A growth curve is an empirical model of the evolution of a quantity over time. Growth curves are widely used in biology

Biology

Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...

 for quantities such as population

Population

A population is all the organisms that both belong to the same group or species and live in the same geographical area. The area that is used to define a sexual population is such that inter-breeding is possible between any pair within the area and more probable than cross-breeding with individuals...

 size, body height or biomass

Biomass

Biomass, as a renewable energy source, is biological material from living, or recently living organisms. As an energy source, biomass can either be used directly, or converted into other energy products such as biofuel....

. Values for the measured property can be plotted on a graph

Graph of a function

In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...

 as a function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 of time; see Figure 1 for an example.

Growth curves are employed in many disciplines besides biology

Biology

Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...

, particularly in statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, which has an extensive literature on growth curves.
In mathematical statistics

Mathematical statistics

Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis...

, growth curves are often modeled as being continuous

Continuous stochastic process

In the probability theory, a continuous stochastic process is a type of stochastic process that may be said to be "continuous" as a function of its "time" or index parameter. Continuity is a nice property for a process to have, since it implies that they are well-behaved in some sense, and,...

 stochastic process

Stochastic process

In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

es, e.g. as being sample paths that almost surely

Almost surely

In probability theory, one says that an event happens almost surely if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory...

 solve stochastic differential equation

Stochastic differential equation

A stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process....

s.

Bacterial growth

In this example (Figure 1, see Lac operon

Lac operon

The lac operon is an operon required for the transport and metabolism of lactose in Escherichia coli and some other enteric bacteria. It consists of three adjacent structural genes, lacZ, lacY and lacA. The lac operon is regulated by several factors including the availability of glucose and of...

 for details) the number of bacteria present in a nutrient-containing broth was measured during the course of an 8 hour cell growth

Cell growth

The term cell growth is used in the contexts of cell development and cell division . When used in the context of cell division, it refers to growth of cell populations, where one cell grows and divides to produce two "daughter cells"...

 experiment

Experiment

An experiment is a methodical procedure carried out with the goal of verifying, falsifying, or establishing the validity of a hypothesis. Experiments vary greatly in their goal and scale, but always rely on repeatable procedure and logical analysis of the results...

. The observed pattern of bacterial growth is bi-phasic because two different sugars were present, glucose and lactose. The bacteria prefer to consume glucose (Phase I) and only use the lactose (Phase II) after the glucose has been depleted. Analysis of the molecular basis for this bi-phasic growth curve led to the discovery of the basic mechanisms that control gene expression

Gene expression

Gene expression is the process by which information from a gene is used in the synthesis of a functional gene product. These products are often proteins, but in non-protein coding genes such as ribosomal RNA , transfer RNA or small nuclear RNA genes, the product is a functional RNA...

.

Cancer cell growth

Cancer

Cancer

Cancer , known medically as a malignant neoplasm, is a large group of different diseases, all involving unregulated cell growth. In cancer, cells divide and grow uncontrollably, forming malignant tumors, and invade nearby parts of the body. The cancer may also spread to more distant parts of the...

 research is an area of biology where growth curve analysis http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Search&db=books&doptcmdl=GenBookHL&term=%22growth+curve%22%5BAll+Fields%5D+AND+cmed6%5Bbook%5D+AND+351845%5Buid%5D&rid=cmed6.section.10278 plays an important role. In many types of cancer, the rate at which tumor

Tumor

A tumor or tumour is commonly used as a synonym for a neoplasm that appears enlarged in size. Tumor is not synonymous with cancer...

s shrink following chemotherapy

Chemotherapy

Chemotherapy is the treatment of cancer with an antineoplastic drug or with a combination of such drugs into a standardized treatment regimen....

 is related to the rate of tumor growth before treatment. Tumors that grow rapidly are generally more sensitive to the toxic effects that conventional anticancer drugs have on the cancer cell

Cell (biology)

The cell is the basic structural and functional unit of all known living organisms. It is the smallest unit of life that is classified as a living thing, and is often called the building block of life. The Alberts text discusses how the "cellular building blocks" move to shape developing embryos....

s. Many conventional anticancer drugs (for example, 5-Fluoro Uracil) interfere with DNA replication

DNA replication

DNA replication is a biological process that occurs in all living organisms and copies their DNA; it is the basis for biological inheritance. The process starts with one double-stranded DNA molecule and produces two identical copies of the molecule...

 and can cause the death of cells that attempt to replicate their DNA and divide. A rapidly growing tumor will have more actively dividing cells and more cell death upon exposure to such anticancer drugs.

In the example shown in Figure 2, a tumor is found after the cell growth rate has slowed. Most of the cancer cells are removed by surgery. The remaining cancer cells begin to proliferate rapidly and cancer chemotherapy is started. Many tumor cells are killed by the chemotherapy, but eventually some cancer cells that are resistant to the chemotherapy drug begin to grow rapidly. The chemotherapy is no longer useful and is discontinued.

The growth of children

Children who fall significantly below the normal range of growth curves for body height http://www.cdc.gov/nchs/data/nhanes/growthcharts/set1clinical/cj41c021.pdf can be tested for growth hormone deficiency

Growth hormone deficiency

Growth hormone deficiency is a medical condition in which the body does not produce enough growth hormone . Growth hormone, also called somatotropin, is a polypeptide hormone which stimulates growth and cell reproduction...

 and might be treatable with hormone injections http://www.nlm.nih.gov/medlineplus/ency/article/001176.htm.

Exponential growth

Some growth curves for certain biological systems display periods of exponential growth

Exponential growth

Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value...

. Typically, periods of exponential growth are of limited duration due to depletion of some rate-limiting resource.