Terminology, the Standard Human, and Scaling

Abstract

This chapter overviews the general features of the body that will be important throughout the text, and this begins with an overview of the terminology of anatomy. This is then related to how the body moves at synovial joints, and is explained in terms of the degrees of freedom of the motion of these joints. The size and features of the body and how they are related are presented for a standard human and then interrelated using scaling relationships, including allometric rules.

Problems

Body Terminology

1.1

(a) Is the heart superior or inferior to the large intestine?

(b) Is the large intestine superior or inferior to the heart?

1.2

(a) Is the navel posterior or anterior to the spine?

(b) Is the spine posterior or anterior to the navel?

1.3

Is the nose lateral or medial to the ears?

1.4

Are the eyes lateral or medial to the nose?

1.5

Is the foot proximal or distal to the knee?

1.6

Is the elbow proximal or distal to the wrist?

1.7

Is the skeleton superficial or deep to the skin?

1.8

The blind spot in the eye retina is said to be nasal to the fovea (center of the retina). What does this mean?

1.9

What would you expect the term cephalid to mean? What would be an equivalent term?

1.10

Which is the anterior part of the heart in Fig. 8.7? Is this a superior or inferior view of the heart?

1.11

Consider the directional terms ipsilateral and contralateral. One means on the same side of the body, while the other means on opposite sides of the body. Which is which?

1.12

The directional term intermediate means “in between.” What is intermediate between the upper and lower legs?

1.13

Is the lower arm more supinated when throwing a baseball or football , and why is this so?

1.14

Encephalitis is the inflammation, i.e., “itis”, of what?

1.15

Presbyopia refers to disorders in vision due to old age, such as lack of accommodation in the crystalline lens (see Chap. 11). Presbycusis refers to old age-related auditory impairments (see Chap. 10). What parts of these two terms mean old age, vision, and hearing?

1.16

What is the difference between presbyphonia and dysphonia?

1.17

The three tiny bones in the middle ear, the malleus, incus, and stapes are interconnected by the incudomallear articulation and the incudostapedial joint. Describe the origin of the names of these connections.

1.18

The quadriceps muscles in the upper leg attach to the kneecap (patella) through the quadriceps tendon. The kneecap is connected to the tibia by connective tissue that is sometimes called the patellar tendon and sometimes the patellar ligament. Explain why both designations have merit and why neither designation completely describes the linkage perfectly well by itself.

1.19

Consider the drawing of the hand skeleton and the schematic of a hand showing joints with one or (labeled 2) two degrees of freedom in Fig. 1.6.

(a) How many degrees of freedom does each hand have? (Ignore the wrist joint.)

(b) Do we need so many degrees of freedom? Why? (There is no right or wrong answer to this part. Just think about what a human hand should be able to do (in clutching, etc.) and try to express your conclusions in terms of degrees of freedom.)

1.20

Estimate the angle of each of the joints in the hand for each of the following functions [29]. (Define the angle of each joint as shown in the left hand in Fig. 1.6b to be $0^{\circ }$. Define rotations into the paper and clockwise motions in the plane of the paper as being positive.)

• (a) lifting a pail (a hook grip)

• (b) holding a cigarette (a scissors grip)

• (c) lifting a coaster (a five-jaw chuck)

• (d) holding a pencil (a three-jaw chuck)

• (e) threading a needle (a two-jaw pad-to-pad chuck)

• (f) turning a key (a two-jaw pad-to-side chuck)

• (g) holding a hammer (a squeeze grip)

• (h) opening a jar (a disc grip)

• (i) holding a ball (a spherical grip)

1.21

We said that the seven DOFs available for arm motion enabled nonunique positioning of the hand, but analogous nonunique positioning of the foot is not possible because the leg has only six DOFs. Use Table 1.10 to explain why this is not exactly correct.

1.22

Use Table 1.3 to show that the coordinated eye motions in Table 1.4 use the muscles listed for primary motion.

1.23

Consider a limb, of length L, composed of upper and lower limbs with respective lengths $r_{1}$ and $r_{2}$, with $L=r_{1}+r_{2}$. There is a total range of motion in the angles the upper limb makes with the torso and the lower limb makes with the upper limb. Assume motion only in two dimensions (see Fig. 1.7).

(a) What area is subtended by the end of the lower limb (hand or foot) when $r_{1}=r_{2}$?

(b) What area is subtended by the end of the lower limb when $r_{1}>r_{2}$? What fraction of that in (a) is this?

(c) What area is subtended by the end of the lower limb when $r_{1}<r_{2}$? What fraction of that in (a) is this?

1.24

Redo Problem 1.23 in three dimensions, finding the volume subtended by the end of the lower limb in each case.

The Standard Human

1.25

Qualitatively explain the differences of density in Table 1.7 in the different segments of the body. The average densities of blood, bone, muscle, fat, and air (in the lungs) can be determined from Table 1.11.

1.26

(a) Use Table 1.7 to determine the average density of the body.

(b) Use this to determine the average volume of a 70 kg body.

Your answers will be a bit different from the rough volume estimate given in Table 1.11.

1.27

(a) Calculate the range of segment masses alternatively using Tables 1.7 and 1.16, for each type of segment listed in the latter table, for people with masses in the range 40–100 kg.

(b) Give several reasons why these ranges seem to be different.

Table 1.16 An alternative set of relations of weights of body segments (all in lb)

1.28

(a) Show that (1.3) becomes

$$\begin{aligned} \rho \text{(in } \text{ kg/L } \text{ or } \text{ g/cm }^{3}) = 0.69 + 0.0297S \, , \end{aligned}$$

(1.7)

when $S=H/W_{\mathrm {b}}^{1/3}$ is expressed with H in inches and $W_{\mathrm {b}}$ in lb.

(b) Show that the average density for an adult of height 5 ft 10 in and weight 170 lb is 1.065 g/cm$^{3}$, with $S=12.64$.

(c) Show that the average density for an adult of height 1.78 m and mass 77.3 kg is 1.066 g/cm$^{3}$, with $S=0.418$.

1.29

What percentage of body mass is fat, skin, the skeleton, blood, liver, the brain, the lungs, heart, kidneys, and eyes?

1.30

Use Table 1.11 to determine the mass density of blood, skin, the lungs, the air in the lungs, fat, liver, hair, eyes, and blood vessels.

1.31

In modeling heat loss in Chap. 6, a typical man is modeled as a cylinder that is 1.65 m high with a 0.234 m diameter. If the human density is 1.1 g/cm$^{3}$, what is the mass (in kg) and weight (in N and lb) of this man?

1.32

(a) If a man has a mass of 70 kg and an average density of 1.1 g/cm$^{3}$, find the man’s volume.

(b) If this man is modeled as a sphere, find his radius and diameter.

(c) If this man is 1.72 m high, and is modeled as a right circular cylinder, find the radius and diameter of this cylinder.

(d) Now model a man of this height and mass as a rectangular solid with square cross-section, and find the length of the square.

(e) Repeat this for a constant rectangular cross-section, and determine the sizes if the long and short rectangle dimensions have a ratio of either 2:1, 3:1, or 4:1.

(f) In each above case calculate his surface area and compare it to that predicted by (1.2) for a 1.72-m tall man. Which of the above models seems best?

1.33

The cylindrical model of a man in Fig. 1.17 was once used in studies of convective cooling. What are the volume, mass, and exposed surface area (including the bottom of the lower limb) for this person? Assume each finger is 3.5 in long and has a diameter of 0.875 in and that the mass density of all components is 1.05 g/cm$^{3}$.

Fig. 1.17

Cylindrical model of a man used in studies of convective cooling (from [5], adapted from [28])

1.34

How much heavier is someone with a totally full stomach, small intestine, large intestine and rectum, than when each system is empty? Assume the mass density of the contents is 1 g/cm$^{3}$. Express your answer in mass (kg) and weight (N and lb). (Use Table 7.4.)

1.35

Use the anthropometric data to determine the average cross-sectional area and diameter of an arm and a leg of a 70 kg man. Assume the cross section of each is circular.

1.36

Compare the surface area of the standard man given in Table 1.5, alternatively as predicted by (1.1) and (1.2). Use the data given in Table 1.13.

1.37

Compare the surface area of a 50 kg, 5 ft 5 in woman, alternatively as predicted by (1.1) and (1.2). Use the data given in Table 1.13.

1.38

For an adult, the average fractional surface area is 9% for the head, 9% for each upper limb, 18% both for front and back of the torso, and 18% for each lower limb. (The remaining 1% is for genitalia.)

(a) This is used to estimate the fraction of damaged area in burn victims. It is known as the “Rule of Nines.” Why?

(b) Use the data given in Table 1.13 to determine the average surface area for each of these parts of the body for the standard man.

1.39

In the system depicted in Fig. 1.18, a body segment is put in the measuring cylinder and the valve is opened to allow flow of water up to the “beginning” of the body segment (giving the “1” heights). The valve is then opened until water flows into the measuring cylinder to the “end” of the body segment (giving the “2” heights). Explain how this can be used to measure the volume of the body segment.

Fig. 1.18

Immersion technique for measuring the volume of various body segments, with the solid lines denoting the initial water level and the dashed lines the final water level (based on [25]). For Problem 1.39

1.40

(a) Calculate the BMI (also known as Quételet’s index) and the specific stature (also known as the ponderal index) of a person of average density $\rho$ modeled as a cube of length L.

(b) How do these change if the person has the same overall mass, but is modeled as a rectangular solid of height H, width 0.20H, and depth 0.15H?

1.41

(a) A person with mass M is modeled as a rectangular solid of height H, width 0.25H, and depth 0.25H. The person loses weight, maintaining the same mass density, and then has a width 0.20H and depth 0.15H. Calculate the BMI and the specific stature of the person before and after the weight loss .

(b) Would you expect the mass density of the person to change during the weight loss? If so, how would you expect it to change?

1.42

(a) Explain why the use of the BMI as a metric of ideal weight would be problematic if body mass and volume scaled as the cube of the body height, because it depends inversely as the square of body height [13, 19].

(b) Specifically, if the BMI of a person who is 1.75 m tall is 25.0 kg/m$^{2}$, find the (usual) BMI for the same person who is either 1.68 or 1.83 m tall, assuming their body mass and volume scale as the cube of body height.

(c) The ideal BMI seems to decrease with body height. Would revising the BMI index so it varies as a power of $m_{\mathrm {b}}/H^{3}$ (such as $1/S^{3}$, where S is the specific stature) make the ideal BMI less sensitive to height?

Fig. 1.19

Determining human body fat content and density by weighing a person in water (photo by Clifton Boutelle, News and Information Service, Bowling Green State University. Used with permission of Brad Phalin. Also see [14]). For Problem 1.43

Table 1.17 Comparison of the density and fat percentage for two men with the same height and mass, but different underwater masses

1.43

You can determine the density and percentage of fat in people by weighing them underwater, as in Fig. 1.19. Data for two men with the same height and mass, but with different underwater masses are given in Table 1.17.

(a) Why are the volumes given as listed?

(b) What assumption has been made about the relative densities of fat and the average of the rest of the body?

(c) Is the value assumed for the density for the rest of the body reasonable? Why or why not?

(d) Table 1.17 uses the Siri formula for the percentage of body fat: $100(4.95/({\text {body density}}) -4.50)$. How do the results for the percentage and mass of body fat differ using the Brozek formula: $100(4.57/({{\text {body density}}}) -4.142)$ for the two cases in the table? (Both formulas may, in fact, give values for body fat percentage that are too high.)

1.44

Explain why in Problem 1.43 you can either measure the weight of the water displaced by the body or the weight of the body when it is completely submerged [14].

1.45

You can measure the location of the anatomical center of mass of the body using the arrangement in Fig. 1.20a. The weight ($w_{1}$) and location of the mass ($x_{1}$) of the balance board are known along with the body weight $w_{2}$. The location of the body center of mass relative to the pivot point is $x_{2}$. The distance from the pivot to the scale is $x_{3}$. With the body center of mass to the left of the pivot point there is a measurable force S on the scale (under the head). Show that

$$\begin{aligned} x_{2}=\frac{Sx_{3}-w_{1}x_{1}}{w_{2}} \, . \end{aligned}$$

(1.8)

Fig. 1.20

In vivo estimation of a body center of mass and b mass of a distal segment, for Problems 1.45 and 1.46 (from [42]. Reprinted with permission of John Wiley & Sons)

1.46

You can determine the weight of the lower part of a limb ($w_{4}$) using the same balance board as in Problem 1.45, using Fig. 1.20b. The center of mass of the limb changes from $x_{4}$ to $x_{5}$ relative to the pivot point when the limb is set vertically; concomitantly the scale reading changes from S to $S^{\prime }$. Show that

$$\begin{aligned} w_{4}=\frac{(S^{\prime }-S)x_{3}}{(x_{4}-x_{5})} \, . \end{aligned}$$

(1.9)

The location of the center of mass of the limb relative the joint near the trunk is assumed to be known. To determine the weight of the entire limb the subject should be lying on his or her back and the entire limb is flexed to a right angle.

1.47

(a) Determine the goal mass (in kg, and find the weight in lb) to achieve 10% fat for the two men described in Table 1.17, by using the fat-free mass.

(b) How much fat mass (and weight) must be lost by each to attain this goal?

1.48

The normalized distances of the segment center of mass from the proximal and distal ends in Table 1.8 always sum to 1. Is this a coincidence, a trivial point, or significant? Why?

4 Allometry and Scaling

1.49

Determine the parameters for a 70 kg person for each set of allometric relation parameters in Table 1.13. How do they compare with similar parameters listed in Tables 1.5 and 1.11?

1.50

Derive the allometric laws for the percentages of the total body mass residing in the brain, heart, muscle, and skeletal mass for mammals (such as humans).

1.51

Compare the prediction of the fat in a standard man using the BMI, with those listed in Table 1.11.

1.52

Compare the % body fat in:

• (a) a male and female who are both 5 ft 6 in, 140 lb.

• (b) males who are 6 ft 2 in and 5 ft 8 in tall, both weighing 190 lb.

1.53

For a 70 kg person living 70 years, determine the person’s total lifetime

• (a) number of heart beats

• (b) number of breaths

• (c) energy consumed

• (d) energy consumed per unit mass.

1.54

Does it make sense that the ratio of the volumetric flow rates in the respiratory and circulatory systems in mammals (first entry line in Table 1.14) is essentially independent of mammal mass? Why?

1.55

Use Table 1.14 to find the allometry parameters for the ratio of the respiratory and cardiac rates (both in 1/s).

1.56

Use Table 1.14 to find the allometry parameters for the ratio of the volumes per breath (tidal volume) and per heart beat (heart stroke volume).

1.57

Use Fig. 1.21 to comment on whether the same mass-dependent-only allometric rules should be used within a species from birth to adulthood.

Fig. 1.21

Human development, showing the change in body shape from birth to adulthood, for Problem 1.57 (from [32])

1.58

(a) Arm length scales as the body height to the 1.0 power for people older that 9 months and to the 1.2 power for those younger. At 9 months of age, a male is 61 cm tall and has an arm length of 23 cm. When that male was 0.42 yr old he had a height of 30 cm and when he will be 25.75 yr old he will have a height of 190 cm. What would the arm length be expected to be at these earlier and later times, alternatively using the age-correct and age-incorrect scaling exponents?

(b) Is this an example of allometric or isometric scaling?

1.59

Scaling arguments can also be used to understand some general trends.

(a) If the linear dimension of an object is L, show that its surface area varies as $L^{2}$, its volume as $L^{3}$, and its surface to volume ratio as 1 / L, by using a sphere (diameter L) and a cube (length L) as examples.

(b) An animal loses heat by loss at the surface, so its rate of losing heat varies as its surface area, whereas its metabolic rate varies as its volume. In cold environments this loss of heat can be devastating. Do scaling arguments suggest animals would be bigger or smaller in cold climates?

(c) A cell receives oxygen and nutrients across its membrane to supply the entire volume of the cell. Do scaling arguments suggest that limitations in supplying oxygen and nutrients place a lower limit or upper limit in the size of cells? Why?

1.60

The strength of bones varies as their cross-sectional area, as we will see in Chap. 4. We have seen that this suggests how the diameter of a long bone scales with its length.

(a) Does this scaling relation mean that smaller creatures have thinner or thicker bones than bigger creatures assuming the same strength criterion?

(b) Does this “static” argument imply a limitation on how small or how large an animal can be?

1.61

(a) The work an animal of dimension L needs to propel itself a distance equal to its dimension is the needed force—which is proportional to its mass—times its dimension. Show this work scales as $L^{4}$.

(b) This force must be supplied by muscles, and the work done by the muscles is this force times the distance the muscles can contract; this distance scales as the length of the muscles, which in turn scales with L. In Chap. 5 we will see that the force exerted by a muscle is proportional to its cross-sectional area. If the lateral dimension of the muscle also scales as L, show that the maximum work that can be done by the muscle scales as $L^{3}$.

(c) For work done by muscles to scale as fast as that needed for locomotion, how must the lateral dimensions of muscles vary?

(d) Do these “dynamic” arguments limit how small or how large an animal can be [24]?