Analysis of cell flux in the parallel plate flow chamber: implications for cell capture studies.

The parallel plate flow chamber provides a controlled environment for determinations of the shear stress at which cells in suspension can bind to endothelial cell monolayers. By decreasing the flow rate of cell-containing media over the monolayer and assessing the number of cells bound at each wall shear stress, the relationship between shear force and binding efficiency can be determined. The rate of binding should depend on the delivery of cells to the surface as well as the intrinsic cell-surface interactions; thus, only if the cell flux to the surface is known can the resulting binding curves be interpreted correctly. We present the development and validation of a mathematical model based on the sedimentation rate and velocity profile in the chamber for the delivery of cells from a flowing suspension to the chamber surface. Our results show that the flux depends on the bulk cell concentration, the distance from the entrance point, and the flow rate of the cell-containing medium. The model was then used in a normalization procedure for experiments in which T cells attach to TNF-alpha-stimulated HUVEC monolayers, showing that a threshold for adhesion occurs at a shear stress of about 3 dyn/cm2.     

Visualization of flow-dependent concentration polarization of macromolecules at the surface of a cultured endothelial cell monolayer by means of fluorescence microscopy

To substantiate the occurrence of flow-dependent concentration or depletion of atherogenic lipoproteins, which has been theoretically predicted to take place at a blood/endothelium boundary, we have studied the effects of perfusion pressure and wall shear rate on the accumulation and uptake of microspheres by cultured vascular endothelial cells in a monolayer. The study was carried out by flowing a cell culture medium containing fetal calf serum and fluorescent microspheres through a parallel-plate flow chamber having a cultured bovine aortic endothelial cell (BAEC) monolayer on one wall of the chamber. The microspheres had a nominal diameter of 19 nm, approximately the same as that of low-density lipoproteins, and thus served as models and tracers of plasma proteins and lipoproteins. Experiments were carried out in steady flow in the physiological range of wall shear rate and water filtration velocity at the monolayer, while monitoring the intensity of fluorescence of the spheres accumulated at and taken up by the endothelial cells. It was found that in a perfusate containing only fluorescent microspheres, due to increased phagocytic activity of the endothelial cells, the intensity of fluorescence which reflected the number of the microspheres taken up by the endothelial cells, increased almost linearly with time and independently of wall shear rate. However, with perfusates containing fetal calf serum, this abnormal phenomenon did not occur, and the intensity of fluorescence increased with increasing perfusion pressure and decreasing wall shear rate. It was also found that the number of fluorescent microspheres accumulated at and taken up by the BAEC monolayer was shear-dependent only at low wall shear rates, and increased sharply when the flow rate was reduced to zero. These results provided solid experimental evidence that flow-dependent concentration or depletion of macromolecules occurs at the luminal surface of the endothelium at physiological wall shear rates and water filtration velocities, and strongly supports the hypothesis that flow-dependent concentration polarization of lipoproteins plays an important role in the localization of atherosclerosis and intimal hyperplasia in man by facilitating the uptake of atherogenic lipoproteins by endothelial cells.     

Influence of hydrodynamic imposed shear stress on the activation of factor X in the presence of tissue factor-factor VIIa complex in a continuous flow reactor

Summary The effect of shear stress on the ability of tissue factor-factor VIIa complex to activate factor X in a continuous flow reactor was studied. Tissue factor immobilized in a phospholipid bilayer on the inner surface of a capillary tube was exposed to a perfusate containing factors VIIa and X flowing at flow rates of 12.7, and 204 l/min, corresponding to wall shear rates of 100, and 1760 sec^-1. The maximum flux (moles formed per unit surface area per unit time) of factor Xa (activated form of factor X) produced at the wall decreased as the shear stress at the wall was increased from 1 to 3 dynes/cm² (3-fold) at a constant shear rate of 100 sec^-1. In contrast, at higher shear rate (1760 sec^-1), increasing shear stress from 16 to 48 dynes/cm² had no significant influence on factor Xa production. The decreased production of factor Xa at higher shear stress (low shear rate 100 sec^-1) probably reflects the transport limitation of factor X to the wall. Apparently shear stress can directly influence the activation of factor X at low shear rates.     

Inhibition of tissue factor-factor VIIa-catalyzed factor X activation by factor Xa-tissue factor pathway inhibitor. A rotating disc study on the effect of phospholipid membrane composition.

The physiological inhibitor of tissue factor (TF).factor VIIa (FVIIa), full-length tissue factor pathway inhibitor (TFPI(FL)) in complex with factor Xa (FXa), has a high affinity for anionic phospholipid membranes. The role of anionic phospholipids in the inhibition of TF.FVIIa-catalyzed FX activation was investigated. FXa generation at a rotating disc coated with TF embedded in a membrane composed of pure phosphatidylcholine (TF.PC) or 25% phosphatidylserine and 75% phosphatidylcholine (TF.PSPC) was measured in the presence of preformed complexes of FXa.TFPI(FL) or FXa.TFPI(1-161) (TFPI lacking the third Kunitz domain and C terminus). At TF.PC, FXa.TFPI(FL) and FXa.TFPI(1-161) showed similar rate constants of inhibition (0.07 x 10(8) M(-1) s(-1) and 0.1 x 10(8) M(-1) s(-1), respectively). With phosphatidylserine present, the rate constant of inhibition for FXa.TFPI(FL) increased 3-fold compared with a 9-fold increase in the rate constant for FXa. TFPI(1-161). Incubation of TF.PSPC with FXa.TFPI(FL) in the absence of FVIIa followed by depletion of solution FXa.TFPI(FL) showed that FXa.TFPI(FL) remained bound at the membrane and pursued its inhibitory activity. This was not observed with FXa.TFPI(1-161) or at TF.PC membranes. These data suggest that the membrane-bound pool of FXa.TFPI(FL) may be of physiological importance in an on-site regulation of TF.FVIIa activity.     

Analysis of temporal shear stress gradients during the onset phase of flow over a backward-facing step

Endothelial cells in blood vessels are exposed to bloodflow and thus fluid shear stress. In arterial bifurcations and stenoses, disturbed flow causes zones of recirculation and stagnation, which are associated with both spatial and temporal gradients of shear stress. Such gradients have been linked to the generation of atherosclerotic plaques. For in-vitro studies of endothelial cell responses, the sudden-expansion flow chamber has been widely used and described. A two-dimensional numerical simulation of the onset phase of flow through the chamber was performed. The wall shear stress action on the bottom plate was computed as a function of time and distance from the sudden expansion. The results showed that depending on the time for the flow to be established, significant temporal gradients occurred close to the second stagnation point of flow. Slowly ramping the flow over 15 s instead of 200 ms reduces the temporal gradients by a factor of 300, while spatial gradients are reduced by 23 percent. Thus, the effects of spatial and temporal gradients can be observed separately. In experiments on endothelial cells, disturbed flow stimulated cell proliferation only when flow onset was sudden. The spatial patterns of proliferation rate match the exposure to temporal gradients. This study provides information on the dynamics of spatial and temporal gradients to which the cells are exposed in a sudden-expansion flow chamber and relates them to changes in the onset phase of flow.     

Flow in the well: computational fluid dynamics is essential in flow chamber construction

A perfusion system was developed to generate well defined flow conditions within a well of a standard multidish. Human vein endothelial cells were cultured under flow conditions and cell response was analyzed by microscopy. Endothelial cells became elongated and spindle shaped. As demonstrated by computational fluid dynamics (CFD), cells were cultured under well defined but time varying shear stress conditions. A damper system was introduced which reduced pulsatile flow when using volumetric pumps. The flow and the wall shear stress distribution were analyzed by CFD for the steady and unsteady flow field. Usage of the volumetric pump caused variations of the wall shear stresses despite the controlled fluid environment and introduction of a damper system. Therefore the use of CFD analysis and experimental validation is critical in developing flow chambers and studying cell response to shear stress. The system presented gives an effortless flow chamber setup within a 6-well standard multidish.     

A miniaturised parallel-plate shearing apparatus for the measurement of cell adhesion

The design and construction of a miniaturised shearing apparatus is described. Cultures (24 h) of an established epithelial cell line (BEB) were exposed to flow conditions in the shearing chamber at 37 degrees C, and subsequently glass coverslip cultures were prepared for photography. The critical shear radius (CSR) was determined by densitometry from a negative film and the minimum distraction force (MDF) at the CSR calculated using predetermined viscosity values of the flow medium. The mean calculated MDF of BEB cells ranged from 1.04-1.36 Nm-2, and was independent of the culture inoculation density (9 to 37 X 10(4) cells cm-2) and the time (5-20 min) of exposure to shearing conditions. The MDF was increased to 3.3 Nm-2 by a 30 min exposure of cultures to concanavalin A (50 micrograms ml-1), and this effect was abolished by treating with mannose (0.05 M). The results demonstrated that the radial flow chamber principle is applicable to the measurement of cell to substratum adhesion of cultured mammalian cells.     

The biased lamellipodium development and microtubule organizing center position in vascular endothelial cells migrating under the influence of fluid flow*

Summary— To analytically study the morphological responses of vascular endothelial cells (ECs) to fluid flow, we designed a parallel plate flow culture chamber in which cells were cultured under fluid shear stress ranging from 0.01 to 2.0 Pa for several days. Via a viewing window of the chamber, EC responses to known levels of fluid shear stress were monitored either by direct observations or by a video-enhanced time-lapse microscopy. Among the responses of cultured ECs to flow, morphological responses take from hours to days to be fully expressed, except for the fluid shear stress-dependent motility pattern change we reported earlier which could be detected within 30 min of flow changes. We report here that ECs exposed to more than 1.0 Pa of fluid shear shear stress have developed lamellipodia in the direction of flow in 10 min. This is the fastest structurally identifiable EC response to fluid shear stress. This was a reversible response. When the flow was stopped or reduced to the level which exerted less than 0.1 Pa of fluid shear stress, the biased lamellipodium development was lost within several minutes. The microtubule organizing center was located posterior to the nucleus in ECs under the influence of flow. However, this position was established only in ECs responding to fluid shear stress for longer than 1 h, indicating that positioning of the microtubule organizing center was not the reason for, but rather the result of, the biased lamellipodium response. Colcemid-treated ECs responded normally to flow, indicating that microtubules were not involved in both flow sensing and the flow-induced, biased lamellipodium development.     

A flow system for the study of shear forces upon cultured endothelial cells

A parallel plate chamber in a flow system has been designed to study the effects of fluid shear stresses on cells. The system was applied to the study of cultured endothelial cells grown on cover slips which were accommodated in recessed wells in the base plate. Dye injection studies in the chamber indicated laminar flow over the cells. Shear rates measured over the cover slips by an electrochemical technique were found to be linear with flow rate. Laser doppler anemometry showed parabolic profiles between the plates. Endothelial cells subjected to flow showed a correlation between the time required for orientation and the magnitude of the shear stress.     

The impact of process stress on suspended anchorage-dependent mammalian cells as an indicator of likely challenges for regenerative medicines

To quantify the engineering shear constraint on processing, the effect of capillary shear stress (pipe flow) on suspended anchorage-dependent mammalian cells has been investigated. Exposure of cultured rat aortic smooth muscle cells to repeated capillary shear stress (2-120 N m(-2)) causes a decrease in total number of cells, number of intact cells and number of cells able to grow. The optimum wall shear stress for cell survival was found to be 10-50 N m(-2) (flowrate 4-20 mL/min, I.D. 0.45 mm). Cell populations which are able to grow after exposure to shear stress do not exhibit reduced growth rate or altered metabolism.     

Inhibition of platelet-surface-bound proteins during coagulation under flow I: TFPI

Blood coagulation is a self-repair process regulated by activated platelet surfaces, clotting factors, and inhibitors. Tissue factor pathway inhibitor (TFPI) is one such inhibitor, well known for its inhibitory action on the active enzyme complex comprising tissue factor (TF) and activated clotting factor VII. This complex forms when TF embedded in the blood vessel wall is exposed by injury and initiates coagulation. A different role for TFPI, independent of TF:VIIa, has recently been discovered whereby TFPI binds a partially cleaved form of clotting factor V (FV-h) and impedes thrombin generation on activated platelet surfaces. We hypothesized that this TF-independent inhibitory mechanism on platelet surfaces would be a more effective platform for TFPI than the TF-dependent one. We examined the effects of this mechanism on thrombin generation by including the relevant biochemical reactions into our previously validated mathematical model. Additionally, we included the ability of TFPI to bind directly to and inhibit platelet-bound FXa. The new model was sensitive to TFPI levels and, under some conditions, TFPI could completely shut down thrombin generation. This sensitivity was due entirely to the surface-mediated inhibitory reactions. The addition of the new TFPI reactions increased the threshold level of TF needed to elicit a strong thrombin response under flow, but the concentration of thrombin achieved, if there was a response, was unchanged. Interestingly, we found that direct binding of TFPI to platelet-bound FXa had a greater anticoagulant effect than did TFPI binding to FV-h alone, but that the greatest effects occurred if both reactions were at play. The model includes activated platelets’ release of FV species, and we explored the impact of varying the FV/FV-h composition of the releasate. We found that reducing the zymogen FV fraction of this pool, and thus increasing the fraction that is FV-h, led to acceleration of thrombin generation.     

The 12 cc Penn State pulsatile pediatric ventricular assist device: fluid dynamics associated with valve selection

The mortality rate for infants awaiting a heart transplant is 40% because of the extremely limited number of donor organs. Ventricular assist devices (VADs), a common bridge-to-transplant solution in adults, are becoming a viable option for pediatric patients. A major obstacle faced by VAD designers is thromboembolism. Previous studies have shown that the interrelated flow characteristics necessary for the prevention of thrombosis in a pulsatile VAD are a strong inlet jet, a late diastolic recirculating flow, and a wall shear rate greater than 500 s(-1). Particle image velocimetry was used to compare the flow fields in the chamber of the 12 cc Penn State pediatric pulsatile VAD using two mechanical heart valves: Bjork-Shiley monostrut (BSM) tilting disk valves and CarboMedics (CM) bileaflet valves. In conjunction with the flow evaluation, wall shear data were calculated and analyzed to help quantify wall washing. The major orifice inlet jet of the device containing BSM valves was more intense, which led to better recirculation and wall washing than the three jets produced by the CM valves. Regurgitation through the CM valve served as a significant hindrance to the development of the rotational flow.     

Numerical model of fluid flow and oxygen transport in a radial-flow microchannel containing hepatocytes

The incorporation of monolayers of cultured hepatocytes into an extracorporeal perfusion system has become a promising approach for the development of a temporary bioartificial liver (BAL) support system. In this paper we present a numerical investigation of the oxygen tension, shear stress, and pressure drop in a bioreactor for a BAL composed of plasma-perfused chambers containing monolayers of porcine hepatocytes. The chambers consist of microfabricated parallel disks with center-to-edge radial flow. The oxygen uptake rate (OUR), measured in vitro for porcine hepatocytes, was curve-fitted using Michaelis-Menten kinetics for simulation of the oxygen concentration profile. The effect of different parameters that may influence the oxygen transport inside the chambers, such as the plasma flow rate, the chamber height, the initial oxygen tension in the perfused plasma, the OUR, and K(m) was investigated. We found that both the plasma flow rate and the initial oxygen tension may have an important effect upon oxygen transport. Increasing the flow rate and/or the inlet oxygen tension resulted in improved oxygen transport to cells in the radial-flow microchannels, and allowed significantly greater diameter reactor without oxygen limitation to the hepatocytes. In the range investigated in this paper (10 microns < H < 100 microns), and for a constant plasma flow rate, the chamber height, H, had a negligible effect on the oxygen transport to hepatocytes. On the contrary, it strongly affected the mechanical stress on the cells that is also crucial for the successful design of the BAL reactors. A twofold decrease in chamber height from 50 to 25 microns produced approximately a fivefold increase in maximal shear stress at the inlet of the reactor from 2 to 10 dyn/cm2. Further decrease in chamber height resulted in shear stress values that are physiologically unrealistic. Therefore, the channel height needs to be carefully chosen in a BAL design to avoid deleterious hydrodynamic effects on hepatocytes.     

Specific adhesion of glycophorin liposomes to a lectin surface in shear flow.

The adhesion of cells to other cells or to surfaces by receptor-ligand binding in a shear field is an important aspect of many different biological processes and various cell separation techniques. The purpose of this study was to observe the adhesion of model cells with receptor molecules embedded in their surfaces to a ligand-coated surface under well-defined flow conditions in a parallel plate flow chamber. Liposomes containing glycophorin were used as the model cells to permit a variation in the adhesion parameters and then to observe the effect on adhesion. A mathematical model for cell sedimentation was created to predict the deposition time and the velocity preceding adhesion for the selection of experimental operating conditions and the methods useful for data analysis. The likelihood of cell attachment was represented by a quantity called the sticking probability which was defined as the inverse of the number of times a liposome made contact with the surface before attachment occurred. The sticking probability decreased as the cell receptor concentration was lowered from approximately 10(4) to 10(2) receptors per 4-microns diam liposome and as the shear rate increased from 5 to 22 s-1. The effect of the wall shear rate and particle diameter on detachment of liposomes from a surface was also observed.     

Probing the interface between factor Xa and tissue factor in the quaternary complex tissue factor-factor VIIa-factor Xa-tissue factor pathway inhibitor.

Blood coagulation is triggered by the formation of a complex between factor VIIa (FVIIa) and its cofactor, tissue factor (TF). TF-FVIIa is inhibited by tissue factor pathway inhibitor (TFPI) in two steps: first TFPI is bound to the active site of factor Xa (FXa), and subsequently FXa-TFPI exerts feedback inhibition of TF-FVIIa. The FXa-dependent inhibition of TF-FVIIa activity by TFPI leads to formation of the quaternary complex TF-FVIIa-FXa-TFPI. We used site-directed fluorescence probing to map part of the region of soluble TF (sTF) that interacts with FXa in sTF-FVIIa-FXa-TFPI. We found that the C-terminal region of sTF, including positions 163, 166, 200 and 201, is involved in binding to FXa in the complex, and FXa, most likely via its Gla domain, is also in contact with the Gla domain of FVIIa in this part of the binding region. Furthermore, a region that includes the N-terminal part of the TF2 domain and the C-terminal part of the TF1 domain, i.e. the residues 104 and 197, participates in the interaction with FXa in the quaternary complex. Moreover, comparisons of the interaction areas between sTF and FX(a) in the quaternary complex sTF-FVIIa-FXa-TFPI and in the ternary complexes sTF-FVII-FXa or sTF-FVIIa-FX demonstrated large similarities.     

An apparatus to study the response of cultured endothelium to shear stress

An apparatus which has been developed to study the response of cultured endothelial cells to a wide range of shear stress levels is described. Controlled laminar flow through a rectangular tube was used to generate fluid shear stress over a cell-lined coverslip comprising part of one wall of the tube. A finite element method was used to calculate shear stresses corresponding to cell position on the coverslip. Validity of the finite element analysis was demonstrated first by its ability to generate correctly velocity profiles and wall shear stresses for laminar flow in the entrance region between infinitely wide parallel plates (two-dimensional flow). The computer analysis also correctly predicted values for pressure difference between two points in the test region of the apparatus for the range of flow rates used in these experiments. These predictions thus supported the use of such an analysis for three-dimensional flow. This apparatus has been used in a series of experiments to confirm its utility for testing applications. In these studies, endothelial cells were exposed to shear stresses of 60 and 128 dynes/cm2. After 12 hr at 60 dynes/cm2, cells became aligned with their longitudinal axes parallel to the direction of flow. In contrast, cells exposed to 128 dynes/cm2 required 36 hr to achieve a similar reorientation. Interestingly, after 6 hr at 128 dynes/cm2, specimens passed through an intermediate phase in which cells were aligned perpendicular to flow direction. Because of its ease and use and the provided documentation of wall shear stress, this flow chamber should prove to be a valuable tool in endothelial research related to atherosclerosis.     

Numerical simulation of fluid field and in vitro three-dimensional fabrication of tissue-engineered bones in a rotating bioreactor and in vivo implantation for repairing segmental bone defects

In this paper, two-dimensional flow field simulation was conducted to determine shear stresses and velocity profiles for bone tissue engineering in a rotating wall vessel bioreactor (RWVB). In addition, in vitro three-dimensional fabrication of tissue-engineered bones was carried out in optimized bioreactor conditions, and in vivo implantation using fabricated bones was performed for segmental bone defects of Zelanian rabbits. The distribution of dynamic pressure, total pressure, shear stress, and velocity within the culture chamber was calculated for different scaffold locations. According to the simulation results, the dynamic pressure, velocity, and shear stress around the surface of cell-scaffold construction periodically changed at different locations of the RWVB, which could result in periodical stress stimulation for fabricated tissue constructs. However, overall shear stresses were relatively low, and the fluid velocities were uniform in the bioreactor. Our in vitro experiments showed that the number of cells cultured in the RWVB was five times higher than those cultured in a T-flask. The tissue-engineered bones grew very well in the RWVB. This study demonstrates that stress stimulation in an RWVB can be beneficial for cell/bio-derived bone constructs fabricated in an RWVB, with an application for repairing segmental bone defects.     

Smooth muscle cells contract in response to fluid flow via a Ca2+-independent signaling mechanism.

Smooth muscle cells (SMC) are exposed to fluid shear stress because of transmural (interstitial) flow across the arterial wall. This shear stress may play a role in the myogenic response and flow-mediated vasomotion. We, therefore, examined the effects of fluid flow on contraction of rat aortic SMC. SMC that had been serum-starved to induce a contractile phenotype were plated on quartz slides and exposed to controlled shear stress levels in a flow chamber. The area of the cells was quantified, and reduction in the cell area was reported as contraction. At 25 dyn/cm(2), significant area reduction was apparent 3 min after the onset of flow and exceeded 30% at 30 min. At 1 dyn/cm(2), significant contraction was not observed at 30 min. The threshold for significant shear-induced contraction appeared to be 11 dyn/cm(2). The signal transduction mechanism was studied at 25 dyn/cm(2). Intracellular calcium was imaged by using the calcium-sensitive fluorescent dye fura 2-AM. There was no detectable change in intracellular calcium during 10 min of exposure to shear stress, even though the cells displayed a significant calcium response to thapsigargin, calcium ionophore, and KCl. Further studies using pathway inhibitors provided evidence that the most important signal transduction pathway mediating calcium-independent contraction in response to fluid flow is the Rho-kinase pathway, although there was a suggestion that protein kinase C plays a secondary role.     

Effects of factor Xa and protein S on the individual activated protein C-mediated cleavages of coagulation factor Va

Activated protein C inhibits the procoagulant function of activated factor V (FVa) through proteolytic cleavages at Arg-306, Arg-506, and Arg-679. The cleavage at Arg-506 is kinetically favored but protected by factor Xa (FXa). Protein S has been suggested to annihilate the inhibitory effect of FXa, a proposal that has been challenged. To elucidate the effects of FXa and protein S on the individual cleavage sites of FVa, we used recombinant FVa:Q306/Q679 and FVa:Q506/Q679 variants, which can only be cleaved at Arg-506 and Arg-306, respectively. In the presence of active site blocked FXa (FXa-1.5-dansyl-Glu-Gly-Arg), the FVa inactivation was followed over time, and apparent second order rate constants were calculated. Consistent with results on record, we observed that FXa-1.5-dansyl-Glu-Gly-Arg decreased the Arg-506 cleavage by 20-fold, with a half-maximum inhibition of approximately 2 nM. Interestingly and in contrast to the inhibitory effect of FXa on the 506 cleavage, FXa stimulated the Arg-306 cleavage. Protein S counteracted the inhibition by FXa of the Arg-506 cleavage, whereas protein S and FXa yielded additive stimulatory effect of the cleavage at Arg-306. This suggests that FXa and protein S interact with distinct sites on FVa, which is consistent with the observed lack of inhibitory effect on FXa binding to FVa by protein S. We propose that the apparent annihilation of the FXa protection of the Arg-506 cleavage by protein S is due to an enhanced rate of Arg-506 cleavage of FVa not bound to FXa, resulting in depletion of free FVa and dissociation of FXa-FVa complexes.     

Dynamic shear stress in parallel-plate flow chambers

An in vitro model using a parallel-plate fluid flow chamber is supposed to simulate in vivo fluid shear stresses on various cell types exposed to dynamic fluid flow in their physiological environment. The metabolic response of cells in vitro is associated with the wall shear stress. However, parallel-plate flow chambers have not been characterized for dynamic fluid flow experiments. We use a dimensionless ratio h / lambda(v), in determining the exact magnitude of the dynamic wall shear stress, with its oscillating components scaled by a shear factor T. It is shown that, in order to expose cells to predictable levels of dynamic fluid shear stress, two conditions have to be met: (1) h / lambda(v) < 2, where h is the distance between the plates and lambda(v) is the viscous penetration depth; and (2) f(0) < f(c) / m, where the critical frequency f(c) is the upper threshold for this flow regime, m is the highest harmonic mode of the flow, and f(0) is the fundamental frequency of fluid flow.