Fig.1 Formula (1) The relationship between time and width and length of each part of paper chip.
T is the time taken for solution to diffuse from one end to another of the paper chip.
V is the volume of paper chip.
μ is the viscosity of solution.
K is the permeability.
△p is the difference in pressure.
L is the length of a part of paper chip.
W is the width of a part of paper chip.
H is the thickness of a part of paper chip.
In practical, as the viscosity of plasma, which is 1.2-1.66, is almost equal to viscosity of water, water can be used instead of plasma. And the effect of concentration of glucose to the plasma viscosity can also be ignored. Thus viscosity can be seen as a constant.
K is a constant and is provided by the producer of filter paper.
△p in a horizontal placed paper chip with height h which is much smaller than width(h<<w) can be calculated using formula (2):
Fig.2 Formula (2)
γ is the surface tension of plasma. When blood keeps evaporating, its surface tension is less than 60mJ/m^2 in the liquid/vapor interface.60 can be taken as an extreme value.
H is the height of solution, which can be seen as the same with the height of tunnels in the paper chip.
Cosθis the contact angle between air and the surface of solution.
From formula (2) we can see that △p is constant with constant temperature, constant thickness of paper chip and within the same interface. Thus we see it as a constant value in the practical.
Fig.3 A sketch of the reaction region of the test paper
This is a sketch of the reaction region of the test paper. Assumed that the length and width of each region are the same, and the length and width of each tubule also are the same, the whole test paper can be seen as a combination of four tubules and four square reaction regions.
Let the length, width and thickness of reaction region be L1, W1 and H1 and those of tubules be L2, W2 and H2. Thus the formula can be simplified to:
In this formula,
And the thickness is the same in every part of the test paper, so H1 = H2. Then we can cancel out the thickness in the formula to get:
μ, k and △p are all constants.
So the time taken t is directly proportional to the product of (L1W1+L2W2)(L1/W1+L2/W2).
And it is also proved that the time taken is reversely proportional to L for reaction regions and W2/W1, but directly proportional to the area of tubules. In consideration of the reader and size of the filter paper (whatman 5), we set up a few experimental groups of test paper with different width and length.
The time taken for water to travel to the end of the test paper is recorded in the experiment and a graph is plotted for time against the product of (L1W1+L2W2)(L1/W1+L2/W2).
Then we do a linear regression to find the best-fit line of the graph. The gradient of this line equals to 16μ/k△p. And the size of test paper which takes the least time is found.
Moreover, in order to read the result more easily, we also have another design of test paper, which coincide with the size of 96-well plates. This kind of test paper can put on the elisa reader and test for the result directly.