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I wouldn’t call it downright rude, but it’s really not the best phrasing. “Here is my LinkedIn profile” puts the onus completely on the recipient to go read it and analyze it. “Please let me know what you have available” implies you haven’t looked into it yourself, which implies either “too lazy to bother” or “too stupid to figure out how”, or of course both, asking them to go do your work for you.On second thought, yeah, that’s inconsiderate enough to call rude. :-)So let’s look at some alternatives. Suppose you’re an Assistant Nosepicker, selecting noses off the conveyor belt to put into boxed sets at the Mr. Turnip factory of Hassis. You want to become a full Component Selector/Assembler (that sounds SO much nicer!) at your competitor, Muttel, makers of Blurby Dolls, the ones with the very short ads. You spot one of their recruiters, Rita “Wreck” Rooter, on LinkedIn! Yay! You write her… what? NOTHING YET, first you scope out the goods! No, not her picture, I mean the jobs!You go to the Careers section of Muttel’s web site. They’ve got openings! Mostly in marketing, distribution, and artificial zombie control, but there are some for CS/As, and even a few in cities you wouldn’t mind moving to. Let’s s’pose those are Pismo Beach CA, Albuquerque NM, and Benson AZ. You also look at their benefits, Mission Statement, etc. and it all looks good. So you decide to go ahead and contact Rita.But wait! There’s more! You happen to notice that Rita is in El Paso, not very far from two of the three. And she’s been at Muttel for several years now. This is not necessary, but useful information. It is however necessary to scour your sources for whatever useful info you can find.So now you can write to Rita. And you say… what? Maybe something like:===8<——cut here——Dear Ms. Rooter:I am interested in the Component Selector/Assembler positions advertised at https://www.muttel.com/careers/csa.html, in Benson AZ, Albuquerque NM, and Pismo Beach CA. I saw the application link but thought it would probably be more productive to contact you directly; I hope you don’t mind. I’d also like your advice on which of those locations is best, seeing as you are fairly close to two of the three, what you think of working for Muttel, and where you think it is headed in the near future.I have twelve years of experience in selecting assorted components at the Mr. Tuber factory here in Taterville ID, and assembling them into the final boxed sets. I worked my way up from sweeping the floors as a Junior Dust Collector, and am ready and anxious for further growth, but there are no higher openings here in tiny little Taterville, be it at the Hassis toy factory or elsewhere. Further details are on my LinkedIn profile (https://www.linkedin.com/in/nosepicker) and my resume (https://www.geocities.com/nosepicker/resume.html). I am also so anxious to get out of this freezing state and into some southwestern sunshine, that I’ll even pay my own relocation expenses! :-)Thank you very much,Hassan Chop===8<——cut here——Note how:You stated what you were interested in, rather than asking what there was. This shows decisiveness, as opposed to lack of consideration for her time.You stated why you were contacting her directly rather than the Usual Channels.You stated your qualifications for the job, rather than making her go read.You supplied links so she doesn’t have to go searching, whether for her jobs or your resume or whatever.You even subtly snuck in there some things that imply you read about her, though not in too creepy/stalkerish a way, and value her opinion (which is sure to flatter anyone, except of course us Quora question-answerers (yeah right!)).You show more decisiveness, and eagerness to grow, rather than being just some dullard satisfied doing the same menial labor over and over, even if it means moving about a thousand miles.You snuck in a little humor, not in an unprofessional way, and made a case whereby you might be seen as cheaper than someone else (who might demand relo expenses).Despite all that, you kept it reasonably concise.So, try it this way, with your own personal spins added, and let us know how it goes!Or alternately, if you’re asking because you’re a recruiter yourself and have received things like what you wrote above, surely you have some innnnteresting stories to tell!

Here is what Wikipedia says about supercapacitor materials. While it seems transition elements involved in pseudocapacitors might be rarer than carbon used in true supercapacitors - no materials are in danger of running out. Things in nature don’t run out - they just get more and more expensive over time as they become progressively scarcer. Energy for example is something like 10x more expensive than it was in the 1950s - and use patterns have adjusted to that.Given the fact that carbon and other common materials make supercapacitors, they will likely be used in supercapacitor design in favour of more expensive transition elements anyway.Wikipedia entry on materials and supercapacitorsThe properties of supercapacitors come from the interaction of their internal materials. Especially, the combination of electrode material and type of electrolyte determine the functionality and thermal and electrical characteristics of the capacitors.A micrograph of activated carbon under bright field illumination on a light microscope. Notice the fractal-like shape of the particles, hinting at their enormous surface area. Each particle in this image, despite being only around 0.1 mm across, has a surface area of several square meters.Supercapacitor electrodes are generally thin coatings applied and electrically connected to a conductive, metallic current collector. Electrodes must have good conductivity, high temperature stability, long-term chemical stability (inertness), high corrosion resistance and high surface areas per unit volume and mass. Other requirements include environmental friendliness and low cost.The amount of double-layer as well as pseudocapacitance stored per unit voltage in a supercapacitor is predominantly a function of the electrode surface area. Therefore, supercapacitor electrodes are typically made of porous, spongy material with an extraordinarily high specific surface area, such as activated carbon. Additionally, the ability of the electrode material to perform faradaic charge transfers enhances the total capacitance.Generally the smaller the electrode's pores, the greater the capacitance and specific energy. However, smaller pores increase equivalent series resistance (ESR) and decrease specific power. Applications with high peak currents require larger pores and low internal losses, while applications requiring high specific energy need small pores.The most commonly used electrode material for supercapacitors is carbon in various manifestations such as activated carbon (AC), carbon fibre-cloth (AFC), carbide-derived carbon (CDC), carbon aerogel, graphite (graphene), graphane, and carbon nanotubes (CNTs).Carbon-based electrodes exhibit predominantly static double-layer capacitance, even though a small amount of pseudocapacitance may also be present depending on the pore size distribution. Pore sizes in carbons typically range from micropores (less than 2 nm) to mesopores (2-50 nm), but only micropores (<2 nm) contribute to pseudocapacitance. As pore size approaches the solvation shell size, solvent molecules are excluded and only unsolvated ions fill the pores (even for large ions), increasing ionic packing density and storage capability by faradaic H₂ intercalation.Activated carbon (AC) was the first material chosen for EDLC electrodes. Even though its electrical conductivity is approximately 0.003% that of metals (1,250 to 2,000 S/m), it is sufficient for supercapacitors.Activated carbon is an extremely porous form of carbon with a high specific surface area — a common approximation is that 1 gram (0.035 oz) (a pencil-eraser-sized amount) has a surface area of roughly 1,000 to 3,000 square metres (11,000 to 32,000 sq ft)—about the size of 4 to 12 tennis courts. The bulk form used in electrodes is low-density with many pores, giving high double-layer capacitance.Solid activated carbon, also termed consolidated amorphous carbon (CAC) is the most-used electrode material for supercapacitors and may be cheaper than other carbon derivatives.It is produced from activated carbon powder pressed into the desired shape, forming a block with a wide distribution of pore sizes. An electrode with a surface area of about 1000 m²/g results in a typical double-layer capacitance of about 10 μF/cm² and a specific capacitance of 100 F/g.As of 2010 virtually all commercial supercapacitors use powdered activated carbon made from coconut shells.Coconut shells produce activated carbon with more micropores than does charcoal made from wood.Activated carbon fibres (ACF) are produced from activated carbon and have a typical diameter of 10 µm. They can have micropores with a very narrow pore-size distribution that can be readily controlled. The surface area of ACF woven into a textile is about 2500 m2/g. Advantages of ACF electrodes include low electrical resistance along the fibre axis and good contact to the collector.As for activated carbon, ACF electrodes exhibit predominantly double-layer capacitance with a small amount of pseudocapacitance due to their micropores.A block of silica aerogel in handCarbon aerogel is a highly porous, synthetic, ultralight material derived from an organic gel in which the liquid component of the gel has been replaced with a gas.Aerogel electrodes are made via pyrolysis of resorcinol-formaldehyde aerogels and are more conductive than most activated carbons. They enable thin and mechanically stable electrodes with a thickness in the range of several hundred micrometres (µm) and with uniform pore size. Aerogel electrodes also provide mechanical and vibration stability for supercapacitors used in high-vibration environments.Researchers have created a carbon aerogel electrode with gravimetric densities of about 400–1200 m²/g and volumetric capacitance of 104 F/cm³, yielding a specific energy of 325 kJ/kg (90 Wh/kg) and specific power of 20 W/g.Standard aerogel electrodes exhibit predominantly double-layer capacitance. Aerogel electrodes that incorporate composite material can add a high amount of pseudocapacitance.Pore size distributions for different carbide precursors.Carbide-derived carbon (CDC), also known as tunable nanoporous carbon, is a family of carbon materials derived from carbide precursors, such as binary silicon carbide and titanium carbide, that are transformed into pure carbon via physical (e.g., thermal decomposition) or chemical (e.g., halogenation) processes.Carbide-derived carbons can exhibit high surface area and tunable pore diameters (from micropores to mesopores) to maximize ion confinement, increasing pseudocapacitance by faradaic H₂ adsorption treatment. CDC electrodes with tailored pore design offer as much as 75% greater specific energy than conventional activated carbons.As of 2015, a CDC supercapacitor offered a specific energy of 10.1 Wh/kg, 3,500 F capacitance and over one million charge-discharge cycles.Graphene is an atomic-scale honeycomb lattice made of carbon atoms.Graphene is a one-atom thick sheet of graphite, with atoms arranged in a regular hexagonal pattern, also called "nanocomposite paper".Graphene has a theoretical specific surface area of 2630 m²/g which can theoretically lead to a capacitance of 550 F/g. In addition, an advantage of graphene over activated carbon is its higher electrical conductivity. As of 2012 a new development used graphene sheets directly as electrodes without collectors for portable applications.In one embodiment, a graphene-based supercapacitor uses curved graphene sheets that do not stack face-to-face, forming mesopores that are accessible to and wettable by ionic electrolytes at voltages up to 4 V. A specific energy of 85.6 Wh/kg (308 kJ/kg) is obtained at room temperature equaling that of a conventional nickel metal hydride battery, but with 100-1000 times greater specific power.The two-dimensional structure of graphene improves charging and discharging. Charge carriers in vertically oriented sheets can quickly migrate into or out of the deeper structures of the electrode, thus increasing currents. Such capacitors may be suitable for 100/120 Hz filter applications, which are unreachable for supercapacitors using other carbon materials.A scanning tunneling microscopy image of single-walled carbon nanotubeSEM image of carbon nanotube bundles with a surface of about 1500 m²/gCarbon nanotubes (CNTs), also called buckytubes, are carbon molecules with a cylindrical nanostructure. They have a hollow structure with walls formed by one-atom-thick sheets of graphite. These sheets are rolled at specific and discrete ("chiral") angles, and the combination of chiral angle and radius controls properties such as electrical conductivity, electrolyte wettability and ion access. Nanotubes are categorized as single-walled nanotubes (SWNTs) or multi-walled nanotubes (MWNTs). The latter have one or more outer tubes successively enveloping a SWNT, much like the Russian matryoshka dolls. SWNTs have diameters ranging between 1 and 3 nm. MWNTs have thicker coaxial walls, separated by spacing (0.34 nm) that is close to graphene's interlayer distance.Nanotubes can grow vertically on the collector substrate, such as a silicon wafer. Typical lengths are 20 to 100 µm.Carbon nanotubes can greatly improve capacitor performance, due to the highly wettable surface area and high conductivity.A SWNT-based supercapacitor with aqueous electrolyte was systematically studied at University of Delaware in Prof. Bingqing Wei's group. Li et al., for the first time, discovered that the ion-size effect and the electrode-electrolyte wettability are the dominant factors affecting the electrochemical behavior of flexible SWCNTs-supercapacitors in different 1 molar aqueous electrolytes with different anions and cations. The experimental results also showed for flexible supercapacitor that it is suggested to put enough pressure between the two electrodes to improve the aqueous electrolyte CNT supercapacitor.CNTs can store about the same charge as activated carbon per unit surface area, but nanotubes' surface is arranged in a regular pattern, providing greater wettability. SWNTs have a high theoretical specific surface area of 1315 m²/g, while that for MWNTs is lower and is determined by the diameter of the tubes and degree of nesting, compared with a surface area of about 3000 m²/g of activated carbons. Nevertheless, CNTs have higher capacitance than activated carbon electrodes, e.g., 102 F/g for MWNTs and 180 F/g for SWNTs.MWNTs have mesopores that allow for easy access of ions at the electrode–electrolyte interface. As the pore size approaches the size of the ion solvation shell, the solvent molecules are partially stripped, resulting in larger ionic packing density and increased faradaic storage capability. However, the considerable volume change during repeated intercalation and depletion decreases their mechanical stability. To this end, research to increase surface area, mechanical strength, electrical conductivity and chemical stability is ongoing.MnO₂ and RuO₂ are typical materials used as electrodes for pseudocapacitors, since they have the electrochemical signature of a capacitive electrode (linear dependence on current versus voltage curve) as well as exhibiting faradaic behavior. Additionally, the charge storage originates from electron-transfer mechanisms rather than accumulation of ions in the electrochemical double layer. Pseudocapacitors were created through faradaic redox reactions that occur within the active electrode materials. More research was focused on transition-metal oxides such as MnO₂ since transition-metal oxides have a lower cost compared to noble metal oxides such as RuO₂. Moreover, the charge storage mechanisms of transition-metal oxides are based predominantly on pseudocapacitance. Two mechanisms of MnO₂ charge storage behavior were introduced. The first mechanism implies the intercalation of protons (H⁺) or alkali metal cations (C⁺) in the bulk of the material upon reduction followed by deintercalation upon oxidation.[math](\text{MnO}_2)+\text{H}^+(\text{C}^+)+\text{e}^-\rightleftharpoons\text{MnOOH(C)}[/math]The second mechanism is based on the surface adsorption of electrolyte cations on MnO₂.[math](\text{MnO}_2)_\text{surface}+\text{C}^++\text{e}^-\rightleftharpoons(\text{MnO}_2^-\text{ C}^+)_\text{surface}[/math]Not every material that exhibits faradaic behavior can be used as an electrode for pseudocapacitors, such as Ni(OH)₂, since it is a battery-type electrode (non-linear dependence on current versus voltage curve).Brian Evans Conway's research described electrodes of transition-metal oxides that exhibited high amounts of pseudocapacitance. Oxides of transition metals including ruthenium (RuO₂), iridium (IrO₂), iron (Fe₃O₄), manganese (MnO₂), or sulfides such as titanium sulfide (TiS₂) alone or in combination generate strong faradaic electron-transferring reactions combined with low resistance.Ruthenium dioxide in combination with H₂SO₄ electrolyte provides specific capacitance of 720 F/g and a high specific energy of 26.7 Wh/kg (96.12 kJ/kg).Charge/discharge takes place over a window of about 1.2 V per electrode. This pseudocapacitance of about 720 F/g is roughly 100 times higher than for double-layer capacitance using activated carbon electrodes. These transition metal electrodes offer excellent reversibility, with several hundred-thousand cycles. However, ruthenium is expensive and the 2.4 V voltage window for this capacitor limits their applications to military and space applications. Das et al. reported highest capacitance value (1715 F/g) for ruthenium-oxide-based supercapacitor with electrodeposited ruthenium oxide onto porous single-wall carbon nanotube film electrode.A high specific capacitance of 1715 F/g has been reported which closely approaches the predicted theoretical maximum RuO₂ capacitance of 2000 F/g.In 2014 a RuO₂ supercapacitor anchored on a graphene foam electrode delivered specific capacitance of 502.78 F/g and areal capacitance of 1.11 F/cm² leading to a specific energy of 39.28 Wh/kg and specific power of 128.01 kW/kg over 8,000 cycles with constant performance. The device was a three-dimensional (3D) sub-5 nm hydrous ruthenium-anchored graphene and carbon nanotube (CNT) hybrid foam (RGM) architecture. The graphene foam was conformally covered with hybrid networks of RuO₂ nanoparticles and anchored CNTs.Less expensive oxides of iron, vanadium, nickel and cobalt have been tested in aqueous electrolytes, but none has been investigated as much as manganese dioxide (MnO₂). However, none of these oxides are in commercial use.Another approach uses electron-conducting polymers as pseudocapacitive material. Although mechanically weak, conductive polymers have high conductivity, resulting in a low ESR and a relatively high capacitance. Such conducting polymers include polyaniline, polythiophene, polypyrrole and polyacetylene. Such electrodes also employ electrochemical doping or dedoping of the polymers with anions and cations. Electrodes made from or coated with conductive polymers have costs comparable to carbon electrodes.Conducting polymer electrodes generally suffer from limited cycling stability.However, polyacene electrodes provide up to 10,000 cycles, much better than batteries.All commercial hybrid supercapacitors are asymmetric. They combine an electrode with high amount of pseudocapacitance with an electrode with a high amount of double-layer capacitance. In such systems the faradaic pseudocapacitance electrode with their higher capacitance provides high specific energy while the non-faradaic EDLC electrode enables high specific power. An advantage of the hybrid-type supercapacitors compared with symmetrical EDLC's is their higher specific capacitance value as well as their higher rated voltage and correspondingly their higher specific energy.Composite electrodes for hybrid-type supercapacitors are constructed from carbon-based material with incorporated or deposited pseudocapacitive active materials like metal oxides and conducting polymers. As of 2013, most research for supercapacitors explores composite electrodes.CNTs give a backbone for a homogeneous distribution of metal oxide or electrically conducting polymers (ECPs), producing good pseudocapacitance and good double-layer capacitance. These electrodes achieve higher capacitances than either pure carbon or pure metal oxide or polymer-based electrodes. This is attributed to the accessibility of the nanotubes' tangled mat structure, which allows a uniform coating of pseudocapacitive materials and three-dimensional charge distribution. The process to anchor pseudocapacitive materials usually uses a hydrothermal process. However, a recent researcher, Li et al., from the University of Delaware found a facile and scalable approach to precipitate MnO₂ on a SWNT film to make an organic-electrolyte based supercapacitor.Another way to enhance CNT electrodes is by doping with a pseudocapacitive dopant as in lithium-ion capacitors. In this case the relatively small lithium atoms intercalate between the layers of carbon.The anode is made of lithium-doped carbon, which enables lower negative potential with a cathode made of activated carbon. This results in a larger voltage of 3.8-4 V that prevents electrolyte oxidation. As of 2007 they had achieved capacitance of 550 F/g and reach a specific energy up to 14 Wh/kg (50.4 kJ/kg).Rechargeable battery electrodes influenced the development of electrodes for new hybrid-type supercapacitor electrodes as for lithium-ion capacitors.Together with a carbon EDLC electrode in an asymmetric construction offers this configuration higher specific energy than typical supercapacitors with higher specific power, longer cycle life and faster charging and recharging times than batteries.Recently some asymmetric hybrid supercapacitors were developed in which the positive electrode were based on a real pseudocapacitive metal oxide electrode (not a composite electrode), and the negative electrode on an EDLC activated carbon electrode.An advantage of this type of supercapacitors is their higher voltage and correspondingly their higher specific energy (up to 10-20 Wh/kg (36-72 kJ/kg)).As far as is known, no commercially-offered supercapacitors with such kind of asymmetric electrodes are on the market.Electrolytes consist of a solvent and dissolved chemicals that dissociate into positive cations and negative anions, making the electrolyte electrically conductive. The more ions the electrolyte contains, the better its conductivity. In supercapacitors electrolytes are the electrically conductive connection between the two electrodes. Additionally, in supercapacitors the electrolyte provides the molecules for the separating monolayer in the Helmholtz double-layer and delivers the ions for pseudocapacitance.The electrolyte determines the capacitor's characteristics: its operating voltage, temperature range, ESR and capacitance. With the same activated carbon electrode an aqueous electrolyte achieves capacitance values of 160 F/g, while an organic electrolyte achieves only 100 F/g.The electrolyte must be chemically inert and not chemically attack the other materials in the capacitor to ensure long time stable behavior of the capacitor's electrical parameters. The electrolyte's viscosity must be low enough to wet the porous, sponge-like structure of the electrodes. An ideal electrolyte does not exist, forcing a compromise between performance and other requirements.Water is a relatively good solvent for inorganic chemicals. Treated with acids such as sulfuric acid (H₂2SO₄), alkalis such as potassium hydroxide (KOH), or salts such as quaternary phosphonium salts, sodium perchlorate (NaClO₄), lithium perchlorate (LiClO₄), or lithium hexafluoride arsenate (LiAsF₆), water offers relatively high conductivity values of about 100 to 1000 mS/cm. Aqueous electrolytes have a dissociation voltage of 1.15 V per electrode (2.3 V capacitor voltage) and a relatively low operating temperature range. They are used in supercapacitors with low specific energy and high specific power.Electrolytes with organic solvents such as acetonitrile, propylene carbonate, tetrahydrofuran, diethyl carbonate, γ-butyrolactone and solutions with quaternary ammonium salts or alkyl ammonium salts such as tetraethylammonium tetrafluoroborate (N(Et)₄BF₄) or triethyl (methyl) tetrafluoroborate (NMe(Et)₃BF₄) are more expensive than aqueous electrolytes, but they have a higher dissociation voltage of typically 1.35 V per electrode (2.7 V capacitor voltage), and a higher temperature range. The lower electrical conductivity of organic solvents (10 to 60 mS/cm) leads to a lower specific power, but since the specific energy increases with the square of the voltage, a higher specific energy.Separators have to physically separate the two electrodes to prevent a short circuit by direct contact. It can be very thin (a few hundredths of a millimeter) and must be very porous to the conducting ions to minimize ESR. Furthermore, separators must be chemically inert to protect the electrolyte's stability and conductivity. Inexpensive components use open capacitor papers. More sophisticated designs use nonwoven porous polymeric films like polyacrylonitrile or Kapton, woven glass fibers or porous woven ceramic fibres.Current collectors connect the electrodes to the capacitor's terminals. The collector is either sprayed onto the electrode or is a metal foil. They must be able to distribute peak currents of up to 100 A.If the housing is made out of a metal (typically aluminum) the collectors should be made from the same material to avoid forming a corrosive galvanic cell.

Rates, Risks, and Probability ModelsSome individuals exposed to environmental carcinogens (e.g., ionizing radiation) develop cancer and some do not; the same is true of unexposed individuals. Thus, cancer is not a necessary consequence of exposure, and exposure is not necessary for cancer. However, the greater incidence of cancer in individuals exposed to known carcinogens indicates that the probability or risk of developing cancer is increased by exposure. Compared to unexposed individuals, the elevated risks of exposed individuals are manifest by increased cancer rates in the latter group. Risks and rates are the basic measures used to compare disease occurrence in exposed and unexposed individuals. This section describes rates and risks and their relationship to one another as a prelude to the sections on modeling and model fitting.Incidence RateA common measure of disease occurrence used in cancer epidemiology is the incidence rate. Incidence refers to new cases of disease occurring among previously unaffected individuals. The population incidence rate is the number of new cases of the disease occurring in the population in a specified time interval divided by the sum of observation times, in that interval, on all individuals who were disease free at the beginning of the time interval. In general an incidence rate is time dependent and depends on both the starting point and the length of the interval.With data from studies in which subjects are followed over time, incidence rates can be estimated by partitioning the following period into intervals of lengths Lj having midpoints tj for j = 1,…,J, and estimating a rate for each interval. Let nj denote the number of individuals who are disease free and still under observation at time tj, and dj the number of new diagnoses during the jth interval. An estimate of the incidence rate at time tj is obtained by dividing dj by the product of nj and Lj:The denominator inis an approximation to the sum of observation times on the nj population members in the jth interval and in practice is usually replaced by the actual observation time, which accounts for the fact that the dj diagnoses of disease did not occur exactly at time tj.Suggested Citation:"11 Risk Assessment Models and Methods." National Research Council. 2006. Health Risks from Exposure to Low Levels of Ionizing Radiation: BEIR VII Phase 2. Washington, DC: The National Academies Press. doi: 10.17226/11340.RiskRisk is defined as the probability that an individual develops a specified disease over a specified interval of time, given that the individual is alive and disease free at the start of the time period. As with the incidence rate, risk is time dependent and depends on both the starting point and the length of the interval. In a longitudinal follow-up study as described above, the proportion of new occurrences dj among nj disease-free individuals still under observation at time tj,is an estimate of the risk or probability of disease occurrence in the jth time interval.Incidence rates and risks are related via the general formula, risk = rate × time. For the longitudinal follow-up study estimates defined above, the relationship is manifest by the equationProbability ModelsThe description of rates and risks in terms of estimates from a longitudinal follow-up study is informative and clearly indicates the relevance of these numerical quantities to the study of disease. However, the development of a general theory of risk and risk estimation requires definitions of rates and risks that are not tied to particular types of studies or methods of estimation. Probability models provide a mathematical framework for studying incidence rates and risks and also are used in defining statistical methods of estimation depending on the type of study and the data available.Models for studying the relationship between disease and exposure are usually formulated in terms of the instantaneous incidence rate, which is the theoretical counterpart of the incidence rate estimate defined below. The instantaneous incident rate is defined in terms of the probability distribution function F(t) of the time to disease occurrence. That is, F(t) represents the probability that an individual develops the disease of interest in the interval of time (0, t). Two functions derived from F(t) are used to define the instantaneous incidence rate. One is the survivor function, which is the probability of being disease free throughout the interval (0, t) and is equal to 1 − F(t). The second is the probability density function, which is the derivative of F(t) with respect to t, that is, f(t) = (d / dt)F(t), and measures the rate of increase in F(t). The instantaneous incidence rate, also known as the hazard function, is the ratioIntegrating the instantaneous incident rate yields the cumulative incidence rateThe cumulative incidence rate and the distribution function satisfy the relationship(11-1)from which it follows that the instantaneous incidence rate completely determines the first-occurrence distribution F(t).The risk of first disease occurrence in the interval (t, t + h), given no previous occurrence, is the conditional probabilityWhen h is not too large, so that the difference quotient {F(t + h) − F(t)} / h approximates f(t) = dF(t)/dt,Thus, among individuals who are disease free at time t, the risk of disease in the interval (t, t + h) is approximately λ(t)h. This approximation is the theoretical counterpart of the relationship between risks and rates described in the discussion of risk. In the remainder of this chapter, incidence rate means instantaneous incidence rate unless explicitly noted otherwise.Incidence Rates and Excess RisksIt is clear that the incidence rate plays an important role in the stochastic modeling of disease occurrence. Consequently, models and methods for studying the dependence of disease occurrence on exposure are generally formulated in terms of incidence rates. In the following it is assumed that individuals have been stratified on the basis of age, sex, calendar time, and possibly other factors related to disease occurrence, and that incidence rates are stratum specific. In the simple case of two exposure categories, exposed and unexposed, let λE(t) and λU(t) denote the incidence rates of the exposed and unexposed groups, respectively. If disease occurrence is unrelated to exposure, one expects that λE(t) = λU(t), whereas lack of equality between these two incidence rates indicates an association between disease occurrence and exposure.A common measure of discrepancy between incidence rates is the differencewhich by convention is called the excess absolute risk (EAR) even though it is, technically, a difference in rates. Rearranging terms results inPage 261Suggested Citation:"11 Risk Assessment Models and Methods." National Research Council. 2006. Health Risks from Exposure to Low Levels of Ionizing Radiation: BEIR VII Phase 2. Washington, DC: The National Academies Press. doi: 10.17226/11340.×showing that EAR(t) describes the additive increase in incidence rate associated with exposure. For example, if the EAR is constant, EAR(t) = b, then the effect of exposure is to increase the incidence rate by the constant amount b for all time periods. Note that b = 0 corresponds to the case of no association.A second common measure of discrepancy is the relative risk (RR), defined asRearranging terms shows thatso that RR(t) describes the multiplicative increase in incidence rate associated with exposure. When the RR is constant, RR(t) = r, the effect of exposure is to alter incidence rate by the factor r. If exposure increases risk, then r > 1; if exposure decreases risk, then r < 1, and r = 1 corresponds to the case of no association. The excess relative risk ERR(t) isThe ERR of the exposed and unexposed incidence rates are related via the equationRISK MODELSDirect Estimates of RiskThe previous section defined the fundamental quantities used in risk estimation: risks, rates, EAR, RR, and ERR, and established their relevance to the study of environmental carcinogens. These measures enable the study of differences in disease occurrence in relationship to time, by studying either EAR(t) or ERR(t) between unexposed and exposed groups. For most carcinogens, exposure is not a simple dichotomy (unexposed, exposed) but occurs on a continuum. That is, the exposure or dose d can vary from no exposure (d = 0) upward. In such cases the relationship between risk—or EAR(t) or ERR(t)—and dose is of fundamental importance. For all carcinogens it is generally agreed that sufficiently large doses increase the risk of cancer. By definition there is no increase in risk in the absence of exposure (d = 0). That is, when d = 0, both EAR(t) = 0 and ERR(t) = 0. Thus, for many carcinogens the only open or unresolved issue is the dependence of risk on small or low doses. Low-dose ranges are often the most relevant in terms of numbers of exposed individuals. They are also the most difficult ranges for which to obtain unequivocal evidence of increased risk. These difficulties result from the fact that small increases in risk associated with low levels of exposure are difficult to detect (using statistical methods) in the presence of background risks.The difficulties can be seen by considering the estimates of risk from the longitudinal follow-up study described in “Rates, Risks, and Probability Models.” For a time period Lj, let nj,E, dj,E and nj,U, dj,Udenote the number of individuals at risk at the start of the interval and the number of occurrences of disease during the interval for the exposed and unexposed subgroups, respectively. A direct estimate of the excess risk for the jth time period is the difference between two proportions (dj,E / nj,E) − (dj,U / nj,U). Even in the favorable situation in which the baseline risk is relatively well estimated compared to the risk of the exposed group (when nj,U is large relative to nj,E), the ability to reliably detect small increases in risk associated with exposure requires a large number of exposed individuals at risk. For example, using the usual criterion for statistical testing in order to detect with probability .80 a 5% increase in risk when the baseline risk is 0.10, the number of individuals at risk in the exposed group would have to be approximately nj,E = 30,000.A key objective of this report is the calculation of quantitative estimates of human health risks (e.g., cancer) associated with exposure to ionizing radiation for specific subpopulations defined by stratification on variables such as sex, age, exposure profile, and smoking history. In theory, such estimates could be derived by identifying a large group of individuals having common exposure profiles within each stratum and following the groups over a long period of time. As described above, the proportion of individuals in each group who develop cancer in specific time periods provides the desired estimates of risk. However, this approach is not feasible because sufficient data are not available. At low levels of exposure, cancer risks associated with exposure are small relative to baseline or background risks. The increases in observed cancer rates associated with exposure are small relative to the natural random fluctuations in baseline cancer rates. Thus, very large groups of individuals would have to be followed for very long periods of time to provide sufficiently precise estimates of risk associated with exposure. Consequently, direct estimates of risk are not possible for stratified subpopulations. The alternative is to use mathematical models for risk as functions of dose and stratifying variables such as sex and age.Estimation via Mathematical Models for RiskModel-based estimation provides a feasible alternative to direct estimation. Model-based estimates efficiently exploit the information in the available data and provide a means of deriving estimates for strata and dose profile combinations for which data are sparse. This is accomplished by exploiting assumptions about the functional form of a risk model. Of course, the validity of estimates derived from models depends on the appropriateness of the model; thus model choice is important. The accepted approach in radiation epi-Page 262Suggested Citation:"11 Risk Assessment Models and Methods." National Research Council. 2006. Health Risks from Exposure to Low Levels of Ionizing Radiation: BEIR VII Phase 2. Washington, DC: The National Academies Press. doi: 10.17226/11340.×demiology is to base models on radiobiological principles and theories of carcinogenesis to the fullest extent possible, keeping in mind statistical limitations imposed by the quantity and quality of data available for model fitting. Biologically based and empirically derived mathematical models for risk are discussed in the next two sections.Biologically Based Risk ModelsBiologically based risk models are designed to describe the fundamental biological processes involved in the transformation of somatic cells into malignant cancer cells. The use of biologically based risk models in epidemiologic analyses can result in a greater understanding of the mechanisms of carcinogenesis. These models can also help to expose the complex interrelationships between different time- and age-dependent exposure patterns and cancer risk. Biologically based risk models provide an analytical method that is complementary to the traditional, well-established, empirical approaches.Armitage and Doll (1954) observed that for many human cancers the log-log plot of age-specific incidence rates versus age is nearly linear, up to moderately old ages. This observation has led to the development of models for carcinogenesis. In brief, Armitage and Doll’s theory postulates that malignant transformation occurs following the kth stage of a series of spontaneous and irreversible changes (Armitage 1985). The corresponding hazard function is of the form λ(t)= atk−1, where t denotes time and a is a constant reflecting the dependence of the hazard on the number of stages, k. These models have been fit to various data sets, leading to the observation that most cancers arise after the occurrence of five to seven stages. Comprehensive reviews of the mathematical theory of carcinogenesis have been given by Armitage and Doll (1961), Whittemore (1978), and Armitage (1985).In response to the multiplicity of parameters produced by their earlier models, Armitage and Doll proposed a simpler two-stage model designed to avoid parameters not readily estimable from available data. A major limitation of these early two-stage models is their failure to address the multiplication and death of normal cells, which was known to occur in tissues undergoing malignant change (Moolgavkar and Knudson 1981). A revised two-stage model was later proposed by Moolgavkar and colleagues, which allowed for the growth of normal tissue and the clonal expansion of intermediate cells (Moolgavkar and Knudson 1981). Numerous two-stage models have since been described in the literature (Fisher 1985; Moolgavkar 1991; Sielken and others 1994; Luebeck and others 1996; Heidenreich and others 1999, 2002a, 2002b; Moolgavkar and others 1999; Heidenreich and Paretzke 2001; Moolgavkar and Luebeck 2003).The two-stage clonal expansion (TSCE) model assumes a normal stem cell population of fixed size X and a rate of first mutation of v(d), depending on the dose d of the carcinogen. The number of initiated cells arising from the normal cell pool is described by a Poisson process with a rate of vX. The initiated cells then divide either symmetrically or nonsymmetrically. Symmetrical division results in two initiated cells, while nonsymmetrical division results in an initiated cell and a differentiated cell. The rate of symmetrical division is designated by α(t), and the death differentiation rate by β(t). The difference α − β is the net proliferation rate for initiated cells. The rate of division into one initiated cell and one malignant cell is designated by μ(t) (Hazleton and others 2001).TSCE models for radiation carcinogenesis have now been applied successfully to a number of important data sets, including atomic bomb survivors (Kai and others 1997) and occupational groups such as nuclear power plant workers and miners (Moolgavkar and others 1993; Luebeck and others 1999; Sont and others 2001). A study of atomic bomb survivors illustrates the usefulness of the two-stage model in radiation epidemiology (Kai and others 1997). Findings from this analysis include the observation of a high excess risk among children that may not be explained by enhanced tissue sensitivity to radiation exposure. The temporal patterns in cancer risk can be explained in part by a radiation-induced increase in the pool of initiated cells, resulting in a direct dose-rate effect (Kai and others 1997). Exact solutions of the two-stage model (Heidenreich and others 1997) and multistage models (Heidenreich and others 2002b) have been applied to atomic bomb survivors’ data.Another data set to which application of the TSCE has been useful is the National Dose Registry (NDR) of Canada. This database contains personal dosimetry records for workers exposed to ionizing radiation since 1951, with current records for more than 500,000 Canadians (Ashmore and others 1998). Application of the TSCE model to the NDR suggests an explanation of the apparently high excess relative risk observed, relative to the A-bomb data (Sont and others 2001). The TCSE model reveals that the dose-response for the NDR cohort is consistent with the lung cancer incidence in the A-bomb survivors’ cohort, provided that proper adjustments are made for the duration of exposure and differences in the background rate parameters.In addition to the TSCE model, the Armitage-Doll model of carcinogenesis has evolved into several other analytic methods, including the general mutagen model (Pierce 2002). The basic assumption of this model is that a malignant cell results from the accumulation of mutations, with k mutations required for malignancy. The effect of exposure is that an increment of dose at age a, at rate d(a), results in a multiplicative increase λr[1 + βd(a)] in the rate of all k mutations. Although this model applies to both recessive and dominant mutations, it does not explicitly allow for selective proliferation of cells having only some of the required mutations. The general mutagen model has been applied successfully to A-bomb survivor data (Pierce and Mendelsohn 1999; Pierce and Preston 2000) and to underground miners exposed to radon (Lubin and others 1995).Page 263Suggested Citation:"11 Risk Assessment Models and Methods." National Research Council. 2006. Health Risks from Exposure to Low Levels of Ionizing Radiation: BEIR VII Phase 2. Washington, DC: The National Academies Press. doi: 10.17226/11340.×Whereas empirical approaches to risk modeling rely on statistical models to describe data, biologically based models depend on fundamental assumptions regarding the mechanisms of radiation carcinogenesis. The parameters created by modern biologically based risk models have direct biological interpretation, provide insight into cancer mechanisms, and generate substantive questions about the pathways by which exposure to ionizing radiation can increase cancer risk. These models also provide a way of describing temporal patterns of exposure and risk.Although biologically based risk models have many strengths, some general limitations are associated with their use. Such models can only approximate biological reality and require an understanding of the complex mechanisms of radiation carcinogenesis for interpretation. In addition, it is difficult to distinguish among alternative models that yield similar dose-response curves without direct information on the fundamental biological processes represented by the model, which are often unknown. Biologically based risk models are generally more complex than empirical models and may require richer databases to develop properly. Despite these limitations, biologically based models have found many applications for important epidemiologic data sets, and the successes achieved to date afford support for the continual development of such models for future analyses that will directly inform the association between radiation exposure and human cancer risk.Biologically based models have not been employed as the primary method of analysis in this report for several reasons. The mechanisms of radiation carcinogenesis are not fully understood, which makes the development of a fully biologically based model difficult. The data required for a biologically based model, such as rates of cell proliferation and mutation, are also generally not available. The availability of empirical risk models that provide a good description of the available data on radiation and cancer permits the preparation of useful risk projection.Empirically Based Risk ModelsThe following symbols are used to describe the variables that enter into risk models based on the Japanese A-bomb survivor data:a: attained age of an individuale: age at exposure to radiationd: dose of radiation receiveds: code for sex (1 if the individual is a female and 0 if male)p:study population-specific factorsModels also sometimes include time since exposure (t). Since t = a − e, models that include a and e implicitly include t.Models for the incidence rate for individuals of age a, exposed to dose d, at age e, generally depend on sex s (1 for females, 0 for males) and other study population-specific factors generically represented by p. For example, the study population-specific parameters for A-bomb survivor data models are city c and calendar year y, that is, p = (c, y). The incidence rate is, in general, a function λ(a, e, d, s, p) of all of these factors. By definition, the background incidence rate does not depend on either d or e, so the EAR formulation of the exposed incidence rate has the formand the ERR formulation iswhere EAR (a, e, d, s, p) and ERR (a, e, d, s, p) are the EAR and ERR, respectively. When the excess risk functions are dependent on the study population—that is, when they depend on the factor p—estimates of risk derived from the models are specific to the study population and therefore of limited utility for estimating risks in other populations. Thus, it is desirable to find suitable models in which either the excess risk or the excess relative risk does not depend on population-specific parameters. Consequently, models used in radiation risk estimation are often of the formorThat is, the excess risk functions depend only on a, e, d, and s, but not p. Note that if t represents time after exposure, then because t = a − e, any two of the variables t, a, and e determine the third, so at the current level of generality, the excess risk functions could also be written as functions of t, e, d, and s. Also, because there is no excess risk at ages prior to exposure (a < e), ER(a, e, d, s) = 0 (a < e), EAR(a, e, d, s) = 0 and ERR(a, e, d, s) = 0 for a < e and thus, λ(a, e, d, s, p) = λ(a, s, p) for a < e. The formulas and equations in the remainder of this chapter are described only for the relevant case a e.Radiobiological considerations suggest that for low-dose, low-LET (linear energy transfer) radiation, the risk of disease for an individual exposed to dose d depends on a linear or quadratic function of d. That is, risk depends on dose d through a function of the formwhere α1 and α2 are parameters to be estimated from the data. At higher doses of radiation, cell sterilization and cell death compete with the process of malignant transformation, thereby attenuating the risk of cancer at higher doses. A more general model applicable to a broader dose range and used extensively in radiation research isThe models for dependence on dose are generally incorporated into risk models by assuming that the excess risk functions are proportional to f(d), where the multiplicative constant (in dose) depends on a, e, and s.Source: 11 Risk Assessment Models and MethodsPredictive Analytics and Modeling in Product Pricing (Personal and Commercial Insurance)