Plateau principle
Encyclopedia
The plateau principle is a mathematical model
or scientific law
originally developed to explain the time course of drug action The principle has wide applicability in pharmacology, physiology, nutrition, biochemistry and system dynamics. It applies whenever a drug or nutrient is infused or ingested at a relatively constant rate and when a constant fraction is eliminated during each time interval. Under these conditions, any change in the rate of infusion leads to an exponential increase or decrease until a new level is achieved. This behavior is also called an approach to steady state
because rather than causing an indefinite increase or decrease, a natural balance is achieved when the rate of infusion or production is balanced by the rate of loss.
An especially important use of the plateau principle is to study the renewal of tissue constituents in the human and animal body. In adults, daily synthesis of tissue constituents is nearly constant, and most constituents are removed with a first order reaction rate
. Applicability of the plateau principle was recognized during radiotracer studies of protein turnover in the 1940s by Rudolph Schoenheimer
and David Rittenberg
. Unlike the case with drugs, the initial amount of tissue or tissue protein is not zero because daily synthesis offsets daily elimination. In this case, the model is also said to approach a steady state
with exponential
or logarithmic
kinetics. Constiutents that change in this manner are said to have a biological half-life
.
A practical application of the plateau principle is that most people have experienced "plateauing" during regimens for weight management or training for sports. After a few weeks of progress, one seems unable to continue gaining in ability or losing weight. This outcome results from the same underlying quantitative model. This entry will describe the popular concepts as well as development of the plateau principle as a scientific, mathematical model.
In the sciences, the broadest application of the plateau principle is creating realistic time signatures for change in kinetic models (see Mathematical model
). One example of this principle is the long time required to effectively change human body composition. Theoretical studies have shown that many months of consistent physical training and food restriction are needed to bring about permanent weight stability in people who were previously Overweight
.
with first order kinetics. For this reason, when a drug is introduced into the body at a constant rate by intravenous therapy
, it approaches a new steady concentration in the blood at a rate defined by its half-life
. Similarly, when the intravenous infusion is ended, the drug concentration decreases exponentially and reaches an undetectable level after 5-6 half-lives have passed. If the same drug is administered as a bolus (medicine)
with a single injection, peak concentration is achieved almost immediately and then the concentration declines exponentially.
Most drugs are taken by mouth. In this case, the assumption of constant infusion is only approximated as doses are repeated over the course of several days. The plateau principle still applies but more complex models are required to account for the route of administration
.
and in scientific literature.,
The relationship between the elimination rate constant and half-life is given by the following equation:
Because ln 2 equals 0.693, the half-life is readily calculated from the elimination rate constant. Half-life has units of time, and the elimination rate constant has units of 1/time, e.g., per hour or per day.
An equation can be used to forecast the concentration of a compound at any future time when the fractional degration rate and steady state concentration are known:
The exponential function in parentheses corresponds to the fraction of total change that has been achieved as time passes and the difference between Css and C0 equals the total amount of change. Finally, at steady state, the concentration is expected to equal the rate of synthesis, production or infusion divided by the first order elimination constant.
Although these equations were derived to assist with predicting the time course of drug action, the same equation can be used for any substance or quantity that is being produced at a measurable rate and degraded with first-order kinetics. Because the equation applies in many instances of mass balance
, it has very broad applicability in addition to pharmacokinetics
. The most important inference derived from the steady state equation and the equation for fractional change over time is that the elimination rate constant (ke) or the sum of rate constants that apply in a model determine the time course for change in mass when a system is perturbed (either by changing the rate of inflow or production, or by changing the elimination rate(s)).
of differences between observed data and values predicted based on initial estimates of the rate constant and steady state value. This can be done using any software package that contains a curve fitting
routine. An example of this methodology implemented with spreadsheet software has been reported. The same article reports a method that requires only 3 equally spaced data points to obtain estimates for kinetic parameters. Spreadsheets that compare these methods are available. Three point method explained
In general, no essential nutrient
is produced in the body. Nutrient kinetics therefore follow the plateau principle with the distinction that most are ingested by mouth and the body must contain an amount adequate for health. The plateau principle is important in determining how much time is needed to produce a deficiency when intake is insufficient. Because of this, pharmacokinetic considerations should be part of the information needed to set a dietary reference intake
for essential nutrients.
or ascorbic acid
as a function of dose attains a plateau with a half-life of about 2 weeks. Bioavailability
of vitamin C is highest at dosages below 200 mg per day. Above 500 mg, nearly all of excess vitamin C is excreted through urine.
can be formed in the skin by ultraviolet irradiation or obtained from the diet. Once hydroxylated, the vitamin has a half-life of about 2 months.
Various studies have suggested that current intakes are inadequate for optimum bone health and much current research is aimed at determining recommendations for obtaining adequate circulating vitamin D3 and calcium while also minimizing potential toxicity.
and spice
as well as in the skins and seeds of apples, onions and grapes.
Investigations into healthful benefits of phytochemicals follow exactly the same principles of pharmacokinetics that are required to study drug therapy. The initial concentration of any non-nutritive phytochemical in the blood plasma is zero unless a person has recently ingested a food or beverage. For example, as increasing amounts of green tea extract are consumed, a graded increase in plasma catechin can be measured, and the major compound is eliminated with a half-life of about 5 hours. Other considerations that must be evaluated include whether the ingested compound interacts favorably or unfavorably with other nutrients or drugs, and whether there is evidence for a threshold or toxicity
at higher levels of intake.
is required to maintain the resting metabolic rate, which makes the initial regimen less effective. The idea of weight plateaus has been discussed for subjects who are participating in a calorie restriction experiment Food energy is expended largely through work done against gravity (see Joule
), so weight reduction lessens the effectiveness of a given workout. In addition, a trained person has greater skill and therefore greater efficiency during a workout. Remedies include increasing the workout intensity or length and reducing portion sizes of meals more than may have been done initially.
The fact that weight loss and dieting reduce the metabolic rate is supported by research. In one study, heat production was reduced 30% in obese men after a weight loss program, and this led to resistance to further lose body weight. Whether body mass increases or decreases, adjustments in the thermic effect of food
, resting energy expenditure, and non-resting energy expenditure all oppose further change.
is in balance between the amount of muscle synthesized or renewed each day and the amount that is degraded. Muscle fibers respond to repetition and load, and increased training causes the quantity of exercised muscle fiber to increase exponentially (simply meaning that the greatest gains are seen during the first weeks of training). Successful training produces hypertrophy
of muscle fibers as an adaptation to the training regimen. In order to make further gains, greater workout intensity is required with heavier loads and more repetitions, although improvement in skill can contribute to gains in ability.
When a bodily constituent adjusts exponentially over time, it usually attains a new stable level as a result of the plateau principle. The new level may be higher than the initial level (hypertrophy
) in the case of strength training or lower in the case of dieting or disuse atrophy
. This adjustment contributes to homeostasis
but does not require feedback
regulation. Gradual, asymptotic approach to a new balance between synthesis and degradation produces a stable level. Because of this, the plateau principle is sometimes called the stability principle. Mathematically, the result is linear
dynamics despite the fact that most biological processes are non-linear (see Nonlinear system) if considered over a very broad range of inputs.
by Ancel Keys
and others demonstrate that during food restriction, total body mass, fat mass and lean body mass follow an exponential approach to a new steady state. The observation that body mass changes exponentially during partial or complete starvation seems to be a general feature of adaptation to energy restriction.
and enzyme
s. One of the key methods of cellular regulation is to change the rate of transcription
of messenger RNA
, which gives rise to a change in the rate of synthesis for the protein that the messenger RNA encodes. The plateau principle explains why the concentration of different enzymes increases at unique rates in response to a single hormone
. Because each enzyme is degraded with at a unique rate (each has a different half-life
), the rate of change differs even when the same stimulus is applied. This principle has been demonstrated for the response of liver enzymes that degrade amino acids to cortisone
, which is a catabolic hormone.
The method of approach to steady state has also been used to analyze the change in messenger RNA levels when synthesis or degradation changes, and a model has also been reported in which the plateau principle is used to connect the change in messenger RNA synthesis to the expected change in protein synthesis and concentration as a function of time.
, which may include elevated fasting blood sugar
(or glucose
), resistance to the action of insulin
, elevated Low-density lipoprotein (LDL cholesterol) or decreased High-density lipoprotein (HDL cholesterol), and elevated blood pressure
. Although obesity
by itself is not considered a disease, it increases the risk for Diabetes mellitus type II. Because body mass, fat mass and fat free mass all change exponentially during weight reduction, it is a reasonable hypothesis to expect that symptoms of metabolic syndrome will also adjust exponentially towards normal values.
, and are then distributed throughout the bodily tissues. In such studies, a Multi-compartment model
is required to analyze turnover by Isotopic labeling
. The isotopic marker is called a tracer and the material being analyzed is the tracee.
In studies with humans, blood plasma is the only tissue that can be easily sampled. A common procedure is to analyze the dynamics by assuming that changes can be attributed to a sum of exponentials. A single mathematical compartment is usually assumed to follow first-order kinetics in accord with the plateau principle. There are many examples of this kind of analysis in nutrition, for example, in the study of metabolism of zinc, and carotenoids.
The commonest assumption in compartmental modeling is that material in a homogeneous compartment behaves exponentially. However, this assumption is sometimes modified to include a saturable response that follows Michaelis-Menten kinetics
or a related model called a Hill equation. When the material in question is present at a concentration near the KM, it often behaves with pseudo first-order kinetics (see Rate equation
) and the plateau principle applies despite the fact that the model is non-linear.
originated as a simple method of developing mathematical models by Jay Wright Forrester
and colleagues. System dynamics represents a compartment or pool as a stock and movement among compartments as flows. In general, the rate of flow depends on the amount of material in the stock to which it is connected. It is common to represent this dependence as a constant proportion (or first order) using a connector element in the model.
System dynamics is one application of the field of Control theory
. In the biomedical field, one of the strongest advocates for computer-based analysis of physiological problems was Dr. Arthur Guyton
. For example, system dynamics has been used to analyze the problem of body weight regulation. Similar methods have been used to study the spread of epidemics (see Compartmental models in epidemiology
).
Software that solves systems of equations required for compartmental modeling and system dynamics makes use of Finite difference
methods to represent a set of Ordinary differential equation
s. An expert appraisal of the different types of dynamic behavior that can be developed by application of the plateau principle to the field of system dynamics has been published.
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...
or scientific law
Scientific law
A scientific law is a statement that explains what something does in science just like Newton's law of universal gravitation. A scientific law must always apply under the same conditions, and implies a causal relationship between its elements. The law must be confirmed and broadly agreed upon...
originally developed to explain the time course of drug action The principle has wide applicability in pharmacology, physiology, nutrition, biochemistry and system dynamics. It applies whenever a drug or nutrient is infused or ingested at a relatively constant rate and when a constant fraction is eliminated during each time interval. Under these conditions, any change in the rate of infusion leads to an exponential increase or decrease until a new level is achieved. This behavior is also called an approach to steady state
Steady state
A system in a steady state has numerous properties that are unchanging in time. This implies that for any property p of the system, the partial derivative with respect to time is zero:...
because rather than causing an indefinite increase or decrease, a natural balance is achieved when the rate of infusion or production is balanced by the rate of loss.
An especially important use of the plateau principle is to study the renewal of tissue constituents in the human and animal body. In adults, daily synthesis of tissue constituents is nearly constant, and most constituents are removed with a first order reaction rate
Reaction rate
The reaction rate or speed of reaction for a reactant or product in a particular reaction is intuitively defined as how fast or slow a reaction takes place...
. Applicability of the plateau principle was recognized during radiotracer studies of protein turnover in the 1940s by Rudolph Schoenheimer
Rudolph Schoenheimer
-Bibliography:*[Anon.] "Schoenheimer, Rudolf", Encyclopaedia Britannica, Deluxe CDROM edition...
and David Rittenberg
David Rittenberg
David Rittenberg was a U.S. biochemist who pioneered the radioactive tagging of molecules, enabling detailed studies of metabolism...
. Unlike the case with drugs, the initial amount of tissue or tissue protein is not zero because daily synthesis offsets daily elimination. In this case, the model is also said to approach a steady state
Steady state
A system in a steady state has numerous properties that are unchanging in time. This implies that for any property p of the system, the partial derivative with respect to time is zero:...
with exponential
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
or logarithmic
Logarithmic
Logarithmic can refer to:* Logarithm, a transcendental function in mathematics* Logarithmic scale, the use of the logarithmic function to describe measurements* Logarithmic growth* Logarithmic distribution, a discrete probability distribution...
kinetics. Constiutents that change in this manner are said to have a biological half-life
Biological half-life
The biological half-life or elimination half-life of a substance is the time it takes for a substance to lose half of its pharmacologic, physiologic, or radiologic activity, as per the MeSH definition...
.
A practical application of the plateau principle is that most people have experienced "plateauing" during regimens for weight management or training for sports. After a few weeks of progress, one seems unable to continue gaining in ability or losing weight. This outcome results from the same underlying quantitative model. This entry will describe the popular concepts as well as development of the plateau principle as a scientific, mathematical model.
In the sciences, the broadest application of the plateau principle is creating realistic time signatures for change in kinetic models (see Mathematical model
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...
). One example of this principle is the long time required to effectively change human body composition. Theoretical studies have shown that many months of consistent physical training and food restriction are needed to bring about permanent weight stability in people who were previously Overweight
Overweight
Overweight is generally defined as having more body fat than is optimally healthy. Being overweight is a common condition, especially where food supplies are plentiful and lifestyles are sedentary...
.
The plateau principle in pharmacokinetics
Most drugs are eliminated from the blood plasmaBlood plasma
Blood plasma is the straw-colored liquid component of blood in which the blood cells in whole blood are normally suspended. It makes up about 55% of the total blood volume. It is the intravascular fluid part of extracellular fluid...
with first order kinetics. For this reason, when a drug is introduced into the body at a constant rate by intravenous therapy
Intravenous therapy
Intravenous therapy or IV therapy is the infusion of liquid substances directly into a vein. The word intravenous simply means "within a vein". Therapies administered intravenously are often called specialty pharmaceuticals...
, it approaches a new steady concentration in the blood at a rate defined by its half-life
Half-life
Half-life, abbreviated t½, is the period of time it takes for the amount of a substance undergoing decay to decrease by half. The name was originally used to describe a characteristic of unstable atoms , but it may apply to any quantity which follows a set-rate decay.The original term, dating to...
. Similarly, when the intravenous infusion is ended, the drug concentration decreases exponentially and reaches an undetectable level after 5-6 half-lives have passed. If the same drug is administered as a bolus (medicine)
Bolus (medicine)
In medicine, a bolus is the administration of a medication, drug or other compound that is given to raise its concentration in blood to an effective level...
with a single injection, peak concentration is achieved almost immediately and then the concentration declines exponentially.
Most drugs are taken by mouth. In this case, the assumption of constant infusion is only approximated as doses are repeated over the course of several days. The plateau principle still applies but more complex models are required to account for the route of administration
Route of administration
A route of administration in pharmacology and toxicology is the path by which a drug, fluid, poison, or other substance is taken into the body.-Classification:Routes of administration are usually classified by application location...
.
Equations for the approach to steady state
Derivation of equations that describe the time course of change for a system with Zero order input and first-order elimination are presented in Exponential decay and Biological half-lifeBiological half-life
The biological half-life or elimination half-life of a substance is the time it takes for a substance to lose half of its pharmacologic, physiologic, or radiologic activity, as per the MeSH definition...
and in scientific literature.,
- Ct is concentration after time t
- C0 is the initial concentration (t=0)
- ke is the elimination rate constant
The relationship between the elimination rate constant and half-life is given by the following equation:
Because ln 2 equals 0.693, the half-life is readily calculated from the elimination rate constant. Half-life has units of time, and the elimination rate constant has units of 1/time, e.g., per hour or per day.
An equation can be used to forecast the concentration of a compound at any future time when the fractional degration rate and steady state concentration are known:
- Css is concentration after the steady state has been achieved.
The exponential function in parentheses corresponds to the fraction of total change that has been achieved as time passes and the difference between Css and C0 equals the total amount of change. Finally, at steady state, the concentration is expected to equal the rate of synthesis, production or infusion divided by the first order elimination constant.
- ks is the rate of synthesis or infusion
Although these equations were derived to assist with predicting the time course of drug action, the same equation can be used for any substance or quantity that is being produced at a measurable rate and degraded with first-order kinetics. Because the equation applies in many instances of mass balance
Mass balance
A mass balance is an application of conservation of mass to the analysis of physical systems. By accounting for material entering and leaving a system, mass flows can be identified which might have been unknown, or difficult to measure without this technique...
, it has very broad applicability in addition to pharmacokinetics
Pharmacokinetics
Pharmacokinetics, sometimes abbreviated as PK, is a branch of pharmacology dedicated to the determination of the fate of substances administered externally to a living organism...
. The most important inference derived from the steady state equation and the equation for fractional change over time is that the elimination rate constant (ke) or the sum of rate constants that apply in a model determine the time course for change in mass when a system is perturbed (either by changing the rate of inflow or production, or by changing the elimination rate(s)).
Estimating values for kinetic rate parameters
When experimental data are available, the normal procedure for estimating rate parameters such as ke and Css is to minimize the sum of squaresSum of squares
The partition of sums of squares is a concept that permeates much of inferential statistics and descriptive statistics. More properly, it is the partitioning of sums of squared deviations or errors. Mathematically, the sum of squared deviations is an unscaled, or unadjusted measure of dispersion...
of differences between observed data and values predicted based on initial estimates of the rate constant and steady state value. This can be done using any software package that contains a curve fitting
Curve fitting
Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a "smooth" function...
routine. An example of this methodology implemented with spreadsheet software has been reported. The same article reports a method that requires only 3 equally spaced data points to obtain estimates for kinetic parameters. Spreadsheets that compare these methods are available. Three point method explained
The plateau principle in nutrition
Dr. Wilbur O. Atwater, who developed the first database of food composition in the United States, recognized that the response to excessive or insufficient nutrient intake included an adjustment in efficiency that would result in a plateau. He observed: "It has been found by numerous experiments that when the nutrients are fed in large excess, the body may continue for a time to store away part of the extra material, but after it has accumulated a certain amount, it refuses to take on more, and the daily consumption equals the supply even when this involves great waste."In general, no essential nutrient
Essential nutrient
An essential nutrient is a nutrient required for normal body functioning that either cannot be synthesized by the body at all, or cannot be synthesized in amounts adequate for good health , and thus must be obtained from a dietary source...
is produced in the body. Nutrient kinetics therefore follow the plateau principle with the distinction that most are ingested by mouth and the body must contain an amount adequate for health. The plateau principle is important in determining how much time is needed to produce a deficiency when intake is insufficient. Because of this, pharmacokinetic considerations should be part of the information needed to set a dietary reference intake
Dietary Reference Intake
The Dietary Reference Intake is a system of nutrition recommendations from the Institute of Medicine of the U.S. National Academy of Sciences. The DRI system is used by both the United States and Canada and is intended for the general public and health professionals...
for essential nutrients.
Vitamin C
The blood plasma concentration of vitamin CVitamin C
Vitamin C or L-ascorbic acid or L-ascorbate is an essential nutrient for humans and certain other animal species. In living organisms ascorbate acts as an antioxidant by protecting the body against oxidative stress...
or ascorbic acid
Ascorbic acid
Ascorbic acid is a naturally occurring organic compound with antioxidant properties. It is a white solid, but impure samples can appear yellowish. It dissolves well in water to give mildly acidic solutions. Ascorbic acid is one form of vitamin C. The name is derived from a- and scorbutus , the...
as a function of dose attains a plateau with a half-life of about 2 weeks. Bioavailability
Bioavailability
In pharmacology, bioavailability is a subcategory of absorption and is used to describe the fraction of an administered dose of unchanged drug that reaches the systemic circulation, one of the principal pharmacokinetic properties of drugs. By definition, when a medication is administered...
of vitamin C is highest at dosages below 200 mg per day. Above 500 mg, nearly all of excess vitamin C is excreted through urine.
Vitamin D
Vitamin D metabolism is complex because the provitaminProvitamin
A provitamin is a substance that may be converted within the body to a vitamin.The term "previtamin" is a synonym.For example, Provitamin B5 is a name for Panthenol, which may be converted in the body to Pantothenic Acid ....
can be formed in the skin by ultraviolet irradiation or obtained from the diet. Once hydroxylated, the vitamin has a half-life of about 2 months.
Various studies have suggested that current intakes are inadequate for optimum bone health and much current research is aimed at determining recommendations for obtaining adequate circulating vitamin D3 and calcium while also minimizing potential toxicity.
Phytochemicals in foods and beverages
Many healthful qualities of foods and beverages may be related to the content of phytochemicals (see List of phytochemicals in food). Prime examples are flavonoids found in green tea, berries, cocoaCocoa solids
Cocoa solids are the low-fat component of chocolate. When sold as an end product, it may also be called cocoa powder, cocoa, and cacao....
and spice
Spice
A spice is a dried seed, fruit, root, bark, or vegetative substance used in nutritionally insignificant quantities as a food additive for flavor, color, or as a preservative that kills harmful bacteria or prevents their growth. It may be used to flavour a dish or to hide other flavours...
as well as in the skins and seeds of apples, onions and grapes.
Investigations into healthful benefits of phytochemicals follow exactly the same principles of pharmacokinetics that are required to study drug therapy. The initial concentration of any non-nutritive phytochemical in the blood plasma is zero unless a person has recently ingested a food or beverage. For example, as increasing amounts of green tea extract are consumed, a graded increase in plasma catechin can be measured, and the major compound is eliminated with a half-life of about 5 hours. Other considerations that must be evaluated include whether the ingested compound interacts favorably or unfavorably with other nutrients or drugs, and whether there is evidence for a threshold or toxicity
Toxicity
Toxicity is the degree to which a substance can damage a living or non-living organisms. Toxicity can refer to the effect on a whole organism, such as an animal, bacterium, or plant, as well as the effect on a substructure of the organism, such as a cell or an organ , such as the liver...
at higher levels of intake.
Plateaus during dieting and weight loss
It is especially common for people who are trying to lose weight to experience plateaus after several weeks of successful weight reduction. The plateau principle suggests that this leveling off is a sign of success. Basically, as one loses weight, less food energyFood energy
Food energy is the amount of energy obtained from food that is available through cellular respiration.Food energy is expressed in food calories or kilojoules...
is required to maintain the resting metabolic rate, which makes the initial regimen less effective. The idea of weight plateaus has been discussed for subjects who are participating in a calorie restriction experiment Food energy is expended largely through work done against gravity (see Joule
Joule
The joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...
), so weight reduction lessens the effectiveness of a given workout. In addition, a trained person has greater skill and therefore greater efficiency during a workout. Remedies include increasing the workout intensity or length and reducing portion sizes of meals more than may have been done initially.
The fact that weight loss and dieting reduce the metabolic rate is supported by research. In one study, heat production was reduced 30% in obese men after a weight loss program, and this led to resistance to further lose body weight. Whether body mass increases or decreases, adjustments in the thermic effect of food
Thermic effect of food
Thermic effect of food , or TEF in shorthand, is the increment in energy expenditure above resting metabolic rate due to the cost of processing food for storage and use. It is one of the components of metabolism along with the resting metabolic rate, and the exercise component...
, resting energy expenditure, and non-resting energy expenditure all oppose further change.
Plateaus during strength training
Any athlete who has trained for a sport has probably experienced plateaus, and this has given rise to various strategies to continue improving. Voluntary Skeletal muscleSkeletal muscle
Skeletal muscle is a form of striated muscle tissue existing under control of the somatic nervous system- i.e. it is voluntarily controlled. It is one of three major muscle types, the others being cardiac and smooth muscle...
is in balance between the amount of muscle synthesized or renewed each day and the amount that is degraded. Muscle fibers respond to repetition and load, and increased training causes the quantity of exercised muscle fiber to increase exponentially (simply meaning that the greatest gains are seen during the first weeks of training). Successful training produces hypertrophy
Hypertrophy
Hypertrophy is the increase in the volume of an organ or tissue due to the enlargement of its component cells. It should be distinguished from hyperplasia, in which the cells remain approximately the same size but increase in number...
of muscle fibers as an adaptation to the training regimen. In order to make further gains, greater workout intensity is required with heavier loads and more repetitions, although improvement in skill can contribute to gains in ability.
When a bodily constituent adjusts exponentially over time, it usually attains a new stable level as a result of the plateau principle. The new level may be higher than the initial level (hypertrophy
Hypertrophy
Hypertrophy is the increase in the volume of an organ or tissue due to the enlargement of its component cells. It should be distinguished from hyperplasia, in which the cells remain approximately the same size but increase in number...
) in the case of strength training or lower in the case of dieting or disuse atrophy
Atrophy
Atrophy is the partial or complete wasting away of a part of the body. Causes of atrophy include mutations , poor nourishment, poor circulation, loss of hormonal support, loss of nerve supply to the target organ, disuse or lack of exercise or disease intrinsic to the tissue itself...
. This adjustment contributes to homeostasis
Homeostasis
Homeostasis is the property of a system that regulates its internal environment and tends to maintain a stable, constant condition of properties like temperature or pH...
but does not require feedback
Feedback
Feedback describes the situation when output from an event or phenomenon in the past will influence an occurrence or occurrences of the same Feedback describes the situation when output from (or information about the result of) an event or phenomenon in the past will influence an occurrence or...
regulation. Gradual, asymptotic approach to a new balance between synthesis and degradation produces a stable level. Because of this, the plateau principle is sometimes called the stability principle. Mathematically, the result is linear
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
dynamics despite the fact that most biological processes are non-linear (see Nonlinear system) if considered over a very broad range of inputs.
Changes in body composition when food is restricted
Data from the Minnesota Starvation ExperimentMinnesota Starvation Experiment
The Minnesota Starvation Experiment, also known as the Minnesota Semi-Starvation Experiment, the Minnesota Starvation-Recovery Experiment and the Starvation Study, was a clinical study performed at the University of Minnesota between November 19, 1944 and December 20, 1945...
by Ancel Keys
Ancel Keys
Ancel Benjamin Keys was an American scientist who studied the influence of diet on health. In particular, he hypothesized that different kinds of dietary fat had different effects on health....
and others demonstrate that during food restriction, total body mass, fat mass and lean body mass follow an exponential approach to a new steady state. The observation that body mass changes exponentially during partial or complete starvation seems to be a general feature of adaptation to energy restriction.
The plateau principle in biochemistry
Each cell produces thousands of different kinds of proteinProtein
Proteins are biochemical compounds consisting of one or more polypeptides typically folded into a globular or fibrous form, facilitating a biological function. A polypeptide is a single linear polymer chain of amino acids bonded together by peptide bonds between the carboxyl and amino groups of...
and enzyme
Enzyme
Enzymes are proteins that catalyze chemical reactions. In enzymatic reactions, the molecules at the beginning of the process, called substrates, are converted into different molecules, called products. Almost all chemical reactions in a biological cell need enzymes in order to occur at rates...
s. One of the key methods of cellular regulation is to change the rate of transcription
Transcription (genetics)
Transcription is the process of creating a complementary RNA copy of a sequence of DNA. Both RNA and DNA are nucleic acids, which use base pairs of nucleotides as a complementary language that can be converted back and forth from DNA to RNA by the action of the correct enzymes...
of messenger RNA
Messenger RNA
Messenger RNA is a molecule of RNA encoding a chemical "blueprint" for a protein product. mRNA is transcribed from a DNA template, and carries coding information to the sites of protein synthesis: the ribosomes. Here, the nucleic acid polymer is translated into a polymer of amino acids: a protein...
, which gives rise to a change in the rate of synthesis for the protein that the messenger RNA encodes. The plateau principle explains why the concentration of different enzymes increases at unique rates in response to a single hormone
Hormone
A hormone is a chemical released by a cell or a gland in one part of the body that sends out messages that affect cells in other parts of the organism. Only a small amount of hormone is required to alter cell metabolism. In essence, it is a chemical messenger that transports a signal from one...
. Because each enzyme is degraded with at a unique rate (each has a different half-life
Half-life
Half-life, abbreviated t½, is the period of time it takes for the amount of a substance undergoing decay to decrease by half. The name was originally used to describe a characteristic of unstable atoms , but it may apply to any quantity which follows a set-rate decay.The original term, dating to...
), the rate of change differs even when the same stimulus is applied. This principle has been demonstrated for the response of liver enzymes that degrade amino acids to cortisone
Cortisone
Cortisone is a steroid hormone. It is one of the main hormones released by the adrenal gland in response to stress. In chemical structure, it is a corticosteroid closely related to corticosterone. It is used to treat a variety of ailments and can be administered intravenously, orally,...
, which is a catabolic hormone.
The method of approach to steady state has also been used to analyze the change in messenger RNA levels when synthesis or degradation changes, and a model has also been reported in which the plateau principle is used to connect the change in messenger RNA synthesis to the expected change in protein synthesis and concentration as a function of time.
The plateau principle in physiology
Excessive gain in body weight contributes to the metabolic syndromeMetabolic syndrome
Metabolic syndrome is a combination of medical disorders that, when occurring together, increase the risk of developing cardiovascular disease and diabetes. It affects one in five people in the United States and prevalence increases with age...
, which may include elevated fasting blood sugar
Blood sugar
The blood sugar concentration or blood glucose level is the amount of glucose present in the blood of a human or animal. Normally in mammals, the body maintains the blood glucose level at a reference range between about 3.6 and 5.8 mM , or 64.8 and 104.4 mg/dL...
(or glucose
Glucose
Glucose is a simple sugar and an important carbohydrate in biology. Cells use it as the primary source of energy and a metabolic intermediate...
), resistance to the action of insulin
Insulin
Insulin is a hormone central to regulating carbohydrate and fat metabolism in the body. Insulin causes cells in the liver, muscle, and fat tissue to take up glucose from the blood, storing it as glycogen in the liver and muscle....
, elevated Low-density lipoprotein (LDL cholesterol) or decreased High-density lipoprotein (HDL cholesterol), and elevated blood pressure
Blood pressure
Blood pressure is the pressure exerted by circulating blood upon the walls of blood vessels, and is one of the principal vital signs. When used without further specification, "blood pressure" usually refers to the arterial pressure of the systemic circulation. During each heartbeat, BP varies...
. Although obesity
Obesity
Obesity is a medical condition in which excess body fat has accumulated to the extent that it may have an adverse effect on health, leading to reduced life expectancy and/or increased health problems...
by itself is not considered a disease, it increases the risk for Diabetes mellitus type II. Because body mass, fat mass and fat free mass all change exponentially during weight reduction, it is a reasonable hypothesis to expect that symptoms of metabolic syndrome will also adjust exponentially towards normal values.
The plateau principle in compartmental modeling
Scientists have evaluated turnover of bodily constituents using radiotracer methods and stable isotopes. If given orally, the tracers are absorbed and move into the blood plasmaBlood plasma
Blood plasma is the straw-colored liquid component of blood in which the blood cells in whole blood are normally suspended. It makes up about 55% of the total blood volume. It is the intravascular fluid part of extracellular fluid...
, and are then distributed throughout the bodily tissues. In such studies, a Multi-compartment model
Multi-compartment model
A multi-compartment model is a type of mathematical model used for describing the way materials or energies are transmitted among the compartments of a system. Each compartment is assumed to be a homogenous entity within which the entities being modelled are equivalent...
is required to analyze turnover by Isotopic labeling
Isotopic labeling
Isotopic labeling is a technique for tracking the passage of a sample of substance through a system. The substance is 'labeled' by including unusual isotopes in its chemical composition...
. The isotopic marker is called a tracer and the material being analyzed is the tracee.
In studies with humans, blood plasma is the only tissue that can be easily sampled. A common procedure is to analyze the dynamics by assuming that changes can be attributed to a sum of exponentials. A single mathematical compartment is usually assumed to follow first-order kinetics in accord with the plateau principle. There are many examples of this kind of analysis in nutrition, for example, in the study of metabolism of zinc, and carotenoids.
The commonest assumption in compartmental modeling is that material in a homogeneous compartment behaves exponentially. However, this assumption is sometimes modified to include a saturable response that follows Michaelis-Menten kinetics
Michaelis-Menten kinetics
In biochemistry, Michaelis–Menten kinetics is one of the simplest and best-known models of enzyme kinetics. It is named after German biochemist Leonor Michaelis and Canadian physician Maud Menten. The model takes the form of an equation describing the rate of enzymatic reactions, by relating...
or a related model called a Hill equation. When the material in question is present at a concentration near the KM, it often behaves with pseudo first-order kinetics (see Rate equation
Rate equation
The rate law or rate equation for a chemical reaction is an equation that links the reaction rate with concentrations or pressures of reactants and constant parameters . To determine the rate equation for a particular system one combines the reaction rate with a mass balance for the system...
) and the plateau principle applies despite the fact that the model is non-linear.
The plateau principle in system dynamics
Compartmental modeling in biomedical sciences primarily originated from the need to study metabolism by using tracers. In contrast, System dynamicsSystem dynamics
System dynamics is an approach to understanding the behaviour of complex systems over time. It deals with internal feedback loops and time delays that affect the behaviour of the entire system. What makes using system dynamics different from other approaches to studying complex systems is the use...
originated as a simple method of developing mathematical models by Jay Wright Forrester
Jay Wright Forrester
Jay Wright Forrester is a pioneer American computer engineer, systems scientist and was a professor at the MIT Sloan School of Management. Forrester is known as the founder of System Dynamics, which deals with the simulation of interactions between objects in dynamic systems.- Biography :Forrester...
and colleagues. System dynamics represents a compartment or pool as a stock and movement among compartments as flows. In general, the rate of flow depends on the amount of material in the stock to which it is connected. It is common to represent this dependence as a constant proportion (or first order) using a connector element in the model.
System dynamics is one application of the field of Control theory
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
. In the biomedical field, one of the strongest advocates for computer-based analysis of physiological problems was Dr. Arthur Guyton
Arthur Guyton
Arthur Clifton Guyton was an American physiologist. He was born in Oxford, Mississippi, to Dr. Billy S. Guyton, a highly respected eye, ear, nose, and throat specialist, who later became Dean of the University of Mississippi Medical School, and Kate Smallwood Guyton, a mathematics and physics...
. For example, system dynamics has been used to analyze the problem of body weight regulation. Similar methods have been used to study the spread of epidemics (see Compartmental models in epidemiology
Compartmental models in epidemiology
In order to model the progress of an epidemic in a large population, comprising many different individuals in various fields, the population diversity must be reduced to a few key characteristics which are relevant to the infection under consideration...
).
Software that solves systems of equations required for compartmental modeling and system dynamics makes use of Finite difference
Finite difference
A finite difference is a mathematical expression of the form f − f. If a finite difference is divided by b − a, one gets a difference quotient...
methods to represent a set of Ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
s. An expert appraisal of the different types of dynamic behavior that can be developed by application of the plateau principle to the field of system dynamics has been published.