Team:Queens Canada/FluidDynamics
Fluid Dynamics
Overview
This page describes and models the fluid portion of the Lateral Flow Immunoassay (LFIA). First, the portion of sample that can be extracted is determined to be approximately 1.256mm3. Next, the fluid flow through the device is modelled using the Lucas-Washborn equation for three very different pore sizes; 0.45๐m, 8๐m, and 15๐m. These models revealed that as pore diameter decreases, the flow within the assay should decrease as well. These models can be applied when selecting or modifying pore size to best optimize flow to allowing binding within the LFIA to occur.
Quantity of Sample
While the lower limits of the assayโs sensitivity was proven on the Assay Design page, the question remains on exactly how large a sample can be extracted from a tick to be applied to the test. An estimation for this quantity is provided below.
The volume of the tick can be calculated as an ellipsoid through the use of the following equation:
where l is the length of the tick, w is the width of the tick, and t is the thickness of the tick. Estimating the average size of an unfed, black-legged tick, the approximate volume of the tick can be calculated as follows:
However, the tick's head and limbs do not contribute to the availability of midgut samples therefore, a volume correction is required. By approximating that the midgut is about 80% of the ticks total volume, the following correction can be made:
Therefore, the volume of the available sample is approximately 1.256 mm3. This sample will then be diluted within a lysis buffer solution to ensure proper flow within the assay.
Characterizing Flow
Liquid moves through a lateral flow assay through the use of capillary action. The best way to characterize the movement of the flow through the assay is by capillary flow time due to the non-uniform pore distribution within materials as a result of manufacturing [1]. Capillary flow time is the time it takes for the solution to transverse the entirety of the assay length and has the units s/cm [1]. The flow can also be characterized by its velocity which can be determined through the use of the Lucas-Washburn equation and is inversely proportionate to the capillary flow time [2]. Normally, the flow within LFIA adheres to the Lucas-Washburn equation and the flow rate is the fastest at the beginning of the strip and declines with time [2]. The equation is based on the assumption that the porous media is comprised of capillary tubes that are arranged in parallel to one another with identical radii, in addition, to complete saturation behind the wetted front [3]. The Lucas-Washburn equation is used to relate time to the position of the visible wetted front (L) or the front of the aqueous flow [3]. The equation is as follows:
where ๐พ is the surface tension of the liquid, ๐ is the contact angle, r is the radius of the pore/capillary tubes, ๐ is the dynamic viscosity of the liquid, and t is the time [3]. The position of the wetted front (L) or length can then be isolated as a function of time as follows:
Since the solution that will be mixed with the tick innards will be aqueous, it was assumed to have the same properties of water at 25โ.
Nitrocellulose
Contact Angle
Before the length can be calculated, the contact angle between the solution, nitrocellulose membrane, and air must be calculated. This can be done through the use of Youngโs Equation which is as follows:
Which is illustrated by the following diagram:
where ๐พLV is the interfacial surface tension between the liquid and vapour, ๐พSV is the interfacial surface tension between the solid (surface) and vapour, and ๐พSL is the interfacial surface tension between the solid and the liquid surfaces. Youngโs equation describes the contact angle at the triple point between the three phases at equilibrium and is an indicator of the wettability of the liquid. If the contact angle is less than 90ยฐ, then the substances are said to be hydrophilic and the droplet will spread out and โwetโ the surface. If the contact angle is greater than 90, the substances are said to be hydrophobic and the liquid droplet will maintain its round shape and not wet the surface.
First, since the vapour in question is air, it can be assumed that ๐พSV=๐พS and ๐พLV=๐พL, meaning interfacial tensions are equal to the surface tensions of the individual substances nitrocellulose and water respectively. The surface tension of water at 25ยฐC is 72.0 mJ/m2 and that for nitrocellulose fiber is 54.9 mJ/m2 [4, 5]. These two free energies can then be used to determine the interfacial tension between the liquid and the solid using the Equation of State which is as follows:
where ๐ฝ is the fitted parameter 0.0001247 (m2/mJ)2. The calculation for the interfacial surface tension between nitrocellulose and the aqueous solution is as follows:
This can then be used in Equation 3 to calculate the resulting contact angle:
Therefore, the contact angle between the aqueous solution and the nitrocellulose membrane is 46.85ยฐ. This value represents the contact angle when the droplet is at equilibrium resting on the surface. When the liquid is moving through the membrane, it is not at equilibrium and therefore the contact angle should be characterized in the form of the advancing and receding contact angles. Since these values cannot be calculated, for the purpose of this calculation, the contact angle at equilibrium is assumed to be a close approximation. This assumption is supported by values for the advancing and receding contact angles for water on porous nitrocellulose membrane with an average pore size of 3.0๐m which are 46ยฐ and 7ยฐ respectively, with the difference between the two being 39ยฐ [6]. This is incredibly close to the calculated equilibrium value which does not account for surface topography. It can be assumed that if surface topography were to be accounted for, the contact angle would decrease as the contact angles between a 0.22๐m and a 3.0๐m decrease by 5ยฐ, increasing the hydrophilicity and wettability of the solution which is ideal [6].
The approximated value for contact angle can then be used in the Lucas-Washburn equation to determine the length the solution will travel over a certain period of time.
Equation Results
Now that the contact angle has been determined, the length, as a function can be determined, first using a pore diameter of 15๐m which corresponds to a radius of 7.5๐m, the previous states surface tension for water, and the corresponding dynamic viscosity of water at 25โ which is 0.000891 Ns/m2. Once the length is calculated for its corresponding time, the solution velocity and capillary time can then be calculated.
The relationship between length, velocity, and capillary flow time and time can be observed in Figure 1, Figure 2, and Figure 3.
As can be observed in the above figures, the flow flows a logarithmic pattern, with the fluid velocity and the capillary flow time (which are inversely proportionate to one another) the fastest at the start and slowing over time which is to be expected. Knowing the rate will assist in determining the optimal length of the membrane so that interaction between the test strip and analyte in the transversing fluid is maximized.
Since the bacteria is going to be lysed, a smaller pore size could potentially be used. Ideally, an 8.0๐m membrane could be used as it would decrease flow rate, increase the interaction times, and increase sensitivity.
The relationship between length, velocity, and capillary flow time and, time can be observed in Figure 4, Figure 5, and Figure 6.
The length, flow times and, capillary flow rates for a pore size of 0.45๐m can be calculated the relationship between length, velocity, and capillary flow time and, time can be observed in Figure 7, Figure 8, and Figure 9.
When comparing the flow rates and distances between the two different pore sizes, the 0.45๐m membrane has a slower flowrate and capillary flow time which is ideal as previously stated, it increases the interaction time and probability of the analyte binding to the immobilized antibody as well increases the sensitivity of the test. The length of the immunoassay will be chosen based on the modelled binding affinity and reaction rate of ospA and the binding antigens. t is assumed that when accounting for resistance and additional pore sizes, the flow time should be within the correct test. It is recommended that future work physically test these models, through flow tests with varying dimensions
References
1. Innova Biosciences, "Guide to Lateral Flow Immunoassays," [Online]. Available: https://fnkprddata.blob.core.windows.net/domestic/download/pdf/IBS_A_guide_to_lateral_flow_immunoassays.pdf. [Accessed 29 May 2021]
2. Z. Xiao, Y. Yang, W. Zhang and W. Guo, "Controlling Capillary Flow Rate on Lateral Flow Test Substrates by Tape," Micromachines, vol. 12, no. 562, 2021.
3. B. Cummins, R. Chinthaptla, F. Ligler and W. M. Glenn, "Time-Dependent Model for Fluid Flow in Porous Materials with Multiple Pore Sizes," Analytical Chemistry, vol. 89, pp. 4377-4381, 2017.
4. Engineering Toolbx, "Water- Dynamic and Kinematic Viscosity," Engineering TooBox, 2004. [Online]. Available: https://www.engineeringtoolbox.com/water-dynamic-kinematic-viscosity-d_596.html. [Accessed 7 June 2021].
5. Q. Shen, "Surface Properties of Cellulose and Cellulose Derivatives: A Review," in Model Cellulose Surface, Oxford, Oxford University Press, 2009, pp. 259-289.
6. Diversified Enterprises, "Surface Energy Data for Nitrocellulose," 2009. [Online]. Available: http://www.accudynetest.com/polymer_surface_data/nitrocellulose.pdf. [Accessed 5 June 2021].