# Diffusion Through Branched Networks

**BE310 SPRING 2004**

BIOENGINEERING LABORATORY IV

## Group R2

**Final Experiment:**

**DIFFUSION THROUGH BRANCHED NETWORKS**

Members:

##### Robby Bowles

Lauren Boyer

Laura Kauer

Justin Mills

Marla Stump

ABSTRACT

Branching networks occur throughout the human body (particularly in the cardiovascular and respiratory system) and have extremely complex velocity profiles around network bifurcations. Because of these complexities, diffusive properties of fluids flowing through the networks cannot be studied by traditional methods. In this experiment, we aimed to study these diffusive properties model the behavior of blood in the circulatory system using water in a branching network. We hypothesized that our results would be comparable to the behavior of gases in the bronchial tree as noted in the journal article, *Measurement of Axial Diffusivities in a Model of the Bronchial Airways.* Effective diffusivity was found to increase linearly with increasing velocity, as was found by Dr. Scherer et. al. The relationship between effective diffusivity and velocity (D/U0) was 6.58 (R2 = 0.7167) for week 1 and 8.07 (R2 = 0.7655) for week 2.

INTRODUCTION

Branching networks are frequently seen in biological systems ranging from the circulatory system to the respiratory system, and an understanding of the fluid dynamics in a general case can help to predict behavior in other instances. This is the case with a prior experiment to observe the diffusion and convection properties of air in a simplified model of branching in the lungs conducted by Scherer et al. Using the same analysis employed to quantify diffusion in the lungs, capillary diffusion can be analyzed to determine the effect of branched networks on tracer concentrations a fluid.

Though this effect is usually called diffusion, in reality the contribution of molecular diffusion to the tracer migration is negligible when compared with convection due to turbulence cause by the network bifurcations. The combination of these two functions is called dispersion and is the experimental intent of the study though the diffusion constant is a main finding.

HYPOTHYSIS

The behavior of blood in the circulatory system as modeled using water in a branching network can be compared to the behavior of gases in the bronchial tree as noted in the journal article, *Measurement of Axial Diffusivities in a Model of the Bronchial Airways.*

MATERIALS

- Tygon Tubing 1/8” I.D.
- Nalgene Needle Valves
- Syringe Pump and Rotary Pump
- One Cole Conductivity Meters
- 0.1 M KCl
- 3-way Injection Valve
- Y-Branches
- Graduated Cylinders
- Deionized Water
- Stopwatch

METHODS

Setup

Week 2

A model of a three generation Y-branching network was built using 1/8” Tygon tubing and Y-branches. A syringe pump was used to facilitate flow with two conductivity metersplaced at zero and third generations as in Figure 1. Flow out of each generation was controlled using Nalgene needle valves at the openings of each generation.

Figure 1- Week 2 Apparatus

Week 3

Figure 2- Week 3 Apparatus

Experiment

Week 2

One mL of KCl injectate was used to study the dispersion through the branching networks. Two trials at 8 different flow rates were run through the branching tube network with mean velocities ranging from 4.21 to 25.26 cm/s. The concentration profiles of KCl were obtained at the zero and third generations using conductivity meters. During each trial, the flows were collected from each generation using graduated cylinders and a stopwatch. The flow rates out of each generation were not accurate to the 50% flow rate decrease after each successive generation, as was desired, because of the experimental set up, but they were held constant throughout week 2 experimentation.

Week 3

As in week 2, 1 mL of KCl injectate was used to study the dispersion through the branching networks. Three trials at 6 different flow rates were run through the branching tube network with mean velocities ranging from 9.68 to 17.50 cm/s. The concentration profiles of KCl were obtained at the zero and third generations using conductivity meters. During each trial, the flows were collected from each generation using graduated cylinders and a stopwatch. The flows out of each generation were modeled as closely to a symmetric branching tube network with a 50% flow rate decrease after each branch.

Data Analysis

Week 2 and 3

The equation

was derived from the one-dimensional convective diffusion equation as seen in the Appendix. In this equation, U0 is the mean velocity at the zero generation and T is the time difference between the two concentration peaks obtained at the zero and third generation. T2 is the time difference between the rise and fall of the concentration peaks at the third generation and T1 is the time difference between the rise and fall of the concentration peaks at the zero generation. The rise and fall times of the peak were defined at the times in which 9/10 of the maximum were obtained.

The effective diffusivities (cm2/s) were graphed versus the mean velocities (cm/s) in the zero generation. A linear regression with an R2 value was fit to the data to test the relationship between diffusivity and mean velocity. The relationship between the Reynold’s number, Peclet Number, effective diffusivity, and molecular Diffusivity was also obtained and compared to similar values obtained by Scherer’s,which studied branched networks in the respiratory system.

RESULTS

Figure 3 represents the concentration profiles of KCl injectate as measured by conductivity meters before and after system bifurcations. As shown on the graph, increasing fluid flow velocities correspond to broader concentration profile peaks. This effect is observable in the profiles measured by first and the second conductivity meters, placed before and after the system bifurcations respectively.

During Week 2, the syringe pump was used to find the diffusivity constant, D, based on the calculated parameters: velocity, U, and the overall change in time and that at 90% rise of the peaks. Diffusivity versus velocity was plotted, and a linear regression was performed with an R2 value of 0.7167. This is graphically displayed in Figure 4 with a best fit line: D=6.5794·Uo-7.9518.

The relationship between the Reynold’s number, Peclet Number, effective diffusivity, and molecular Diffusivity for Week 2 is shown at right.

Figure 5 displays diffusivity constants calculated in week 3, when the rotary pump was used. Data again showed an approximate linear relationship between the mean velocity and the diffusivity of the KCl flowing through the system. A best fit line through the data revealed the following relationship: D=8.0711·Uo-32.245, with an R2 relationship of 0.7655.

Using the same comparison as mentioned earlier, the relationship between Reynold’s number, Peclet number was found as seen to the left.

DISCUSSION

Previous experiments have shown that in branching networks velocity fields around bifurcations are extremely complex, having Reynolds Numbers above 10 and spiraling secondary flows. Because of these complexities, mixing within the fluid in the tubes cannot be quantified using traditional methods. To study this mixing, the dispersion of a solute needed to be experimentally measured in a branching system. Dr. Scherer et. al. studied the effect of bifurcations on dispersion in branching networks. Although his studies dealt with gas diffusivity through the bronchial tree, the experiment and results are analogous to the flow of fluid through branching networks, which we studied. Results will be compared later.

Because of the lack of prior examples of the lab, much of the time spent in the lab consisted of optimizing the apparatus and experimental protocol to produce repeatable results.

The apparatus underwent numerous alterations throughout the course of the experiment in order to optimize and produce acceptable results. To provide inherent stability to the apparatus, from the first week all of the components were attached to an L-bracket which provided a foundation to build on and eliminated errors due to varying heights and slopes. This bar could be checked for levelness each week for consistency. The first resistance option explored was crimp valves which pinched the tubing to provide increased resistance. This system problems with stability as each trial required clamp adjustment and an unintentional bump changed the whole flow of the system and the data set was ruined. Thus, other systems were investigated. The pipette tips were cut at increasing lengths to model increasing flows. However, this system also showed poor performance which was attributed to leakage around the pipette tips due to insufficient attachment. Another drawback to this system was to attain optimization due to the trial and error method of pipet cutting (which yielded the flow). The final valve choice was for needle valves which allowed for sensitive yet consistent increase or decrease of resistance. One further addition to improve exit from these valves were elbow fittings that were also aligned in a horizontal plane reduce the drainage when the pump was disconnected.

Weeks two and three saw further optimization through changing the pumps responsible for the fluid flow. Initially, the syringe pump was used with the two syringes connected to double the pump output. Even so, the pump still required frequent refilling and disconnection of the 0th generation line. This introduced bubbles into the branching system which required most of the syringe pump capacity to eject as to not affect the flow especially at low rates. The benefits this pump did offer were a known, selectable output at the onset of the trial. We advanced to a rotary pump the final week which allowed for continuous flow without the need for refilling or the nuisance of expelling air bubbles. Unfortunately this pump did not allow for the range of flows or the benefit of selecting flows that the syringe pump did. However, it did streamline the data gathering which allowed for more trials.

Finally, during the data analysis stage of the experiment, we discovered that the calibration of the conductivity probes was unnecessary due to the accuracy of the probes and the nature of the equation used. Comparing the calibration equations from week to week, the probes varied 1.20% and 1.57% which would not introduce significant error if the same calibration was used each week. When compared with each other, the probes varied by 3.70% and 6.33%. Additionally, the format of the equation used to calculate apparent diffusivity involved differences in time with no concern for the actual magnitude. Because time-shifts and signal amplification do not affect relative values in the time domain, it is unnecessary to convert raw signals into actual concentrations by a linear calibration curve. This fact coupled with the inherent consistency of the electrodes makes calibration in the future unnecessary.

In our experimentation, we found that y-branching tubes yielded effective diffusivities highly sensitive to the mean velocity. Since the molecular diffusivity, 0.0000199 cm2/s, is significantly smaller than the diffusivity due to the velocity, this term is considered negligible. Based off this assumption, we focused our efforts on proving the linear relationship between the effective diffusivity of KCl in the water solution and the manually varied velocity between trials. A report of our findings is displayed for weeks 2 and 3.

For week 2, the use of syringe pump caused more air bubbles to enter the system as it constantly had to be refilled. However it still yielded a decent R2 of .7167 considering the assumptions made in the diffusivity equation. For week 3, when the rotary pump was used instead of the syringe pumps to run the experiments, precision of the diffusivity constants improved. The correlation between D and Uo increased from an R2 value of 0.7167 to an R2 value of 0.7655. The increase in precision is credited to the improved apparatus, the rotary pump, for reasons previously described.

As mentioned before, our results were analogous to results obtained by Dr. Scherer et. al. in their study of gas dispersion in the branching bronchial network, as values for effective diffusivity increased linearly with increasing velocity. In his experiment, Dr. Scherer found this for inspiration and expiration of benzene vapor and cigarette tar vapor. The effective diffusivity versus velocity ratio (D/U0) was 3.33cm for inspiration and expiration 1.1cm. Although our results and Dr. Scherer’s results both exhibited a linear relationship between D and U0, the magnitudes of this relationship (slope U0/D) were not equivalent, because experiments were run in different phases (liquid and gas) using different model parameters.

The study of dispersion through branched networks has clinical relevance because of the prevalence of branched networks throughout the body, particularly in the cardiovascular and respiratory systems. As previously stated, bifurcations change the dispersive properties of fluids flowing through them, and it is important to be able to quantify these changes when studying hemodynamics and respiratory function. Research in this field has already been conducted and has positive implications for tumor treatment in terms of drug (chemotherapeutic agent) delivery via vessel networks created by angiogenesis.

##### APPENDIX

REFERENCES

Scherer et. al., “Measurement of Axial Diffusivities in a Model of the Bronchial Airways.” Journal of Applied Physiology, Volume 38, no. 4, April 1975, pp. 719-23.

McDougall et. al., “Mathematical Modeling of Flow Through Vascular Networks: Implications for Tumor-induced Angiogenesis and Chemotherapy Strategies.” Bulletin of Mathematical Biology (2002). Issue 64, pp 673-702.