1 Physical Foundations for Biology and Medicine

Quantities, units, and dimensions

A Physical quantity is a measurable property described by a number and a unit. The number alone is incomplete: a length of $2.0$ has no clear physical meaning until its unit is specified. For example, a vessel radius might be written as $r=2.0\times10^{-3}\ \text{m}$.

The SI units provide a common measurement system. Its seven base units are the meter ($\text{m}$), kilogram ($\text{kg}$), second ($\text{s}$), ampere ($\text{A}$), kelvin ($\text{K}$), mole ($\text{mol}$), and candela ($\text{cd}$). Many quantities in biology and medicine use derived units: speed is measured in $\text{m}\,\text{s}^{-1}$, force in newtons ($\text{N}$), and pressure in pascals ($\text{Pa}$).

Prefixes help express very large or small measurements. A millimeter is $10^{-3}\ \text{m}$, a micrometer is $10^{-6}\ \text{m}$, and a nanometer is $10^{-9}\ \text{m}$. Check prefixes carefully in clinical and laboratory settings: a milligram and a microgram differ by a factor of $1000$.

Dimensions describe how a quantity depends on base quantities, regardless of the particular units used. With length $L$, mass $M$, and time $T$, speed has dimension $LT^{-1}$, while pressure has dimension $ML^{-1}T^{-2}$. Dimensional checks can reveal incompatible equations: both sides of an equation must have the same dimensions, and quantities with different dimensions cannot be added.

Takeaway: Include units with measurements, convert prefixes carefully, and use dimensions to check whether equations are physically consistent.

Measurement and uncertainty

A measurement is an estimate rather than an exact description of a biological system. Instrument resolution, variation between repeated measurements, calibration, sampling, and environmental conditions can all contribute to Measurement uncertainty.

Report precision in a way that reflects the measurement. For example, $r=(2.00\pm0.05)\ \text{mm}$ communicates an estimated radius together with its uncertainty, rather than implying unsupported precision. Uncertainty in measured inputs can also affect a value calculated from them.

Takeaway: A result is most informative when its precision reflects the limits of the measurement and the uncertainty in its inputs.

How size changes biological function

Size can affect biological function because surface area and volume do not change at the same rate. If a structure keeps the same shape while its characteristic length is multiplied by a factor $k$, its surface area scales as $k^2$, while its volume scales as $k^3$. Doubling a cell’s radius therefore multiplies its surface area by $4$ and its volume by $8$; its surface-area-to-volume ratio is halved. This difference can affect exchange across membranes, heat transfer, and diffusion distances.

An Allometric relationship describes how a trait varies with a size measure, such as body mass:

Here, $M$ is the size measure, $y$ is the trait, $a$ is a fitted coefficient, and $b$ is the scaling exponent. On logarithmic axes, the relationship becomes $\log y=\log a+b\log M$, so the exponent is the slope. The exponent must be estimated for the trait and population being studied; biological scaling relationships are not necessarily universal. Physiology, shape, behavior, and evolutionary history can all affect observed patterns.

Scaling can help compare people, organs, or experimental systems, but a conversion based only on body mass or surface area may miss physiological differences. State what is being scaled and why the chosen scaling approach is appropriate.

Takeaway: Geometric scaling gives useful expectations, but biological scaling relationships should be tested rather than assumed to apply universally.

Building and evaluating physical models

A Physical model is a simplified representation designed to answer a particular question. Its usefulness depends on defining the system carefully and making its assumptions explicit. For example, a model might represent a vessel as a rigid tube or a membrane as an electrical circuit; each simplification makes analysis possible while limiting where the result applies.

A practical modeling workflow is:

1. Define the system and question. Specify what is included, what will be measured, and the conditions of interest, such as flow through a vessel or heat loss from skin.

2. Choose variables and scales. Identify relevant quantities and characteristic lengths, times, pressures, or energies, and use consistent units.

3. State assumptions. Decide, for example, whether a vessel can be treated as rigid or a tissue as uniform.

4. Select governing relationships. Apply suitable physical principles, such as conservation of mass or energy, force balance, or electrical relationships. Include relevant properties such as viscosity, elasticity, or conductivity.

5. Check dimensions and limiting cases. Confirm dimensional consistency and consider whether the model behaves sensibly when a parameter becomes very small or very large.

6. Compare with measurements. Check predictions against relevant experimental or clinical data, examine uncertainty, and revise the model if important behavior is missed.

Models may be lumped, using a few average variables, or distributed, representing how quantities vary across space or time. A lumped flow model can estimate average circulation, while a more detailed fluid model may be needed to examine local flow near a narrowing. Likewise, a simple electrical-circuit model may approximate charge storage and conduction across a membrane, but a spatial model is needed when voltage varies substantially across tissue.

Model complexity should match the question. A useful model need not include every detail; its assumptions must suit its purpose, and its predictions should be checked. For high-consequence uses, such as medical-device applications, evidence supporting a computational model’s credibility should be proportionate to how strongly a decision depends on it.

Takeaway: Define the question, expose assumptions, check the mathematics, and test predictions against relevant measurements.