- Email: [email protected]
Mass transfer in vascular access ports
International Journal of Heat and Mass Transfer 54 (2011) 949–958
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Mass transfer in vascular access ports S. Chelikani a, E.M. Sparrow a, J.P. Abraham b,⇑, W.J. Minkowycz c a
Laboratory for Heat Transfer and Fluid Flow Practice, Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455-0111, USA Laboratory for Heat Transfer and Fluid Flow Practice, School of Engineering, University of St. Thomas, St. Paul, MN 55105-1079, USA c Department of Mechanical Engineering, University of Illinois at Chicago, 842 W. Taylor Street, Chicago, IL 60607-7022, USA b
a r t i c l e
i n f o
Article history: Received 23 August 2010 Received in revised form 11 October 2010 Accepted 11 October 2010 Available online 29 November 2010 Keywords: Vascular access Computational fluid dynamics Biological mass transfer Non-Newtonian flow
a b s t r a c t Numerical simulations have been performed to evaluate the fluid flow and mass transfer processes that occur in a human-body vascular-access port. Such ports are used to facilitate the frequent introduction of cleansed blood and other drugs into the body from external sources. Each of the infusion ports studied here consists of a reservoir and an attached tube-like catheter which delivers the infused medium to its point of use. All told, three unique infusion ports were investigated. Each had a particular geometry characterized by the shape of the reservoir and the mode of attachment of the catheter to the reservoir. The numerical simulations were three-dimensional and unsteady. Both Newtonian and non-Newtonian constitutive equations were employed for the fluid flow solutions and for the subsequent mass transfer solutions. The initiation of fluid motion was the injection of a controlled volume of fluid into the reservoir. In some cases, the injected fluid was the same as that in the reservoir, and in others it was different. For all the investigated infusion systems, no hemolysis (red blood cell destruction) was in evidence when blood was passed through the catheter. Potential hemolysis was averted in two of the systems due to the very low fluid velocities. As witnessed by the mass transfer results, the use of the reservoir as a chamber to mix a secondary liquid with blood is a viable strategy. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Mass transfer processes are ubiquitous and of diverse function in the human body, as witnessed by an extensive literature. Representative highlights of that literature will now be conveyed. In an analytical solution for macromolecule mass transfer in an artery, the transport within the wall and in the lumen were coupled in [1]. There, mass transfer was found to play a significant role in the development of atherosclerosis, a progressive disease of the artery wall [2]. Antifreeze proteins affect the heat and mass transport processes in the cryopreservation of human tissue and organs [3]. Hemodiafiltration depends strongly on mass transfer coefficients as well as on concentration polarization and water filtration flux [4]. The dispersion of angiographic contrast fluid in arteries can be used as a means of evaluating arterial geometry [5]. Dissolution of a gas bubble in blood is controlled by the diffusion of the gas from the bubble surface into the surrounding blood [6]. The foregoing diversity of mass transfer processes can be extended by further reference to the literature. The mass transfer of hemodialyzers was successfully modeled by a double-porousmedium construct consisting of two interpenetrating porous media, respectively for the blood flow and for the dialysate flow [7]. A ⇑ Corresponding author. Tel.: +1 651 962 5766. E-mail address: [email protected] (J.P. Abraham). 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.10.008
four-layer model of an artery encompassing the endothelium, intima, elastic lamina, and media was used to determine the transfer of LDL across the wall [8]. Electroporation is an approach used to enhance transdermal transport of large molecules [9,10]. A mixture of saline and plasma replaced blood in in-vitro experiments for the development of membranes [11]. The development of an impedance method of measuring blood flow rate was carried by means of experiments in which mixtures of saline and blood were the working fluids. The publications cited in the preceding paragraphs were encountered as the authors sought prior information about the focus of the present research: mass transfer in human-body vascular-access ports (or infusion ports). There are several medical therapies that are administered to patients frequently. To facilitate such therapies, it is convenient to be able to enter the body at a designated location without having to create a local aperture at each session of administration. The designated access location is often termed vascular access. An example of this situation is the need for application of dialysis therapy. This need arises in situations when kidneys are unable to fulfill their function of waste removal from the blood stream. It is common that the use of dialysis machines is required several times a week. In that case, it is clearly advantageous that a vascular access exists. Another example of the need for an established access port is during the frequent administration of drugs.
950
S. Chelikani et al. / International Journal of Heat and Mass Transfer 54 (2011) 949–958
Nomenclature cp C d D f k n m p q Re t T
specific heat concentration diameter binary diffusion coefficient friction factor thermal conductivity constant in power-law viscosity model constant used in Carreau viscosity model pressure constant used in Carreau viscosity model Reynolds number time temperature
The common factor of these procedures is the flow of fluids. With respect to dialysis, the cleansed blood is returned to the body by means of a pump or other fluid which serves to push the blood. When drugs are administered, the administration is almost always in fluid form. The injected drug may, in certain circumstances, interface with blood and mix with it. In such a circumstance, mass transfer occurs. The literature of vascular access was found to be both sparse and uninformative. History and current status of the modality are discussed in [12–15], but there appears to be no prior information available on mass transfer in vascular access ports. The focus of this research is the fluid mechanics and mass transfer of fluids that are passing through a permanent vascular access port in the human body. The work will be subdivided into two parts. In the first part, the flow of a homogenous fluid through several infusion port geometries will be analyzed. The second part will consider both fluid flow and mass transfer, the latter due to the presence of a different species. The numerical simulations will be performed by means of ANSYS – CFX software. This is a commercial code based on a finite-volume discretization of the Navier–Stokes equations, the conservation of mass equation and, when appropriate, a species conservation equation. The solutions encompass unsteady three-dimensional flows. 2. Fluid-flow evaluation of three vascular access ports Views of the three vascular access ports to be considered are conveyed in Fig. 1(a)–(c). These three configurations, respectively designated as A, B and C, correspond to different candidate ports. The first two of these are close approximations of those already used in practice, while the third is a new design whose evaluation is an important goal of this investigation. A common feature of all three configurations is an inlet aperture through which a fluid is introduced into a relatively large reservoir. The inflowing fluid serves to displace fluid that is already in the reservoir, the result of the displacement being an outflow into a catheter which delivers fluid to the blood stream. Geometrical differences among the three ports are related to the shape and size of the reservoir as well as the location of the exit. As will be demonstrated when the results are presented, these differences give rise to potential hemolysis-creating flow patterns. Fig. 1(a) shows Port A. In essence, it is a flat disc with a cylindrical depression in its upper surface. The disc is hollow and serves the purpose of a reservoir. The aperture shown in the figure and labeled inlet is created by the user when inserting the needle of a syringe. The catheter is attached to the reservoir along a radial line extending outward from its surface. The second of the currently available ports, to be designated as Port B, is exhibited in Fig. 1(b). Once again, the reservoir is a hollow
u, v, w U x, y, z X
velocity components cross-sectional average velocity coordinates mass fraction
Greek symbols density dynamic viscosity kinematic viscosity constant use in power-law viscosity model constants used in Carreau viscosity model shear strain rate
q l m j l0, l1, k c
disc-shaped body. The catheter is joined to the reservoir as a tangent to its outer rim. The last of the ports, shown below, is to be designated as Port C. It differs from Port B in that the attachment of the catheter to the reservoir is accomplished by means of a gently converging transition piece. In Port C, the capability to build a gentle convergence into the catheter is enabled by the height of the reservoir being considerably larger than the diameter of the catheter. It can be expected that the gradual contraction would give rise to a much smoother inflow than would occur for Ports A and B which do not have this feature. To reinforce the foregoing discussion, Fig. 2(a)–(c) have been prepared. These figures are planar cuts which show the geometry of the transitions from the reservoir to the catheter. Fig. 2(a), which corresponds to Port A, has a transition which includes a sharp-edged inlet. It is well known that sharp-edged inlets give rise to flow separation because fluids are unable to turn sharp corners. The inlet shape shown in Fig. 2(b) (Port B) includes a turn that is greater than 90° (re-entrant). Such a turn is even more difficult for a fluid to negotiate than would be a 90° turn. Again, flow separation is to be expected. Another feature of the geometry of the transition section of Port B is the abrupt change of cross section that can be seen at the top of the diagram (Fig. 2(b)). Such an abrupt change is difficult for the fluid to cope with, and separation is likely. The just-identified two geometric errors in the design of Port B have been corrected in Port C, as can be seen from Fig. 2(c). Not only has the more than 90° turn been eliminated, but also a rounding of the transition geometry has been introduced. Furthermore the abrupt change of cross section noted at the end of the last paragraph no longer exists in Port C. 3. Governing equations of fluid flow The equations which will be solved for the fluid-flow problem are the three-dimensional, unsteady, incompressible form of the Navier–Stokes equations and the equation of continuity. In these equations, two constitutive equations were employed. The first is the Newtonian model which makes use of a constant value of the viscosity. The other is a non-Newtonian model, two candidate forms of which were used. 3.1. Navier–Stokes equations
" # @u @ 2 u @ðuv Þ @ðuwÞ @p @ @u ¼ þ 2þ þ þ l @t @x @y2 @z2 @x @x @x @ @u @ @u l l þ þ @y @y @z @z
q
ð1Þ
951
S. Chelikani et al. / International Journal of Heat and Mass Transfer 54 (2011) 949–958
3.2. Continuity equation
@u @ v @w þ ¼0 þ @x @y @z
ð2Þ
The boundary conditions for the velocity problem are: (a) all velocities are zero on all solid surfaces, (b) the fluid entering the solution domain is assigned a steady velocity that is uniformly distributed across the inlet section, and (c) standard weak boundary conditions at the exit of the catheter and a pressure prescribed there. Since the problem is unsteady, an initial condition is also needed. The condition of zero velocity and uniform pressure throughout the solution domain was imposed as the initial condition. 4. Numerical simulations For each of the port configurations, the numerical simulation mimicked an experiment which was initiated with both the reservoir and the catheter filled with blood. The activation of the fluid motion was accomplished by the steady injection of a second fluid (sometimes the same as the fluid already in the system and sometimes a different fluid). The operating conditions for the executed numerical simulations are listed in Table 1. It can be seen from the table, nine of the investigated cases were characterized by the same injected volumetric flow, 10 ml/min. To implement a sensitivity study, the other three cases were accorded injection flow rates that were four times higher. The period of injection was the same for all cases. The injected fluids were either blood or saline. The numerical solutions made use of two constitutive models for blood, either Newtonian or non-Newtonian. For the latter, two different algebraic models were employed. These equations are: Carreau equation [16] (non-Newtonian 1)
h
m1 l ¼ l1 þ ðl0 l1 Þð1 þ ðckÞq Þð q Þ
i
ð3Þ
Power law (non-Newtonian 2)
l ¼ j cð 10 Þ ; for 0:001 6 c 6 1000
ð4aÞ
n1 l ¼ j 10ð 10 Þ ;
ð4bÞ
n1
for c P 1000
n1 l ¼ j 0:001ð 10 Þ ;
for c 6 0:001
ð4cÞ
where l is the viscosity (kg/m s), and c is the shear rate (1/s). The quantities m, j, q, k, n, l1, and l0 are constants. Values of the Carreau constants taken from [17] were: k = 0.11, m = 0.392, l0 = 0.022 kg/m s, l1 = 0.0022 kg/m s, and q = 0.644. For the power-law model, j = 0.42 and n = 0.61.
Fig. 1. Three-dimensional isometric view of (a) Port A (close facsimile of the PowerPort device from Bard Access Systems), (b) of Port B (close proximity of the VortexPort device from Angiodynamics), and (c) C (under development from R4 Vascular).
This equation applies for the x-coordinate direction. The quantities u, v, and w are the velocity components in the x, y, and z directions; q is the density, p is the pressure, and l is the viscosity. The symbol t represents time. The two complementary equations are obtained by cycling u, v, and w and x, y, and z. Note that the viscosity is contained within the differentiation operators to accommodate the spatial variations of the non-Newtonian model. For the Newtonian model, the viscosity passes through these operators.
A graph of these equations is displayed in Fig. 3. In that figure, the viscosity is plotted as function of the strain rate. Also appearing in the figure is a horizontal line that represents a Newtonian model for blood. It can be seen from Fig. 3 that at higher shear rates, both the above models become very close to the Newtonian model. The calculations were carried out using CFX 11.0, a commercial finite-volume-based CFD program. A false-transient, time-stepping approach is employed to enable convergence to the steady-state solution. While the fully implicit, backward-Euler, time-stepping algorithm exhibits first-order accuracy in time, its use does not affect the accuracy of the final, converged solution. Coupling of the velocity-pressure equations was achieved on a non-staggered, collocated grid using the techniques developed by Rhie and Chow [18] and Majumdar [19]. The inclusion of pressure-smoothing terms in the mass conservation equation
952
S. Chelikani et al. / International Journal of Heat and Mass Transfer 54 (2011) 949–958
Fig. 2. Planar view of (a) Port A, (b) Port B, and (c) Port C.
suppresses oscillations which can occur when both the velocity and pressure are evaluated at coincident locations. The advection terms in the momentum and energy equations were evaluated by using the upwind values of the momentum flux, supplemented with an advection-correction term. The correction term reduces the occurrence of numerical diffusion and is of second-order accuracy. Details of the advection treatment can be found in [20].
5. Diagnostic fluid flow results Figs. 4 and 5, respectively show the variation of the shear strain rate and pressure gradient along the center line of the catheter extending from the center of the reservoir to the end of the catheter. The center line of the catheter lies along the z coordinate. The section z = 0 to z = 0.011 (m) extends from the center of the reservoir to the outlet of the reservoir which is also the inlet of the catheter.
Table 1 Detailed information for the numerical simulation parameters. Run no.
Port type
Injected fluid
Rate of injected fluid (ml/min)
Injection time (s)
Total domain volume (m3)
Constitutive law
1 2 3 4 5 6 7 8 9 10 11 12
A B C A B C B C C A B C
Blood Blood Blood Blood Blood Blood Blood Blood Blood Saline Saline Saline
10 10 10 40 40 40 10 10 10 10 10 10
2 2 2 2 2 2 2 2 2 2 2 2
7.51e7 8.85e7 9.96e7 7.51e7 8.85e7 9.96e7 8.85e7 9.96e7 9.96e7 7.51e7 8.85e7 9.96e7
Newtonian Newtonian Newtonian Newtonian Newtonian Newtonian Non–Newtonian 1 Non–Newtonian 1 Non–Newtonian 2 Newtonian Newtonian Newtonian
953
S. Chelikani et al. / International Journal of Heat and Mass Transfer 54 (2011) 949–958
Table 2 Pressure drop and maximum shear stress in the fluid domain at different flow rates.
Fig. 3. Viscosity of blood according to the Carreau model, the power-law model, and the Newtonian model.
No.
Port
Flow rate (ml/min)
Maximum shear stress (Pa)
Pressure drop (Pa)
1 2 3 4 5 6
A A B B C C
10 40 10 40 10 40
3.9 24 4.5 25 7.2 30
113 586 113 588 89 490
is to be simulated, it would be sufficient to use a Newtonian viscosity model in conjunction with a mixing rule to determine the viscosity of saline-blood mixtures. For simplicity, an empirical formula which is commonly used to determine the viscosity of fluid blends was used to determine the viscosity of the salineblood mixture [21].
m¼e
V
blend 10:975 14:524
0:8
ð5Þ
where
V blend ¼ ½X a VBNa þ ½X b VBNb
ð5aÞ
and
VBN ¼ 14:534 ln½lnðm þ 0:8Þ þ 10:975
Fig. 4. Shear strain rate along the center of the catheter.
ð5bÞ
where, m is the kinematic viscosity (centistokes), and X is the mass fraction. The term VBN refers to the Viscosity Blend Number. The duration of each simulation run was equal to the duration of injection period. In all cases, the flow was regarded as laminar with calculated Reynolds numbers of 47 and 188 when a Newtonian viscosity was used with the given flow rates of 10 ml/min and 40 ml/min, respectively. As a check on the accuracy of the numerical simulation, it may first be observed that for fully developed laminar flow of a Newtonian fluid, the friction factor in the catheter is expected to be f = 64/ Re. To compare this friction factor with that from the simulation, the definition of the friction factor is given by
@p 1 d f ¼ qU 2 @x 2
ð6Þ
where @p is the pressure gradient in the catheter, d is the catheter @x diameter, q is the fluid density, and U is the mean fluid velocity. By making use of the known values of the quantities appearing on the right-hand side of this equation, it is found that f = 1.37 in the catheter proper for a flow rate of 10 ml/min. From the theoretical solution, f = 64/Re = 1.36. The agreement between these two values is within 0.74%. 5.1. Pressure drop and shear stress
Fig. 5. Pressure gradient along the center of the catheter.
It can be seen that both Newtonian and non-Newtonian flows give similar profiles for the pressure gradient and for the strain rate. The Newtonian model gives an upper bound for the strain rate. It is well established that higher strain rates give rise to higher rates of hemolysis. So, for the given conditions, a Newtonian model gives a conservative representation of the flow field with respect to hemolysis and has been chosen for further simulations because of its simplicity. This finding also suggests that when saline injection
The effects of flow rate and port geometry on the pressure drop and maximum shear stress are summarized in Table 2. The listed results correspond to the steady state that was achieved prior to the termination of the two-second run period. The pressure drop corresponds to end-to-end pressure variations for the access port. An inspection of the pressure drop results listed in Table 2 shows a remarkable similarity between those for Ports A and B. On the other hand, the pressure drops for Port C are approximately 80% of those for the other ports. This reduction in pressure drop can be attributed to an improved geometric design for Port C. In truth, however, this reduction in pressure does not provide any practical advantage. Further insights into the pressure results can be obtained by noting that an increase in flow rate by a factor of four gives rise
954
S. Chelikani et al. / International Journal of Heat and Mass Transfer 54 (2011) 949–958
to a pressure drop increase that is larger than a factor of four. Since the flow in the catheter is more or less fully developed throughout (horizontal lines in Fig. 4) and fully developed pressure drops scale linearly with the flow rate for laminar flow, the loss mechanism in the reservoir does not follow a simple flow rate scaling rule. Next, attention may be turned to the maximum shear stress results of Table 2. The maximum shear stress in the fluid domain is found to occur within the reservoir near the fluid injection region. When considering these results, it is important to recognize that they are locally calculated. Therefore, they are subject to the local nature of the mesh. As a consequence, they should not be regarded as having the same level of fidelity as would characterize average results such as the end-to-end pressure drops that are also listed in Table 2. However, the issue of the accuracy level is moot when looked at with respect to the context of hemolysis. It can be demonstrated that the shear stress values listed in the table are far below the threshold at which hemolysis occurs. This demonstration will be implemented with the aid of Fig. 6. Fig. 6 shows a summary of results available in the literature [22–35] which give the relationship between shear stress level and hemolysis. It shows the time required to induce hemolysis at different magnitudes of shear stress. The figure indicates that hemolysis is unlikely to occur at shear stresses below 100 Pa. Since the values of shear listed in Table 2 are considerably lower than 100 Pa, it may be concluded that none of the ports investigated give rise to a high-enough shear stress to cause hemolysis at the given flow rates.
5.2. Streamlines and vector diagrams Streamlines and vector diagrams provide a means for visualizing the flow pattern in the reservoirs. Fig. 7(a)–(c) show the threedimensional streamlines originating from the inlets of ports A, B and C, respectively. Since the flow becomes steady very quickly, these lines also represent the pathlines. It can be seen from these figures that the streamlines are twisted near the inlet of the catheter for Ports A and B due the sudden change of cross section. On the other hand, the twist is much less pronounced for Port C. Another feature that distinguishes Port C from the others is that its coiled flow pattern in the reservoir is not as tightly knotted as the rest. The degree of smoothness experienced by the flow passing from the reservoir into the catheter is clearly displayed by the vector diagrams shown in Fig. 8(a)–(c). Examination of Fig. 8(a), Port A, shows the streamlines bending sharply as they negotiate what is nearly a 90° change of direction. Since the Reynolds number of the flow is very low, such a sharp change is able to occur without
Fig. 7. Three-dimensional streamlines originating from the inlet of (a) Port A, (b) Port B, and (c) Port C.
Fig. 6. Relationship between shear stress level and hemolysis.
flow separation. Fig. 8(b) indicates the presence of a recirculation zone for Port B. In general, recirculation zones may bring on hemolysis. However, in the present situation, the flow velocities are so low that hemolysis does not occur. In addition, a recirculation zone is filled with a captive fluid which does not exit the catheter. It may be speculated that particulates could congregate in the captive fluid and cause agglomeration. Fig. 8(c) corresponds to Port C. The passage of the flow from the reservoir into the catheter follows smoothly converging streamlines and may be regarded as ideal.
S. Chelikani et al. / International Journal of Heat and Mass Transfer 54 (2011) 949–958
" # @T @T @T @CT @2T @2T @2T þu þv þw ¼k qcp þ þ @t @x @y @z @x2 @y2 @z2
955
ð8Þ
In this equation, T is the temperature, k is the thermal conductivity, and cp is the specific heat at constant pressure. The two foregoing equations can be brought into congruence by a simple transformation:
T ! C;
cp ! 1=q;
and k ! D
ð9Þ
By this transformation, the species conservation equation can be solved by CFX as if it were a heat convection equation. The value of the blood-saline diffusion coefficient was taken to be 1.5e9 m2/s. The boundary conditions for the mass transfer problem are: (a) no mass transfer at any solid surface, (b) given value of C at the inlet of the solution domain, and (c) standard weak boundary conditions at the exit of the catheter. Initially, C = 0 throughout the solution domain. 6.1. Concentration contour diagrams
Fig. 8. Velocity vectors at the entrance of the catheter for (a) Port A, (b) Port B, and (c) Port C.
6. Mass transfer model If the quantity C represents the concentration of a diffusing species in another medium, the species conservation equation is:
" # @C @C @C @2C @2C @2C þv þw ¼D u þ þ @x @y @z @x2 @y2 @z2
ð7Þ
where D is the binary diffusion coefficient. A simplified treatment of mass transfer is accomplished by an extension of the thermal energy equation, which is
The results of the mass transfer simulations will be conveyed by means of concentration contour diagrams. The displayed concentration is that of the injected fluid. A concentration C = 1 at a point means that the fluid at that point is totally injected fluid. On the other hand, C = 0 signifies the total absence of the injected fluid. In that case, the fluid is that which originally occupied the reservoir and the catheter, hereafter referred to as the incumbent fluid. The presented concentration information is for the steady state. The color strip at the left of each contour diagram provides the relationship between the grey tones and the concentration values. The in-reservoir results for each of the ports are presented in a series of figures, respectively Fig. 9(a)–(c) for Port A, Port B, and Port C. The left and right images of each figure are focused on the reservoir, but display different views. The left parts show the concentration contours in a plane that is perpendicular to the top and bottom surfaces of the reservoir and parallel to the geometric axis of the catheter. The right part of each figure displays contours in a plane that is perpendicular to that used in the left-hand image. The plane of the right-hand images lies at the half-height of the reservoir. Inspection and comparison of the three left-hand figures show the degree of coherence of the injected fluid as it mixes with the incumbent fluid. For Port B, the injected fluid experiences little mixing as it moves downward along the centerline of the reservoir. For Port A, there is slightly more mixing along this path. In contrast, there is considerably more mixing between the injected and incumbent fluids in Port C as the former makes its way downward along the centerline of the reservoir. The injected fluid dead-ends against the bottom of the reservoir and then fans outward along the bottom surface. The flaring is quite symmetric for Port A, but less so for Ports B and C. The main observation from the left-hand figures is the degree of mixing of the injected and incumbent fluids. In this regard, it is clear that the geometry of Port C gives rise to the best mixing. The right-hand figures confirm the hierarchy of mixing for the various ports. From most mixing to least mixing, the ports are ordered as C, A and B. Further observation of these images reveals an arrow-shaped contour formation, with the arrow pointing from the center of the reservoir toward the exit. This formation is clearly in evidence for Ports B and C and to a lesser extent for Port A. The legend identifies these zones as containing primarily incumbent fluid. Three additional figures, Figs. 10–12, will now be presented to convey mass concentration information for the catheter. Each of these figures is made up of three circles, each of which represents a cross section of the catheter. From left to right, the respective circles
956
S. Chelikani et al. / International Journal of Heat and Mass Transfer 54 (2011) 949–958
Fig. 9. Concentration contour along a plane perpendicular to the reservoir top and bottom surfaces (left image) and along a plane perpendicular to the fluid injection and located at the center of reservoir (right image) for (a) Port A, (b) Port B, and (c) Port C.
Fig. 10. Concentration contours in cross-sectional planes near the inlet (a), center (b) and outlet (c) of the catheter for Port A.
S. Chelikani et al. / International Journal of Heat and Mass Transfer 54 (2011) 949–958
957
Fig. 11. Concentration contours in cross-sectional planes near the inlet (a), center (b) and outlet (c) of the catheter for Port B.
Fig. 12. Concentration contours in cross-sectional planes near the inlet (a), center (b) and outlet (c) of the catheter for Port C.
correspond to the catheter inlet, the catheter half length, and the catheter exit. For Port B, Fig. 11, there is clear evidence that mixing occurs along the length of the catheter. At the exit cross section, the concentration is generally homogeneous, but the incumbent fluid still dominates adjacent to the wall. Progressive mixing is also in evidence for Port A (Fig. 10), but the concentration is less uniform at the exit cross section of the reservoir (inlet of the catheter) than that observed for Port B. The results for Port C, Fig. 12, display little lengthwise evolution. The concentration at the exit cross closely resembles that for Port B.
7. Concluding remarks To the best knowledge of the author, the work presented here is the first attempt to quantify the fluid flow and mass transfer characteristics of human-body infusion ports. The work was motivated by the need to characterize competing versions of such therapyfacilitating systems. Two of the systems (A and B in the designation of the report) are already available in the marketplace. The other, system C, is still in the developmental stage. The method of investigation used here was based on numerical simulation. Threedimensional, unsteady solutions of the Navier–Stokes, continuity, and species conservation were performed. The general configuration of the three systems is the same. A reservoir containing the medium to be infused is attached to a tube-like catheter which delivers the medium to its point of use. Fluid motion is activated by the introduction of a fixed volume of liquid into the reservoir. The liquid may be either the same or different from that in the reservoir. The differences among the systems are geometrical, including reservoirs of different sizes and in the mode of attachment of the
catheter to the reservoir. In particular, system C had an optimal attachment which guaranteed the absence of flow separation. The attachment mode of system A gave rise to flow separation, and that of B gave rise to incipient separation. The flow velocities were so small that hemolysis did not occur in either of the ports, regardless of the attachment mode. The successful use of the reservoir as a mixing chamber for different fluids has also been validated. References [1] M. Khakpour, K. Vafai, A comprehensive analytical solution of macromolecular transport within an artery, Int. J. Heat Mass Transfer 51 (2008) 2905–2913. [2] C.R. Ethier, M.R. Kaazempur-Mofrad, S. Wada, J.G. Myers, Mass transport and fluid flow in stenotic arteries, Int. J. Heat Mass Transfer 48 (2005) 4510–4517. [3] H. Ishiguro, B. Rubinsky, Influence of fish antifreeze proteins on the freezing of cell suspensions with cryoprotectant penetrating cells, Int. J. Heat Mass Transfer 41 (1998) 1907–1915. [4] C. Legallais, G. Catapano, B.V. Harten, U. Baurmeiters, A theoretical model to predict the in vitro performance of hemodiafilters, J. Membrane Sci. 168 (2000) 3–15. [5] B.B. Lieber, C. Sadasivan, H. Wing, S. Jaehoon, L. Cesar, The mixability of angiographic contrast with arterial blood, Med. Phys. 36 (2009) 5064–5078. [6] M. Fischer, I. Zinovik, D. Poulikakos, Diffusion and reaction controlled dissolution of oxygen microbubbles in blood, Int. J. Heat Mass Transfer 52 (2009) 5013–5019. [7] D. Weiping, H. Liqun, Z. Gang, Z. Haifeng, S. Zhiquan, G. Dayong, Double porous media model for mass transfer of hemodialyzers, Int. J. Heat Mass Transfer 47 (2004) 4849–4855. [8] N. Yang, K. Vafai, Low-density lipoprotein (LDL) transport in an artery – a simplified analytical solution, Int. J. Heat Mass Transfer 51 (2008) 497–505. [9] A.V. Kuznetsov, S.M. Becker, Thermal damage reduction associated with in vivo skin electroporation: a numerical investigation justifying aggressive precooling, Int. J. Heat Mass Transfer 50 (2007) 105–116. [10] Y. Granot, B. Rubinsky, Mass transfer model for drug delivery in tissue cells with reversible electroporation, Int. J. Heat Mass Transfer 51 (2008) 5610– 5616. [11] S.E. Locke, T.J. Gale, D. Kilpatrick, Development of an implantable blood flow and pressure monitor for pulmonary hypertension, Comput. Cardiol. 31 (2004) 713–716.
958
S. Chelikani et al. / International Journal of Heat and Mass Transfer 54 (2011) 949–958
[12] S.B. Palder, R.L. Kirkman, A.B. Whittemore, R.M. Hakin, J.M. Lazarus, N.L. Tilnet, Vascular access for hemodialysis, Ann. Surg. 202 (1985) 235–239. [13] J. Ethier, D.C. Mendelssohn, S.J. Elder, T. Hasegara, T. Akizawa, T. Akida, B.J. Canaud, R.L. Pisoni, Vascular access use and outcomes: an international perspective from the dialysis outcomes and practice patterns study, Nephrol. Dial. Transplant. 23 (2008) 3219–3226. [14] S.J. Schwab, Hemodialysis vascular access: the Achilles’ heel remains, Kidney Int. 72 (2007) 665–666. [15] S.J. Dudrick, History of vascular access, J. Parenter. Enter. Nutr. 30 (2006) S47– S56. [16] F.J.H. Gijsen, F.N. van de Vosse, J.D. Janssen, The influence of the nonNewtonian properties of blood on the flow in large arteries: steady flow in a carotid bifurcation model, J. Biomech. 32 (1999) 601–608. [17] N. Benard, R. Perrault, D. Coisne, Computational approach to estimating the effects of blood properties on changes in intra-stent flow, Ann. Biomed. Eng. 34 (2006) 1259–1271. [18] C. Rhie, W. Chow, A numerical study of the turbulent flow past an isolated airfoil with trailing edge separation, AIAA Paper No. 82-0998, 1982. [19] S. Majumdar, Role of underrelaxation in momentum interpolation for calculation of flow with nonstaggered grids, Numer. Heat Transfer 13 (1988) 125–132. [20] T. Barth, D. Jesperson, The design and applications of upwind schemes on unstructured meshes, AIAA Paper No. 89-0366, 1989. [21] J.M. Akl-Besharah, C.J. Mumford, S.A. Akashah, O. Salman, Prediction of the viscosity of lubricating oil blends, Fuel 68 (1989) 809–811. [22] P. Blackshear, Hemolysis at prosthetic surfaces, in: Y. Fung (Ed.), Biomechanics, Its Foundations and Objectives, Prentice Hall, Englewood Cliffs, CA, 1972.
[23] P. Blackshear, G. Blackshear, Mechanical hemolysis, in: R. Skalak, S. Chien (Eds.), Handbook of Bioengineering, McGraw-Hill, New York, NY, 1987. [24] P. Lu, H. Lai, J. Liu, A re-evaluation and discussion on the threshold limit for hemolysis in a turbulent shear flow, J. Biomech. 34 (2001) 1361–1364. [25] M. Grigioni, C. Daniele, G. D’Avenio, V. Barbaro, A discussion on the threshold limit for hemolysis related to Reynolds shear stress, J. Biomech. 32 (1999) 1107–1112. [26] A. Williams, D. Hughes, W. Nyborg, Hemolysis near a transversely oscillating wire, Science 169 (1970) 871–873. [27] R. Bacher, M. Williams, Hemolysis in capillary flow, J. Lab. Clin. Med. 76 (1970) 485–496. [28] J. Rooney, Hemolysis near an ultrasonically pulsating gas bubble, Science 169 (1970) 869–871. [29] R. Paul, J. Apel, S. Klaus, F. Schugner, P. Schwindke, H. Reul, Shear stress related blood damage in laminar Couette flow, Artif. Organs 27 (2003) 517– 529. [30] A. Sallam, N. Hwang, Human red blood cell hemolysis in a turbulent shear flow. Contribution of Reynolds shear stress, Biorheology 21 (1984) 783–797. [31] L. Wurzinger, R. Optiz, P. Blasberg, H. Schmid-Schonbein, Platelet and coagulation parameters following millisecond exposure to laminar shear stress, Thromb. Haemostasis 54 (1985) 381–386. [32] S. Sutera, M. Mehrjardi, Deformation and fragmentation of human RBC in turbulent shear flow, Biophys. J. 15 (1975) 1–10. [33] L. Leverett, J. Hellums, C. Alfrey, B. Lynch, Red blood cell damage by shear stress, Biophys. J. 12 (1972) 257–273. [34] S. Shapiro, M. Williams, Hemolysis in simple shear flows, Am. Inst. Chem. Eng. 16 (1970) 575–580. [35] J. Steinbach, PhD Thesis, University of Minnesota, Minneapolis, 1970.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Mass transfer in vascular access ports S. Chelikani a, E.M. Sparrow a, J.P. Abraham b,⇑, W.J. Minkowycz c a
Laboratory for Heat Transfer and Fluid Flow Practice, Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455-0111, USA Laboratory for Heat Transfer and Fluid Flow Practice, School of Engineering, University of St. Thomas, St. Paul, MN 55105-1079, USA c Department of Mechanical Engineering, University of Illinois at Chicago, 842 W. Taylor Street, Chicago, IL 60607-7022, USA b
a r t i c l e
i n f o
Article history: Received 23 August 2010 Received in revised form 11 October 2010 Accepted 11 October 2010 Available online 29 November 2010 Keywords: Vascular access Computational fluid dynamics Biological mass transfer Non-Newtonian flow
a b s t r a c t Numerical simulations have been performed to evaluate the fluid flow and mass transfer processes that occur in a human-body vascular-access port. Such ports are used to facilitate the frequent introduction of cleansed blood and other drugs into the body from external sources. Each of the infusion ports studied here consists of a reservoir and an attached tube-like catheter which delivers the infused medium to its point of use. All told, three unique infusion ports were investigated. Each had a particular geometry characterized by the shape of the reservoir and the mode of attachment of the catheter to the reservoir. The numerical simulations were three-dimensional and unsteady. Both Newtonian and non-Newtonian constitutive equations were employed for the fluid flow solutions and for the subsequent mass transfer solutions. The initiation of fluid motion was the injection of a controlled volume of fluid into the reservoir. In some cases, the injected fluid was the same as that in the reservoir, and in others it was different. For all the investigated infusion systems, no hemolysis (red blood cell destruction) was in evidence when blood was passed through the catheter. Potential hemolysis was averted in two of the systems due to the very low fluid velocities. As witnessed by the mass transfer results, the use of the reservoir as a chamber to mix a secondary liquid with blood is a viable strategy. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Mass transfer processes are ubiquitous and of diverse function in the human body, as witnessed by an extensive literature. Representative highlights of that literature will now be conveyed. In an analytical solution for macromolecule mass transfer in an artery, the transport within the wall and in the lumen were coupled in [1]. There, mass transfer was found to play a significant role in the development of atherosclerosis, a progressive disease of the artery wall [2]. Antifreeze proteins affect the heat and mass transport processes in the cryopreservation of human tissue and organs [3]. Hemodiafiltration depends strongly on mass transfer coefficients as well as on concentration polarization and water filtration flux [4]. The dispersion of angiographic contrast fluid in arteries can be used as a means of evaluating arterial geometry [5]. Dissolution of a gas bubble in blood is controlled by the diffusion of the gas from the bubble surface into the surrounding blood [6]. The foregoing diversity of mass transfer processes can be extended by further reference to the literature. The mass transfer of hemodialyzers was successfully modeled by a double-porousmedium construct consisting of two interpenetrating porous media, respectively for the blood flow and for the dialysate flow [7]. A ⇑ Corresponding author. Tel.: +1 651 962 5766. E-mail address: [email protected] (J.P. Abraham). 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.10.008
four-layer model of an artery encompassing the endothelium, intima, elastic lamina, and media was used to determine the transfer of LDL across the wall [8]. Electroporation is an approach used to enhance transdermal transport of large molecules [9,10]. A mixture of saline and plasma replaced blood in in-vitro experiments for the development of membranes [11]. The development of an impedance method of measuring blood flow rate was carried by means of experiments in which mixtures of saline and blood were the working fluids. The publications cited in the preceding paragraphs were encountered as the authors sought prior information about the focus of the present research: mass transfer in human-body vascular-access ports (or infusion ports). There are several medical therapies that are administered to patients frequently. To facilitate such therapies, it is convenient to be able to enter the body at a designated location without having to create a local aperture at each session of administration. The designated access location is often termed vascular access. An example of this situation is the need for application of dialysis therapy. This need arises in situations when kidneys are unable to fulfill their function of waste removal from the blood stream. It is common that the use of dialysis machines is required several times a week. In that case, it is clearly advantageous that a vascular access exists. Another example of the need for an established access port is during the frequent administration of drugs.
950
S. Chelikani et al. / International Journal of Heat and Mass Transfer 54 (2011) 949–958
Nomenclature cp C d D f k n m p q Re t T
specific heat concentration diameter binary diffusion coefficient friction factor thermal conductivity constant in power-law viscosity model constant used in Carreau viscosity model pressure constant used in Carreau viscosity model Reynolds number time temperature
The common factor of these procedures is the flow of fluids. With respect to dialysis, the cleansed blood is returned to the body by means of a pump or other fluid which serves to push the blood. When drugs are administered, the administration is almost always in fluid form. The injected drug may, in certain circumstances, interface with blood and mix with it. In such a circumstance, mass transfer occurs. The literature of vascular access was found to be both sparse and uninformative. History and current status of the modality are discussed in [12–15], but there appears to be no prior information available on mass transfer in vascular access ports. The focus of this research is the fluid mechanics and mass transfer of fluids that are passing through a permanent vascular access port in the human body. The work will be subdivided into two parts. In the first part, the flow of a homogenous fluid through several infusion port geometries will be analyzed. The second part will consider both fluid flow and mass transfer, the latter due to the presence of a different species. The numerical simulations will be performed by means of ANSYS – CFX software. This is a commercial code based on a finite-volume discretization of the Navier–Stokes equations, the conservation of mass equation and, when appropriate, a species conservation equation. The solutions encompass unsteady three-dimensional flows. 2. Fluid-flow evaluation of three vascular access ports Views of the three vascular access ports to be considered are conveyed in Fig. 1(a)–(c). These three configurations, respectively designated as A, B and C, correspond to different candidate ports. The first two of these are close approximations of those already used in practice, while the third is a new design whose evaluation is an important goal of this investigation. A common feature of all three configurations is an inlet aperture through which a fluid is introduced into a relatively large reservoir. The inflowing fluid serves to displace fluid that is already in the reservoir, the result of the displacement being an outflow into a catheter which delivers fluid to the blood stream. Geometrical differences among the three ports are related to the shape and size of the reservoir as well as the location of the exit. As will be demonstrated when the results are presented, these differences give rise to potential hemolysis-creating flow patterns. Fig. 1(a) shows Port A. In essence, it is a flat disc with a cylindrical depression in its upper surface. The disc is hollow and serves the purpose of a reservoir. The aperture shown in the figure and labeled inlet is created by the user when inserting the needle of a syringe. The catheter is attached to the reservoir along a radial line extending outward from its surface. The second of the currently available ports, to be designated as Port B, is exhibited in Fig. 1(b). Once again, the reservoir is a hollow
u, v, w U x, y, z X
velocity components cross-sectional average velocity coordinates mass fraction
Greek symbols density dynamic viscosity kinematic viscosity constant use in power-law viscosity model constants used in Carreau viscosity model shear strain rate
q l m j l0, l1, k c
disc-shaped body. The catheter is joined to the reservoir as a tangent to its outer rim. The last of the ports, shown below, is to be designated as Port C. It differs from Port B in that the attachment of the catheter to the reservoir is accomplished by means of a gently converging transition piece. In Port C, the capability to build a gentle convergence into the catheter is enabled by the height of the reservoir being considerably larger than the diameter of the catheter. It can be expected that the gradual contraction would give rise to a much smoother inflow than would occur for Ports A and B which do not have this feature. To reinforce the foregoing discussion, Fig. 2(a)–(c) have been prepared. These figures are planar cuts which show the geometry of the transitions from the reservoir to the catheter. Fig. 2(a), which corresponds to Port A, has a transition which includes a sharp-edged inlet. It is well known that sharp-edged inlets give rise to flow separation because fluids are unable to turn sharp corners. The inlet shape shown in Fig. 2(b) (Port B) includes a turn that is greater than 90° (re-entrant). Such a turn is even more difficult for a fluid to negotiate than would be a 90° turn. Again, flow separation is to be expected. Another feature of the geometry of the transition section of Port B is the abrupt change of cross section that can be seen at the top of the diagram (Fig. 2(b)). Such an abrupt change is difficult for the fluid to cope with, and separation is likely. The just-identified two geometric errors in the design of Port B have been corrected in Port C, as can be seen from Fig. 2(c). Not only has the more than 90° turn been eliminated, but also a rounding of the transition geometry has been introduced. Furthermore the abrupt change of cross section noted at the end of the last paragraph no longer exists in Port C. 3. Governing equations of fluid flow The equations which will be solved for the fluid-flow problem are the three-dimensional, unsteady, incompressible form of the Navier–Stokes equations and the equation of continuity. In these equations, two constitutive equations were employed. The first is the Newtonian model which makes use of a constant value of the viscosity. The other is a non-Newtonian model, two candidate forms of which were used. 3.1. Navier–Stokes equations
" # @u @ 2 u @ðuv Þ @ðuwÞ @p @ @u ¼ þ 2þ þ þ l @t @x @y2 @z2 @x @x @x @ @u @ @u l l þ þ @y @y @z @z
q
ð1Þ
951
S. Chelikani et al. / International Journal of Heat and Mass Transfer 54 (2011) 949–958
3.2. Continuity equation
@u @ v @w þ ¼0 þ @x @y @z
ð2Þ
The boundary conditions for the velocity problem are: (a) all velocities are zero on all solid surfaces, (b) the fluid entering the solution domain is assigned a steady velocity that is uniformly distributed across the inlet section, and (c) standard weak boundary conditions at the exit of the catheter and a pressure prescribed there. Since the problem is unsteady, an initial condition is also needed. The condition of zero velocity and uniform pressure throughout the solution domain was imposed as the initial condition. 4. Numerical simulations For each of the port configurations, the numerical simulation mimicked an experiment which was initiated with both the reservoir and the catheter filled with blood. The activation of the fluid motion was accomplished by the steady injection of a second fluid (sometimes the same as the fluid already in the system and sometimes a different fluid). The operating conditions for the executed numerical simulations are listed in Table 1. It can be seen from the table, nine of the investigated cases were characterized by the same injected volumetric flow, 10 ml/min. To implement a sensitivity study, the other three cases were accorded injection flow rates that were four times higher. The period of injection was the same for all cases. The injected fluids were either blood or saline. The numerical solutions made use of two constitutive models for blood, either Newtonian or non-Newtonian. For the latter, two different algebraic models were employed. These equations are: Carreau equation [16] (non-Newtonian 1)
h
m1 l ¼ l1 þ ðl0 l1 Þð1 þ ðckÞq Þð q Þ
i
ð3Þ
Power law (non-Newtonian 2)
l ¼ j cð 10 Þ ; for 0:001 6 c 6 1000
ð4aÞ
n1 l ¼ j 10ð 10 Þ ;
ð4bÞ
n1
for c P 1000
n1 l ¼ j 0:001ð 10 Þ ;
for c 6 0:001
ð4cÞ
where l is the viscosity (kg/m s), and c is the shear rate (1/s). The quantities m, j, q, k, n, l1, and l0 are constants. Values of the Carreau constants taken from [17] were: k = 0.11, m = 0.392, l0 = 0.022 kg/m s, l1 = 0.0022 kg/m s, and q = 0.644. For the power-law model, j = 0.42 and n = 0.61.
Fig. 1. Three-dimensional isometric view of (a) Port A (close facsimile of the PowerPort device from Bard Access Systems), (b) of Port B (close proximity of the VortexPort device from Angiodynamics), and (c) C (under development from R4 Vascular).
This equation applies for the x-coordinate direction. The quantities u, v, and w are the velocity components in the x, y, and z directions; q is the density, p is the pressure, and l is the viscosity. The symbol t represents time. The two complementary equations are obtained by cycling u, v, and w and x, y, and z. Note that the viscosity is contained within the differentiation operators to accommodate the spatial variations of the non-Newtonian model. For the Newtonian model, the viscosity passes through these operators.
A graph of these equations is displayed in Fig. 3. In that figure, the viscosity is plotted as function of the strain rate. Also appearing in the figure is a horizontal line that represents a Newtonian model for blood. It can be seen from Fig. 3 that at higher shear rates, both the above models become very close to the Newtonian model. The calculations were carried out using CFX 11.0, a commercial finite-volume-based CFD program. A false-transient, time-stepping approach is employed to enable convergence to the steady-state solution. While the fully implicit, backward-Euler, time-stepping algorithm exhibits first-order accuracy in time, its use does not affect the accuracy of the final, converged solution. Coupling of the velocity-pressure equations was achieved on a non-staggered, collocated grid using the techniques developed by Rhie and Chow [18] and Majumdar [19]. The inclusion of pressure-smoothing terms in the mass conservation equation
952
S. Chelikani et al. / International Journal of Heat and Mass Transfer 54 (2011) 949–958
Fig. 2. Planar view of (a) Port A, (b) Port B, and (c) Port C.
suppresses oscillations which can occur when both the velocity and pressure are evaluated at coincident locations. The advection terms in the momentum and energy equations were evaluated by using the upwind values of the momentum flux, supplemented with an advection-correction term. The correction term reduces the occurrence of numerical diffusion and is of second-order accuracy. Details of the advection treatment can be found in [20].
5. Diagnostic fluid flow results Figs. 4 and 5, respectively show the variation of the shear strain rate and pressure gradient along the center line of the catheter extending from the center of the reservoir to the end of the catheter. The center line of the catheter lies along the z coordinate. The section z = 0 to z = 0.011 (m) extends from the center of the reservoir to the outlet of the reservoir which is also the inlet of the catheter.
Table 1 Detailed information for the numerical simulation parameters. Run no.
Port type
Injected fluid
Rate of injected fluid (ml/min)
Injection time (s)
Total domain volume (m3)
Constitutive law
1 2 3 4 5 6 7 8 9 10 11 12
A B C A B C B C C A B C
Blood Blood Blood Blood Blood Blood Blood Blood Blood Saline Saline Saline
10 10 10 40 40 40 10 10 10 10 10 10
2 2 2 2 2 2 2 2 2 2 2 2
7.51e7 8.85e7 9.96e7 7.51e7 8.85e7 9.96e7 8.85e7 9.96e7 9.96e7 7.51e7 8.85e7 9.96e7
Newtonian Newtonian Newtonian Newtonian Newtonian Newtonian Non–Newtonian 1 Non–Newtonian 1 Non–Newtonian 2 Newtonian Newtonian Newtonian
953
S. Chelikani et al. / International Journal of Heat and Mass Transfer 54 (2011) 949–958
Table 2 Pressure drop and maximum shear stress in the fluid domain at different flow rates.
Fig. 3. Viscosity of blood according to the Carreau model, the power-law model, and the Newtonian model.
No.
Port
Flow rate (ml/min)
Maximum shear stress (Pa)
Pressure drop (Pa)
1 2 3 4 5 6
A A B B C C
10 40 10 40 10 40
3.9 24 4.5 25 7.2 30
113 586 113 588 89 490
is to be simulated, it would be sufficient to use a Newtonian viscosity model in conjunction with a mixing rule to determine the viscosity of saline-blood mixtures. For simplicity, an empirical formula which is commonly used to determine the viscosity of fluid blends was used to determine the viscosity of the salineblood mixture [21].
m¼e
V
blend 10:975 14:524
0:8
ð5Þ
where
V blend ¼ ½X a VBNa þ ½X b VBNb
ð5aÞ
and
VBN ¼ 14:534 ln½lnðm þ 0:8Þ þ 10:975
Fig. 4. Shear strain rate along the center of the catheter.
ð5bÞ
where, m is the kinematic viscosity (centistokes), and X is the mass fraction. The term VBN refers to the Viscosity Blend Number. The duration of each simulation run was equal to the duration of injection period. In all cases, the flow was regarded as laminar with calculated Reynolds numbers of 47 and 188 when a Newtonian viscosity was used with the given flow rates of 10 ml/min and 40 ml/min, respectively. As a check on the accuracy of the numerical simulation, it may first be observed that for fully developed laminar flow of a Newtonian fluid, the friction factor in the catheter is expected to be f = 64/ Re. To compare this friction factor with that from the simulation, the definition of the friction factor is given by
@p 1 d f ¼ qU 2 @x 2
ð6Þ
where @p is the pressure gradient in the catheter, d is the catheter @x diameter, q is the fluid density, and U is the mean fluid velocity. By making use of the known values of the quantities appearing on the right-hand side of this equation, it is found that f = 1.37 in the catheter proper for a flow rate of 10 ml/min. From the theoretical solution, f = 64/Re = 1.36. The agreement between these two values is within 0.74%. 5.1. Pressure drop and shear stress
Fig. 5. Pressure gradient along the center of the catheter.
It can be seen that both Newtonian and non-Newtonian flows give similar profiles for the pressure gradient and for the strain rate. The Newtonian model gives an upper bound for the strain rate. It is well established that higher strain rates give rise to higher rates of hemolysis. So, for the given conditions, a Newtonian model gives a conservative representation of the flow field with respect to hemolysis and has been chosen for further simulations because of its simplicity. This finding also suggests that when saline injection
The effects of flow rate and port geometry on the pressure drop and maximum shear stress are summarized in Table 2. The listed results correspond to the steady state that was achieved prior to the termination of the two-second run period. The pressure drop corresponds to end-to-end pressure variations for the access port. An inspection of the pressure drop results listed in Table 2 shows a remarkable similarity between those for Ports A and B. On the other hand, the pressure drops for Port C are approximately 80% of those for the other ports. This reduction in pressure drop can be attributed to an improved geometric design for Port C. In truth, however, this reduction in pressure does not provide any practical advantage. Further insights into the pressure results can be obtained by noting that an increase in flow rate by a factor of four gives rise
954
S. Chelikani et al. / International Journal of Heat and Mass Transfer 54 (2011) 949–958
to a pressure drop increase that is larger than a factor of four. Since the flow in the catheter is more or less fully developed throughout (horizontal lines in Fig. 4) and fully developed pressure drops scale linearly with the flow rate for laminar flow, the loss mechanism in the reservoir does not follow a simple flow rate scaling rule. Next, attention may be turned to the maximum shear stress results of Table 2. The maximum shear stress in the fluid domain is found to occur within the reservoir near the fluid injection region. When considering these results, it is important to recognize that they are locally calculated. Therefore, they are subject to the local nature of the mesh. As a consequence, they should not be regarded as having the same level of fidelity as would characterize average results such as the end-to-end pressure drops that are also listed in Table 2. However, the issue of the accuracy level is moot when looked at with respect to the context of hemolysis. It can be demonstrated that the shear stress values listed in the table are far below the threshold at which hemolysis occurs. This demonstration will be implemented with the aid of Fig. 6. Fig. 6 shows a summary of results available in the literature [22–35] which give the relationship between shear stress level and hemolysis. It shows the time required to induce hemolysis at different magnitudes of shear stress. The figure indicates that hemolysis is unlikely to occur at shear stresses below 100 Pa. Since the values of shear listed in Table 2 are considerably lower than 100 Pa, it may be concluded that none of the ports investigated give rise to a high-enough shear stress to cause hemolysis at the given flow rates.
5.2. Streamlines and vector diagrams Streamlines and vector diagrams provide a means for visualizing the flow pattern in the reservoirs. Fig. 7(a)–(c) show the threedimensional streamlines originating from the inlets of ports A, B and C, respectively. Since the flow becomes steady very quickly, these lines also represent the pathlines. It can be seen from these figures that the streamlines are twisted near the inlet of the catheter for Ports A and B due the sudden change of cross section. On the other hand, the twist is much less pronounced for Port C. Another feature that distinguishes Port C from the others is that its coiled flow pattern in the reservoir is not as tightly knotted as the rest. The degree of smoothness experienced by the flow passing from the reservoir into the catheter is clearly displayed by the vector diagrams shown in Fig. 8(a)–(c). Examination of Fig. 8(a), Port A, shows the streamlines bending sharply as they negotiate what is nearly a 90° change of direction. Since the Reynolds number of the flow is very low, such a sharp change is able to occur without
Fig. 7. Three-dimensional streamlines originating from the inlet of (a) Port A, (b) Port B, and (c) Port C.
Fig. 6. Relationship between shear stress level and hemolysis.
flow separation. Fig. 8(b) indicates the presence of a recirculation zone for Port B. In general, recirculation zones may bring on hemolysis. However, in the present situation, the flow velocities are so low that hemolysis does not occur. In addition, a recirculation zone is filled with a captive fluid which does not exit the catheter. It may be speculated that particulates could congregate in the captive fluid and cause agglomeration. Fig. 8(c) corresponds to Port C. The passage of the flow from the reservoir into the catheter follows smoothly converging streamlines and may be regarded as ideal.
S. Chelikani et al. / International Journal of Heat and Mass Transfer 54 (2011) 949–958
" # @T @T @T @CT @2T @2T @2T þu þv þw ¼k qcp þ þ @t @x @y @z @x2 @y2 @z2
955
ð8Þ
In this equation, T is the temperature, k is the thermal conductivity, and cp is the specific heat at constant pressure. The two foregoing equations can be brought into congruence by a simple transformation:
T ! C;
cp ! 1=q;
and k ! D
ð9Þ
By this transformation, the species conservation equation can be solved by CFX as if it were a heat convection equation. The value of the blood-saline diffusion coefficient was taken to be 1.5e9 m2/s. The boundary conditions for the mass transfer problem are: (a) no mass transfer at any solid surface, (b) given value of C at the inlet of the solution domain, and (c) standard weak boundary conditions at the exit of the catheter. Initially, C = 0 throughout the solution domain. 6.1. Concentration contour diagrams
Fig. 8. Velocity vectors at the entrance of the catheter for (a) Port A, (b) Port B, and (c) Port C.
6. Mass transfer model If the quantity C represents the concentration of a diffusing species in another medium, the species conservation equation is:
" # @C @C @C @2C @2C @2C þv þw ¼D u þ þ @x @y @z @x2 @y2 @z2
ð7Þ
where D is the binary diffusion coefficient. A simplified treatment of mass transfer is accomplished by an extension of the thermal energy equation, which is
The results of the mass transfer simulations will be conveyed by means of concentration contour diagrams. The displayed concentration is that of the injected fluid. A concentration C = 1 at a point means that the fluid at that point is totally injected fluid. On the other hand, C = 0 signifies the total absence of the injected fluid. In that case, the fluid is that which originally occupied the reservoir and the catheter, hereafter referred to as the incumbent fluid. The presented concentration information is for the steady state. The color strip at the left of each contour diagram provides the relationship between the grey tones and the concentration values. The in-reservoir results for each of the ports are presented in a series of figures, respectively Fig. 9(a)–(c) for Port A, Port B, and Port C. The left and right images of each figure are focused on the reservoir, but display different views. The left parts show the concentration contours in a plane that is perpendicular to the top and bottom surfaces of the reservoir and parallel to the geometric axis of the catheter. The right part of each figure displays contours in a plane that is perpendicular to that used in the left-hand image. The plane of the right-hand images lies at the half-height of the reservoir. Inspection and comparison of the three left-hand figures show the degree of coherence of the injected fluid as it mixes with the incumbent fluid. For Port B, the injected fluid experiences little mixing as it moves downward along the centerline of the reservoir. For Port A, there is slightly more mixing along this path. In contrast, there is considerably more mixing between the injected and incumbent fluids in Port C as the former makes its way downward along the centerline of the reservoir. The injected fluid dead-ends against the bottom of the reservoir and then fans outward along the bottom surface. The flaring is quite symmetric for Port A, but less so for Ports B and C. The main observation from the left-hand figures is the degree of mixing of the injected and incumbent fluids. In this regard, it is clear that the geometry of Port C gives rise to the best mixing. The right-hand figures confirm the hierarchy of mixing for the various ports. From most mixing to least mixing, the ports are ordered as C, A and B. Further observation of these images reveals an arrow-shaped contour formation, with the arrow pointing from the center of the reservoir toward the exit. This formation is clearly in evidence for Ports B and C and to a lesser extent for Port A. The legend identifies these zones as containing primarily incumbent fluid. Three additional figures, Figs. 10–12, will now be presented to convey mass concentration information for the catheter. Each of these figures is made up of three circles, each of which represents a cross section of the catheter. From left to right, the respective circles
956
S. Chelikani et al. / International Journal of Heat and Mass Transfer 54 (2011) 949–958
Fig. 9. Concentration contour along a plane perpendicular to the reservoir top and bottom surfaces (left image) and along a plane perpendicular to the fluid injection and located at the center of reservoir (right image) for (a) Port A, (b) Port B, and (c) Port C.
Fig. 10. Concentration contours in cross-sectional planes near the inlet (a), center (b) and outlet (c) of the catheter for Port A.
S. Chelikani et al. / International Journal of Heat and Mass Transfer 54 (2011) 949–958
957
Fig. 11. Concentration contours in cross-sectional planes near the inlet (a), center (b) and outlet (c) of the catheter for Port B.
Fig. 12. Concentration contours in cross-sectional planes near the inlet (a), center (b) and outlet (c) of the catheter for Port C.
correspond to the catheter inlet, the catheter half length, and the catheter exit. For Port B, Fig. 11, there is clear evidence that mixing occurs along the length of the catheter. At the exit cross section, the concentration is generally homogeneous, but the incumbent fluid still dominates adjacent to the wall. Progressive mixing is also in evidence for Port A (Fig. 10), but the concentration is less uniform at the exit cross section of the reservoir (inlet of the catheter) than that observed for Port B. The results for Port C, Fig. 12, display little lengthwise evolution. The concentration at the exit cross closely resembles that for Port B.
7. Concluding remarks To the best knowledge of the author, the work presented here is the first attempt to quantify the fluid flow and mass transfer characteristics of human-body infusion ports. The work was motivated by the need to characterize competing versions of such therapyfacilitating systems. Two of the systems (A and B in the designation of the report) are already available in the marketplace. The other, system C, is still in the developmental stage. The method of investigation used here was based on numerical simulation. Threedimensional, unsteady solutions of the Navier–Stokes, continuity, and species conservation were performed. The general configuration of the three systems is the same. A reservoir containing the medium to be infused is attached to a tube-like catheter which delivers the medium to its point of use. Fluid motion is activated by the introduction of a fixed volume of liquid into the reservoir. The liquid may be either the same or different from that in the reservoir. The differences among the systems are geometrical, including reservoirs of different sizes and in the mode of attachment of the
catheter to the reservoir. In particular, system C had an optimal attachment which guaranteed the absence of flow separation. The attachment mode of system A gave rise to flow separation, and that of B gave rise to incipient separation. The flow velocities were so small that hemolysis did not occur in either of the ports, regardless of the attachment mode. The successful use of the reservoir as a mixing chamber for different fluids has also been validated. References [1] M. Khakpour, K. Vafai, A comprehensive analytical solution of macromolecular transport within an artery, Int. J. Heat Mass Transfer 51 (2008) 2905–2913. [2] C.R. Ethier, M.R. Kaazempur-Mofrad, S. Wada, J.G. Myers, Mass transport and fluid flow in stenotic arteries, Int. J. Heat Mass Transfer 48 (2005) 4510–4517. [3] H. Ishiguro, B. Rubinsky, Influence of fish antifreeze proteins on the freezing of cell suspensions with cryoprotectant penetrating cells, Int. J. Heat Mass Transfer 41 (1998) 1907–1915. [4] C. Legallais, G. Catapano, B.V. Harten, U. Baurmeiters, A theoretical model to predict the in vitro performance of hemodiafilters, J. Membrane Sci. 168 (2000) 3–15. [5] B.B. Lieber, C. Sadasivan, H. Wing, S. Jaehoon, L. Cesar, The mixability of angiographic contrast with arterial blood, Med. Phys. 36 (2009) 5064–5078. [6] M. Fischer, I. Zinovik, D. Poulikakos, Diffusion and reaction controlled dissolution of oxygen microbubbles in blood, Int. J. Heat Mass Transfer 52 (2009) 5013–5019. [7] D. Weiping, H. Liqun, Z. Gang, Z. Haifeng, S. Zhiquan, G. Dayong, Double porous media model for mass transfer of hemodialyzers, Int. J. Heat Mass Transfer 47 (2004) 4849–4855. [8] N. Yang, K. Vafai, Low-density lipoprotein (LDL) transport in an artery – a simplified analytical solution, Int. J. Heat Mass Transfer 51 (2008) 497–505. [9] A.V. Kuznetsov, S.M. Becker, Thermal damage reduction associated with in vivo skin electroporation: a numerical investigation justifying aggressive precooling, Int. J. Heat Mass Transfer 50 (2007) 105–116. [10] Y. Granot, B. Rubinsky, Mass transfer model for drug delivery in tissue cells with reversible electroporation, Int. J. Heat Mass Transfer 51 (2008) 5610– 5616. [11] S.E. Locke, T.J. Gale, D. Kilpatrick, Development of an implantable blood flow and pressure monitor for pulmonary hypertension, Comput. Cardiol. 31 (2004) 713–716.
958
S. Chelikani et al. / International Journal of Heat and Mass Transfer 54 (2011) 949–958
[12] S.B. Palder, R.L. Kirkman, A.B. Whittemore, R.M. Hakin, J.M. Lazarus, N.L. Tilnet, Vascular access for hemodialysis, Ann. Surg. 202 (1985) 235–239. [13] J. Ethier, D.C. Mendelssohn, S.J. Elder, T. Hasegara, T. Akizawa, T. Akida, B.J. Canaud, R.L. Pisoni, Vascular access use and outcomes: an international perspective from the dialysis outcomes and practice patterns study, Nephrol. Dial. Transplant. 23 (2008) 3219–3226. [14] S.J. Schwab, Hemodialysis vascular access: the Achilles’ heel remains, Kidney Int. 72 (2007) 665–666. [15] S.J. Dudrick, History of vascular access, J. Parenter. Enter. Nutr. 30 (2006) S47– S56. [16] F.J.H. Gijsen, F.N. van de Vosse, J.D. Janssen, The influence of the nonNewtonian properties of blood on the flow in large arteries: steady flow in a carotid bifurcation model, J. Biomech. 32 (1999) 601–608. [17] N. Benard, R. Perrault, D. Coisne, Computational approach to estimating the effects of blood properties on changes in intra-stent flow, Ann. Biomed. Eng. 34 (2006) 1259–1271. [18] C. Rhie, W. Chow, A numerical study of the turbulent flow past an isolated airfoil with trailing edge separation, AIAA Paper No. 82-0998, 1982. [19] S. Majumdar, Role of underrelaxation in momentum interpolation for calculation of flow with nonstaggered grids, Numer. Heat Transfer 13 (1988) 125–132. [20] T. Barth, D. Jesperson, The design and applications of upwind schemes on unstructured meshes, AIAA Paper No. 89-0366, 1989. [21] J.M. Akl-Besharah, C.J. Mumford, S.A. Akashah, O. Salman, Prediction of the viscosity of lubricating oil blends, Fuel 68 (1989) 809–811. [22] P. Blackshear, Hemolysis at prosthetic surfaces, in: Y. Fung (Ed.), Biomechanics, Its Foundations and Objectives, Prentice Hall, Englewood Cliffs, CA, 1972.
[23] P. Blackshear, G. Blackshear, Mechanical hemolysis, in: R. Skalak, S. Chien (Eds.), Handbook of Bioengineering, McGraw-Hill, New York, NY, 1987. [24] P. Lu, H. Lai, J. Liu, A re-evaluation and discussion on the threshold limit for hemolysis in a turbulent shear flow, J. Biomech. 34 (2001) 1361–1364. [25] M. Grigioni, C. Daniele, G. D’Avenio, V. Barbaro, A discussion on the threshold limit for hemolysis related to Reynolds shear stress, J. Biomech. 32 (1999) 1107–1112. [26] A. Williams, D. Hughes, W. Nyborg, Hemolysis near a transversely oscillating wire, Science 169 (1970) 871–873. [27] R. Bacher, M. Williams, Hemolysis in capillary flow, J. Lab. Clin. Med. 76 (1970) 485–496. [28] J. Rooney, Hemolysis near an ultrasonically pulsating gas bubble, Science 169 (1970) 869–871. [29] R. Paul, J. Apel, S. Klaus, F. Schugner, P. Schwindke, H. Reul, Shear stress related blood damage in laminar Couette flow, Artif. Organs 27 (2003) 517– 529. [30] A. Sallam, N. Hwang, Human red blood cell hemolysis in a turbulent shear flow. Contribution of Reynolds shear stress, Biorheology 21 (1984) 783–797. [31] L. Wurzinger, R. Optiz, P. Blasberg, H. Schmid-Schonbein, Platelet and coagulation parameters following millisecond exposure to laminar shear stress, Thromb. Haemostasis 54 (1985) 381–386. [32] S. Sutera, M. Mehrjardi, Deformation and fragmentation of human RBC in turbulent shear flow, Biophys. J. 15 (1975) 1–10. [33] L. Leverett, J. Hellums, C. Alfrey, B. Lynch, Red blood cell damage by shear stress, Biophys. J. 12 (1972) 257–273. [34] S. Shapiro, M. Williams, Hemolysis in simple shear flows, Am. Inst. Chem. Eng. 16 (1970) 575–580. [35] J. Steinbach, PhD Thesis, University of Minnesota, Minneapolis, 1970.