Disclosure
Introduction: an idealized fluid flow model for mathematical simplicity The science of fluid flow applies to many facets in the field of radiology, from arteriography and angioplasty to percutaneous drainage. Precise modeling of physiologic fluid flows under real-life flow conditions requires mathematics too complex for practical use in clinical medicine. A workable model may be devised from the first principles of engineering to create a first-order approximation that suits most purposes. A steady-state laminar flow model of an ideal (Newtonian) fluid in a rigid circular pipe of uniform diameter is used by the author. Frictional (viscosity) energy losses are found for all fluids traveling through a length of conduit (pipe). This causes a pressure drop along the length of the pipe as a function of the pipe size, the type of fluid, and the mean flow velocity or flow rate. The central formula in the laminar flow model is the Poiseuille equation: R = (8hL)/pr4. or the Poiseuille law: Q = (pDPr4)/(8hL), where P = pressure, n = mean flow velocity, Q = volumetric flow rate, L = tube length, h = fluid viscosity, and r = tube radius. Origins of the Poiseuille equation The underlying assumption of laminar flow is the condition of uniform viscosity across the diameter of the conduit, ie, each fluid molecule within the pipe is exerting a similar force against its immediate neighbor towards the periphery. This yields the following:
Another way to understand laminar flow conditions is to imagine concentric circles of fluid flow. Flow velocity is uniform within each annulus. Velocity increases as you approach the center of the conduit. Further simplification: the Ohm law For clinical applications fluid properties may be taken as constant. The Poiseuille equation can be simplified to resemble the Ohm law of electrical resistance. This law relates voltage drop across an electric circuit (DV) to electric resistance (R) and electric current (I), as follows: DV = IR. The greater the current (electron flow) or resistance, the higher the voltage drop required. For the sake of simplicity, DV often is written as V, although it is the change in voltage and not the absolute voltage that matters. Fluid-flow calculations can be modelled as simple electric circuits. For this analogy (model) we relate DV to DP, defining flow resistance as R = 8Lh/pr4 to obtain the following: DP= QR. Resistive losses (pressure drops) linearly are related to flow rate and flow resistance. While flow resistance linearly relates to conduit length, it is inversely related to the fourth power of the radius (or diameter). For example, a 1-cm–diameter pipe has 16 times the flow resistance of a 2-cm–diameter pipe of the same length and carrying the same fluid. Stated another way, 16 conduits of 1-cm diameter are needed to handle the same fluid flow at the same pressure as a single 2-cm–diameter conduit; this is an amazing concept. Related equations From the Ohm law, equations for series and parallel circuits can be derived (see Image 3). For a series circuit: R = R1 + R2 + R3 + .... For a parallel circuit: 1/Rtotal = 1/R1 + 1/R2 + 1/R3 + .... For the relative resistance of 2 conduits of equal length for the same fluid: R1/R2 = (D2/D1)4, where d corresponds to the diameter of the conduit. For conduits of unequal length: R1/R2 = (L1/L2)(D2/D1)4. For the relative flow rate for the same pressure decrease: Q1/Q2 = (D1/D2)4.
Laminar flow cannot be maintained under high-flow, small-diameter conditions. When these conditions occur, the symmetric parabolic flow distribution is disrupted, and turbulent or disordered flow follows. The transition to turbulent flow conditions always results in an increase in flow resistance; laminar flow is always the most efficient. On a macroscopic level, turbulent flow may be considered to have uniform velocity across the conduit diameter, hence the term plug flow. Under turbulent flow conditions, the fluid follows a random microscopic flow pattern, with swirling eddies and currents. The shear stresses against the conduit wall are increased greatly. Likely, these stresses are related to the atherosclerotic patterns found at the carotid bifurcation. The Reynolds number (Re) can be used to predict whether flow will be laminar or turbulent, as follows: Re = 2rnr/h = Dnr/h, where r = density and D = diameter. No fixed Reynolds number exists for the transition from laminar flow to turbulent flow. The transition occurs at Re = 2000-2500, depending on various factors outside the scope of this discussion. For clinical purposes, turbulent flow can often be approximated by using the laminar flow equations as long as the resistance values are adjusted appropriately.
The Poiseuille equation is related to steady-state flow conditions in a pipe of uniform diameter. The pressure drop corresponds to the viscous energy loss of fluid flow. By applying the Poiseuille equation, estimated pressure drops can be calculated across arterial stenoses, bypass-diameter requirements, and abscess fluid flow through drainage needles and catheters. These viscous losses represent a minimum estimate of resistance to fluid flow. The second term in the energy-loss equation is related to inertial losses, ie, the loss of energy resulting from changes in velocity (speed and direction). In cases of complex or tandem atherosclerotic stenoses, inertial losses can exceed viscous losses. The pressure drop may be significantly greater than what the Poiseuille equation predicts. Energy is expended as fluid accelerates and decelerates through the diseased vessel. The shape of the leading and trailing edges of the stenosis appears to be most important. The fluid dynamics of energy losses through nozzles may be applicable. An engineering simplified approach is to measure pressure losses across various shapes in vitro and apply them to in vivo estimates. |
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A discrepancy is seen in conventional radiology literature regarding the requirements for a successful percutaneous drainage. Although most drainage fluids can be aspirated readily through an 18-gauge needle, many authors recommend the use of drainage catheters with an outer diameter of 10F or greater. However, as indicated previously, fluid flow through a drainage catheter predominantly depends on fluid viscosity and the inner diameter (ID) of the catheter. Any fluid that can be aspirated readily through an 18-gauge needle can be drained with a 7.1F polyethylene pigtail catheter (a low-cost catheter) (See Percutaneous Abscess Drainage). Gobien et al reported similar good results with an 8.3F catheter termed the Gobien-minimal catheter, which has a smaller ID than the 7.1F catheter used by the current author. Aside from a report by Park et al, little has been published regarding the viscosity of abscess fluid. To address this need, the present author published fluid viscosity data for common food and petroleum products ranging from water to crank-case oil. Intuitive understanding of these fluids can be adapted for clinical decision-making. Relative resistance values were calculated for 3 needle and 5 catheter designs, including the Gobien catheter with Park abscess fluid. The catheter ID required for each fluid viscosity to drain was calculated by using the following equation: Gobien-equivalent ID = 0.284(h0.25) cm. The author concluded that, on a theoretical basis, drainage catheters with a small ID should be adequate for typical drainage fluids.
Following are examples of application of the laminar flow equation to vascular disease.
Hemodynamic significance
Conventional radiology wisdom states that an arterial stenosis is hemodynamically significant if either a 50% or greater diameter reduction or a 10% or greater peak systolic pressure gradient is noted. As an example, consider these conditions: a smooth 2-cm–long 50% stenosis of a 20-cm–long portion of the iliac system with a diameter reduction from 8-4 mm and a systemic pressure of 100 mm Hg, a pressure drop across the normal segment of 1, and equal flow across the normal and diseased segment. Because the pressure drop along the diseased segment is predominantly across the stenosis, the normal segment of the diseased vessel can be ignored. The relative resistance equation can be applied as follows:
Rstenosis/Rnormal=(2/20)(8/4)4 = (0.1)(24) = 1.6.
Applying the Poiseuille equation, Qstenosis = Qnormal:
DPstenosis/DPnormal = Rstenosis/Rnormal = 1.6, as calculated.
However, in the model, DPstenosis/DPnormal = 10/1 = 10. The most likely cause of the discrepancy is absence of inertial losses in the calculations.
The laminar-flow equation is likely to be more accurate in flow calculations for longer conduits, such as bypass grafts. In addition, it also can be useful in making relative comparisons when inertial losses are similar.
Tandem lesions
Reliance on pressure gradients to assess the significance of stenosis encounters difficulty when tandem lesions (significant lesions distal to the stenosis) are evaluated. Examples include (1) sequential stenoses in the common iliac and external iliac arteries and (2) significant iliac stenosis in a patient with significant infrainguinal disease, prior to bypass surgery.
The tandem stenosis model may be used to better understand this issue (see Table 4).
Table 4. Tandem stenosis model Note.—Calculations were performed to 2 significant figures.
*Rtotal = R1 + R2 + R3, and R3 = 1, where R is in units of mm Hg/s/cm3. Depending on the severity of the distal stenosis (R2), a substantial variance in pressure drop is seen across a proximal stenosis (R1 = 5). For moderate stenosis, the pressure drop varies in the range of 15-64 mm Hg, from moderately significant to substantially significant.
The pressure drop across a stenosis significantly depends on the severity of any distal stenosis (R2). For moderate stenosis (R1=2), a distal lesion causes the pressure drop to vary in the range of 15-64, which is a change from moderate to substantial significance. The milder the proximal R1 stenosis, the more the distal tandem lesion R2 affects the pullback pressure. If not taken into consideration, this can result in underdiagnosis of significant stenotic disease.
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