tubing. Steady state cellular P(O(2)) is regulated to an accuracy of approximately 1 Torr. The response time for cycling of P(O(2)) is less than 1.6 sec. This system is ideally suited for studies of intermittent hypoxia in adherent cells.

Design engineering of a bioartificial renal tubule cell therapy device

The use of a bioartificial renal tubule device composed of renal proximal tubule cells grown within a hollow fiber cartridge is a first step in engineering a bioartificial kidney to provide more complete replacement therapy of renal function than is available today. In this study, the feasibility of two designs for a tubule device were investigated: one with cells grown on microcarrier beads densely packed within the extracapillary space of a hollow fiber cartridge, and the other with cells grown as a confluent monolayer within the hollow fibers themselves. First, the oxygen requirements of porcine renal proximal tubule cells were determined, both attached to microcarriers and in suspension and compared to that of proximal tubule segments. The basal rate of cell respiration was found to be 2.29 +/- 0.53 nmol O2/10(6) cells/min for our cultured proximal tubule cells in suspension and no significant difference was seen with attached cells. Proximal tubule segments displayed significantly higher respiratory rates. Cells were also found to be responsive in the presence of mitochondrial inhibitors or uncouplers, and their respiratory rates remained constant, despite multiple passaging. The resultant cell oxygen consumption parameter was used in models describing oxygen concentration profiles within the two device configurations. From these models, it was found that cells within our proposed device designs could theoretically be sustained and remain viable, with respect to oxygen limitations. Finally, flow visualization studies were performed to assess fluid flow distribution and determine optimal device configuration and geometry to decrease areas of low or stagnant flow.

Temperature uniformity during hyperthermia: the impact of large vessels

During hyperthermia the presence of a large vessel entering the heated volume and carrying blood at the systemic temperature can be an important source of temperature non-uniformity and possible underdosage. The minimal tumour temperature near a large vessel is determined by the vessel wall temperature: a number of factors influencing the vessel wall temperature are considered--effective tissue conductivity, flow type, vessel size, entrance effects and counter-current flow. In some specific cases, especially when tissue perfusion is high, the vessel wall temperature may reach therapeutic levels when the mean blood temperature is still low. In general, well perfused tumours have a better chance of being heated uniformly. Regional heating improves temperature uniformity by reducing entrance and equilibration effects as blood is heated before entering the tumour. Raising the core temperature also reduces temperature inhomogeneity. Spatial SAR resolution should preferably be of the order of magnitude of a centimetre or better.

Relationship between release rate and surface concentration for heparinized materials

Mathematical models are used to predict surface concentrations that result from the release of heparin into flowing blood and stagnant or well-mixed plasma. Two release rates--4 X 10(-2) and 3 X 10(-5) micrograms/cm2 min--are considered, which describe elution from an ionically heparinized material and from an immobilized heparin-PVA hydrogel, respectively. When heparin is released at the higher rate into blood flowing in cylindrical tubes with dimensions characteristic of the vasculature, or annular tubes representative of catheter experiments, a minimum surface concentration of 0.5 micrograms/mL is attained virtually at the tube inlet. Release at the lower rate requires tube lengths of several thousand meters to attain the same critical value. Similarly, heparin released from a suspension of beads at the higher rate leads to critical surface concentrations of 0.2 micrograms/mL within a fraction of a second in stagnant plasma, or ca. 5 s in a well-mixed environment. At the lower release rate, 45 or 100 min must elapse before the same level is achieved. These results support the validity of 4 X 10(-2) micrograms/cm2 min as a reasonable minimum release rate to produce a heparin microenvironment sufficient to prevent thrombosis. The lower rate is shown to be insufficient to generate a critical concentration, thus supporting the argument that heparin-PVA does not owe its biological activity to a heparin microenvironment. The model equations can be applied to the release of any material to determine surface concentrations.

MASS TRANSFER IN THE ENTRANCE REGION OF A CIRCULAR TUBE

A solution in the form of an asymptotic expansion is obtained for the problem of mass transfer in the entrance region of a circular tube or flat channel for arbitrary hydrodynamically-developed velocity profile and arbitrary dependence of the diffusion coefficient on the coordinate perpendicular to the flow. Boundary conditions of the first, second and third kind are considered. The results of the analysis are compared with known approximate and numerical solutions of similar problems.