Steady Flow Through a Symmetric Sacular Aneurysm Model

BE 310 Final Project May 4, 1998 Group M6Parametric Analysis of the Correction Factor and Optimal Injecting Length for use in ThermodilutionPenelope Siraj · Jennifer Russert · Rupesh Patel · Vesal Dini · Mariza Clement Abstract A mathematical model was developed to establish the relationship between optimal injectant length and the parameters of the Thermodilution Experiment in the Bioengineering Undergraduate Laboratory. The model indicates that the optimal length occurs between 13cm –8m. By experimentally varying the injectant length between 3.5 – 44 cm, the optimal length was determined to be 15.5cm. Furthermore, a discrepancy was found between the existing clinical correction factor and the correction factor needed in the simulated experiments. According to the constraints of the simulations, the correction factor was found to be a function of the injectant length. Background Cardiac output is the volume of blood pumped by the heart per unit time. The heart acts as a pump and ejects a volume of blood called the stroke volume. Cardiac output is defined as the stroke volume times the heart rate. A single measurement of cardiac output reflects the effects of many interacting physiological systems. The thermodilution technique is commonly used for measurement as it employs the Schwan-Ganz pulmonary artery catheter. Cold liquid is injected through the pulmonary artery catheter, into the right atrium, where it mixes with venus blood and causes the blood to cool slightly. Then it passes through the remaining heart pump and passes by a thermistor near the tip of the pulmonary artery. The extent of cooling is inversely proportional to cardiac output (Bowdle). In thermodilution, cardiac output is measured by the Stewart-Hamilton equation. VI  TB  TI  SI  CI  60  CT Q   SB  CB  TBtdt Equation 1 0 where Q is the cardiac output (L/min), VI is the volume of injectant (ml), TB is the blood temperature (°C), SI is the specific gravity of injectant, CI is the specific heat of injectant, 60 is a constant for number of seconds per minute, CT is the correction factor, SB is the  specificity gravity of blood, CB is the specific heat of blood, and TBtdt is the 0 integral of blood temperature change (°C/sec).Determining the appropriate correction factor, CT, for the Bioengineering laboratory simulations is the focus of this project. In order determine CT, a mathematical model was developed which correlated the optimal length (dx) between the thermistor and the initial injectant location. At this optimal length the correction factor will be at its maximum, closest to unity.Mathematical Model: During thermodilution, convective heat transfer takes place when a Schwan-Ganz catheter placed within a tube model of the superior vena cava. A bolus of cold water is injected at Time t=0 and travels a distance x before reaching a themistor at the end of the catheter where the temperature is monitored. Assuming the bolus of water is well mixed, temperature of water in the tube is homogeneous and constant at 37degrees Celsius, the flow (cardiac output) is steady state and in the x direction only at 5 L/min. Also assuming the temperature is not a function of radius, only a function of the x direction. The control volume consists of the segment of tubing including where the bolus is injected and continues to where the thermistor reads the temperature Injectant entrance Temp. Measurement by CatheterTi q2 Tf Flow q1 q3 T(water)Figure 1: Mathematical Model From the first law of thermodynamics the rate of heat transfer out of the control volume due to the flow (cardiac output) is equal to the rate of heat transfer into the control volume due the flow plus the rate of heat transfer in due to convection 1. These three heat transfer rates are as follows.For the heat transfer rate into the control volume due to fluid flow D 2 q   vCpT 1 4 i For the heat transfer rate into the control volume due the fluid flow q2  hDx(Twater  Ti ) For the heat transfer rate out of the control volume due to convection D 2 q   vCpT 3 4 f Then taking q1=q2+q3 and simplifying yields the following equation D  T  T  h  f i     (Ti  Twater   0 4  x  Cp Which simplifies to: DvCp  T  T  x   f i  4h  T  T   water i  Equation 2 Where r = 1 g/mL v = q/A = to be determined during experiment h = from CRC 1 Welty J, Wicks C, Wilson R: Fundamentals of Momentum, Heat, and Mass Transfer, Third Edition 1984 Cp = 4175 J*K/Kg 0 Ti = 0 C 0 Twater = 37 C Tf = found from catheter reading D = 13mm Equation 2 presents a mathematical relationship between the optimal length and the parameters in the bioengineering laboratory experiment. Methods and MaterialsFigure 2: Re-Design of the ApparatusThe apparatus was assembled according to Figure 2. The valve coming off the tank was connected with 13mm inner diameter tygon tubing, this tubing size is estimated to be half of the diameter of the vena cava. This therefore requires scaling down the experiment to half of the physiological model. The Schwan-Ganz catheter (thermistor), and the injectant syringe are feed into the 13mm inner diameter tubing, see Figure 2.The flow rate of 5 liters/min is the flow in the aortic arch thus in this model the flow rate will be fixed at 2.5liters/min. The exact water height, depth and volume to induce this flow was determined. The tank was filled to a designated height; this tank level was kept constant throughout all trials in order to obtain a constant flowrate. A collection bucket was placed at the end of the tubing to collect the water as it is displaced. To qualitatively observe the mixing of the two fluids a dye will be used to color the injectant. This is a check to ensure that the cold injectant is mixing with the flowing water instead of resting in a steam on the bottom of the Tygon tubing and causing an inaccurate reading by the thermistor. About one liter of Evans blue was prepared from concentrate (0.05g per 500mL) to act as the visible injectant. Evans blue dye was chosen for this experiment since it has a specific heat capacity and relative density approaching that of water while coloring the stream a vivid blue. The Evans Blue was then chilled in an ice/water bath to a temperature around 0-5 degrees Celsius. The thermistor, within the Schwan-Ganz Catheter, was hooked up to LabView, which then monitored the temperature change in the water flowing through the tubing. A calibration was done for the thermistor by placing it in a few water baths of known temperature and recording the voltage reading given by each temperature, from this data a calibration curve was plotted.Experimental trials: For each trial the temperature of the water in the tank and the temperature of the injectant were recorded. The length, dx, was set at 44cm, this is the length from the injecting area to the end of the thermistor where the temperature will be recorded, see Figure 2. The three-way valve (at injecting area) was rotated so that it was open to the dye input tube where 7.5 ml of the Evans Blue injectant was drawn into the injecting syringe. The flow valve was turned to the on position, and the flowrate was taken by measuring the volume of water displaced over a time recorded by a stopwatch. LabView was run, and the cold bolus of Evans Blue was injected into the tubing so that it passed the thermistor during the 10 seconds that LabView is monitoring the fluid temperature within the tube. Flow was then stopped, and all water that had left the system during the experiment was collected and recycled back to the tank for further trials. The thermistor was then moved to vary length dx to 34cm, 24 cm, 14cm, and 3.5 cm. The minimum of 3.5cm (7.0cm divided by 2 from biological scaling) since 7 cm is the distance that the vena cava extends into the right atrium. For each set of lengths (44cm to 3.5cm) the temperature of the tank water was also varied. The first tank of water was at 30 degrees Celsius. With the addition of a circulating water heater, another experiment was done with the tank water at 37 degrees Celsius, average body temperature. The temperature of the tank water was then increased once more to 43 degrees Celsius for a final set of experiments, this being the approximate temperature of the body with an extreme fever, thus ensuring that trials were done in for a logical physiological range of temperatures. Results Calibration: The thermistor was calibrated by placing it in water baths of 40C, 230C, and 430C and recording the voltage readings. The equation of the calibration curve, derived from linear regression, was used to convert voltages from the LabView readings to temperatures when calculating Reimann sums (described below).Voltage-Temperature Calibration )C 50 s 45 e e r 40 y = -24.164x + 108.78 g e 35 2 d( R = 0.9992 30 e r 25 u t 20 a r e 15 p 10 m e 5 T 0 2 3 4 5 Voltage (V) Figure 3: Calibration curve converting voltage to temperatureCalculation of the Correction Factor: The Stewart-Hamilton equation, Equation 1, was manipulated in order to solve for the correction factor, and the various parameters therein were calculated. Flow rates (Q) were measured by hand using the “Bucket/Stop-Watch” method, and the injectant temperature and flowing water temperature were measured for each trial (four trials for each of five different lengths, each at three different temperatures). Several steps were taken in order to accurately calculate the area under the curve of the Temperature verses Time graph. Initially, a baseline voltage was calculated from the first 500 points of the raw LabView voltage readings from each of the trials. These 500 points corresponded to the interval before the bolus of cold solution was injected and thus accurately describes the initial temperature of the fluid in the tube. Using this baseline voltage, the raw data (voltages) obtained from the entire experimental run of 10 seconds were subtracted from the baseline voltage. These new values were then used to find the total time (time) taken for the water temperature to return to the baseline temperature. Subsequently, the calibration curve of the catheter (Figure 3) was used to convert the voltage readings, measured by LabView, to temperature. Figure 4 is an example of a LabView image, converted form voltage to tempurature, that was used to find the are under the curve.Temperature vs. Time35.5 ) C 35 s e e 34.5 r g e 34 d (33.5 e r u 33 a r e 32.5 p m 32 eT 31.5 0 20 40 60 80 100 Time (1/1000th sec) sdfjsf kjgdgf Figure 4: Temperature versus Time, finding are under the curve The integral of the temperature change in the blood was calculated using the classical calculus method of Riemann sums (time multiplied by temperature). Thus, the area under the curve was found using Riemann sums as described, and that was subtracted from the area of large rectangle (baseline temperature multiplied by number of intervals and further multiplied by 1/1000 rectangle, yielding the area in the nipple of the curve, see Figure 4. The correction factor needed to satisfy the cardiac output in the Stewart-Hamilton equation was plotted for each length at each temperature in Figure 5. Our Correction Factor vs. Length 1.00 0.90 r 0.80 o t c 0.70 aF 0.60 n 0.50 o i t c 0.40 e r r 0.30 oC 0.20 0.10 0.00 0 0.1 0.2 0.3 0.4 0.5 Length /m OCT 30 Degrees OCT 37 Degrees OCT 43 DegreesFigure 5: Correction Factor Needed as a Function of Injectant LengthFrom Figure 5, the correction factor is closest to unity at 15.5 cm for all three temperatures. This indicates that the injectant length reaches an optimum as predicted by the mathematical model in Equation 2. Furthermore, this optimal length is independent of the initial temperature with constant flow rate. The error bars represent the precision error found by executing the differential technique. The calculated correction factors and associated error are summarized in table 1 (below). In the clinical setting, the correction factor used in the Stewart-Hamilton equation is described by Equation 3, TB  TID CCT  TB  TI Equation 3 where TI is the initial temperature of the injectant, TB is the temperature of the blood and TID is the mean temperature of the injectant delivered to the right atrium. Applying thisTB - TI TID = TB - 2 relationship in the laboratory set-up implies that TB is the baseline temperature of the water, TI is the initial injectant temperature and . The correction factor given in Equation 3 was graphed as a function of the distance of the injectant in Figure 6. Clinical Correction Factor Vs. Length0.700.65r o t c0.60a FCCT 30 n Degrees o0.55i CCT 37 t c Degrees e CCT 43 0.50r r o Degrees0.45C 0.40 0 0.1 0.2 0.3 0.4 0.5 Length (m)Figure 6: Clinical Correction Factor as a Function of Injectant LengthThis figure implies that the correction factor should not vary as the length of the injectant changes. Instead, the correction factor remains constant as shown in Figure 6. The calculations of the clinical correction factor and its error also determined by the differential technique are summarized in Table 1. Length (m) 0.44 0.34 0.24 0.14 0.035 Our Correction Factor (OCT) for 30 Degrees 0.40 0.41 0.52 0.78 0.08% Error of OCT 30 Degrees 6.64 6.68 7.08 7.19 7.51 Clinical Correction Factor (CCT) for 30 Degrees 0.54 0.54 0.54 0.54 0.54% Error of CCT 30 Degrees 0.99 0.99 1.00 1.00 1.00 Our Correction Factor (OCT) for 37 Degrees 0.59 0.69 0.78 0.92 0.19% Error of OCT 37 Degrees 3.43 3.93 3.92 4.36 4.62 Clinical Correction Factor (CCT) for 37 Degrees 0.59 0.59 0.58 0.59 0.59% Error of CCT 37 Degrees 0.31 0.31 0.34 0.30 0.30 Our Correction Factor (OCT) for 43 Degrees 0.47 0.45 0.70 0.97 0.07% Error of OCT 43 Degrees 2.98 3.06 2.85 2.57 2.33 Clinical Correction Factor (CCT) for 43 Degrees 0.55 0.58 0.55 0.55 0.56% Error of CCT 43 Degrees 0.34 0.24 0.42 0.34 0.32 Table 1: Comparison of OCT and CCTDiscussionDiscussion of the Results: The optimal length found through experimentation was 15.5 cm. From Equation 2 presented in the mathematical model, x can be calculated according the specifications of the experiment. Taking an arbitrary trial at 37 degrees Celsius the following information is taken from the data files, each variable is explained in the background. DvCp  T f  Ti  x    4h  Twater  Ti Tf = 33.14 C Ti = 2.8 C Tw = 35.47 C  = .001kg/ml Q = (43ml/s)( 1/(D/2)^2) D= .013 m Cp = 4175 J*K/kg2 h = 250 W/m^2*K – 15000 W/m^2*K1Inputting these values into Equation 2 and dividing by a factor of 2 to scale it according to the model, x is 8.145m using the lower end of the range given for the convection coefficient, h. Using the higher end of the convection coefficient, x is 0.1358m. The range is then 8.145m -13.58 cm. The optimal length found through experimentation as 15.5 cm corresponds with the mathematical model by fitting into the given range. The optimal length has thus been proved both mathematically and experimentally.The optimal length, determined by its correction factor being closest to unity, was found to be the same (15.5 cm) for all three temperature readings. It was found that when only the length varied and all else was kept constant, the correction factor for the readings at 300C, 370C, and 430C were found to vary as a function of injectant length. When the correction factor was calculated using the clinical method, correction factor was not observed to be a function of length. A relationship between temperature and the correction factor can be seen from Figure 5, as the temperature increases the correction factor at optimal length is closer to unity. This can be described through the difference in temperature between the injectant and the flowing water. The difference in temperature increases with increased water temp, since injectant temperature is keep relatively constant throughout the experiment. With the increased temperature difference the deflection in the temperature vs. time curve obtained 2 Handbook of Physics and Chemistry CRC 73rd Edition from LabView is greater. With a greater deflection there is less error and therefore less need for a correction factor to come into play, bringing the correction factor value closer to one.Error Analysis: The error associated with the correction factors was greatest at 30C. From the aforementioned relationship between temperature and error, at 30C the error is expected to be largest since the deflection measured will be the smallest. However, this error remained below 8%. The errors for correction factors at 370C,and 430C were 4.6% and 2.3%. Previously mentioned error has come from the area under the temperature vs. time curve. Error can also be attributed to the method in which flow rate was measured. The “Bucket/Stopwatch” method is prone to human error. For this reason, it is assumed that the flow rate measurement contained a maximum of 10% error. 10% was used since the measured flow rates deviated 10% from the target flow rate (2.5L/min). Even with this large estimated error in the measurement of flow rate, the error in the correction factor is mostly due to the temperature drop. This is evidenced in Table 1 by a decrease in error with an increase in temperature. Improvements and Extensions to the Lab: One of the other goals of this experiment was to improve the method of delivering the injectant into the flowing fluid. In the original apparatus, the injectant was rarely released in less than 3 seconds and a great amount of continuous force was required to release a large amount of the injectant through a relatively tiny outlet hole. In the redesigned apparatus, the injectant was released into the flow stream almost instantaneously and with minimal force and the bolus of injectant was mixed with the flowing fluid without it being isolated in any one part of the flow stream as shown by the completely blue bolus.A more accurate prediction of the optimal length can be obtained by taking more readings around the region of 15.5cm away from the injectant input (x) to determine which length (x) yields a correction factor closest to unity.In order to find an actual mathematical relationship between the correction factor and the experimental parameters, more experimentation needs to be done to observe the effects of changes in flow rate on the correction factor. In a biological system, fluctuations in temperature and flow rate are most likely, whereas the viscosity and specific heat fluctuations may not be as common or dramatic. Conclusion: The optimal length between the injectant site and thermistor (x) was found via mathematical modeling and experimentation (where the optimal length was defined as the length x where the correction factor is closest to unity). From experimentation, it was also concluded that the clinical correction factor (Equation 3) was not accurate for the Thermodilution lab set up (Experiment 5, BE 310 Laboratory Manual). Finally, through further research, this experiment would culminate in a mathematical relationship for the Correction Factor (OCT); this might be possible with further experimentation involving varying of the flow rate as well as the length for specific temperature trials.

Steady Flow Through a Symmetric Sacular Aneurysm Model

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