Sphere

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Sphere

Sphere
A perspective projection of a sphere
Type	Smooth surface Algebraic surface
Euler char.	2
Symmetry group	O(3)
Surface area	4πr2
Volume	4/3πr3

A sphere (from Ancient Greek σφαῖρα (sphaîra) 'globe, ball') is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. Formally, a sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. That given point is the centre of the sphere, and r is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians.

The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings.

Geometrically, a sphere can be formed by rotating a circle one half revolution around an axis that intersects the center of the circle, or by rotating a semicircle one full revolution around the axis that is coincident (or concurrent) with the straight edge of the semicircle.

Basic terminology

Two orthogonal radii of a sphere

As mentioned earlier r is the sphere's radius; any line from the center to a point on the sphere is also called a radius.

If a radius is extended through the center to the opposite side of the sphere, it creates a diameter. Like the radius, the length of a diameter is also called the diameter, and denoted d. Diameters are the longest line segments that can be drawn between two points on the sphere: their length is twice the radius, d = 2r. Two points on the sphere connected by a diameter are antipodal points of each other.

A unit sphere is a sphere with unit radius (r = 1). For convenience, spheres are often taken to have their center at the origin of the coordinate system, and spheres in this article have their center at the origin unless a center is mentioned.

A great circle on the sphere has the same center and radius as the sphere, and divides it into two equal hemispheres.

Although the Earth is not perfectly spherical, terms borrowed from geography are convenient to apply to the sphere. If a particular point on a sphere is (arbitrarily) designated as its north pole, its antipodal point is called the south pole. The great circle equidistant to each is then the equator. Great circles through the poles are called lines of longitude or meridians. A line connecting the two poles may be called the axis of rotation. Small circles on the sphere that are parallel to the equator are lines of latitude. In geometry unrelated to astronomical bodies, geocentric terminology should be used only for illustration and noted as such, unless there is no chance of misunderstanding.

Mathematicians consider a sphere to be a two-dimensional closed surface embedded in three-dimensional Euclidean space. They draw a distinction a sphere and a ball, which is a three-dimensional manifold with boundary that includes the volume contained by the sphere. An open ball excludes the sphere itself, while a closed ball includes the sphere: a closed ball is the union of the open ball and the sphere, and a sphere is the boundary of a (closed or open) ball. The distinction between ball and sphere has not always been maintained and especially older mathematical references talk about a sphere as a solid. The distinction between "circle" and "disk" in the plane is similar.

Small spheres or balls are sometimes called spherules, e.g. in Martian spherules.

Equations

In analytic geometry, a sphere with center (x₀, y₀, z₀) and radius r is the locus of all points (x, y, z) such that

${\displaystyle (x-x_0{)}^2+(y-y_0{)}^2+(z-z_0{)}^2=r^2.}$

Since it can be expressed as a quadratic polynomial, a sphere is a quadric surface, a type of algebraic surface.

Let a, b, c, d, e be real numbers with a ≠ 0 and put

${\displaystyle x_0=\frac{-b}{a},y_0=\frac{-c}{a},z_0=\frac{-d}{a},\rho =\frac{b^2+c^2+d^2-ae}{a^2}.}$

Then the equation

${\displaystyle f(x,y,z)=a(x^2+y^2+z^2)+2(bx+cy+dz)+e=0}$

has no real points as solutions if ${\displaystyle \rho <0}$ and is called the equation of an imaginary sphere. If ${\displaystyle \rho =0}$, the only solution of ${\displaystyle f(x,y,z)=0}$ is the point ${\displaystyle P_0=(x_0,y_0,z_0)}$ and the equation is said to be the equation of a point sphere. Finally, in the case ${\displaystyle \rho >0}$, ${\displaystyle f(x,y,z)=0}$ is an equation of a sphere whose center is ${\displaystyle P_0}$ and whose radius is ${\displaystyle \sqrt{\rho }}$.

If a in the above equation is zero then f(x, y, z) = 0 is the equation of a plane. Thus, a plane may be thought of as a sphere of infinite radius whose center is a point at infinity.

Parametric

A parametric equation for the sphere with radius ${\displaystyle r>0}$ and center ${\displaystyle (x_0,y_0,z_0)}$ can be parameterized using trigonometric functions.

${\displaystyle \begin{array}{rl}x& =x_0+r\sin \theta \cos \phi \\ y& =y_0+r\sin \theta \sin \phi \\ z& =z_0+r\cos \theta \end{array}}$^[5]

The symbols used here are the same as those used in spherical coordinates. r is constant, while θ varies from 0 to π and ${\displaystyle \phi }$ varies from 0 to 2π.

Properties

Enclosed volume

Sphere and circumscribed cylinder

In three dimensions, the volume inside a sphere (that is, the volume of a ball, but classically referred to as the volume of a sphere) is

${\displaystyle V=\frac{4}{3}\pi r^3=\frac{\pi }{6}{d}^3\approx 0.5236\cdot d^3}$

where r is the radius and d is the diameter of the sphere. Archimedes first derived this formula by showing that the volume inside a sphere is twice the volume between the sphere and the circumscribed cylinder of that sphere (having the height and diameter equal to the diameter of the sphere). This may be proved by inscribing a cone upside down into semi-sphere, noting that the area of a cross section of the cone plus the area of a cross section of the sphere is the same as the area of the cross section of the circumscribing cylinder, and applying Cavalieri's principle. This formula can also be derived using integral calculus, i.e. disk integration to sum the volumes of an infinite number of circular disks of infinitesimally small thickness stacked side by side and centered along the x-axis from x = −r to x = r, assuming the sphere of radius r is centered at the origin.

Proof of sphere volume, using calculus

For most practical purposes, the volume inside a sphere inscribed in a cube can be approximated as 52.4% of the volume of the cube, since V = π/6 d³, where d is the diameter of the sphere and also the length of a side of the cube and π/6 ≈ 0.5236. For example, a sphere with diameter 1 m has 52.4% the volume of a cube with edge length 1 m, or about 0.524 m³.

Surface area

The surface area of a sphere of radius r is:

${\displaystyle A=4\pi r^2.}$

Archimedes first derived this formula from the fact that the projection to the lateral surface of a circumscribed cylinder is area-preserving. Another approach to obtaining the formula comes from the fact that it equals the derivative of the formula for the volume with respect to r because the total volume inside a sphere of radius r can be thought of as the summation of the surface area of an infinite number of spherical shells of infinitesimal thickness concentrically stacked inside one another from radius 0 to radius r. At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is infinitesimal, and the elemental volume at radius r is simply the product of the surface area at radius r and the infinitesimal thickness.

Proof of surface area, using calculus

The sphere has the smallest surface area of all surfaces that enclose a given volume, and it encloses the largest volume among all closed surfaces with a given surface area. The sphere therefore appears in nature: for example, bubbles and small water drops are roughly spherical because the surface tension locally minimizes surface area.

The surface area relative to the mass of a ball is called the specific surface area and can be expressed from the above stated equations as

${\displaystyle \mathrm{S}\mathrm{S}\mathrm{A}=\frac{A}{V\rho }=\frac{3}{r\rho }}$

where ρ is the density (the ratio of mass to volume).

Other geometric properties

A sphere can be constructed as the surface formed by rotating a circle one half revolution about any of its diameters; this is very similar to the traditional definition of a sphere as given in Euclid's Elements. Since a circle is a special type of ellipse, a sphere is a special type of ellipsoid of revolution. Replacing the circle with an ellipse rotated about its major axis, the shape becomes a prolate spheroid; rotated about the minor axis, an oblate spheroid.

A sphere is uniquely determined by four points that are not coplanar. More generally, a sphere is uniquely determined by four conditions such as passing through a point, being tangent to a plane, etc. This property is analogous to the property that three non-collinear points determine a unique circle in a plane.

Consequently, a sphere is uniquely determined by (that is, passes through) a circle and a point not in the plane of that circle.

By examining the common solutions of the equations of two spheres, it can be seen that two spheres intersect in a circle and the plane containing that circle is called the radical plane of the intersecting spheres. Although the radical plane is a real plane, the circle may be imaginary (the spheres have no real point in common) or consist of a single point (the spheres are tangent at that point).

The angle between two spheres at a real point of intersection is the dihedral angle determined by the tangent planes to the spheres at that point. Two spheres intersect at the same angle at all points of their circle of intersection. They intersect at right angles (are orthogonal) if and only if the square of the distance between their centers is equal to the sum of the squares of their radii.

Pencil of spheres

Main article: Pencil (mathematics) § Pencil of spheres

If f(x, y, z) = 0 and g(x, y, z) = 0 are the equations of two distinct spheres then

${\displaystyle sf(x,y,z)+tg(x,y,z)=0}$

is also the equation of a sphere for arbitrary values of the parameters s and t. The set of all spheres satisfying this equation is called a pencil of spheres determined by the original two spheres. In this definition a sphere is allowed to be a plane (infinite radius, center at infinity) and if both the original spheres are planes then all the spheres of the pencil are planes, otherwise there is only one plane (the radical plane) in the pencil.

Properties of the sphere

A normal vector to a sphere, a normal plane and its normal section. The curvature of the curve of intersection is the sectional curvature. For the sphere each normal section through a given point will be a circle of the same radius: the radius of the sphere. This means that every point on the sphere will be an umbilical point.

In their book Geometry and the Imagination, David Hilbert and Stephan Cohn-Vossen describe eleven properties of the sphere and discuss whether these properties uniquely determine the sphere. Several properties hold for the plane, which can be thought of as a sphere with infinite radius. These properties are:

1. The points on the sphere are all the same distance from a fixed point. Also, the ratio of the distance of its points from two fixed points is constant.

   The first part is the usual definition of the sphere and determines it uniquely. The second part can be easily deduced and follows a similar result of Apollonius of Perga for the circle. This second part also holds for the plane.

2. The contours and plane sections of the sphere are circles.

   This property defines the sphere uniquely.

3. The sphere has constant width and constant girth.

   The width of a surface is the distance between pairs of parallel tangent planes. Numerous other closed convex surfaces have constant width, for example the Meissner body. The girth of a surface is the circumference of the boundary of its orthogonal projection on to a plane. Each of these properties implies the other.

4. All points of a sphere are umbilics.

   At any point on a surface a normal direction is at right angles to the surface because on the sphere these are the lines radiating out from the center of the sphere. The intersection of a plane that contains the normal with the surface will form a curve that is called a normal section, and the curvature of this curve is the normal curvature. For most points on most surfaces, different sections will have different curvatures; the maximum and minimum values of these are called the principal curvatures. Any closed surface will have at least four points called umbilical points. At an umbilic all the sectional curvatures are equal; in particular the principal curvatures are equal. Umbilical points can be thought of as the points where the surface is closely approximated by a sphere.

   For the sphere the curvatures of all normal sections are equal, so every point is an umbilic. The sphere and plane are the only surfaces with this property.

5. The sphere does not have a surface of centers.

   For a given normal section exists a circle of curvature that equals the sectional curvature, is tangent to the surface, and the center lines of which lie along on the normal line. For example, the two centers corresponding to the maximum and minimum sectional curvatures are called the focal points, and the set of all such centers forms the focal surface.

   For most surfaces the focal surface forms two sheets that are each a surface and meet at umbilical points. Several cases are special:

   * For channel surfaces one sheet forms a curve and the other sheet is a surface

   * For cones, cylinders, tori and cyclides both sheets form curves.

   * For the sphere the center of every osculating circle is at the center of the sphere and the focal surface forms a single point. This property is unique to the sphere.

6. All geodesics of the sphere are closed curves.

   Geodesics are curves on a surface that give the shortest distance between two points. They are a generalization of the concept of a straight line in the plane. For the sphere the geodesics are great circles. Many other surfaces share this property.

7. Of all the solids having a given volume, the sphere is the one with the smallest surface area; of all solids having a given surface area, the sphere is the one having the greatest volume.

   It follows from isoperimetric inequality. These properties define the sphere uniquely and can be seen in soap bubbles: a soap bubble will enclose a fixed volume, and surface tension minimizes its surface area for that volume. A freely floating soap bubble therefore approximates a sphere (though such external forces as gravity will slightly distort the bubble's shape). It can also be seen in planets and stars where gravity minimizes surface area for large celestial bodies.

8. The sphere has the smallest total mean curvature among all convex solids with a given surface area.

   The mean curvature is the average of the two principal curvatures, which is constant because the two principal curvatures are constant at all points of the sphere.

9. The sphere has constant mean curvature.

   The sphere is the only imbedded surface that lacks boundary or singularities with constant positive mean curvature. Other such immersed surfaces as minimal surfaces have constant mean curvature.

10. The sphere has constant positive Gaussian curvature.

    Gaussian curvature is the product of the two principal curvatures. It is an intrinsic property that can be determined by measuring length and angles and is independent of how the surface is embedded in space. Hence, bending a surface will not alter the Gaussian curvature, and other surfaces with constant positive Gaussian curvature can be obtained by cutting a small slit in the sphere and bending it. All these other surfaces would have boundaries, and the sphere is the only surface that lacks a boundary with constant, positive Gaussian curvature. The pseudosphere is an example of a surface with constant negative Gaussian curvature.

11. The sphere is transformed into itself by a three-parameter family of rigid motions.

    Rotating around any axis a unit sphere at the origin will map the sphere onto itself. Any rotation about a line through the origin can be expressed as a combination of rotations around the three-coordinate axis (see Euler angles). Therefore, a three-parameter family of rotations exists such that each rotation transforms the sphere onto itself; this family is the rotation group SO(3). The plane is the only other surface with a three-parameter family of transformations (translations along the x- and y-axes and rotations around the origin). Circular cylinders are the only surfaces with two-parameter families of rigid motions and the surfaces of revolution and helicoids are the only surfaces with a one-parameter family.

Treatment by area of mathematics

Spherical geometry

Great circle on a sphere

Main article: Spherical geometry

The basic elements of Euclidean plane geometry are points and lines. On the sphere, points are defined in the usual sense. The analogue of the "line" is the geodesic, which is a great circle; the defining characteristic of a great circle is that the plane containing all its points also passes through the center of the sphere. Measuring by arc length shows that the shortest path between two points lying on the sphere is the shorter segment of the great circle that includes the points.

Many theorems from classical geometry hold true for spherical geometry as well, but not all do because the sphere fails to satisfy some of classical geometry's postulates, including the parallel postulate. In spherical trigonometry, angles are defined between great circles. Spherical trigonometry differs from ordinary trigonometry in many respects. For example, the sum of the interior angles of a spherical triangle always exceeds 180 degrees. Also, any two similar spherical triangles are congruent.

Any pair of points on a sphere that lie on a straight line through the sphere's center (i.e. the diameter) are called antipodal points—on the sphere, the distance between them is exactly half the length of the circumference. Any other (i.e. not antipodal) pair of distinct points on a sphere

• lie on a unique great circle,
• segment it into one minor (i.e. shorter) and one major (i.e. longer) arc, and
• have the minor arc's length be the shortest distance between them on the sphere.

Spherical geometry is a form of elliptic geometry, which together with hyperbolic geometry makes up non-Euclidean geometry.

Differential geometry

The sphere is a smooth surface with constant Gaussian curvature at each point equal to 1/r². As per Gauss's Theorema Egregium, this curvature is independent of the sphere's embedding in 3-dimensional space. Also following from Gauss, a sphere cannot be mapped to a plane while maintaining both areas and angles. Therefore, any map projection introduces some form of distortion.

A sphere of radius r has area element ${\displaystyle dA=r^2\sin \theta d\theta d\phi }$. This can be found from the volume element in spherical coordinates with r held constant.

A sphere of any radius centered at zero is an integral surface of the following differential form:

${\displaystyle xdx+ydy+zdz=0.}$

This equation reflects that the position vector and tangent plane at a point are always orthogonal to each other. Furthermore, the outward-facing normal vector is equal to the position vector scaled by 1/r.

In Riemannian geometry, the filling area conjecture states that the hemisphere is the optimal (least area) isometric filling of the Riemannian circle.

Topology

Remarkably, it is possible to turn an ordinary sphere inside out in a three-dimensional space with possible self-intersections but without creating any creases, in a process called sphere eversion.

The antipodal quotient of the sphere is the surface called the real projective plane, which can also be thought of as the Northern Hemisphere with antipodal points of the equator identified.

Curves on a sphere

Plane section of a sphere: 1 circle

Coaxial intersection of a sphere and a cylinder: 2 circles

Circles

Main article: Circle of a sphere

Circles on the sphere are, like circles in the plane, made up of all points a certain distance from a fixed point on the sphere. The intersection of a sphere and a plane is a circle, a point, or empty. Great circles are the intersection of the sphere with a plane passing through the center of a sphere: others are called small circles.

More complicated surfaces may intersect a sphere in circles, too: the intersection of a sphere with a surface of revolution whose axis contains the center of the sphere (are coaxial) consists of circles and/or points if not empty. For example, the diagram to the right shows the intersection of a sphere and a cylinder, which consists of two circles. If the cylinder radius were that of the sphere, the intersection would be a single circle. If the cylinder radius were larger than that of the sphere, the intersection would be empty.

Loxodrome

Loxodrome

Main article: Rhumb line

In navigation, a rhumb line or loxodrome is an arc crossing all meridians of longitude at the same angle. Loxodromes are the same as straight lines in the Mercator projection. A rhumb line is not a spherical spiral. Except for some simple cases, the formula of a rhumb line is complicated.

Clelia curves

Main article: Clélie

spherical spiral with ${\displaystyle c=8}$

A Clelia curve is a curve on a sphere for which the longitude ${\displaystyle \phi }$ and the colatitude ${\displaystyle \theta }$ satisfy the equation

${\displaystyle \phi =c\theta ,c>0}$.

Special cases are: Viviani's curve (${\displaystyle c=1}$) and spherical spirals (${\displaystyle c>2}$) such as Seiffert's spiral. Clelia curves approximate the path of satellites in polar orbit.

Spherical conics

Main article: Spherical conic

The analog of a conic section on the sphere is a spherical conic, a quartic curve which can be defined in several equivalent ways, including:

• as the intersection of a sphere with a quadratic cone whose vertex is the sphere center;
• as the intersection of a sphere with an elliptic or hyperbolic cylinder whose axis passes through the sphere center;
• as the locus of points whose sum or difference of great-circle distances from a pair of foci is a constant.

Many theorems relating to planar conic sections also extend to spherical conics.

Intersection of a sphere with a more general surface

General intersection sphere-cylinder

If a sphere is intersected by another surface, there may be more complicated spherical curves.

Example

sphere – cylinder

Main article: Sphere–cylinder intersection

The intersection of the sphere with equation ${\displaystyle x^2+y^2+z^2=r^2}$ and the cylinder with equation ${\displaystyle (y-y_0{)}^2+z^2=a^2,y_0\ne 0}$ is not just one or two circles. It is the solution of the non-linear system of equations

${\displaystyle x^2+y^2+z^2-r^2=0}$

${\displaystyle (y-y_0{)}^2+z^2-a^2=0.}$

(see implicit curve and the diagram)

Generalizations

Ellipsoids

An ellipsoid is a sphere that has been stretched or compressed in one or more directions. More exactly, it is the image of a sphere under an affine transformation. An ellipsoid bears the same relationship to the sphere that an ellipse does to a circle.

Dimensionality

Main article: n-sphere

Spheres can be generalized to spaces of any number of dimensions. For any natural number n, an n-sphere, often denoted Sⁿ, is the set of points in (n + 1)-dimensional Euclidean space that are at a fixed distance r from a central point of that space, where r is, as before, a positive real number. In particular:

• S⁰: a 0-sphere consists of two discrete points, −r and r
• S¹: a 1-sphere is a circle of radius r
• S²: a 2-sphere is an ordinary sphere
• S³: a 3-sphere is a sphere in 4-dimensional Euclidean space.

Spheres for n > 2 are sometimes called hyperspheres.

The n-sphere of unit radius centered at the origin is denoted Sⁿ and is often referred to as "the" n-sphere. The ordinary sphere is a 2-sphere, because it is a 2-dimensional surface which is embedded in 3-dimensional space.

In topology, the n-sphere is an example of a compact topological manifold without boundary. A topological sphere need not be smooth; if it is smooth, it need not be diffeomorphic to the Euclidean sphere (an exotic sphere).

The sphere is the inverse image of a one-point set under the continuous function ‖x‖, so it is closed; Sⁿ is also bounded, so it is compact by the Heine–Borel theorem.

Metric spaces

Main article: Metric space

More generally, in a metric space (E,d), the sphere of center x and radius r > 0 is the set of points y such that d(x,y) = r.

If the center is a distinguished point that is considered to be the origin of E, as in a normed space, it is not mentioned in the definition and notation. The same applies for the radius if it is taken to equal one, as in the case of a unit sphere.

Unlike a ball, even a large sphere may be an empty set. For example, in Zⁿ with Euclidean metric, a sphere of radius r is nonempty only if r² can be written as sum of n squares of integers.

An octahedron is a sphere in taxicab geometry, and a cube is a sphere in geometry using the Chebyshev distance.

History

The geometry of the sphere was studied by the Greeks. Euclid's Elements defines the sphere in book XI, discusses various properties of the sphere in book XII, and shows how to inscribe the five regular polyhedra within a sphere in book XIII. Euclid does not include the area and volume of a sphere, only a theorem that the volume of a sphere varies as the third power of its diameter, probably due to Eudoxus of Cnidus. The volume and area formulas were first determined in Archimedes's On the Sphere and Cylinder by the method of exhaustion. Zenodorus was the first to state that, for a given surface area, the sphere is the solid of maximum volume.

Archimedes wrote about the problem of dividing a sphere into segments whose volumes are in a given ratio, but did not solve it. A solution by means of the parabola and hyperbola was given by Dionysodorus. A similar problem — to construct a segment equal in volume to a given segment, and in surface to another segment — was solved later by al-Quhi.

Biomaterial

 

From Wikipedia, the free encyclopedia

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

A hip implant is an example of an application of biomaterials

A biomaterial is a substance that has been engineered to interact with biological systems for a medical purpose, either a therapeutic (treat, augment, repair, or replace a tissue function of the body) or a diagnostic one. The corresponding field of study, called biomaterials science or biomaterials engineering, is about fifty years old. It has experienced steady and strong growth over its history, with many companies investing large amounts of money into the development of new products. Biomaterials science encompasses elements of medicine, biology, chemistry, tissue engineering and materials science.

Note that a biomaterial is different from a biological material, such as bone, that is produced by a biological system. Additionally, care should be exercised in defining a biomaterial as biocompatible, since it is application-specific. A biomaterial that is biocompatible or suitable for one application may not be biocompatible in another.

IUPAC definition

Material exploited in contact with living tissues, organisms, or microorganisms.

Introduction

Biomaterials can be derived either from nature or synthesized in the laboratory using a variety of chemical approaches utilizing metallic components, polymers, ceramics or composite materials. They are often used and/or adapted for a medical application, and thus comprise the whole or part of a living structure or biomedical device which performs, augments, or replaces a natural function. Such functions may be relatively passive, like being used for a heart valve, or maybe bioactive with a more interactive functionality such as hydroxy-apatite coated hip implants. Biomaterials are also used every day in dental applications, surgery, and drug delivery. For example, a construct with impregnated pharmaceutical products can be placed into the body, which permits the prolonged release of a drug over an extended period of time. A biomaterial may also be an autograft, allograft or xenograft used as a transplant material.

Bioactivity

The ability of an engineered biomaterial to induce a physiological response that is supportive of the biomaterial's function and performance is known as bioactivity. Most commonly, in bioactive glasses and bioactive ceramics this term refers to the ability of implanted materials to bond well with surrounding tissue in either osteo conductive or osseo productive roles. Bone implant materials are often designed to promote bone growth while dissolving into surrounding body fluid. Thus for many biomaterials good biocompatibility along with good strength and dissolution rates are desirable. Commonly, bioactivity of biomaterials is gauged by the surface biomineralization in which a native layer of hydroxyapatite is formed at the surface. These days, the development of clinically useful biomaterials is greatly enhanced by the advent of computational routines that can predict the molecular effects of biomaterials in a therapeutic setting based on limited in vitro experimentation.

Self-assembly

Self-assembly is the most common term in use in the modern scientific community to describe the spontaneous aggregation of particles (atoms, molecules, colloids, micelles, etc.) without the influence of any external forces. Large groups of such particles are known to assemble themselves into thermodynamically stable, structurally well-defined arrays, quite reminiscent of one of the seven crystal systems found in metallurgy and mineralogy (e.g., face-centered cubic, body-centered cubic, etc.). The fundamental difference in equilibrium structure is in the spatial scale of the unit cell (lattice parameter) in each particular case.

Molecular self assembly is found widely in biological systems and provides the basis of a wide variety of complex biological structures. This includes an emerging class of mechanically superior biomaterials based on microstructural features and designs found in nature. Thus, self-assembly is also emerging as a new strategy in chemical synthesis and nanotechnology. Molecular crystals, liquid crystals, colloids, micelles, emulsions, phase-separated polymers, thin films and self-assembled monolayers all represent examples of the types of highly ordered structures, which are obtained using these techniques. The distinguishing feature of these methods is self-organization.

Structural hierarchy

Nearly all materials could be seen as hierarchically structured, since the changes in spatial scale bring about different mechanisms of deformation and damage. However, in biological materials, this hierarchical organization is inherent to the microstructure. One of the first examples of this, in the history of structural biology, is the early X-ray scattering work on the hierarchical structure of hair and wool by Astbury and Woods. In bone, for example, collagen is the building block of the organic matrix, a triple helix with diameter of 1.5 nm. These tropocollagen molecules are intercalated with the mineral phase (hydroxyapatite, calcium phosphate) forming fibrils that curl into helicoids of alternating directions. These "osteons" are the basic building blocks of bones, with the volume fraction distribution between organic and mineral phase being about 60/40.

In another level of complexity, the hydroxyapatite crystals are mineral platelets that have a diameter of approximately 70 to 100 nm and thickness of 1 nm. They originally nucleate at the gaps between collagen fibrils.

Similarly, the hierarchy of abalone shell begins at the nanolevel, with an organic layer having a thickness of 20 to 30 nm. This layer proceeds with single crystals of aragonite (a polymorph of CaCO₃) consisting of "bricks" with dimensions of 0.5 and finishing with layers approximately 0.3 mm (mesostructure).

Crabs are arthropods, whose carapace is made of a mineralized hard component (exhibits brittle fracture) and a softer organic component composed primarily of chitin. The brittle component is arranged in a helical pattern. Each of these mineral "rods" (1 μm diameter) contains chitin–protein fibrils with approximately 60 nm diameter. These fibrils are made of 3 nm diameter canals that link the interior and exterior of the shell.

Applications

Biomaterials are used in:

1. Joint replacements
2. Bone plates 
3. Intraocular lenses (IOLs) for eye surgery
4. Bone cement
5. Artificial ligaments and tendons
6. Dental implants for tooth fixation
7. Blood vessel prostheses
8. Heart valves
9. Skin repair devices (artificial tissue)
10. Cochlear replacements
11. Contact lenses
12. Breast implants
13. Drug delivery mechanisms
14. Sustainable materials
15. Vascular grafts
16. Stents
17. Nerve conduits
18. Surgical sutures, clips, and staples for wound closure
19. Pins and screws for fracture stabilisation
20. Surgical mesh

Biomaterials must be compatible with the body, and there are often issues of biocompatibility, which must be resolved before a product can be placed on the market and used in a clinical setting. Because of this, biomaterials are usually subjected to the same requirements as those undergone by new drug therapies. All manufacturing companies are also required to ensure traceability of all of their products, so that if a defective product is discovered, others in the same batch may be traced.

Bone grafts

Calcium sulfate (its α- and β-hemihydrates) is a well known biocompatible material that is widely used as a bone graft substitute in dentistry or as its binder.

Heart valves

In the United States, 49% of the 250,000 valve replacement procedures performed annually involve a mechanical valve implant. The most widely used valve is a bileaflet disc heart valve or St. Jude valve. The mechanics involve two semicircular discs moving back and forth, with both allowing the flow of blood as well as the ability to form a seal against backflow. The valve is coated with pyrolytic carbon and secured to the surrounding tissue with a mesh of woven fabric called Dacron (du Pont's trade name for polyethylene terephthalate). The mesh allows for the body's tissue to grow, while incorporating the valve.

Skin repair

Main article: Tissue engineering

Most of the time, artificial tissue is grown from the patient's own cells. However, when the damage is so extreme that it is impossible to use the patient's own cells, artificial tissue cells are grown. The difficulty is in finding a scaffold that the cells can grow and organize on. The characteristics of the scaffold must be that it is biocompatible, cells can adhere to the scaffold, mechanically strong and biodegradable. One successful scaffold is a copolymer of lactic acid and glycolic acid.

Properties

As discussed previously, biomaterials are used in medical devices to treat, assist, or replace a function within the human body. The application of a specific biomaterial must combine the necessary composition, material properties, structure, and desired in vivo reaction in order to perform the desired function. Categorizations of different desired properties are defined in order to maximize functional results.

Host response

Host response is defined as the "response of the host organism (local and systemic) to the implanted material or device". Most materials will have a reaction when in contact with the human body. The success of a biomaterial relies on the host tissue's reaction with the foreign material. Specific reactions between the host tissue and the biomaterial can be generated through the biocompatibility of the material.

Biomaterial and tissue interactions

The in vivo functionality and longevity of any implantable medical device is affected by the body's response to the foreign material. The body undergoes a cascade of processes defined under the foreign body response (FBR) in order to protect the host from the foreign material. The interactions between the device upon the host tissue/blood as well as the host tissue/blood upon the device must be understood in order to prevent complications and device failure.

Tissue injury caused by device implantation causes inflammatory and healing responses during FBR. The inflammatory response occurs within two time periods: the acute phase, and the chronic phase. The acute phase occurs during the initial hours to days of implantation, and is identified by fluid and protein exudation along with a neutrophilic reaction. During the acute phase, the body attempts to clean and heal the wound by delivering excess blood, proteins, and monocytes are called to the site. Continued inflammation leads to the chronic phase, which can be categorized by the presence of monocytes, macrophages, and lymphocytes. In addition, blood vessels and connective tissue form in order to heal the wounded area.

Compatibility

Biocompatibility is related to the behavior of biomaterials in various environments under various chemical and physical conditions. The term may refer to specific properties of a material without specifying where or how the material is to be used. For example, a material may elicit little or no immune response in a given organism, and may or may not able to integrate with a particular cell type or tissue. Immuno-informed biomaterials that direct the immune response rather than attempting to circumvent the process is one approach that shows promise. The ambiguity of the term reflects the ongoing development of insights into "how biomaterials interact with the human body" and eventually "how those interactions determine the clinical success of a medical device (such as pacemaker or hip replacement)". Modern medical devices and prostheses are often made of more than one material, so it might not always be sufficient to talk about the biocompatibility of a specific material. Surgical implantation of a biomaterial into the body triggers an organism-inflammatory reaction with the associated healing of the damaged tissue. Depending upon the composition of the implanted material, the surface of the implant, the mechanism of fatigue, and chemical decomposition there are several other reactions possible. These can be local as well as systemic. These include immune response, foreign body reaction with the isolation of the implant with a vascular connective tissue, possible infection, and impact on the lifespan of the implant. Graft-versus-host disease is an auto- and alloimmune disorder, exhibiting a variable clinical course. It can manifest in either acute or chronic form, affecting multiple organs and tissues and causing serious complications in clinical practice, both during transplantation and implementation of biocompatible materials.

Toxicity

A biomaterial should perform its intended function within the living body without negatively affecting other bodily tissues and organs. In order to prevent unwanted organ and tissue interactions, biomaterials should be non-toxic. The toxicity of a biomaterial refers to the substances that are emitted from the biomaterial while in vivo. A biomaterial should not give off anything to its environment unless it is intended to do so. Nontoxicity means that biomaterial is: noncarcinogenic, nonpyrogenic, nonallergenic, blood compatible, and noninflammatory. However, a biomaterial can be designed to include toxicity for an intended purpose. For example, application of toxic biomaterial is studied during in vivo and in vitro cancer immunotherapy testing. Toxic biomaterials offer an opportunity to manipulate and control cancer cells. One recent study states: "Advanced nanobiomaterials, including liposomes, polymers, and silica, play a vital role in the codelivery of drugs and immunomodulators. These nanobiomaterial-based delivery systems could effectively promote antitumor immune responses and simultaneously reduce toxic adverse effects." This is a prime example of how the biocompatibility of a biomaterial can be altered to produce any desired function.

Biodegradable biomaterials

Biodegradable biomaterials refers to materials that are degradable through natural enzymatic reactions. The application of biodegradable synthetic polymers began in the later 1960s. Biodegradable materials have an advantage over other materials, as they have lower risk of harmful effects long term. In addition to ethical advancements using biodegradable materials, they also improve biocompatibility for materials used for implantation. Several properties including biocompatibility are important when considering different biodegradable biomaterials. Biodegradable biomaterials can be synthetic or natural depending on their source and type of extracellular matrix (ECM).

Biocompatible plastics

Some of the most commonly-used biocompatible materials (or biomaterials) are polymers due to their inherent flexibility and tunable mechanical properties. Medical devices made of plastics are often made of a select few including: cyclic olefin copolymer (COC), polycarbonate (PC), polyetherimide (PEI), medical grade polyvinylchloride (PVC), polyethersulfone (PES), polyethylene (PE), polyetheretherketone (PEEK) and even polypropylene (PP). To ensure biocompatibility, there are a series of regulated tests that material must pass to be certified for use. These include the United States Pharmacopoeia IV (USP Class IV) Biological Reactivity Test and the International Standards Organization 10993 (ISO 10993) Biological Evaluation of Medical Devices. The main objective of biocompatibility tests is to quantify the acute and chronic toxicity of material and determine any potential adverse effects during use conditions, thus the tests required for a given material are dependent on its end-use (i.e. blood, central nervous system, etc.).

Surface and bulk properties

Two properties that have a large effect on the functionality of a biomaterial is the surface and bulk properties.

Bulk properties refers to the physical and chemical properties that compose the biomaterial for its entire lifetime. They can be specifically generated to mimic the physiochemical properties of the tissue that the material is replacing. They are mechanical properties that are generated from a material's atomic and molecular construction.

Important bulk properties:

• Chemical Composition
• Microstructure
• Elasticity
• Tensile Strength
• Density
• Hardness
• Electrical Conductivity
• Thermal Conductivity

Surface properties refers to the chemical and topographical features on the surface of the biomaterial that will have direct interaction with the host blood/tissue. Surface engineering and modification allows clinicians to better control the interactions of a biomaterial with the host living system.

Important surface properties:

• Wettability (surface energy)
• Surface chemistry
• Surface textures (smooth/rough)
  • Topographical factors including: size, shape, alignment, structure determine the roughness of a material.
• Surface Tension
• Surface Charge

Mechanical properties

In addition to a material being certified as biocompatible, biomaterials must be engineered specifically to their target application within a medical device. This is especially important in terms of mechanical properties which govern the way that a given biomaterial behaves. One of the most relevant material parameters is the Young's Modulus, E, which describes a material's elastic response to stresses. The Young's Moduli of the tissue and the device that is being coupled to it must closely match for optimal compatibility between device and body, whether the device is implanted or mounted externally. Matching the elastic modulus makes it possible to limit movement and delamination at the biointerface between implant and tissue as well as avoiding stress concentration that can lead to mechanical failure. Other important properties are the tensile and compressive strengths which quantify the maximum stresses a material can withstand before breaking and may be used to set stress limits that a device may be subject to within or external to the body. Depending on the application, it may be desirable for a biomaterial to have high strength so that it is resistant to failure when subjected to a load, however in other applications it may be beneficial for the material to be low strength. There is a careful balance between strength and stiffness that determines how robust to failure the biomaterial device is. Typically, as the elasticity of the biomaterial increases, the ultimate tensile strength will decrease and vice versa. One application where a high-strength material is undesired is in neural probes; if a high-strength material is used in these applications the tissue will always fail before the device does (under applied load) because the Young's Modulus of the dura mater and cerebral tissue is on the order of 500 Pa. When this happens, irreversible damage to the brain can occur, thus the biomaterial must have an elastic modulus less than or equal to brain tissue and a low tensile strength if an applied load is expected.

For implanted biomaterials that may experience temperature fluctuations, e.g., dental implants, ductility is important. The material must be ductile for a similar reason that the tensile strength cannot be too high, ductility allows the material to bend without fracture and also prevents the concentration of stresses in the tissue when the temperature changes. The material property of toughness is also important for dental implants as well as any other rigid, load-bearing implant such as a replacement hip joint. Toughness describes the material's ability to deform under applied stress without fracturing and having a high toughness allows biomaterial implants to last longer within the body, especially when subjected to large stress or cyclically loaded stresses, like the stresses applied to a hip joint during running.

For medical devices that are implanted or attached to the skin, another important property requiring consideration is the flexural rigidity, D. Flexural rigidity will determine how well the device surface can maintain conformal contact with the tissue surface, which is especially important for devices that are measuring tissue motion (strain), electrical signals (impedance), or are designed to stick to the skin without delaminating, as in epidermal electronics. Since flexural rigidity depends on the thickness of the material, h, to the third power (h³), it is very important that a biomaterial can be formed into thin layers in the previously mentioned applications where conformality is paramount.

Structure

The molecular composition of a biomaterial determines the physical and chemical properties of a biomaterial. These compositions create complex structures that allow the biomaterial to function, and therefore are necessary to define and understand in order to develop a biomaterial. biomaterials can be designed to replicate natural organisms, a process known as biomimetics. The structure of a biomaterial can be observed at different at different levels to better understand a materials properties and function.

Atomic structure

Rutherford model of lithium-7's atomic structure

The arrangement of atoms and ions within a material is one of the most important structural properties of a biomaterial. The atomic structure of a material can be viewed at different levels, the sub atomic level, atomic or molecular level, as well as the ultra-structure created by the atoms and molecules. Intermolecular forces between the atoms and molecules that compose the material will determine its material and chemical properties.

The sub atomic level observes the electrical structure of an individual atom to define its interactions with other atoms and molecules. The molecular structure observes the arrangement of atoms within the material. Finally the ultra-structure observes the 3-D structure created from the atomic and molecular structures of the material. The solid-state of a material is characterized by the intramolecular bonds between the atoms and molecules that comprise the material. Types of intramolecular bonds include: ionic bonds, covalent bonds, and metallic bonds. These bonds will dictate the physical and chemical properties of the material, as well as determine the type of material (ceramic, metal, or polymer).

Microstructure

A unit cell shows the locations of lattice points repeating in all directions.

The microstructure of a material refers to the structure of an object, organism, or material as viewed at magnifications exceeding 25 times. It is composed of the different phases of form, size, and distribution of grains, pores, precipitates, etc. The majority of solid microstructures are crystalline, however some materials such as certain polymers will not crystallize when in the solid state.

Crystalline structure

Crystalline structure is the composition of ions, atoms, and molecules that are held together and ordered in a 3D shape. The main difference between a crystalline structure and an amorphous structure is the order of the components. Crystalline has the highest level of order possible in the material where amorphous structure consists of irregularities in the ordering pattern. One way to describe crystalline structures is through the crystal lattice, which is a three-dimensional representation of the location of a repeating factor (unit cell) in the structure denoted with lattices. There are 14 different configurations of atom arrangement in a crystalline structure, and are all represented under Bravais lattices.

Defects of crystalline structure

Main article: Crystallographic defect

During the formation of a crystalline structure, different impurities, irregularities, and other defects can form. These imperfections can form through deformation of the solid, rapid cooling, or high energy radiation. Types of defects include point defects, line defects, as well as edge dislocation.

Macrostructure

Macrostructure refers to the overall geometric properties that will influence the force at failure, stiffness, bending, stress distribution, and the weight of the material. It requires little to no magnification to reveal the macrostructure of a material. Observing the macrostructure reveals properties such as cavities, porosity, gas bubbles, stratification, and fissures. The material's strength and elastic modulus are both independent of the macrostructure.

Natural biomaterials

Biomaterials can be constructed using only materials sourced from plants and animals in order to alter, replace, or repair human tissue/organs. Use of natural biomaterials were used as early as ancient Egypt, where indigenous people used animal skin as sutures. A more modern example is a hip replacement using ivory material which was first recorded in Germany 1891.

Valuable criteria for viable natural biomaterials:

• Biodegradable
• Biocompatible
• Able to promote cell attachment and growth
• Non-toxic

Examples of natural biomaterials:

• Alginate
• Matrigel
• Fibrin
• Collagen
• Myocardial tissue engineering

Biopolymers

Main article: Biopolymer

Biopolymers are polymers produced by living organisms. Cellulose and starch, proteins and peptides, and DNA and RNA are all examples of biopolymers, in which the monomeric units, respectively, are sugars, amino acids, and nucleotides. Cellulose is both the most common biopolymer and the most common organic compound on Earth. About 33% of all plant matter is cellulose. On a similar manner, silk (proteinaceous biopolymer) has garnered tremendous research interest in a myriad of domains including tissue engineering and regenerative medicine, microfluidics, drug delivery.