Problem 2 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

The Visible Human Project at the National Institutes of Health has provided numerous images of cross sections of the human body. They are being integrated into the medical education and research community through a program based at the University of Michigan. Shown in Exercise Figure 11.1 .2 are eight cross sections of the right side of the female brain. Your job is to estimate the volume of the brain. Assume that the sections are at \(1 \mathrm{~cm}\) separation, that the first section only shows brain membrane, and that the scale of the cross sections is \(1: 4 .\) Include only the brain and not the membrane which is apparent as white tissue. We found it useful to make a \(5 \mathrm{~mm}\) grid on clear plastic (the cover of a CD box), each square of which would be equivalent to \(4 \mathrm{~cm}^{2}\). The average human brain volume is \(1450 \mathrm{~cm}^{3}\).

Step-by-Step Solution

Answer

Use the grid to find areas, apply the scale to get real areas, sum up for volume, and double the result for both hemispheres.

Step 1: Understanding the Problem

We need to estimate the volume of the brain using cross-sectional images taken at 1 cm intervals. We will use a grid to measure areas of the brain from each cross section and then sum these to approximate the volume. The sections represent areas in a scaled down format of 1:4.

Step 2: Set Up the Grid

To measure area, use a clear plastic grid with squares measuring 5 mm \((0.5 \, \text{cm})\) on each side. Each square on this grid, when adjusted for the 1:4 scale, corresponds to an area of 4 cm² in real life.

Step 3: Measure Areas from Cross Sections

Place the grid over each of the 8 cross sections. Count the number of 5 mm squares covering the brain in each section, ignoring the membrane (white tissue). Multiply the count for each section by 4 to get the area in cm² for each section.

Step 4: Estimate Total Volume

Since the cross sections are 1 cm apart, multiply the calculated area from each section by 1 cm to find the volume for that slice. Sum these volumes to get the total estimated volume of the right side of the brain.

Step 5: Final Calculation

Double the volume obtained from the right side only, to estimate the total brain volume, since we only measured one hemisphere. Compare this volume to the average human brain volume of 1450 cm³ for context.

Key Concepts

Cross Sections

Cross sections are two-dimensional slices through a three-dimensional object, which help us understand the structure of that object by examining it piece by piece. In the context of this exercise, we're looking at cross sections of the human brain, each sliced at 1 cm intervals. Each cross section provides a detailed view of the brain's structure at that particular level. By studying cross sections, scientists and medical professionals can accurately assess and analyze the volume and areas of different parts of the brain.
Imagine taking a loaf of bread and slicing it—each slice reveals a different cross section. Similarly, in medical imaging, cross-sectional images are crucial for diagnosing and studying the anatomy and pathology of organs like the brain. By combining information from these cross sections, we can construct a 3D understanding and estimate the volume of the brain.

Human Brain

The human brain is a complex organ as integral to our body's functions as it is intricate in structure. Although it generally weighs about 3 pounds (or approximately 1.4 kg), its volume can vary. The average human brain volume is about 1450 cm³.
In scientific studies and educational exercises, understanding the brain's volume is crucial because it reflects overall size and helps in comparative anatomy studies and various medical evaluations. By focusing only on the brain tissue and excluding the non-functional membranes, this exercise helps provide a clear approximation of actual brain volume. Grasping the brain's structure through cross sections, we gain insights into both normal and pathological states, enabling further advancements in neuroscientific research.

Area Measurement

Measuring the area of irregular shapes, such as those found in cross sections of the brain, often requires creative methods. In this exercise, a grid overlay method is used. This involves placing a clear plastic grid over each cross-sectional image of the brain. Each grid square represents a specific area, scaled down to provide an accurate measurement.
The grid consists of 5 mm squares that, when scaled (due to the image's 1:4 ratio), equal 4 cm² of actual brain area. By counting the number of grid squares covering the brain, and converting these into actual dimensions using the scale factor, we find the area of each cross section.
This method is both practical and educational, illustrating mathematical concepts such as scaling and area calculation in real-world scientific contexts. Calculating areas of each section lays the groundwork for approximating the brain's volume by summing these areas over the length of the brain.