# Fetal Body Composition

## Summary

This is a comprehensive and practical project in Calculus for life science majors, which helps students gain a deeper understanding of fundamental concepts of Calculus and enables them to apply those concepts to solve a practical and real-life problem.

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## Learning Goals

The main objective of this project is to support students to:
1. build the essential skills to mathematically rationalize a biomedical-related scenario
2. develop independent analytical thinking
3. cultivate an "appreciation of mathematics" as an integral part of our world today.

This multi-step group project integrates the following key mathematical concepts through a fetal body composition problem:

• How clinical data is collected, graphs are presented, and data points are regenerated.
• How to interpret graph of a sequence.
• How to apply building blocks of calculus such as differentiation and integration to discrete data structures
• How to interpret the mathematically-derived results and relate them to actual real-world phenomena
This comprehensive project is very beneficial to students, since it inspires them to think "outside the box" and draw upon their full scientific knowledge in approaching the problem, as opposed to narrowly compartmentalizing the topics they have learned in different parts of the course.

## Context for Use

This project was developed for MATH-116 (Calculus A), which is a 4-credit Calculus course specifically designed for life science majors at Montclair State University. In this course, students expand their knowledge from the prerequisite PreCalculus and learn fundamental concepts of Functions and Sequences, Limits, Single variable Differentiation, and Integration. The primary learning goal for students is to acquire the ability to understand the importance of these mathematical concepts and apply them to solve problems arising in life sciences. This course is particularly important and challenging, since it is a terminal math course (students are not required to take any additional math courses) and they are all non-math majors (mainly Biology and Information Technology). Therefore, it is crucial to connect the subject matter to students' interest in real world applications. By doing so, the underlying relationship between apparently disparate areas of science can be illuminated, which offers students a glimpse of the bigger picture.

As a part of this effort, in Spring 2016, we developed and implemented a comprehensive and practical project across all sections of Calculus A (5 sections of approximately 25-40 students each). The project was resulted from several brainstorming discussions with colleagues both in the Biology and Math Departments, as well as health professionals at a local medical center, on how to make Calculus a more relevant and useful course for our life science students.

The project is introduced to students after they learn about the applications of differentiation (~ mid-semester). Students are placed in groups of 3-4, where they work on the problems, report their progress/any challenges to their instructors on a regular basis, and submit the final report in both written and audiovisual formats at the conclusion of the semester.

Originally, this project was designed to be completed using Microsoft Excel (or an equivalent spreadsheet application) in addition to Desmos graphing calculator (or an equivalent graphing application). However, it would be beneficial to use MATLAB as a scientific platform where both mathematical analysis and graphing capabilities can be applied. Regardless, students should be able to (or learn how to) perform basic data analysis as well as plot functions in appropriate applications such as Microsoft Excel, MATLAB, etc.

### Prerequisite/Requisite Skills:

• Units conversion
• Basic data representation and analysis in Microsoft Excel or MATLAB
• Fundamental concepts of differentiation and integration and their applications
• Ability to find the roots of a polynomial using their graphs

## Description and Teaching Materials

### Project Description

Oftentimes we have to critically think about the data provided. The data on the attached pages leads to an interesting result. Throughout this project, you should think about reasons why the fetal and placental weight/volume could be growing the way they show here.

1. First, read carefully the attached document, which has been extracted from a medical text, Human Body Composition, by G. Forbes (Fetal Mass by Forbes (Acrobat (PDF) 244kB Aug8 18)), where the collected data is shown and explained. Pay particular attention to the graph on page 103 and determine whether the data is presented in the form of w(t), w'(t), or w''(t), given that w(t) represents the fetal/placental weight as a function of time. Explain why.
2. Refer to the Figure 3.1 on page 103 and use a grid paper and a ruler to estimate the coordinates of data points for each set (fetus and placenta). Make the units of measurement consistent and then transfer each set to MATLAB arrays that could be used to replicate the scatter plots in Figure 3.1. Make sure to use proper legends and labels (including units) on the horizontal (t) and vertical axes and name the plots Fig. 1 and Fig. 2, respectively for fetus and placental growth data.
3. Next, use MATLAB Basic Fitting tool from the pull-down menu of the figure window to fit a polynomial of degree 1 (linear) to each set of data. Display the fitted equation and calculate the R-squared values. How good is the fitting?
4. Repeat part (3) using a polynomial of degree 4 with at least 10 decimal digits. Display the results on the same graphs. How good is the new fitting? Explain why?
5. Using the fitted polynomial in part 4 and what you have learned in calculus, calculate Wf''(t) and Wp''(t) for fetal and placental weights respectively, and plot them on two new graphs, named Fig. 3 and Fig. 4. You can also use MATLAB symbolic toolbox to calculate the derivatives. Make sure to use proper legends and labels (including units) on the horizontal (t) and vertical axes. Then, find the inflection points of fetal and placental weight data. Verify your answers on Figs. 1 and 2. What is the actual physical/physiological interpretation of these inflection points?
6. What do you notice about the inflection points of the placental weight and the inflection point of the fetal weight? Do they occur around the same week?
7. Using the fitted polynomial in part (4) and what you have learned in calculus, calculate two polynomials Wf (t) and Wp (t) that describe respectively the fetal and placental weight (grams) as a function of time (days), assuming at 112 days (16 weeks), a regular fetus is about 100 grams and a regular placenta weighs 120 grams. You can also use MATLAB symbolic toolbox to calculate the antiderivatives. Plot the fetal and placental weight as a function of time on two new graphs, named Fig. 5 and Fig. 6. Make sure to use proper legends and labels (including units) on the horizontal (t) and vertical axes.
8. In a different study, researchers have looked at the effects of exercise on pregnancy. The results, which are summarized in the following table, show the average volume of placenta at several weeks of gestation in non-exercising mothers.
 Week of Gestation (t) Placental Volume (Vp) in mL 20 181 23 220 25 262 27 289 31 302 35 350 37 385 40 414
Enter the above data points as MATLAB arrays, and starting with timet=0 , plot the placental volume Vp(t) (mL) as a function of time (weeks). Repeat part (4) using a polynomial of degree 4. Make sure to use proper legends and labels (including units) on the horizontal (t) and vertical axes and name this plot Fig. 7. Using the fitted polynomial, estimate the placental volume at time t=0, and explain the procedure. Does this value make sense? What is the actual physical/physiological interpretation of the extrapolated value? Elaborate in your own words.
9. Suppose you want to compare the inflection point of placental volume data to that of the placental weight data from Forbes. How do you find the inflection point for the placental volume data? How did you find it from Forbes' data? Why can't you compare the data directly? Find approximately at what week the inflection point occurs for both sets of data (placental volume and placental weight).
10. MAKE A MEDICAL CONCLUSION: Both sets of data were collected by different experiments.
1. Does the placental volume's inflection point match the same time period (approximately) for the placental weight for Forbes data?
2. What does this say about the data? Do you trust the data in both experiments? Why or why not?
3. What possible medical reasons can you provide for the inflection points occurring at different times for fetal weight and placental weight? List all you can think of. Remember to look up the fetal weight data from Forbes and check how this data was obtained.
All plots, calculations, and comments should be included in the project report.

## Teaching Notes and Tips

• The project is introduced to students after they learn about the applications of differentiation (~ mid-semester). Students are placed in groups of 3 (min) or 4 (max), where they work on the problems, report their progress/any challenges to their instructors on a regular basis, and submit the final report in both written and audiovisual formats at the conclusion of the semester.
• As described in the following section (Assessment), it is very important that the instructor checks in with students about the status of their current group project on a regular basis. One of the common errors committed by students early in the project is not making units of measurement consistent (refer to part 2 in the Project Description (Acrobat (PDF) 80kB Dec28 18)). As a result, the fitted polynomial and all subsequently calculated functions would be incorrect. Another common error is not using enough decimal digits (refer to part 4 in the Project Description (Acrobat (PDF) 80kB Dec28 18)), which may result in a loss of precision, given the small magnitude of the leading coefficients.
• The completed project is submitted electronically through a learning management system (in our case, Canvas).
• One person in each group should be in charge of uploading the project reports and activity logs. If a video is too large to be uploaded to Canvas, students are given an option to post it somewhere else on the web (Google Drive, YouTube, etc.) and upload a link instead.

## Assessment

• Progress Reports: 2 progress reports (on the first 4 questions and then on questions 5-8) are due respectively 2 and 4 weeks after the introduction of the project. Students are referred to the Project Description (Acrobat (PDF) 80kB Dec28 18) for all details on what needs to be included. Each group must also submit an activity log explaining what each member has contributed to the team. The purpose of this progress report is simply making sure students have started to work on the project, they are on the right track, and all members of the team are contributing equally.
• Final Report: At the conclusion of the semester, each group submits a written (.pdf) and an audiovisual report (video), where the problems are explained and the results are discussed. In the written report, all equations must be typed in Microsoft Equations (or LaTeX). For the audiovisual report, students can use many different tools such as their smartphone, Prezi, PowerPoint, Keynote, Camtasia, iMovie, Quick Time Player, and Windows Movie Maker to create a video.
• Activity Log: Each group is required submit a separate file containing a table of all individual and group activity logs that include each member's contribution to the project, how much time has been spent on each problem, and also when the group has spent time together on the project.
• Grading: The group project counts as 10% of the final grade and it is graded based on the content as well students' creativity. Therefore, students are encouraged to create audiovisual reports that are informing, entertaining, and appealing.

### Rubric to assess the learning outcomes

Criteria

Ratings

Pts

This criterion is linked to a Learning Outcome

Creativity and Presentation

 2.0 pts Excellent The project is very creative, fun, engaging, and interesting. 1.0 pts Good The project is somewhat creative, but not very engaging or interesting. 0 pts Unacceptable The project lacks creativity and it is not engaging.

2.0 pts

This criterion is linked to a Learning Outcome

Following Directions/Guidelines

 2.0 pts Complete All directions and guidelines are followed. 1.0 pts Incomplete Some directions and guidelines are not followed. This includes late submission. 0.0 pts Unacceptable The directions and guidelines are not followed.

2.0 pts

This criterion is linked to a Learning Outcome

Elaboration/Discussion Quality

 4.0 pts Excellent Both the breadth and depth of the project are appropriate. All questions are answered completely and correctly. 3.0 pts Good The breath and depth of the project are acceptable, but could be improved. All questions are answered completely, but some of them are not correct. 2.0 pts Average Either the breadth or depth of the project is not appropriate. Some questions are not answered completely and/or the answers are not all correct. 1.0 pts Below Average The breadth or depth of the project is not appropriate. Some questions are not answered completely and/or most of the answers are not correct. 0.0 pts Unacceptable No submission

4.0 pts

This criterion is linked to a Learning Outcome

Teamwork

 2.0 pts Meets expectation Contribution to the group effort meets expectation. 1.0 pts Meets expectation Contribution to the group effort is not enough. 0.0 pts Unacceptable Contribution to the group effort is not acceptable.

2.0 pts

## References and Resources

• Basic Fitting in MATLAB: https://www.mathworks.com/help/matlab/data_analysis/interactive-fitting.html
• Video tutorial on the Basic Fitting Tool in MATLAB: https://www.youtube.com/watch?v=Yyhs_n-imjI
• Desmos Graphing Calculator: https://www.desmos.com/calculator
• Forbes GB. Human Body Composition: Growth, Aging, Nutrition, and Activity [Internet]. Springer New York; 2012. Available from: https://books.google.com.br/books?id=A9XTBwAAQBAJ