The element Nitrogen (N2) was discovered in 1772 by Scott physician Daniel Rutherfield, who called it “ noxious air” or “fixed air”. It is the fifth most abundant element in the universe and makes up the major portion (78.03% by volume and 75.5% by weight) of the earths atmosphere. Nitrogen is inert and will not support combustion. It occurs in all living organisms. It is an important constituent element of amino acids which make up proteins which form nucleic acids (DNA and RNA); resides in the chemical structure of almost all neurotransmitters and is a defining component of alkaloids. The name Nitrogen is derived from Greek words “Nitron” which means ‘saltpeter’ and “genes” which means ‘forming. Gaseous Nitrogen is referred to as N2. Nitrogen is pronounced as NYE-Treh - gen.
Properties of Nitrogen
Nitrogen is a nonmetal with an electronegativity of 3.0. It has five electrons in its outer shell, therefore trivalent in most compounds. The triple bond in molecular nitrogen (N2) is one of the strongest in nature. The resulting difficulty of converting (N2) into other compounds, and the ease (and associated high energy release) of converting nitrogen compounds into elemental N2, have dominated the role of nitrogen in both nature and human economic activities.
At atmospheric pressure, molecular nitrogen condenses (liquefies) at 77 K (−195.8°C) and freezes at 63 K (−210.0°C) into the beta hexagonal close-packed crystal allotropic form. Below 35.4 K (−237.6°C), nitrogen assumes the alpha cubic crystal allotropic form. Liquid nitrogen, a fluid resembling water, but with 80.8% of the density, is a common cryogen. Under extremely high pressures (1.1 million atm) and high temperatures (2000 K), as produced under diamond anvil conditions, Nitrogen polymerizes into the single bonded diamond crystal structure, an allotrope nicknamed “Nitrogen Diamond”.
Occurrence of Nitrogen
Nitrogen is the largest single constituent of the earth's atmosphere (78.082% by volume of dry air, 75.3% by weight in dry air). It is created by fusion processes in stars and is estimated to be the 7th most abundant chemical element by mass in the universe.
Nitrogen is present in all living organisms in proteins, nucleic acids and other molecules. It typically makes up around 4% of the dry weight of plant matter, and around 3% of the weight of the human body. It is a large component of animal waste usually in the form of urea, uric acid, ammonium compounds and derivatives of these nitrogenous products, which are essential nutrients for all plants that are unable to fix atmospheric nitrogen. Nitrogen occurs naturally in a number of minerals, such as saltpeter (potassium nitrate), Chile saltpeter (sodium nitrate) and salt ammoniac (ammonium chloride). Most of these are relatively uncommon, partly because of the minerals' ready solubility in water.
LIQUID NITROGEN (LN2)
Liquid Nitrogen is nitrogen gas from the atmosphere in which the molecules of N2 have been pressed together using pressure.
This is a form of chilled, condensed gaseous nitrogen. It is a colorless, odorless, nonflammable, noncorrosive and non-toxic liquid that is lighter than water.
Liquid Nitrogen is normally referred to as LN2. When nitrogen is converted to liquid form it becomes a cryogenic fluid. A cryogen is a substance used to cool other substances.
Definition: Liquid nitrogen is a cold, liquefied gas with a temperature of −321° F.
UNIQUE PROPERTIES OF LN2
A. Has a cold boiling point…does such a thing exist?
At standard temperature and pressure, nitrogen exists in a gaseous state. At −320° F, nitrogen changes its gaseous form to a liquid state, thus making liquid nitrogen extremely cold. The same temperature at which a substance condenses is the same temperature at which it boils, so nitrogen has a cold boiling point…. It may sound weird but is true logically.
Most of us tend to think of the temperature at which something boils to be a high one… but not in the case with nitrogen, which has a unique “Cold boiling point”. It is extremely cold (LN2 boils at −3204° F or −195.8°C) hereby handling of liquid nitrogen requires the use of protective gloves and goggles.
B. Leidenfrost effect
This is the ability of liquid nitrogen to boil immediately on contact with a warmer object, enveloping the object in the insulating nitrogen gas.
The negating consequence of this Leidenfrost effect is reduction in the effective coolant property of liquid nitrogen.
C. Smoking of liquid nitrogen
When LN2 rapidly boils and vaporizes on a smooth surface with a temperature that is much higher than nitrogen's boiling point, a physical phenomenon known as the “Leidenfrost effect” is observed. The liquid nitrogen vaporizes quickly and lifts itself above the surface. It hovers, producing little or no friction on the surface. If the surface is irregular, this effect cannot occur and the vaporization is even more rapid. The nitrogen vapor spreads itself out through the air picking up water vapor along the way. LN2 smokes because of the presence of water in the air. The amount of water present in the atmosphere is dependent upon the temperature of the air. Warmer air can hold more water vapor than cooler air. LN2 cools the air to condense the water out of it. These water droplets scatter light and produce the “smoking” effect. The ability of liquid nitrogen to do this makes it very popular at parties, magic shows and in classroom demonstrations.
Physical and Chemical Properties of Liquid Nitrogen (LN2)
Physical state: Liquid
Appearance and odor: Colorless, odorless
Odor threshold (PPM): Odorless
Vapor pressure: Gas@ 70°F (21°C)
Vapor sp. gravity (air = 1): 0.967 @ 70°F (21°C)
Volatiles (% by volume): 100%
-195.8°C (760 mmHg)
Solubility in water (%): Slight.
Stability and Reactivity of Liquid Nitrogen (LN2)
Chemical stability: Stable
Conditions of reactivity: Heat
Hazardous polymerization: Will not occur
Fatty substances in cryogenic grinding
Hazardous decomposition products: None
Hazard Identification of Liquid Nitrogen (LN2)
Hazard overview: Nitrogen gas is colorless, odorless and non-flammable.
The primary health hazard is asphyxiation by displacement of oxygen. Maintain oxygen levels above 19.5%. Contact with the liquid or cold gas can cause freezing of exposed tissue.
Route of entry: Inhalation, skin and eye contact
Effects of acute exposure:
Can cause frostbite (liquid form).
Vapor may cause a stinging sensation
Can cause frostbite (liquid form).
No adverse effects from gas
May cause dizziness.
Can cause vomiting.
May result in unconsciousness
May cause excitation, excess salivation, rapid breathing
May cause stinging of the nose and throat
Ingestion: Not a likely route of exposure
Effects of chronic exposure: Damage to retinal ganglion cells and central nervous system may occur due to the presence of carbon dioxide.
Production of Liquid Nitrogen
Liquid nitrogen (liquid density at the triple point is 0.707 g/ml) is the liquid produced industrially in large quantities by fractional distillation of liquid air. A common method for the production of LN2 is the liquefaction of air. Liquefaction is the phase change of a substance from the gas phase to the liquid phase. In liquid nitrogen compressors or generators air is compressed, expanded and cooled via the Joule-Thompson effect. Since nitrogen boils at a different temperature than oxygen, the nitrogen can be distilled out of the liquid air, re-compressed and then re-liquefied. Once liquid nitrogen is removed from the distillation chamber it is stored in special containers. It is then made available for commercial distribution.
Containers for Liquid Nitrogen
Typical cryogenic liquid cylinder is an insulated, vacuum-jacketed pressure vessels, which are specifically designed and made of materials that can withstand the rapid changes and extreme temperature differences encountered in working with liquid nitrogen.
Equipped with (1) safety relief valves, (2) Rupture disks, (3) Loose fitting lids to protect the cylinders from pressure build-up. These containers operate at pressures up to 350 psig and have capacities between 80 and 450 liters of liquid. Product may be withdrawn as a gas by passing liquid through an internal vaporizer or as a liquid under its own vapor pressure. Even these special containers should be filled slowly to minimize the internal stresses that occur when any material is cooled. Excessive internal stresses can damage the container. Use only containers designed for low-temperature liquids (Figs 1.1 and 1.2).
Dewars are designed for storing and dispensing small quantities of liquid nitrogen. Easy to operate, the snap on cap and neck-tube core assure positive closure and easy access without unnecessary exposures to the cryogen. Constructed from materials of the highest performance and quality including an aluminum exterior, these containers are not only rugged, robust, and dependable, but also have very high thermal efficiencies.
The safe handling and use of liquid nitrogen in liquid nitrogen dewars or flasks is possible only by knowing the potential hazards. There are two important properties of liquid nitrogen that present potential hazards:
- It is extremely cold.
- At atmospheric pressure, liquid nitrogen boils at − 320° F/-196° C. Very small amounts of liquid vaporize into large amounts of gas. One liter of liquid nitrogen becomes 24.6 ft3/0.7 m3 of gas.
The safety precautions as outlined must be followed to avoid potential injury or damage, which could result from these two characteristics. Any attempt to handle liquid nitrogen must be done only after fully understanding the potential hazards, their consequences, and the related safety precautions.
Safety Measures while Handling Liquid Nitrogen
- Precautions while transferring liquid nitrogen: Use a phase separator or special filing funnel to prevent splashing and spilling when transferring liquid nitrogen into or from a dewar. The top of the funnel should be partly covered to reduce splashing. Use only small, easily handled Dewars for pouring liquid. For the larger, heavier containers, use a cryogenic liquid withdrawal device to transfer liquid from one container to another. Be sure to follow instructions supplied with the withdrawal devices.
- Do not overfill containers: Filling above the bottom of the neck tube (or specified maximum level) can result in overflow and spillage of liquid when the neck tube core or cover is placed in the opening. Therefore avoid overfilling the cryocans.
- Special kind of dipstick to be used to measure the level: When a warm tube is inserted into liquid nitrogen, liquid will spout from the bottom of the tube due to gasification and rapid expansion of liquid inside the tube. Wooden or solid metal dipsticks are recommended; avoid using plastics that may become very brittle at cryogenic temperatures, which then become prone to shatter like a fragile piece of glass. Never use hollow rods or tubes.
- Store and use liquid nitrogen only in a well-ventilated place: As the liquid evaporates, the resulting gas tends to displace the normal air from the area. In closed areas, excessive amounts of nitrogen gas reduce the concentration of oxygen and can result in asphyxiation. Because nitrogen gas is colorless, odorless and tasteless, it cannot be detected by the human senses and will be inhaled as if it were air. Inhaling air that contains less than 18 percent oxygen can cause dizziness and quickly result in unconsciousness and death.
- Disposal of liquid nitrogen: Never dispose of liquid nitrogen in confined areas or places where others may enter. It should be done outdoors in a safe place. Pour the liquid slowly on gravel or bare earth where it can evaporate without causing damage.
- Handling liquid nitrogen dewar: Keep the unit upright at all times. Tipping the container or laying it on its side can cause spillage of liquid nitrogen. It may also damage the container and any materials stored in it. Rough handling can cause serious damage to dewar's such as, allowing it to fall over on its side, or subjecting it to sharp impact or severe vibration can result in partial or complete loss of vacuum. To protect the vacuum insulation system, handle containers carefully.Do not drag or roll these units across a floor. Use a dolly or handcart when moving containers. Large units are heavy enough to cause personal injury or damage to equipment if proper lifting and handling techniques are not used.When transporting a liquid nitrogen dewar, maintain adequate ventilation and protect the unit from damage:Do not place these units in closed vehicles where the nitrogen gas that is continuously vented from unit can accumulate. Prevent spillage of liquids and damage to unit by securing it in the upright position so that it cannot be tipped over. Protect the unit from severe jolting and impact that could cause damage, especially to the vacuum seal.
- Keep the unit clean and dry: Do not store it in wet, dirty areas. Moisture, animal waste, chemicals, strong cleaning agents and other substances, which could promote corrosion, should be removed promptly. Use water or mild detergent for cleaning and dry the surface thoroughly. Do not use strong alkaline or acid cleaners that could damage the finish and corrode the metal shell.
- Protect dewar contents: Materials stored in a liquid nitrogen dewar with a wide mouth are protected by the extremely low temperature of the liquid nitrogen or the gas that issues from the evaporating liquid nitrogen. When all of the liquid nitrogen has evaporated, the temperature inside the unit will rise slowly to ambient. The rate at which the liquid nitrogen will evaporate depends upon the pattern of container use and the age and condition of the container. Evaporation increases as insulation efficiency deteriorates with age and rough handling. Opening and closing to insert and remove materials and moving the unit will also increase the evaporation rate.To protect valuable material stored in a liquid nitrogen cryocan, check the liquid level in unit frequently. It is of the greatest importance to check the liquid nitrogen level periodically in order to prevent critical samples from deteriorating when no liquid nitrogen is in the cryocan.
- When to replace existing cryocans: Condensed moisture or frost on the outer shell of a cryocan and abnormally rapid evaporation of the liquid nitrogen are indications of vacuum loss. If vacuum loss is evident or suspected, start thinking immediately about the procurement of a replacement dewar. It is just not cost effective to continue to use a dewar with a bad vacuum and waste valuable liquid nitrogen in the process. There is also the safety issue of excessive boil-off in an enclosed area that is not large enough to “absorb” the higher rate of nitrogen boil off.
APPLICATION OF LIQUID NITROGEN
Liquid nitrogen is valued for its coldness as well as inertness:
This combination is employed to rapidly chill and freeze food items (semen, oocytes, embryos, meat, fruit, vegetables, baked goods, dairy products). Rapid freezing results in very small ice crystals, less cellular damage, and better-quality products after thawing, i.e. the process of “Vitrification”.
The intense cold produced by these products can also be used to make normally soft and flexible materials hard and rigid, allowing them to be ground, machined or fractured.
When liquid nitrogen is vaporized and warmed to ambient temperature, it absorbs a large quantity of heat. The combination of inertness and its intensely cold initial state makes liquid nitrogen an ideal coolant for certain applications such as freezing.
Main Uses of LN2
- Freezing and preservation of food: Foods can be packed, sealed and then sprayed with liquid nitrogen as it evaporates upon contact with many surfaces (including food). Thus, evaporation process allows for the absorption of heat and energy from the food as a result the molecules in the food slow down and the food freezes
- Cryopreservation: Liquid nitrogen is widely used in the preservation of medical specimens it is particularly useful for very long-term preservation of cells and tissues. It may be used for the rapid freezing of different tissues such as bone marrow, blood, sperms, ovarian tissue, embryos, etc. It is useful in the preservation of animal embryos, bacteria and fungi.
- Cryotherapy: Extremely cold temperatures can kill human tissue. Liquid nitrogen may be used in standard medical procedures such as wart removal and treatment of certain skin cancers.
- Cryonic preservation of humans and pets in the hope of future reanimation.
- Research laboratories: Many research labs around the world use liquid nitrogen to aid in research requiring cryogenic conditions (i.e. superconductivity labs).
- Classroom demonstrations: Liquid nitrogen is commonly used in the classroom to demonstrate its amazing chemical properties and to illustrate its cooling effects on other materials.
HAZARDS OF NITROGEN GAS
Rapid release of nitrogen gas into an enclosed space can displace oxygen, and therefore represents an asphyxiation hazard. This may happen with few warning symptoms, since the human carotid body is a relatively slow and a poor low oxygen (hypoxia) sensing system. When inhaled at high partial pressures (more than about 3 atmospheres, encountered at depths below about 30 m in scuba diving) nitrogen begins to act as an anesthetic agent. It can cause nitrogen narcosis, a temporary semi-anesthetized state of mental impairment similar to that caused by nitrous oxide. Nitrogen also dissolves in the bloodstream and body fats. Rapid decompression (particularly in the case of divers ascending too quickly, or astronauts decompressing too quickly from cabin pressure to spacesuit pressure) can lead to a potentially fatal condition called decompression sickness (formerly known as caisson sickness or more commonly, the “bends”), when nitrogen bubbles form in the bloodstream, nerves, joints, and other sensitive or vital areas.
- Cryovials stored in liquid nitrogen may “explode” when removed from the dewar. Cryo vials are not guaranteed to be leak tight if stored in the liquid nitrogen (LN2). Due to “superfluidity” of LN2 it can leak into sealed cyrovials. Although these cryovials are made of a tough plastic which prevents cracking at the liquid nitrogen temperatures of −196°C (−321°F), the vials in which the liquid nitrogen has leaked when removed from the dewar causes expansion of liquid nitrogen and vials can easily explode. To site a hypothecal situation where in simple calculation shows that 0.5 g of liquid nitrogen in a 1.5 ml cryovial will generate a pressure of 4,053 psi when it evaporates. Failure of the screw threads can turn the cap into a projectile with an initial velocity of up to 296 miles per hour (132 meters/sec), and up to 8.4% as much kinetic energy as a 22-caliber bullet! The rapidly expanding gas can be just as dangerous.Precautions when storing samples in LN2 wear cryogloves, face sheild, safety glasses when removing samples.Various ways for liquid nitrogen to enter into a cryovial:
- Defective vials made of inferior quality silicone or plastic.Always use standard cryovials such as Nunc cryovials.
- The screw cap of the cryovial is overtightened or undertightened or there are droplets of water on the threads.
- Using expired cryovials.Nunc tubes must be used within three years of their date of sterilization because the silicone gasket deteriorates over time.
- Frostbite hazard: Contact of sufficient quality of liquid nitrogen with the body, results in a “cold burn” or a “cryogenic burn” within seconds, though not instantly on contact, depending on form of liquid nitrogen (liquid vs. mist) and surface area of the nitrogen-soaked material (soaked clothing or cotton causing more rapid damage than a spill of direct liquid to skin, which for a few seconds is protected by the Leidenfrost effect).Small amounts of LN2 will rapidly evaporate and will give a small sensation similar to a pin prick. The danger comes from larger quantities which do not evaporate quickly and cause more harm. Should a larger quantity come in contact with a person, the person should immediately get away.Prevent frostbite hazard by: Donning protective clothing to keep LN2 from contacting the body.Precautions when LN2 spillage occurs:
- If a large spill occurs, discontinue filling and leave the room until the liquid evaporates.
- If clothing becomes soaked, hold it away from the body until it warms, or remove the clothing.
- If a cold burn occurs, once warmed up, it will appear very similar to, and should be treated the same as a sunburn. Warm affected skin slowly using cold (NOT HOT) water only.
- Any serious cold burn should be treated by a doctor. Ample care should be taken while handling liquid nitrogen avoid spilling the liquid on clothing, since the clothes can easily become saturated with the liquid and being in contact next to the skin for a significant period of time, leading to serious burns. Thus for this reason, cloth gloves may be worse than nothing at all.
- Asphyxiation: Liquid nitrogen rapidly evaporates giving nitrogen gas. Just one liter of liquid produces around 700 liters of gas at atmospheric pressure, displacing significant quantities of breathable air if the gas is released in a confined space, e.g. in a laboratory, cold room, or storage area. The problem is compounded by nitrogen's tendency to accumulate at low levels where it is less easily dispersed than the ambient atmosphere. Thus, an apparently small spillage could lead to dangerously low oxygen levels, presenting a serious hazard to personnel working in the confined area. This condition is called Oxygen Deficiency Hazard (ODH). Gas sensing monitors are available to insure that this O2 level is maintained at all times is a preventive measure to avoid ODH for personnel working with liquid nitrogen (Fig. 1.3).The human body does not detect oxygen deficiencies very well. The feeling of being suffocated is due to excess carbon dioxide, not from a lack of oxygen, so symptoms may not be evident. The normal oxygen level in the atmosphere is 20.9%. Physical and intellectual performance may be inhibited if oxygen levels fall below 17%, and at levels less than 17%, symptoms of asphyxia, such as gasping, vomiting or collapse, will set in. Victims may well not be aware of their condition, especially if the oxygen level falls rapidly, and in the event of a major leak or spillage could fall unconscious without ever being aware of the danger. When the oxygen content of air is reduced to around 10%, unconsciousness can be immediate with virtually no warning thus creating an unsafe environment for workers. If there should be a large spill anywhere in the building, don't wait around notify everyone that they need to evacuate the area.Liquid nitrogen should never be carried in an elevator, because of the remote possibility that the dewar containing the liquid will go “soft” (i.e., the vacuum in the dewar flask will be lost). Never put liquid nitrogen, in styrofoam coffee cups. Although the styrofoam will keep the liquid from evaporating, the risk is too great that someone will walk by and mistake it for limca or sprite and try to take a sip.
- Fire and explosion hazard: Liquid nitrogen is not flammable however it is capable of condensing oxygen out of air creating oxygen rich environment. Flammable materials can ignite in the presence of condensed oxygen.
- Property damage: On spillage of liquid nitrogen on floor tiles it can cause cracking and damage to any rubber tubing. Cracked floors can cause a tipping hazard to personnel working in the laboratory.
First Aid for Liquid Nitrogen Exposure
If a person complains of dizziness or loses consciousness while working with liquid nitrogen, move to a well-ventilated area immediately. If breathing has stopped, apply artificial respiration, give oxygen and call a physician. Keep warm and at rest.
If skin is exposed to liquid nitrogen, restore tissue to normal body temperature 98.6° F (37°C) as rapidly as possible, followed by protection of the injured tissue from further damage and infection. Remove or loosen clothing that may constrict blood circulation to the frozen area. Call a physician. Rapid warming of the affected part is best achieved by running the exposed part under water at 108° F (42° C) for fifteen minutes. Under no circumstances should the water be over 112° F (44°C), nor should the frozen part be rubbed either before or after rewarming as it causes further tissue damage. If the finger is burned do not place it in your mouth. This could burn your mouth. The patient should neither smoke, nor drink alcohol.
- Wear safety goggles at all times while handling liquid nitrogen.
- Never place liquid nitrogen in a sealed container or any object that could cause entrapment of the gas.
- Never mix dry ice or liquid nitrogen with water or water ice or pour it down the sink. Ice can solidify around it, trapping the gas at a high pressure.
- Vials to be stored in contact with liquid phase nitrogen must always be sealed with Nunc CryoFlex, which is a type of heat-shrinkable tubing.
- When cooling objects using liquid nitrogen always do so slowly using tongs to prevent boiling and splashing.
- Use a cryoclaw to retrieve the samples that have fallen into the dewar.
- Liquid nitrogen should never be carried in an elevator, because of the remote possibility that the dewar containing the liquid will go “soft” (i.e. the vacuum in the dewar flask will be lost).
- Never put liquid nitrogen, in styrofoam coffee cups. Although the styrofoam will keep the liquid from evaporating, the risk is too great that someone will walk by and mistake it for limca or sprite and try to take a sip.
Personal Protective Equipment (PPE)
One must be thoroughly familiar with the properties and safety considerations before handling a cryogenic liquid and its associated equipment. The eyes are the most sensitive body part to the extreme cold of the liquid and vapors of liquid nitrogen.
The recommended personal protective equipment (PPE) for handling cryogens includes:
- Full face shield over safety glasses.
- Long sleeve shirts, and trousers without cuffs.
- Safety shoes are recommended for people involved in the handling of containers. Depending on the application, special clothing suitable for that application may be advisable.
- Cryogloves of thermal insulated or leather gloves which are loose fitting so they can be easily and quickly removed if cryogenic liquid is spilled on them. Insulated gloves are not made to permit the hands to be put into a cryogenic liquid. They will only provide short-term protection from accidental contact with the liquid.
Subcooling of the LN2 is accomplished by direct or indirect evaporative cooling which relies on the thermodynamic principle that the boiling point of a liquid of a pure substance in equilibrium with its gaseous phase is a function of pressure. Therefore, as the pressure exerted on liquid nitrogen is lowered, its boiling point decreases resulting in evaporative cooling of the liquid nitrogen. That is, the energy required for the liquid nitrogen to evaporate is supplied by the liquid nitrogen, therefore, resulting in a lowering of the liquid's temperature. For example, for LN2, at a pressure of about 1.8 pounds per square inch (psia), it will l have a temperature of about −210°C. In the receptacle of the vacuum chamber of the refrigeration system, a large volume of LN2 is subjected to a partial vacuum of from about 1.8 psia up to about 14 psia, preferably from about 1.82 to about 3 psia, such that the temperature of the LN2 subcooled refrigerant is lowered to a temperature below its normal boiling point near its triple point at which the liquid and solid nitrogen are in equilibrium and form a thickened mixture referred to as “Nitrogen slush” in which liquid and solid nitrogen coexist.
By applying negative pressure with a vacuum, LN2 will freeze and will be transformed into a slush state with a lower internal temperature of −210°C without vaporization, thus SN2 may offer high-speed cooling rates with a possibility to increase the survival rate of mammalian oocytes as well as other characteristics after vitrifying/warming. As it is seen boiling of liquid nitrogen (LN2) occurs when a sample is immersed and results in gas bubbles around the specimen which, in turn, results in poor heat transfer.
The nitrogen slush or subcooled liquid nitrogen is commonly referred to as SN2. However, it is still controversial whether SN2 has a beneficial effect on the development of embryos after cryopreservation.
- National Science Foundation's Graduate Teaching Fellows in K-12 Education (GK-12) program (DGE # 0231913 and # 9979516) and Cornell University Office of environmental health and Safety University of Pennsylvania
- Physics of human Body with illustrative examples from Medicine and Biology (Springer, Verlag, New York, 2000).