Dimension

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Dimension 

From left to right: the square, the cube and the tesseract. The two-dimensional (2D) square is bounded by one-dimensional (1D) lines; the three-dimensional (3D) cube by two-dimensional areas; and the four-dimensional (4D) tesseract by three-dimensional volumes. For display on a two-dimensional surface such as a screen, the 3D cube and 4D tesseract require projection.

 

The first four spatial dimensions, represented in a two-dimensional picture.

1. Two points can be connected to create a line segment.
2. Two parallel line segments can be connected to form a square.
3. Two parallel squares can be connected to form a cube.
4. Two parallel cubes can be connected to form a tesseract.

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two (2D) because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces.

In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism. The four dimensions (4D) of spacetime consist of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an observer. Minkowski space first approximates the universe without gravity; the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. 10 dimensions are used to describe superstring theory (6D hyperspace + 4D), 11 dimensions can describe supergravity and M-theory (7D hyperspace + 4D), and the state-space of quantum mechanics is an infinite-dimensional function space.

The concept of dimension is not restricted to physical objects. High-dimensional spaces frequently occur in mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics; these are abstract spaces, independent of the physical space in which we live.

In mathematics

In mathematics, the dimension of an object is, roughly speaking, the number of degrees of freedom of a point that moves on this object. In other words, the dimension is the number of independent parameters or coordinates that are needed for defining the position of a point that is constrained to be on the object. For example, the dimension of a point is zero; the dimension of a line is one, as a point can move on a line in only one direction (or its opposite); the dimension of a plane is two, etc.

The dimension is an intrinsic property of an object, in the sense that it is independent of the dimension of the space in which the object is or can be embedded. For example, a curve, such as a circle, is of dimension one, because the position of a point on a curve is determined by its signed distance along the curve to a fixed point on the curve. This is independent from the fact that a curve cannot be embedded in a Euclidean space of dimension lower than two, unless it is a line.

The dimension of Euclidean n-space Eⁿ is n. When trying to generalize to other types of spaces, one is faced with the question "what makes Eⁿ n-dimensional?" One answer is that to cover a fixed ball in Eⁿ by small balls of radius ε, one needs on the order of ε⁻ⁿ such small balls. This observation leads to the definition of the Minkowski dimension and its more sophisticated variant, the Hausdorff dimension, but there are also other answers to that question. For example, the boundary of a ball in Eⁿ looks locally like E^n-1 and this leads to the notion of the inductive dimension. While these notions agree on Eⁿ, they turn out to be different when one looks at more general spaces.

A tesseract is an example of a four-dimensional object. Whereas outside mathematics the use of the term "dimension" is as in: "A tesseract has four dimensions", mathematicians usually express this as: "The tesseract has dimension 4", or: "The dimension of the tesseract is 4" or: 4D.

Although the notion of higher dimensions goes back to René Descartes, substantial development of a higher-dimensional geometry only began in the 19th century, via the work of Arthur Cayley, William Rowan Hamilton, Ludwig Schläfli and Bernhard Riemann. Riemann's 1854 Habilitationsschrift, Schläfli's 1852 Theorie der vielfachen Kontinuität, and Hamilton's discovery of the quaternions and John T. Graves' discovery of the octonions in 1843 marked the beginning of higher-dimensional geometry.

The rest of this section examines some of the more important mathematical definitions of dimension.

Vector spaces

Main article: Dimension (vector space)

The dimension of a vector space is the number of vectors in any basis for the space, i.e. the number of coordinates necessary to specify any vector. This notion of dimension (the cardinality of a basis) is often referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension.

For the non-free case, this generalizes to the notion of the length of a module.

Manifolds

The uniquely defined dimension of every connected topological manifold can be calculated. A connected topological manifold is locally homeomorphic to Euclidean n-space, in which the number n is the manifold's dimension.

For connected differentiable manifolds, the dimension is also the dimension of the tangent vector space at any point.

In geometric topology, the theory of manifolds is characterized by the way dimensions 1 and 2 are relatively elementary, the high-dimensional cases n > 4 are simplified by having extra space in which to "work"; and the cases n = 3 and 4 are in some senses the most difficult. This state of affairs was highly marked in the various cases of the Poincaré conjecture, in which four different proof methods are applied.

Complex dimension

The dimension of a manifold depends on the base field with respect to which Euclidean space is defined. While analysis usually assumes a manifold to be over the real numbers, it is sometimes useful in the study of complex manifolds and algebraic varieties to work over the complex numbers instead. A complex number (x + iy) has a real part x and an imaginary part y, in which x and y are both real numbers; hence, the complex dimension is half the real dimension.

Conversely, in algebraically unconstrained contexts, a single complex coordinate system may be applied to an object having two real dimensions. For example, an ordinary two-dimensional spherical surface, when given a complex metric, becomes a Riemann sphere of one complex dimension.

Varieties

Main article: Dimension of an algebraic variety

The dimension of an algebraic variety may be defined in various equivalent ways. The most intuitive way is probably the dimension of the tangent space at any Regular point of an algebraic variety. Another intuitive way is to define the dimension as the number of hyperplanes that are needed in order to have an intersection with the variety that is reduced to a finite number of points (dimension zero). This definition is based on the fact that the intersection of a variety with a hyperplane reduces the dimension by one unless if the hyperplane contains the variety.

An algebraic set being a finite union of algebraic varieties, its dimension is the maximum of the dimensions of its components. It is equal to the maximal length of the chains ${\displaystyle V_{0}\subsetneq V_{1}\subsetneq \cdots \subsetneq V_{d}}$ of sub-varieties of the given algebraic set (the length of such a chain is the number of "${\displaystyle \subsetneq }$").

Each variety can be considered as an algebraic stack, and its dimension as variety agrees with its dimension as stack. There are however many stacks which do not correspond to varieties, and some of these have negative dimension. Specifically, if V is a variety of dimension m and G is an algebraic group of dimension n acting on V, then the quotient stack [V/G] has dimension m − n.

Krull dimension

The Krull dimension of a commutative ring is the maximal length of chains of prime ideals in it, a chain of length n being a sequence ${\displaystyle {\mathcal {P}}_{0}\subsetneq {\mathcal {P}}_{1}\subsetneq \cdots \subsetneq {\mathcal {P}}_{n}}$ of prime ideals related by inclusion. It is strongly related to the dimension of an algebraic variety, because of the natural correspondence between sub-varieties and prime ideals of the ring of the polynomials on the variety.

For an algebra over a field, the dimension as vector space is finite if and only if its Krull dimension is 0.

Topological spaces

For any normal topological space X, the Lebesgue covering dimension of X is defined to be the smallest integer n for which the following holds: any open cover has an open refinement (a second open cover in which each element is a subset of an element in the first cover) such that no point is included in more than n + 1 elements. In this case dim X = n. For X a manifold, this coincides with the dimension mentioned above. If no such integer n exists, then the dimension of X is said to be infinite, and one writes dim X = ∞. Moreover, X has dimension −1, i.e. dim X = −1 if and only if X is empty. This definition of covering dimension can be extended from the class of normal spaces to all Tychonoff spaces merely by replacing the term "open" in the definition by the term "functionally open".

An inductive dimension may be defined inductively as follows. Consider a discrete set of points (such as a finite collection of points) to be 0-dimensional. By dragging a 0-dimensional object in some direction, one obtains a 1-dimensional object. By dragging a 1-dimensional object in a new direction, one obtains a 2-dimensional object. In general one obtains an (n + 1)-dimensional object by dragging an n-dimensional object in a new direction. The inductive dimension of a topological space may refer to the small inductive dimension or the large inductive dimension, and is based on the analogy that, in the case of metric spaces, (n + 1)-dimensional balls have n-dimensional boundaries, permitting an inductive definition based on the dimension of the boundaries of open sets. Moreover, the boundary of a discrete set of points is the empty set, and therefore the empty set can be taken to have dimension -1.

Similarly, for the class of CW complexes, the dimension of an object is the largest n for which the n-skeleton is nontrivial. Intuitively, this can be described as follows: if the original space can be continuously deformed into a collection of higher-dimensional triangles joined at their faces with a complicated surface, then the dimension of the object is the dimension of those triangles.

See also: dimension of a scheme

Hausdorff dimension

The Hausdorff dimension is useful for studying structurally complicated sets, especially fractals. The Hausdorff dimension is defined for all metric spaces and, unlike the dimensions considered above, can also have non-integer real values. The box dimension or Minkowski dimension is a variant of the same idea. In general, there exist more definitions of fractal dimensions that work for highly irregular sets and attain non-integer positive real values. Fractals have been found useful to describe many natural objects and phenomena.

Hilbert spaces

Every Hilbert space admits an orthonormal basis, and any two such bases for a particular space have the same cardinality. This cardinality is called the dimension of the Hilbert space. This dimension is finite if and only if the space's Hamel dimension is finite, and in this case the two dimensions coincide.

In physics

Spatial dimensions

Classical physics theories describe three physical dimensions: from a particular point in space, the basic directions in which we can move are up/down, left/right, and forward/backward. Movement in any other direction can be expressed in terms of just these three. Moving down is the same as moving up a negative distance. Moving diagonally upward and forward is just as the name of the direction implies; i.e., moving in a linear combination of up and forward. In its simplest form: a line describes one dimension, a plane describes two dimensions, and a cube describes three dimensions. (See Space and Cartesian coordinate system.)

Number of dimensions	Example co-ordinate systems
1	Number line Angle
2	Cartesian (two-dimensional) Polar Latitude and longitude
3	Cartesian (three-dimensional) Cylindrical Spherical

Time

A temporal dimension, or time dimension, is a dimension of time. Time is often referred to as the "fourth dimension" for this reason, but that is not to imply that it is a spatial dimension. A temporal dimension is one way to measure physical change. It is perceived differently from the three spatial dimensions in that there is only one of it, and that we cannot move freely in time but subjectively move in one direction.

The equations used in physics to model reality do not treat time in the same way that humans commonly perceive it. The equations of classical mechanics are symmetric with respect to time, and equations of quantum mechanics are typically symmetric if both time and other quantities (such as charge and parity) are reversed. In these models, the perception of time flowing in one direction is an artifact of the laws of thermodynamics (we perceive time as flowing in the direction of increasing entropy).

The best-known treatment of time as a dimension is Poincaré and Einstein's special relativity (and extended to general relativity), which treats perceived space and time as components of a four-dimensional manifold, known as spacetime, and in the special, flat case as Minkowski space. Time is different from other spatial dimensions as time operates in all spatial dimensions. Time operates in the first, second and third as well as theoretical spatial dimensions such as a fourth spatial dimension. Time is not however present in a single point of absolute infinite singularity as defined as a geometric point, as an infinitely small point can have no change and therefore no time. Just as when an object moves through positions in space, it also moves through positions in time. In this sense the force moving any object to change is time.

Additional dimensions

In physics, three dimensions of space and one of time is the accepted norm. However, there are theories that attempt to unify the four fundamental forces by introducing extra dimensions/hyperspace. Most notably, superstring theory requires 10 spacetime dimensions, and originates from a more fundamental 11-dimensional theory tentatively called M-theory which subsumes five previously distinct superstring theories. Supergravity theory also promotes 11D spacetime = 7D hyperspace + 4 common dimensions. To date, no direct experimental or observational evidence is available to support the existence of these extra dimensions. If hyperspace exists, it must be hidden from us by some physical mechanism. One well-studied possibility is that the extra dimensions may be "curled up" at such tiny scales as to be effectively invisible to current experiments. Limits on the size and other properties of extra dimensions are set by particle experiments such as those at the Large Hadron Collider.

In 1921, Kaluza–Klein theory presented 5D including an extra dimension of space. At the level of quantum field theory, Kaluza–Klein theory unifies gravity with gauge interactions, based on the realization that gravity propagating in small, compact extra dimensions is equivalent to gauge interactions at long distances. In particular when the geometry of the extra dimensions is trivial, it reproduces electromagnetism. However at sufficiently high energies or short distances, this setup still suffers from the same pathologies that famously obstruct direct attempts to describe quantum gravity. Therefore, these models still require a UV completion, of the kind that string theory is intended to provide. In particular, superstring theory requires six compact dimensions (6D hyperspace) forming a Calabi–Yau manifold. Thus Kaluza-Klein theory may be considered either as an incomplete description on its own, or as a subset of string theory model building.

In addition to small and curled up extra dimensions, there may be extra dimensions that instead aren't apparent because the matter associated with our visible universe is localized on a (3 + 1)-dimensional subspace. Thus the extra dimensions need not be small and compact but may be large extra dimensions. D-branes are dynamical extended objects of various dimensionalities predicted by string theory that could play this role. They have the property that open string excitations, which are associated with gauge interactions, are confined to the brane by their endpoints, whereas the closed strings that mediate the gravitational interaction are free to propagate into the whole spacetime, or "the bulk". This could be related to why gravity is exponentially weaker than the other forces, as it effectively dilutes itself as it propagates into a higher-dimensional volume.

Some aspects of brane physics have been applied to cosmology. For example, brane gas cosmology attempts to explain why there are three dimensions of space using topological and thermodynamic considerations. According to this idea it would be since three is the largest number of spatial dimensions in which strings can generically intersect. If initially there are many windings of strings around compact dimensions, space could only expand to macroscopic sizes once these windings are eliminated, which requires oppositely wound strings to find each other and annihilate. But strings can only find each other to annihilate at a meaningful rate in three dimensions, so it follows that only three dimensions of space are allowed to grow large given this kind of initial configuration.

Extra dimensions are said to be universal if all fields are equally free to propagate within them.

In computer graphics and spatial data

Main article: Geometric primitive

Several types of digital systems are based on the storage, analysis, and visualization of geometric shapes, including illustration software, Computer-aided design, and Geographic information systems. Different vector systems use a wide variety of data structures to represent shapes, but almost all are fundamentally based on a set of geometric primitives corresponding to the spatial dimensions:

• Point (0-dimensional), a single coordinate in a Cartesian coordinate system.
• Line or Polyline (1-dimensional), usually represented as an ordered list of points sampled from a continuous line, whereupon the software is expected to interpolate the intervening shape of the line as straight or curved line segments.
• Polygon (2-dimensional), usually represented as a line that closes at its endpoints, representing the boundary of a two-dimensional region. The software is expected to use this boundary to partition 2-dimensional space into an interior and exterior.
• Surface (3-dimensional), represented using a variety of strategies, such as a polyhedron consisting of connected polygon faces. The software is expected to use this surface to partition 3-dimensional space into an interior and exterior.

Frequently in these systems, especially GIS and Cartography, a representation of a real-world phenomena may have a different (usually lower) dimension than the phenomenon being represented. For example, a city (a two-dimensional region) may be represented as a point, or a road (a three-dimensional volume of material) may be represented as a line. This dimensional generalization correlates with tendencies in spatial cognition. For example, asking the distance between two cities presumes a conceptual model of the cities as points, while giving directions involving travel "up," "down," or "along" a road imply a one-dimensional conceptual model. This is frequently done for purposes of data efficiency, visual simplicity, or cognitive efficiency, and is acceptable if the distinction between the representation and the represented is understood, but can cause confusion if information users assume that the digital shape is a perfect representation of reality (i.e., believing that roads really are lines).

Networks and dimension

Some complex networks are characterized by fractal dimensions. The concept of dimension can be generalized to include networks embedded in space. The dimension characterize their spatial constraints.

In literature

Main article: Fourth dimension in literature

Science fiction texts often mention the concept of "dimension" when referring to parallel or alternate universes or other imagined planes of existence. This usage is derived from the idea that to travel to parallel/alternate universes/planes of existence one must travel in a direction/dimension besides the standard ones. In effect, the other universes/planes are just a small distance away from our own, but the distance is in a fourth (or higher) spatial (or non-spatial) dimension, not the standard ones.

One of the most heralded science fiction stories regarding true geometric dimensionality, and often recommended as a starting point for those just starting to investigate such matters, is the 1884 novella Flatland by Edwin A. Abbott. Isaac Asimov, in his foreword to the Signet Classics 1984 edition, described Flatland as "The best introduction one can find into the manner of perceiving dimensions."

The idea of other dimensions was incorporated into many early science fiction stories, appearing prominently, for example, in Miles J. Breuer's The Appendix and the Spectacles (1928) and Murray Leinster's The Fifth-Dimension Catapult (1931); and appeared irregularly in science fiction by the 1940s. Classic stories involving other dimensions include Robert A. Heinlein's —And He Built a Crooked House (1941), in which a California architect designs a house based on a three-dimensional projection of a tesseract; Alan E. Nourse's Tiger by the Tail and The Universe Between (both 1951); and The Ifth of Oofth (1957) by Walter Tevis. Another reference is Madeleine L'Engle's novel A Wrinkle In Time (1962), which uses the fifth dimension as a way for "tesseracting the universe" or "folding" space in order to move across it quickly. The fourth and fifth dimensions are also a key component of the book The Boy Who Reversed Himself by William Sleator.

In philosophy

Immanuel Kant, in 1783, wrote: "That everywhere space (which is not itself the boundary of another space) has three dimensions and that space in general cannot have more dimensions is based on the proposition that not more than three lines can intersect at right angles in one point. This proposition cannot at all be shown from concepts, but rests immediately on intuition and indeed on pure intuition a priori because it is apodictically (demonstrably) certain."

"Space has Four Dimensions" is a short story published in 1846 by German philosopher and experimental psychologist Gustav Fechner under the pseudonym "Dr. Mises". The protagonist in the tale is a shadow who is aware of and able to communicate with other shadows, but who is trapped on a two-dimensional surface. According to Fechner, this "shadow-man" would conceive of the third dimension as being one of time. The story bears a strong similarity to the "Allegory of the Cave" presented in Plato's The Republic (c. 380 BC).

Simon Newcomb wrote an article for the Bulletin of the American Mathematical Society in 1898 entitled "The Philosophy of Hyperspace". Linda Dalrymple Henderson coined the term "hyperspace philosophy", used to describe writing that uses higher dimensions to explore metaphysical themes, in her 1983 thesis about the fourth dimension in early-twentieth-century art. Examples of "hyperspace philosophers" include Charles Howard Hinton, the first writer, in 1888, to use the word "tesseract"; and the Russian esotericist P. D. Ouspensky.

Tablet (pharmacy)

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Tablet_(pharmacy) 

 

Common disk-shaped tablets

A tablet is a pharmaceutical oral dosage form (oral solid dosage, or OSD) or solid unit dosage form. Tablets may be defined as the solid unit dosage form of medicament or medicaments with suitable excipients. It comprises a mixture of active substances and excipients, usually in powder form, pressed or compacted from a powder into a solid dose.

Tablets are prepared either by molding or by compression. The excipients can include diluents, binders or granulating agents, glidants (flow aids) and lubricants to ensure efficient tabletting; disintegrants to promote tablet break-up in the digestive tract; sweeteners or flavours to enhance taste; and pigments to make the tablets visually attractive or aid in visual identification of an unknown tablet. A polymer coating is often applied to make the tablet smoother and easier to swallow, to control the release rate of the active ingredient, to make it more resistant to the environment (extending its shelf life), or to enhance the tablet's appearance. Medicinal tablets were originally made in the shape of a disk of whatever color their components determined, but are now made in many shapes and colors to help distinguish different medicines. Tablets are often stamped with symbols, letters, and numbers, which enable them to be identified. Sizes of tablets to be swallowed range from a few millimetres to about a centimetre.

The compressed tablet is the most popular dosage form in use today. About two-thirds of all prescriptions are dispensed as solid dosage forms, and half of these are compressed tablets. A tablet can be formulated to deliver an accurate dosage to a specific site; it is usually taken orally, but can be administered sublingually, buccally, rectally or intravaginally. The tablet is just one of the many forms that an oral drug can take such as syrups, elixirs, suspensions, and emulsions.

History

Pills are thought to date back to around 1500 BC. Earlier medical recipes, such as those from 4000 BC, were for liquid preparations rather than solids. The first references to pills were found on papyruses in ancient Egypt, and contained bread dough, honey or grease. Medicinal ingredients, such as plant powders or spices, were mixed in and formed by hand to make little balls, or pills. In ancient Greece, such medicines were known as katapotia ("something to be swallowed"), and the Roman scholar Pliny, who lived from 23-79 AD, first gave a name to what we now call pills, calling them pilula.

Pills have always been difficult to swallow and efforts long have been made to make them go down easier. In medieval times, people coated pills with slippery plant substances. Another approach, used as recently as the 19th century, was to gild them in gold and silver, although this often meant that they would pass through the digestive tract with no effect. In the 1800s sugar-coating and gelatin-coating was invented, as were gelatin capsules.

In 1843, the British painter and inventor William Brockedon was granted a patent for a machine capable of "Shaping Pills, Lozenges and Black Lead by Pressure in Dies". The device was capable of compressing powder into a tablet without use of an adhesive.

Types

Pills

Combined oral contraceptive pills were nicknamed "the pill" in the 1960s

A pill was originally defined as a small, round, solid pharmaceutical oral dosage form of medication. Today, pills include tablets, capsules, and variants thereof like caplets — essentially, any solid form of medication colloquially falls into the pill category.

An early example of pills came from Ancient Rome. They were made of the zinc carbonates hydrozincite and smithsonite. The pills were used for sore eyes, and were found aboard a Roman ship Relitto del Pozzino which wrecked in 140 BC. However, these tablets were meant to be pressed on the eyes, not swallowed.

Caplets

Variations on a common tablet design, which can be distinguished by both colour and shape

A caplet is a smooth, coated, oval-shaped medicinal tablet in the general shape of a capsule. Many caplets have an indentation running down the middle so they may be split in half more easily. Since their inception, capsules have been viewed by consumers as the most efficient method of taking medication. For this reason, producers of drugs such as OTC analgesics wanting to emphasize the strength of their product developed the "caplet", a portmanteau of capsule-shaped tablet, in order to tie this positive association to more efficiently-produced tablet pills, as well as being an easier-to-swallow shape than the usual disk-shaped tablet.

Olanzapine tablets

Orally disintegrating tablets (ODT)

An orally disintegrating tablet or orodispersible tablet (ODT), is a drug dosage form available for a limited range of over-the-counter (OTC) and prescription medications

Film coated tablets (FCT)

A film coated tablet is a drug dosage form available for a limited range of over-the-counter (OTC) and prescription medications. The used films protect the drug substance against denaturation by stomach acid and/or support a delayed (modified) release of the drug substance ("retard effect"). Such tablets should not be damaged or broken.

Tabletting formulations

In the tablet-pressing process, it is important that all ingredients be fairly dry, powdered or granular, somewhat uniform in particle size, and freely flowing. Mixed particle sized powders segregate during manufacturing operations due to different densities, which can result in tablets with poor drug or active pharmaceutical ingredient (API) content uniformity, but granulation should prevent this. Content uniformity ensures that the same API dose is delivered with each tablet.

Some APIs may be tableted as pure substances, but this is rarely the case; most formulations include excipients. Normally, a pharmacologically inactive ingredient (excipient) termed a binder is added to help hold the tablet together and give it strength. A wide variety of binders may be used, some common ones including lactose, dibasic calcium phosphate, sucrose, corn (maize) starch, microcrystalline cellulose, povidone polyvinylpyrrolidone and modified cellulose (for example hydroxypropyl methylcellulose and hydroxyethylcellulose).

Often, an ingredient is also needed to act as a disintegrant to aid tablet dispersion once swallowed, releasing the API for absorption. Some binders, such as starch and cellulose, are also excellent disintegrants.

Advantages and disadvantages

Tablets are simple and convenient to use. They provide an accurately measured dosage of the active ingredient in a convenient portable package, and can be designed to protect unstable medications or disguise unpalatable ingredients. Colored coatings, embossed markings and printing can be used to aid tablet recognition. Manufacturing processes and techniques can provide tablets with special properties, for example, sustained release or fast dissolving formulations.

Some drugs may be unsuitable for administration by the oral route. For example, protein drugs such as insulin may be denatured by stomach acids. Such drugs cannot be made into tablets. Some drugs may be deactivated by the liver when they are carried there from the gastrointestinal tract by the hepatic portal vein (the "first pass effect"), making them unsuitable for oral use. Drugs which can be taken sublingually are absorbed through the oral mucosa, so that they bypass the liver and are less susceptible to the first pass effect. The oral bioavailability of some drugs may be low due to poor absorption from the gastrointestinal tract. Such drugs may need to be given in very high doses or by injection. For drugs that need to have rapid onset, or that have severe side effects, the oral route may not be suitable. For example, salbutamol, used to treat problems in the respiratory system, can have effects on the heart and circulation if taken orally; these effects are greatly reduced by inhaling smaller doses direct to the required site of action. A proportion of the population have difficulties swallowing tablets either because they just do not like taking them or because their medical condition makes it difficult for them (dysphagia, vomiting). In such instances it may be better to consider alternative dosage form or administration route.

Tablet properties

Tablets can be made in virtually any shape, although requirements of patients and tableting machines mean that most are round, oval or capsule shaped. More unusual shapes have been manufactured but patients find these harder to swallow, and they are more vulnerable to chipping or manufacturing problems.

Tablet diameter and shape are determined by the machine tooling used to produce them - a die plus an upper and a lower punch are required. This is called a station of tooling. The thickness is determined by the amount of tablet material and the position of the punches in relation to each other during compression. Once this is done, we can measure the corresponding pressure applied during compression. The shorter the distance between the punches, thickness, the greater the pressure applied during compression, and sometimes the harder the tablet. Tablets need to be hard enough that they do not break up in the bottle, yet friable enough that they disintegrate in the gastric tract.

Tablets need to be strong enough to resist the stresses of packaging, shipping and handling by the pharmacist and patient. The mechanical strength of tablets is assessed using a combination of simple failure and erosion tests, and more sophisticated engineering tests. The simpler tests are often used for quality control purposes, whereas the more complex tests are used during the design of the formulation and manufacturing process in the research and development phase. Standards for tablet properties are published in the various international pharmacopeias (USP/NF, EP, JP, etc.). The hardness of tablets is the principal measure of mechanical strength. Hardness is tested using a tablet hardness tester. The units for hardness have evolved since the 1930s, but are commonly measured in kilograms per square centimetre. Models of tester include the Monsanto (or Stokes) Hardness Tester from 1930, the Pfizer Hardness Tester from 1950, the Strong Cob Hardness Tester and the Heberlain (or Schleeniger) Hardness Tester.

Lubricants prevent ingredients from clumping together and from sticking to the tablet punches or capsule filling machine. Lubricants also ensure that tablet formation and ejection can occur with low friction between the solid and die wall, as well as between granules, which helps in uniform filling of the die.

Common minerals like talc or silica, and fats, e.g. vegetable stearin, magnesium stearate or stearic acid are the most frequently used lubricants in tablets or hard gelatin capsules. 

Manufacturing

Manufacture of the tableting blend

In the tablet pressing process, the appropriate amount of active ingredient must be in each tablet. Hence, all the ingredients should be well-mixed. If a sufficiently homogenous mix of the components cannot be obtained with simple blending processes, the ingredients must be granulated prior to compression to assure an even distribution of the active compound in the final tablet. Two basic techniques are used to granulate powders for compression into a tablet: wet granulation and dry granulation. Powders that can be mixed well do not require granulation and can be compressed into tablets through direct compression ("DC"). Direct Compression is desirable as it is quicker. There is less processing, equipment, labor, and energy consumption. However, DC is difficult when a formulation has a high content of poorly compressible active ingredient.

Wet granulation

Wet granulation is a process of using a liquid binder to lightly agglomerate the powder mixture. The amount of liquid has to be properly controlled, as over-wetting will cause the granules to be too hard and under-wetting will cause them to be too soft and friable. Aqueous solutions have the advantage of being safer to deal with than solvent-based systems but may not be suitable for drugs which are degraded by hydrolysis.

Procedure

1. The active ingredient and excipients are weighed and mixed.
2. The wet granulate is prepared by adding the liquid binder–adhesive to the powder blend and mixing thoroughly. Examples of binders/adhesives include aqueous preparations of cornstarch, natural gums such as acacia, cellulose derivatives such as methyl cellulose, gelatin, and povidone.
3. Screening the damp mass through a mesh to form pellets or granules.
4. Drying the granulation. A conventional tray-dryer or fluid-bed dryer are most commonly used.
5. After the granules are dried, they are passed through a screen of smaller size than the one used for the wet mass to create granules of uniform size.

Low shear wet granulation processes use very simple mixing equipment, and can take a considerable time to achieve a uniformly mixed state. High shear wet granulation processes use equipment that mixes the powder and liquid at a very fast rate, and thus speeds up the manufacturing process. Fluid bed granulation is a multiple-step wet granulation process performed in the same vessel to pre-heat, granulate, and dry the powders. It is used because it allows close control of the granulation process.

Dry granulation

Dry granulation processes create granules by light compaction of the powder blend under low pressures. The compacts so-formed are broken up gently to produce granules (agglomerates). This process is often used when the product to be granulated is sensitive to moisture and heat. Dry granulation can be conducted on a tablet press using slugging tooling or on a roll press called a roller compactor. Dry granulation equipment offers a wide range of pressures to attain proper densification and granule formation. Dry granulation is simpler than wet granulation, therefore the cost is reduced. However, dry granulation often produces a higher percentage of fine granules, which can compromise the quality or create yield problems for the tablet. Dry granulation requires drugs or excipients with cohesive properties, and a 'dry binder' may need to be added to the formulation to facilitate the formation of granules.

Hot melt extrusion

Hot melt extrusion is utilized in pharmaceutical solid oral dose processing to enable delivery of drugs with poor solubility and bioavailability. Hot melt extrusion has been shown to molecularly disperse poorly soluble drugs in a polymer carrier increasing dissolution rates and bioavailability. The process involves the application of heat, pressure and agitation to mix materials together and ‘extrude’ them through a die. Twin-screw high shear extruders blend materials and simultaneously break up particles. The extruded particles can then be blended and compressed into tablets or filled into capsules.

Granule lubrication

After granulation, a final lubrication step is used to ensure that the tableting blend does not stick to the equipment during the tableting process. This usually involves low shear blending of the granules with a powdered lubricant, such as magnesium stearate or stearic acid.

Manufacture of the tablets

Tablets that failed due to capping and lamination compared to a normal tablet

Whatever process is used to make the tableting blend, the process of making a tablet by powder compaction is very similar. First, the powder is filled into the die from above. The mass of powder is determined by the position of the lower punch in the die, the cross-sectional area of the die, and the powder density. At this stage, adjustments to the tablet weight are normally made by repositioning the lower punch. After die filling, the upper punch is lowered into the die and the powder is uniaxially compressed to a porosity of between 5 and 20%. The compression can take place in one or two stages (main compression, and, sometimes, pre-compression or tamping) and for commercial production occurs very fast (500–50 ms per tablet). Finally, the upper punch is pulled up and out of the die (decompression), and the tablet is ejected from the die by lifting the lower punch until its upper surface is flush with the top face of the die. This process is repeated for each tablet.

Common problems encountered during tablet manufacturing operations include:

• Fluctuations in tablet weight, usually caused by uneven powder flow into the die due to poor powder flow properties.
• Fluctuations in dosage of the Active Pharmaceutical Ingredient, caused by uneven distribution of the API in the tableting blend (either due to poor mixing or separation in process).
• Sticking of the powder blend to the tablet tooling, due to inadequate lubrication, worn or dirty tooling, or a sticky powder formulation
• Capping, lamination or chipping. This is caused by air being compressed with the tablet formulation and then expanding when the punch is released: if this breaks the tablet apart, it can be due to incorrect machine settings, or due to incorrect formulation: either because the tablet formulation is too brittle or not adhesive enough, or because the powder being fed to the tablet press contains too much air (has too low bulk density).
• Capping can also occur due to high moisture content.

Tablet compaction simulator

Tablet formulations are designed and tested using a laboratory machine called a Tablet Compaction Simulator or Powder Compaction Simulator. This is a computer controlled device that can measure the punch positions, punch pressures, friction forces, die wall pressures, and sometimes the tablet internal temperature during the compaction event. Numerous experiments with small quantities of different mixtures can be performed to optimise a formulation. Mathematically corrected punch motions can be programmed to simulate any type and model of production tablet press. Initial quantities of active pharmaceutical ingredients are very expensive to produce, and using a Compaction Simulator reduces the amount of powder required for product development.

Tablet presses

The tablet pressing operation (click image to enlarge)

 

An old Cadmach rotary tablet press

Tablet presses, also called tableting machines, range from small, inexpensive bench-top models that make one tablet at a time (single-station presses), with only around a half-ton pressure, to large, computerized, industrial models (multi-station rotary presses) that can make hundreds of thousands to millions of tablets an hour with much greater pressure. The tablet press is an essential piece of machinery for any pharmaceutical and nutraceutical manufacturer. Tablet presses must allow the operator to adjust the position of the lower and upper punches accurately, so that the tablet weight, thickness and density/hardness can each be controlled. This is achieved using a series of cams, rollers, or tracks that act on the tablet tooling (punches). Mechanical systems are also incorporated for die filling, and for ejecting and removing the tablets from the press after compression. Pharmaceutical tablet presses are required to be easy to clean and quick to reconfigure with different tooling, because they are usually used to manufacture many different products. There are two main standards of tablet tooling used in pharmaceutical industry: American standard TSM and European standard EU. TSM and EU configurations are similar to each other but cannot be interchanged.

Modern tablet presses reach output volumes of up to 1,700,000 tablets per hour. These huge volumes require frequent in-process quality control for the tablet weight, thickness and hardness. Due to reduce rejects rates and machine down-time, automated tablet testing devices are used on-line with the tablet press or off-line in the IPC-labs.

Tablet coating

See also: Film coating

Many tablets today are coated after being pressed. Although sugar-coating was popular in the past, the process has many drawbacks. Modern tablet coatings are polymer and polysaccharide based, with plasticizers and pigments included. Tablet coatings must be stable and strong enough to survive the handling of the tablet, must not make tablets stick together during the coating process, and must follow the fine contours of embossed characters or logos on tablets. Coatings are necessary for tablets that have an unpleasant taste, and a smoother finish makes large tablets easier to swallow. Tablet coatings are also useful to extend the shelf-life of components that are sensitive to moisture or oxidation. Special coatings (for example with pearlescent effects) can enhance brand recognition.

If the active ingredient of a tablet is sensitive to acid, or is irritant to the stomach lining, an enteric coating can be used, which is resistant to stomach acid, and dissolves in the less acidic area of the intestines. Enteric coatings are also used for medicines that can be negatively affected by taking a long time to reach the small intestine, where they are absorbed. Coatings are often chosen to control the rate of dissolution of the drug in the gastrointestinal tract. Some drugs are absorbed better in certain parts of the digestive system. If this part is the stomach, a coating is selected that dissolves quickly and easily in acid. If the rate of absorption is best in the large intestine or colon, a coating is used that is acid resistant and dissolves slowly to ensure that the tablet reaches that point before dispersing. To measure the disintegration time of the tablet coating and the tablet core, automatic disintegration testers are used which are able to determine the complete disintegration process of a tablet by measuring the rest height of the thickness with every upward stroke of the disintegration tester basket.

There are two types of coating machines used in the pharmaceutical industry: coating pans and automatic coaters. Coating pans are used mostly to sugar coat pellets. Automatic coaters are used for all kinds of coatings; they can be equipped with a remote control panel, a dehumidifier, and dust collectors. An explosion-proof design is required for applying coatings that contain alcohol.

Pill-splitters

It is sometimes necessary to split tablets into halves or quarters. Tablets are easier to break accurately if scored, but there are devices called pill-splitters which cut unscored and scored tablets. Tablets with special coatings (for example enteric coatings or controlled-release coatings) should not be broken before use, as this exposes the tablet core to the digestive juices, circumventing the intended delayed-release effect.

Modified-release dosage

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Modified-release_dosage

Modified-release dosage is a mechanism that (in contrast to immediate-release dosage) delivers a drug with a delay after its administration (delayed-release dosage) or for a prolonged period of time (extended-release [ER, XR, XL] dosage) or to a specific target in the body (targeted-release dosage).

Sustained-release dosage forms are dosage forms designed to release (liberate) a drug at a predetermined rate in order to maintain a constant drug concentration for a specific period of time with minimum side effects. This can be achieved through a variety of formulations, including liposomes and drug-polymer conjugates (an example being hydrogels). Sustained release's definition is more akin to a "controlled release" rather than "sustained".

Extended-release dosage consists of either sustained-release (SR) or controlled-release (CR) dosage. SR maintains drug release over a sustained period but not at a constant rate. CR maintains drug release over a sustained period at a nearly constant rate.

Sometimes these and other terms are treated as synonyms, but the United States Food and Drug Administration has in fact defined most of these as different concepts. Sometimes the term "depot tablet" is used by non-native speakers, but this is not found in any English dictionaries and is a literal translation of the term used in Swedish and some other languages.

Modified-release dosage and its variants are mechanisms used in tablets (pills) and capsules to dissolve a drug over time in order to be released slower and steadier into the bloodstream while having the advantage of being taken at less frequent intervals than immediate-release (IR) formulations of the same drug. For example, extended-release morphine enables people with chronic pain to only take one or two tablets per day.

Most commonly it refers to time dependent release in oral dose formulations. Timed release has several distinct variants such as sustained release where prolonged release is intended, pulse release, delayed release (e.g. to target different regions of the GI tract) etc. A distinction of controlled release is that not only it prolongs action but it attempts to maintain drug levels within the therapeutic window to avoid potentially hazardous peaks in drug concentration following ingestion or injection and to maximize therapeutic efficiency.

In addition to pills, the mechanism can also apply to capsules and injectable drug carriers (that often have an additional release function), forms of controlled release medicines include gels, implants and devices (e.g. the vaginal ring and contraceptive implant) and transdermal patches.

Examples for cosmetic, personal care, and food science applications often centre on odour or flavour release.

The release technology scientific and industrial community is represented by the Controlled Release Society (CRS). The CRS is the worldwide society for delivery science and technologies. CRS serves more than 1,600 members from more than 50 countries. Two-thirds of CRS membership is represented by industry and one-third represents academia and government. CRS is affiliated with the Journal of Controlled Release and Drug Delivery and Translational Research scientific journals.

List of abbreviations

There is no industry standard for these abbreviations, and confusion and misreading have sometimes caused prescribing errors. Clear handwriting is necessary. For some drugs with multiple formulations, putting the meaning in parentheses is advisable.

Abbreviation	Meaning	Notes
CD	controlled delivery
CR	controlled release
DR	delayed release
ER	extended release
IR	immediate release
LA	long-acting
LAR	long-acting release
MR	modified release
PR	prolonged release
SA	sustained action	Ambiguous, can sometimes mean short-acting
SR	sustained release
TR	timed release
XL	extended release
XR	extended release
XT	extended release

A few other abbreviations are similar to these (in that they may serve as suffixes) but refer to dose rather than release rate. They include ES and XS (extra strength).

Methods

Today, most time-release drugs are formulated so that the active ingredient is embedded in a matrix of insoluble substance(s) (various: some acrylics, even chitin; these substances are often patented) such that the dissolving drug must find its way out through the holes.

In some SR formulations, the drug dissolves into the matrix, and the matrix physically swells to form a gel, allowing the drug to exit through the gel's outer surface.

Micro-encapsulation is also regarded as a more complete technology to produce complex dissolution profiles. Through coating an active pharmaceutical ingredient around an inert core, and layering it with insoluble substances to form a microsphere one can obtain more consistent and replicable dissolution rates in a convenient format that can be mixed and matched with other instant release pharmaceutical ingredients in to any two piece gelatin capsule.

There are certain considerations for the formation of sustained-release formulation:

• If the pharmacological activity of the active compound is not related to its blood levels, time releasing has no purpose except in some cases, such as bupropion, to reduce possible side effects.
• If the absorption of the active compound involves an active transport, the development of a time-release product may be problematic.

The half-life of the drug refers to the drug's elimination from the bloodstream which can be caused by metabolism, urine, and other forms of excretion. If the active compound has a long half-life (over 6 hours), it is sustained on its own. If the active compound has a short half-life, it would require a large amount to maintain a prolonged effective dose. In this case, a broad therapeutic window is necessary to avoid toxicity; otherwise, the risk is unwarranted and another mode of administration would be recommended. Appropriate half-lifes used to apply sustained methods are typically 3–4 hours and a drug greater than 0.5 grams is too big.

The therapeutic index also factors whether a drug can be used as a time release drug. A drug with a thin therapeutic range, or small therapeutic index, will be determined unfit for a sustained release mechanism in partial fear of dose dumping which can prove fatal at the conditions mentioned. For a drug that is made to be released over time, the general goals is to stay within the therapeutic range as long as needed.

There are many different methods used to obtain a sustained release.

Diffusion systems

Diffusion systems rate release is dependent on the rate at which the drug dissolves through a barrier which is usually a type of polymer. Diffusion systems can be broken into two subcategories, reservoir devices and matrix devices.

• Reservoir devices coat the drug with polymers and in order for the reservoir devices to have sustained release effects, the polymer must not dissolve and let the drug be released through diffusion. The rate of reservoir devices can be altered by changing the polymer and is possible be made to have zero-order release; however, drugs with higher molecular weight have difficulty diffusing through the membrane.
• Matrix devices forms a matrix (drug(s) mixed with a gelling agent) where the drug is dissolved/dispersed. The drug is usually dispersed within a polymer and then released by undergoing diffusion. However, to make the drug SR in this device, the rate of dissolution of the drug within the matrix needs to be higher than the rate at which it is released. The matrix device cannot achieve a zero-order release but higher molecular weight molecules can be used. The diffusion matrix device also tends to be easier to produce and protect from changing in the gastrointestinal tract but factors such as food can affect the release rate.

Dissolution systems

Dissolution systems must have the system dissolved slowly in order for the drug to have sustained release properties which can be achieved by using appropriate salts and/or derivatives as well as coating the drug with a dissolving material. It is used for drug compounds with high solubility in water. When the drug is covered with some slow dissolving coat, it will eventually release the drug. Instead of diffusion, the drug release depends on the solubility and thickness of the coating. Because of this mechanism, the dissolution will be the rate limiting factor here for drug release. Dissolution systems can be broken down to subcategories called reservoir devices and matrix devices.

• The reservoir device coats the drug with an appropriate material which will dissolve slowly. It can also be used to administer beads as a group with varying thickness, making the drug release in multiple times creating a SR.
• The matrix device has the drug in a matrix and the matrix is dissolved instead of a coating. It can come either as drug impregnated spheres or drug impregnated tablets.

Osmotic systems

Main article: Osmotic controlled-release oral delivery system

 

A 54mg tablet of Concerta, which uses OROS technology.

Osmotic controlled-release oral delivery systems (OROS) have the form of a rigid tablet with a semi-permeable outer membrane and one or more small laser drilled holes in it. As the tablet passes through the body, water is absorbed through the semipermeable membrane via osmosis, and the resulting osmotic pressure is used to push the active drug through the opening(s) in the tablet. OROS is a trademarked name owned by ALZA Corporation, which pioneered the use of osmotic pumps for oral drug delivery.

Osmotic release systems have a number of major advantages over other controlled-release mechanisms. They are significantly less affected by factors such as pH, food intake, GI motility, and differing intestinal environments. Using an osmotic pump to deliver drugs has additional inherent advantages regarding control over drug delivery rates. This allows for much more precise drug delivery over an extended period of time, which results in much more predictable pharmacokinetics. However, osmotic release systems are relatively complicated, somewhat difficult to manufacture, and may cause irritation or even blockage of the GI tract due to prolonged release of irritating drugs from the non-deformable tablet.

Ion-exchange resin

Main article: Ion-exchange resin

In the ion-exchange method, the resins are cross-linked water-insoluble polymers that contain ionisable functional groups that form a repeating pattern of polymers, creating a polymer chain. The drug is attached to the resin and is released when an appropriate interaction of ions and ion exchange groups occur. The area and length of the drug release and number of cross-link polymers dictate the rate at which the drug is released, determining the SR effect.

Floating systems

A floating system is a system where it floats on gastric fluids due to low-density. The density of the gastric fluids is about 1 g/mL; thus, the drug/tablet administered must have a smaller density. The buoyancy will allow the system to float to the top of the stomach and release at a slower rate without worry of excreting it. This system requires there are enough gastric fluids present as well as food. Many types of forms of drugs use this method such as powders, capsules, and tablets.

Bio-adhesive systems

Bio-adhesive systems generally are meant to stick to mucus and can be favorable for mouth based interactions due to high mucus levels in the general area but not as simple for other areas. Magnetic materials can be added to the drug so another magnet can hold it from outside the body to assist in holding the system in place. However, there is low patient compliance with this system.

Matrix systems

The matrix system is the mixture of materials with the drug, which will cause the drug to slow down. However, this system has several subcategories: hydrophobic matrices, lipid matrices, hydrophilic matrices, biodegradable matrices, and mineral matrices.

• A hydrophobic matrix is a drug mixed with a hydrophobic polymer. This causes SR because the drug, after being dissolved, will have to be released by going through channels made by the hydrophilic polymer.
• A hydrophilic matrix will go back to the matrix as discussed before where a matrix is a mixture of a drug or drugs with a gelling agent. This system is well liked because of its cost and broad regulatory acceptance. The polymers used can be broken down into categories: cellulose derivatives, non-cellulose natural, and polymers of acrylic acid.
• A lipid matrix uses wax or similar materials. Drug release happens through diffusion through, and erosion of, the wax and tends to be sensitive to digestive fluids.
• Biodegradable matrices are made with unstable, linked monomers that will erode by biological compounds such as enzymes and proteins.
• A mineral matrix which generally means the polymers used are obtained in seaweed.

Stimuli inducing release

Examples of stimuli that may be used to bring about release include pH, enzymes, light, magnetic fields, temperature, ultrasonics, osmosis, cellular traction forces, and electronic control of MEMS and NEMS.

Spherical hydrogels, in micro-size (50-600 µm diameter) with 3-dimensional cross-linked polymer, can be used as drug carrier to control the release of the drug. These hydrogels called microgels. They may possess a negative charge as example DC-beads. By ion-exchange mechanism, a large amount of oppositely charged amphiphilic drugs can be loaded inside these microgels. Then, the release of these drugs can be controlled by a specific triggering factor like pH, ionic strength or temperature.

Pill splitting

Empty half-shell of a split bupropion XL 150mg manufactured by Anchen Pharmaceuticals that was soaked in water overnight and then shaken.

Some time release formulations do not work properly if split, such as controlled-release tablet coatings, while other formulations such as micro-encapsulation still work if the microcapsules inside are swallowed whole.

Among the health information technology (HIT) that pharmacists use are medication safety tools to help manage this problem. For example, the ISMP "do not crush" list can be entered into the system so that warning stickers can be printed at the point of dispensing, to be stuck on the pill bottle.

Pharmaceutical companies that do not supply a range of half-dose and quarter-dose versions of time-release tablets can make it difficult for patients to be slowly tapered off their drugs.

History

The earliest SR drugs are associated with a patent in 1938 by Israel Lipowski, who coated pellets which led to coating particles. The science of controlled release developed further with more oral sustained-release products in the late 1940s and early 1950s, the development of controlled release of marine anti-foulants in the 1950s and controlled release fertilizer in the 1970s where sustained and controlled delivery of nutrients following a single application to the soil. Delivery is usually effected by dissolution, degradation or disintegration of an excipient in which the active compound is formulated. Enteric coating and other encapsulation technologies can further modify release profiles.