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Assignment 1 By DIVYESH SOLANKI (20020446) Submitted to: Kishore Pochiraju Subject: MODELLING AND SIMULATION Date: 01-26-2024
DEPARTMENT OF MECHANICAL ENGINEERING STEVENS INSTITUTE OF TECHNOLOGY HOBOKEN, UNITED STATES
Table of Contents Problem 1 ..............................................................................................
3
Model: .................................................................................................................................................
4 Simulation: ..........................................................................................................................................
4 Verification and Validation: ................................................................................................................
6 Table of Figure Figure 1 Processed image of cell membrane ..........................................................................................
5 Figure 2 Area found by polygon ..............................................................................................................
6
Problem 1 Prof. C studies cell growth on patterned substrates. Her assistant obtained a set of pictures and one of them is shown below. If the image swatch shows a 100
m x 100
m view port. Determine, with an appropriate model, the area of occupied by the cell - m
arked “A” in the image
. The cell membrane is the darker/thick outline that separates the cell from its environment.
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Model: 1.
List all the assumptions and sources of error. •
In order to approximate the area of cell membrane I have generalized the geometry of the cell to be circle so I have used Area (
𝐴) = 𝜋𝑟
2
formula to calculate the area of the cell membrane. •
To approximate the radius of the cell membrane I have made use of image processing software called Imagej
. Firstly, I have calibrated the image pixels to the given scale of 100
m, and the scale came out to be 3.5850 pixels/
m. •
I have considered the cell wall (black thick outline) to be part of its area. Source of Errors •
One source of error could be as I am approximating the shape of a cell membrane to be a circle there might be some error in the actual area of the cell. •
Another source of error could be length of diameter as I have manually drawn lines along the circumference of the cell membrane to find the radius. I might not be able to point out the lines exactly onto the circumference of the cell membrane. 2.
Describe your model. How do you approximate the cell geometry? •
My model comprises of using image processing software to approximate the diameter of the cell membrane and then using that data received form the image processing software to approximate the area of the cell membrane and calculating error. Simulation: 3.
What are the inputs to the model? •
Gathering values of diameters of the cell membrane using image processing software from different angles. •
Finding maximum, minimum, mean and standard deviation of the diameters. •
The diameter length I received from the image processing software is as follows: Label Length Mean Min Max Angle 1 70.673 171.219 58.744 235.657 0.643
2 67.63 159.661 66.531 237.636 -90 3 68.278 161.932 47.935 249.681 -45.235 4 69.437 184.122 59.175 253.228 43.073 5 67.454 177.612 55.342 255 -68.157 6 68.149 179.668 60.535 255 -112.688 7 68.264 183.441 60.739 253.856 22.048 8 73.443 186.514 52.667 255 -23.983 9 72.483 181.58 61.759 255 -12.846 10 68.7 185.914 59.807 253.637 54.232 11 Mean 69.451 177.166 58.324 250.369 -23.291 12 SD 2.081 9.727 5.183 7.423 55.865 13 Min 67.454 159.661 47.935 235.657 -112.688 14 Max 73.443 186.514 66.531 255 54.232 Figure 1 Processed image of cell membrane •
To approximate the area of cell geometry I have used the function of Microsoft excel to determine the area which can be seen in excel file. •
Finally approximating the area of cell membrane using (
𝐴) = 𝜋𝑟
2
for 10 different diameters, and finding an average value for the area. 4.
How did you solve for the area and what is the result?
•
Once I have enough data to approximate the area of cell membrane, I have calculated the area with the (
𝐴) = 𝜋𝑟
2
to approximate the area with 10 different diameter values, as shown in the table below. Diameter Area (A) in μm^2 1 70.673 3922.691 2 67.63 3592.161 3 68.278 3661.328 4 69.437 3786.683 5 67.454 3573.489 6 68.149 3647.506 7 68.264 3659.827 8 73.443 4236.214 9 72.483 4126.192 10 68.7 3706.727 Average 3791.282 Verification and Validation: 5.
How did you verify the result? (How do you know the calculations are right?) •
To verify the results, I have made use of another tool available in the Imagej software called polygon. I have drawn a polygon using that tool along on the edge of the cell membrane (as shown in the below image), and then using measure command the software calculates the area occupied by that cell. I have then compared the results of areas found by diameter and polygon to check for the error and verify my model. Figure 2 Area found by polygon
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•
Area found using this polygon tool is as follow. Area Mean Min Max Angle Length 1 3837.701 162.376 30 252 0 0 •
Error found in the calculation. Error = (
𝐴??𝑎 ????? ??𝑖?? ???𝑦??? ????−𝐴??𝑎 ????? ??𝑖?? ?𝑖𝑎?????
𝐴??𝑎 ????? ??𝑖?? ???𝑦??? ????
) × 100
= (
3837.701−3791.282
3837.701
) × 100
= 1.20
% 6.
How can you validate the result? (How do you know if its correct?) •
In order to validate the results, I have first set scale in the image processing software using given scale of 100 μm x 100 μm. In that manner I have set a global scale of 3.5850 pixels/
m. •
But now in order to check that the scale I have found is precise or not I have then drawn a several lines of the following pixel length and measured the value in μm, that matched with the actual value. Lines Length in pixel Length in μm 1 100 27.89 2 200 55.79 3 358 99.86 ≈
100 •
So, I can say all the calculation that I have made in my model are correct and it validates my model.
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