Functional near-infrared spectroscopy

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Functional_near-infrared_spectroscopy 

 

fNIRS with a Gowerlabs NTS system

Functional near-infrared spectroscopy (fNIRS) is an optical brain monitoring technique which uses near-infrared spectroscopy for the purpose of functional neuroimaging. Using fNIRS, brain activity is measured by using near-infrared light to estimate cortical hemodynamic activity which occur in response to neural activity. Alongside EEG, fNIRS is one of the most common non-invasive neuroimaging techniques which can be used in portable contexts. The signal is often compared with the BOLD signal measured by fMRI and is capable of measuring changes both in oxy- and deoxyhemoglobin concentration, but can only measure from regions near the cortical surface. fNIRS may also be referred to as Optical Topography (OT) and is sometimes referred to simply as NIRS.

Description

Oxygenated and deoxygenated hemoglobin

fNIRS estimates the concentration of hemoglobin from changes in absorption of near infrared light. As light moves or propagates through the head, it is alternately scattered or absorbed by the tissue through which it travels. Because hemoglobin is a significant absorber of near-infrared light, changes in absorbed light can be used to reliably measure changes in hemoglobin concentration. Different fNIRS techniques can also use the way in which light propagates to estimate blood volume and oxygenation. The technique is safe, non-invasive, and can be used with other imaging modalities.

fNIRS is a non-invasive imaging method involving the quantification of chromophore concentration resolved from the measurement of near infrared (NIR) light attenuation or temporal or phasic changes. The technique takes advantage of the optical window in which (a) skin, tissue, and bone are mostly transparent to NIR light (700–900 nm spectral interval) and (b) hemoglobin (Hb) and deoxygenated-hemoglobin (deoxy-Hb) are strong absorbers of light.

Absorption spectra for oxy-Hb and deoxy-Hb for near-infrared wavelengths

There are six different ways for infrared light to interact with the brain tissue: direct transmission, diffuse transmission, specular reflection, diffuse reflection, scattering, and absorption. fNIRS focuses primarily on absorption: differences in the absorption spectra of deoxy-Hb and oxy-Hb allow the measurement of relative changes in hemoglobin concentration through the use of light attenuation at multiple wavelengths. Two or more wavelengths are selected, with one wavelength above and one below the isosbestic point of 810 nm—at which deoxy-Hb and oxy-Hb have identical absorption coefficients. Using the modified Beer-Lambert law (mBLL), relative changes in concentration can be calculated as a function of total photon path length.

Typically, the light emitter and detector are placed ipsilaterally (each emitter/detector pair on the same side) on the subject's skull so recorded measurements are due to back-scattered (reflected) light following elliptical pathways. fNIRS is most sensitive to hemodynamic changes which occur nearest to the scalp and these superficial artifacts are often addressed using additional light detectors located closer to the light source (short-separation detectors).

Modified Beer–Lambert law

Changes in light intensity can be related to changes in relative concentrations of hemoglobin through the modified Beer–Lambert law (mBLL). The Beer lambert-law has to deal with concentration of hemoglobin. This technique also measures relative changes in light attenuation as well as using mBLL to quantify hemoglobin concentration changes.

Basic functional near infrared spectroscopy (fNIRS) abbreviations BFi = blood flow index CBF = cerebral blood flow CBV = cerebral blood volume CMRO2= metabolic rate of oxygen CW= continuous wave DCS = diffuse correlation spectroscopy FD = frequency-domain Hb, HbR= deoxygenated hemoglobin HbO, HbO2= oxygenated hemoglobin HbT= total hemoglobin concentration HGB = blood hemoglobin SaO2= arterial saturation SO2= hemoglobin saturation SvO2= venous saturation TD=time-domain

History

US & UK

In 1977, Jöbsis reported that brain tissue transparency to NIR light allowed a non-invasive and continuous method of tissue oxygen saturation using transillumination. Transillumination (forward-scattering) was of limited utility in adults because of light attenuation and was quickly replaced by reflectance-mode based techniques - resulting in development of NIRS systems proceeding rapidly. Then, by 1985, the first studies on cerebral oxygenation were conducted by M. Ferrari. Later, in 1989, following work with David Delpy at University College London, Hamamatsu developed the first commercial NIRS system: NIR-1000 cerebral oxygenation monitor. NIRS methods were initially used for cerebral oximetry in the 1990s. In 1993, four publications by Chance et al. PNAS, Hoshi & Tamura J Appl Physiol,  Kato et al. JCBFM, Villringer et al Neuros. Lett. demonstrated the feasibility of fNIRS in adult humans. NIRS techniques were further expanded on by the work of Randall Barbour, Britton Chance, Arno Villringer, M. Cope, D. T. Delpy, Enrico Gratton, and others. Currently, wearable fNIRS are being developed.

Hitachi ETG-4000

Japan

Meanwhile, in the mid-80's, Japanese researchers at the central research laboratory of Hitachi Ltd set out to build a NIRS-based brain monitoring system using a pulse of 70-picosecond rays. This effort came into light when the team, along with their leading expert, Dr Hideaki Koizumi (小泉 英明), held an open symposium to announce the principle of "Optical Topography" in January 1995. In fact, the term "Optical Topography" derives from the concept of using light on "2-Dimensional mapping combined with 1-Dimensional information", or topography. The idea had been successfully implemented in launching their first fNIRS (or Optical Topography, as they call it) device based on Frequency Domain in 2001: Hitachi ETG-100. Later, Harumi Oishi (大石 晴美), a PhD-to-be at Nagoya University, published her doctoral dissertation in 2003 with the subject of "language learners' cortical activation patterns measured by ETG-100" under the supervision of Professor Toru Kinoshita (木下 微)—presenting a new prospect on the use of fNIRS. The company has been advancing the ETG series ever since.

Spectroscopic techniques

Currently, there are three modalities of fNIR spectroscopy:

1. Continuous wave

2. Frequency domain

3. Time-domain

Continuous wave

Continuous wave (CW) system uses light sources with constant frequency and amplitude. In fact, to measure absolute changes in HbO concentration with the mBLL, we need to know photon path-length. However, CW-fNIRS does not provide any knowledge of photon path-length, so changes in HbO concentration are relative to an unknown path-length. Many CW-fNIRS commercial systems use estimations of photon path-length derived from computerized Monte-Carlo simulations and physical models, to approximate absolute quantification of hemoglobin concentrations.

${\displaystyle {\text{OD}}=\operatorname {log} _{10}(I_{0}/I)=\epsilon \cdot [X]\cdot l\cdot {\text{DPF}}+G}$

Where ${\displaystyle {\text{OD}}}$ is the optical density or attenuation, ${\displaystyle I_{0}}$ is emitted light intensity, ${\displaystyle I}$ is measured light intensity, ${\displaystyle \epsilon }$ is the attenuation coefficient, ${\displaystyle [X]}$ is the chromophomore concentration, ${\displaystyle l}$ is the distance between source and detector and ${\displaystyle {\text{DPF}}}$ is the differential path length factor, and ${\displaystyle G}$ is a geometric factor associated with scattering.

When the attenuation coefficients ${\displaystyle \epsilon }$ are known, constant scattering loss is assumed, and the measurements are treated differentially in time, the equation reduces to:

${\displaystyle \Delta [X]=\Delta {\frac {\text{OD}}{\epsilon d}}}$

Where ${\displaystyle d}$ is the total corrected photon path-length.

Using a dual wavelength system, measurements for HbO₂ and Hb can be solved from the matrix equation:

${\displaystyle {\begin{pmatrix}\Delta {\text{OD}}_{\lambda _{1}}\\\Delta {\text{OD}}_{\lambda _{2}}\end{pmatrix}}={\begin{pmatrix}\epsilon _{\lambda _{1}}^{\text{Hb}}d&\epsilon _{\lambda _{1}}^{{\text{HbO}}_{2}}d\\\epsilon _{\lambda _{2}}^{\text{Hb}}d&\epsilon _{\lambda _{2}}^{{\text{HbO}}_{2}}d\end{pmatrix}}{\begin{pmatrix}\Delta [X]^{\text{Hb}}\\\Delta [X]^{{\text{HbO}}_{2}}\end{pmatrix}}}$

Due to their simplicity and cost-effectiveness, CW-fNIRS is by far the most common form of functional NIRS since it is the cheapest to make, applicable with more channels, and ensures a high temporal resolution. However, it does not distinguish between absorption and scattering changes, and cannot measure absolute absorption values: which means that it is only sensitive to relative change in HbO concentration.

Still, the simplicity and cost-effectiveness of CW-based devices prove themselves to be the most favorable for a number of clinical applications: neonatal care, patient monitoring systems, diffuse optical tomography, and so forth. Moreover, thanks to its portability, wireless CW systems have been developed—allowing individuals to be monitored in ambulatory, clinical and sports environments.

Frequency domain

Frequency domain (FD) system comprises NIR laser sources which provide an amplitude-modulated sinusoid at frequencies near 100 MHz. FD-fNIRS measures attenuation, phase shift and the average path length of light through the tissue. Multi-Distance, which is a part of the FD-fNIRS, is insensitive to differences in skin color—giving constant results regardless of subject variation.

Changes in the back-scattered signal's amplitude and phase provide a direct measurement of absorption and scattering coefficients of the tissue, thus obviating the need for information about photon path-length; and from the coefficients we determine the changes in the concentration of hemodynamic parameters.

Because of the need for modulated lasers as well as phasic measurements, FD system-based devices are more technically complex (therefore more expensive and much less portable) than CW-based ones. However, the system is capable of providing absolute concentrations of HbO and HbR.

Time domain

Time domain (TD) system introduces a short NIR pulse with a pulse length usually in the order of picoseconds—around 70 ps. Through time-of-flight measurements, photon path-length may be directly observed by dividing resolved time by the speed of light. Information about hemodynamic changes can be found in the attenuation, decay, and time profile of the back-scattered signal. For this photon-counting technology is introduced, which counts 1 photon for every 100 pulses to maintain linearity. TD-fNIRS does have a slow sampling rate as well as a limited number of wavelengths. Because of the need for a photon-counting device, high-speed detection, and high-speed emitters, time-resolved methods are the most expensive and technically complicated.

TD-based devices are totally immobile, space-consuming, the most difficult to make, costliest, hugest, and heaviest. Even so, they have the highest depth sensitivity and are capable of presenting most accurate values of baseline hemoglobin concentration and oxygenation.

Diffuse correlation spectroscopy

Diffuse correlation spectroscopy (DCS) systems use localized gradients in light attenuation to determine absolute ratios of oxy-Hb and deoxy-Hb. Using a spatial measurement, DCS systems do not require knowledge of photon path-length to make this calculation, however measured concentrations of oxy-Hb and deoxy-Hb are relative to the unknown coefficient of scattering in the media. This technique is most commonly used in cerebral oxymetry systems that report a Tissue Oxygenation Index (TOI) or Tissue Saturation Index (TSI).

System design

At least two open-source fNIRS models are available online:

• http://www.opennirs.org
• https://openfnirs.org/

Data analysis software

HOMER3

HOMER3 allows users to obtain estimates and maps of brain activation. It is a set of matlab scripts used for analyzing fNIRS data. This set of scripts has evolved since the early 1990s first as the Photon Migration Imaging toolbox, then HOMER1 and HOMER2, and now HOMER3.

NIRS toolbox

This toolbox is a set of Matlab-based tools for the analysis of functional near-infrared spectroscopy (fNIRS). This toolbox defines the +nirs namespace and includes a series of tools for signal processing, display, and statistics of fNIRS data. This toolbox is built around an object-oriented framework of Matlab classes and namespaces.

AtlasViewer

AtlasViewer allows fNIRS data to be visualized on a model of the brain. In addition, it also allows the user to design probes which can eventually be placed onto a subject.

Application

Brain–computer interface

fNIRS has been successfully implemented as a control signal for brain–computer interface systems.

Hypoxia & altitude studies

With our constant need for oxygen, our body has developed multiple mechanisms that detect oxygen levels, which in turn can activate appropriate responses to counter hypoxia and generate a higher oxygen supply. Moreover, understanding the physiological mechanism underlying the bodily response to oxygen deprivation is of major importance and NIRS devices have shown to be a great tool in this field of research.

Measurement of brain oxyhemoglobin and deoxyhemoglobin concentration changes at high alltitude induced hypoxia with a portable fNIRS device (PortaLite, Artinis Medical Systems)

Brain mapping

Functional connectivity

fNIRS measurements can be used to calculate functional connectivity. Multi-channel fNIRS measurements create a topographical map of neural activation, whereby temporal correlation between spatially separated events can be analyzed. Functional connectivity is typically assessed in terms correlations between the hemodynamic responses of spatially distinct regions of interest (ROIs). In brain studies, functional connectivity measurements are commonly taken for resting state patient data, as well as data recorded over stimulus paradigms. The low cost, portability and high temporal resolution of fNIRS, with respect to fMRI, have proven to be highly advantageous in studies of this nature.

Cerebral oximetry

NIRS monitoring is helpful in a number of ways. Preterm infants can be monitored reducing cerebral hypoxia and hyperoxia with different patterns of activities. It is an effective aid in Cardiopulmonary bypass, is strongly considered to improve patient outcomes and reduce costs and extended stays.

There are inconclusive results for use of NIRS with patients with traumatic brain injury, so it has been concluded that it should remain a research tool.

Diffuse optical tomography

Diffuse optical tomography is the 3D version of Diffuse optical imaging. Diffuse optical images are obtained using NIRS or fluorescence-based methods. These images can be used to develop a 3D volumetric model which is known as the Diffuse Optical Tomography.

10-20 system

fNIRS cap

fNIRS electrode locations can be defined using a variety of layouts, including names and locations that are specified by the International 10–20 system as well as other layouts that are specifically optimized to maintain a consistent 30mm distance between each location. In addition to the standard positions of electrodes, short separation channels can be added. Short separation channels allow the measurement of scalp signals. Since the short separation channels measure the signal coming from the scalp, they allow the removal of the signal of superficial layers. This leaves behind the actual brain response. Short separation channel detectors are usually placed 8mm away from a source. They do not need to be in a specific direction or in the same direction as a detector.

Functional neuroimaging

The use of fNIRS as a functional neuroimaging method relies on the principle of neuro-vascular coupling also known as the haemodynamic response or blood-oxygen-level dependent (BOLD) response. This principle also forms the core of fMRI techniques. Through neuro-vascular coupling, neuronal activity is linked to related changes in localized cerebral blood flow. fNIRS and fMRI are sensitive to similar physiologic changes and are often comparative methods. Studies relating fMRI and fNIRS show highly correlated results in cognitive tasks. fNIRS has several advantages in cost and portability over fMRI, but cannot be used to measure cortical activity more than 4 cm deep due to limitations in light emitter power and has more limited spatial resolution. fNIRS includes the use of diffuse optical tomography (DOT/NIRDOT) for functional purposes. Multiplexing fNIRS channels can allow 2D topographic functional maps of brain activity (e.g. with Hitachi ETG-4000, Artinis Oxymon, NIRx NIRScout, etc.) while using multiple emitter spacings may be used to build 3D tomographic maps.

Play media

fNIRS hyperscanning with two violinists

Hyperscanning

Hyperscanning involves two or more brains monitored simultaneously to investigate interpersonal (across-brains) neural correlates in various social situations, which proves fNIRS to be a suitable modality for investigating live brain-to-brain social interactions.

Virtual and augmented reality

Modern fNIRS systems are combined with virtual or augmented reality in studies on brain-computer interfaces, neurorehabilitation or social perception.

Mobile and wireless fNIRS and EEG systems synchronized with all-in-one head mounted display (PhotonCap, Cortivision)

Music and the brain

Play media

fNIRS with a pianist

fNIRS can be used to monitor musicians' brain activity while playing musical instruments.

Pros and cons

The advantages of fNIRS are, among other things: noninvasiveness, low-cost modalities, perfect safety, high temporal resolution, full compatibility with other imaging modalities, and multiple hemodynamic biomarkers.

However, no system is without limitations. For fNIRS those include: low brain sensitivity, low spatial resolution, and shallow penetration depth.

Future directions

Despite a few limitations, fNIRS devices are relatively small, lightweight, portable and wearable. Thanks to these features, applications for the devices are astounding—which make them easily accessible in many different scenarios. For example, they have the potential to be used in clinics, a global health situation, a natural environment, and as a health tracker.

Ultimately, future at-risk individuals in hospitals could benefit from neuromonitoring and neurorehabilitation that fNIRS can offer.

Now there are fully wireless research grade fNIRS systems in the market.

fNIRS compared with other neuroimaging techniques

Comparing and contrasting other neuroimaging devices is an important thing to take into consideration. When comparing and contrasting these devices it is important to look at the temporal resolution, spatial resolution, and the degree of immobility. EEG (electroencephalograph) and MEG (magnetoencephalography) have high temporal resolution, but a low spatial resolution. EEG also has a higher degree of mobility than MEG has. When looking at fNIRS, they are similar to an EEG. They have a high degree of mobility as well as temporal resolution, and they have low spatial resolution. PET scans and fMRIs are grouped together, however they are distinctly different from the other neuroimaging scans. They have a high degree of immobility, medium/high spatial resolution, and a low temporal resolution. All of these neuroimaging scans have important characteristics and are valuable, however they have distinct characteristics.

Among all other facts, what makes fNIRS a special point of interest is that it is compatible with some of these modalities, including: MRI, EEG, and MEG.

Near-infrared spectroscopy

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Near-infrared_spectroscopy 

 

Near-IR absorption spectrum of dichloromethane showing complicated overlapping overtones of mid IR absorption features.

Near-infrared spectroscopy (NIRS) is a spectroscopic method that uses the near-infrared region of the electromagnetic spectrum (from 780 nm to 2500 nm). Typical applications include medical and physiological diagnostics and research including blood sugar, pulse oximetry, functional neuroimaging, sports medicine, elite sports training, ergonomics, rehabilitation, neonatal research, brain computer interface, urology (bladder contraction), and neurology (neurovascular coupling). There are also applications in other areas as well such as pharmaceutical, food and agrochemical quality control, atmospheric chemistry, combustion research and astronomy.

Theory

Near-infrared spectroscopy is based on molecular overtone and combination vibrations. Such transitions are forbidden by the selection rules of quantum mechanics. As a result, the molar absorptivity in the near-IR region is typically quite small. One advantage is that NIR can typically penetrate much further into a sample than mid infrared radiation. Near-infrared spectroscopy is, therefore, not a particularly sensitive technique, but it can be very useful in probing bulk material with little or no sample preparation.

The molecular overtone and combination bands seen in the near-IR are typically very broad, leading to complex spectra; it can be difficult to assign specific features to specific chemical components. Multivariate (multiple variables) calibration techniques (e.g., principal components analysis, partial least squares, or artificial neural networks) are often employed to extract the desired chemical information. Careful development of a set of calibration samples and application of multivariate calibration techniques is essential for near-infrared analytical methods.

History

Near-infrared spectrum of liquid ethanol.

The discovery of near-infrared energy is ascribed to William Herschel in the 19th century, but the first industrial application began in the 1950s. In the first applications, NIRS was used only as an add-on unit to other optical devices that used other wavelengths such as ultraviolet (UV), visible (Vis), or mid-infrared (MIR) spectrometers. In the 1980s, a single-unit, stand-alone NIRS system was made available, but the application of NIRS was focused more on chemical analysis. With the introduction of light-fiber optics in the mid-1980s and the monochromator-detector developments in the early 1990s, NIRS became a more powerful tool for scientific research.

This optical method can be used in a number of fields of science including physics, physiology, or medicine. It is only in the last few decades that NIRS began to be used as a medical tool for monitoring patients, with the first clinical application of so-called fNIRS in 1994.

Instrumentation

Instrumentation for near-IR (NIR) spectroscopy is similar to instruments for the UV-visible and mid-IR ranges. There is a source, a detector, and a dispersive element (such as a prism, or, more commonly, a diffraction grating) to allow the intensity at different wavelengths to be recorded. Fourier transform NIR instruments using an interferometer are also common, especially for wavelengths above ~1000 nm. Depending on the sample, the spectrum can be measured in either reflection or transmission.

Common incandescent or quartz halogen light bulbs are most often used as broadband sources of near-infrared radiation for analytical applications. Light-emitting diodes (LEDs) can also be used. For high precision spectroscopy, wavelength-scanned lasers and frequency combs have recently become powerful sources, albeit with sometimes longer acquisition timescales. When lasers are used, a single detector without any dispersive elements might be sufficient.

The type of detector used depends primarily on the range of wavelengths to be measured. Silicon-based CCDs are suitable for the shorter end of the NIR range, but are not sufficiently sensitive over most of the range (over 1000 nm). InGaAs and PbS devices are more suitable and have higher quantum efficiency for wavelengths above 1100 nm. It is possible to combine silicon-based and InGaAs detectors in the same instrument. Such instruments can record both UV-visible and NIR spectra 'simultaneously'.

Instruments intended for chemical imaging in the NIR may use a 2D array detector with an acousto-optic tunable filter. Multiple images may be recorded sequentially at different narrow wavelength bands.

Many commercial instruments for UV/vis spectroscopy are capable of recording spectra in the NIR range (to perhaps ~900 nm). In the same way, the range of some mid-IR instruments may extend into the NIR. In these instruments, the detector used for the NIR wavelengths is often the same detector used for the instrument's "main" range of interest.

Applications

Typical applications of NIR spectroscopy include the analysis of food products, pharmaceuticals, combustion products, and a major branch of astronomical spectroscopy.

Astronomical spectroscopy

Near-infrared spectroscopy is used in astronomy for studying the atmospheres of cool stars where molecules can form. The vibrational and rotational signatures of molecules such as titanium oxide, cyanide, and carbon monoxide can be seen in this wavelength range and can give a clue towards the star's spectral type. It is also used for studying molecules in other astronomical contexts, such as in molecular clouds where new stars are formed. The astronomical phenomenon known as reddening means that near-infrared wavelengths are less affected by dust in the interstellar medium, such that regions inaccessible by optical spectroscopy can be studied in the near-infrared. Since dust and gas are strongly associated, these dusty regions are exactly those where infrared spectroscopy is most useful. The near-infrared spectra of very young stars provide important information about their ages and masses, which is important for understanding star formation in general. Astronomical spectrographs have also been developed for the detection of exoplanets using the Doppler shift of the parent star due to the radial velocity of the planet around the star.

Agriculture

Near-infrared spectroscopy is widely applied in agriculture for determining the quality of forages, grains, and grain products, oilseeds, coffee, tea, spices, fruits, vegetables, sugarcane, beverages, fats, and oils, dairy products, eggs, meat, and other agricultural products. It is widely used to quantify the composition of agricultural products because it meets the criteria of being accurate, reliable, rapid, non-destructive, and inexpensive. Abeni and Bergoglio 2001 apply NIRS to chicken breeding as the assay method for characteristics of fat composition.

Remote monitoring

Techniques have been developed for NIR spectroscopic imaging. Hyperspectral imaging has been applied for a wide range of uses, including the remote investigation of plants and soils. Data can be collected from instruments on airplanes or satellites to assess ground cover and soil chemistry.

Remote monitoring or remote sensing from the NIR spectroscopic region can also be used to study the atmosphere. For example, measurements of atmospheric gases are made from NIR spectra measured by the OCO-2, GOSAT, and the TCCON.

Materials science

Techniques have been developed for NIR spectroscopy of microscopic sample areas for film thickness measurements, research into the optical characteristics of nanoparticles and optical coatings for the telecommunications industry.

Medical uses

The application of NIRS in medicine centres on its ability to provide information about the oxygen saturation of haemoglobin within the microcirculation. Broadly speaking, it can be used to assess oxygenation and microvascular function in the brain (cerebral NIRS) or in the peripheral tissues (peripheral NIRS).

Cerebral NIRS

When a specific area of the brain is activated, the localized blood volume in that area changes quickly. Optical imaging can measure the location and activity of specific regions of the brain by continuously monitoring blood hemoglobin levels through the determination of optical absorption coefficients.

Infrascanner 1000, a NIRS scanner used to detect intracranial bleeding.

NIRS can be used as a quick screening tool for possible intracranial bleeding cases by placing the scanner on four locations on the head. In non-injured patients the brain absorbs the NIR light evenly. When there is an internal bleeding from an injury, the blood may be concentrated in one location causing the NIR light to be absorbed more than other locations, which the scanner detects.

So-called functional NIRS can be used for non-invasive assessment of brain function through the intact skull in human subjects by detecting changes in blood hemoglobin concentrations associated with neural activity, e.g., in branches of cognitive psychology as a partial replacement for fMRI techniques. NIRS can be used on infants, and NIRS is much more portable than fMRI machines, even wireless instrumentation is available, which enables investigations in freely moving subjects. However, NIRS cannot fully replace fMRI because it can only be used to scan cortical tissue, whereas fMRI can be used to measure activation throughout the brain. Special public domain statistical toolboxes for analysis of stand alone and combined NIRS/MRI measurement have been developed (NIRS-SPM).

Example of data acquisition using fNIRS (Hitachi ETG-4000)

The application in functional mapping of the human cortex is called functional NIRS (fNIRS) or diffuse optical tomography (DOT). The term diffuse optical tomography is used for three-dimensional NIRS. The terms NIRS, NIRI, and DOT are often used interchangeably, but they have some distinctions. The most important difference between NIRS and DOT/NIRI is that DOT/NIRI is used mainly to detect changes in optical properties of tissue simultaneously from multiple measurement points and display the results in the form of a map or image over a specific area, whereas NIRS provides quantitative data in absolute terms on up to a few specific points. The latter is also used to investigate other tissues such as, e.g., muscle, breast and tumors. NIRS can be used to quantify blood flow, blood volume, oxygen consumption, reoxygenation rates and muscle recovery time in muscle.

By employing several wavelengths and time resolved (frequency or time domain) and/or spatially resolved methods blood flow, volume and absolute tissue saturation (${\displaystyle StO_{2}}$ or Tissue Saturation Index (TSI)) can be quantified. Applications of oximetry by NIRS methods include neuroscience, ergonomics, rehabilitation, brain-computer interface, urology, the detection of illnesses that affect the blood circulation (e.g., peripheral vascular disease), the detection and assessment of breast tumors, and the optimization of training in sports medicine.

The use of NIRS in conjunction with a bolus injection of indocyanine green (ICG) has been used to measure cerebral blood flow and cerebral metabolic rate of oxygen consumption (CMRO2). It has also been shown that CMRO2 can be calculated with combined NIRS/MRI measurements. Additionally metabolism can be interrogated by resolving an additional mitochondrial chromophore, cytochrome-c-oxidase, using broadband NIRS.

NIRS is starting to be used in pediatric critical care, to help manage patients following cardiac surgery. Indeed, NIRS is able to measure venous oxygen saturation (SVO2), which is determined by the cardiac output, as well as other parameters (FiO2, hemoglobin, oxygen uptake). Therefore, examining the NIRS provides critical care physicians with an estimate of the cardiac output. NIRS is favoured by patients, because it is non-invasive, painless, and does not require ionizing radiation.

Optical coherence tomography (OCT) is another NIR medical imaging technique capable of 3D imaging with high resolution on par with low-power microscopy. Using optical coherence to measure photon pathlength allows OCT to build images of live tissue and clear examinations of tissue morphology. Due to technique differences OCT is limited to imaging 1–2 mm below tissue surfaces, but despite this limitation OCT has become an established medical imaging technique especially for imaging of the retina and anterior segments of the eye, as well as coronaries.

A type of neurofeedback, hemoencephalography or HEG, uses NIR technology to measure brain activation, primarily of the frontal lobes, for the purpose of training cerebral activation of that region.

The instrumental development of NIRS/NIRI/DOT/OCT has proceeded tremendously during the last years and, in particular, in terms of quantification, imaging and miniaturization.

Peripheral NIRS

Peripheral microvascular function can be assessed using NIRS. The oxygen saturation of haemoglobin in the tissue (StO2) can provide information about tissue perfusion. A vascular occlusion test (VOT) can be employed to assess microvascular function. Common sites for peripheral NIRS monitoring include the thenar eminence, forearm and calf muscles.

Particle measurement

NIR is often used in particle sizing in a range of different fields, including studying pharmaceutical and agricultural powders.

Industrial uses

As opposed to NIRS used in optical topography, general NIRS used in chemical assays does not provide imaging by mapping. For example, a clinical carbon dioxide analyzer requires reference techniques and calibration routines to be able to get accurate CO₂ content change. In this case, calibration is performed by adjusting the zero control of the sample being tested after purposefully supplying 0% CO₂ or another known amount of CO₂ in the sample. Normal compressed gas from distributors contains about 95% O₂ and 5% CO₂, which can also be used to adjust %CO₂ meter reading to be exactly 5% at initial calibration.

Countable set

From Wikipedia, the free encyclopedia

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

In mathematics, a set is countable if it has the same cardinality (the number of elements of the set) as some subset of the set of natural numbers N = {0, 1, 2, 3, ...}. Equivalently, a set S is countable if there exists an injective function f : S → N from S to N; it simply means that every element in S has the correspondence to a different element in N.

A countable set is either a finite set or a countably infinite set. Whether finite or infinite, the elements of a countable set can always be counted one at a time and — although the counting may never finish due to the infinite number of the elements to be counted — every element of the set is associated with a unique natural number.

Georg Cantor introduced the concept of countable sets, contrasting sets that are countable with those that are uncountable. Today, countable sets form the foundation of a branch of mathematics called discrete mathematics.

A note on terminology

Although the terms "countable" and "countably infinite" as defined here are quite common, the terminology is not universal. An alternative style uses countable to mean what is here called countably infinite, and at most countable to mean what is here called countable. To avoid ambiguity, one may limit oneself to the terms "at most countable" and "countably infinite", although with respect to concision this is the worst of both worlds. The reader is advised to check the definition in use when encountering the term "countable" in the literature.

The terms enumerable and denumerable may also be used, e.g. referring to countable and countably infinite respectively, but as definitions vary the reader is once again advised to check the definition in use.

Definition

The most concise definition is in terms of cardinality. A set S is countable if its cardinality |S| is less than or equal to ${\displaystyle \aleph _{0}}$ (aleph-null), the cardinality of the set of natural numbers N. A set S is countably infinite if |S| = ${\displaystyle \aleph _{0}}$. A set is uncountable if it is not countable, i.e. its cardinality is greater than ${\displaystyle \aleph _{0}}$; the reader is referred to Uncountable set for further discussion.

For every set S, the following propositions are equivalent:

• S is countable.
• There exists an injective function from S to N.
• S is empty or there exists a surjective function from N to S.
• There exists a bijective mapping between S and a subset of N.
• S is either finite or countably infinite.

Similarly, the following propositions are equivalent:

• S is countably infinite.
• There is an injective and surjective (and therefore bijective) mapping between S and N.
• S has a one-to-one correspondence with N.
• The elements of S can be arranged in an infinite sequence ${\displaystyle a_{0},a_{1},a_{2},\ldots }$, where ${\displaystyle a_{i}}$ is distinct from ${\displaystyle a_{j}}$ for ${\displaystyle i\neq j}$ and every element of S is listed.

History

In 1874, in his first set theory article, Cantor proved that the set of real numbers is uncountable, thus showing that not all infinite sets are countable. In 1878, he used one-to-one correspondences to define and compare cardinalities. In 1883, he extended the natural numbers with his infinite ordinals, and used sets of ordinals to produce an infinity of sets having different infinite cardinalities.

Introduction

A set is a collection of elements, and may be described in many ways. One way is simply to list all of its elements; for example, the set consisting of the integers 3, 4, and 5 may be denoted {3, 4, 5}, called roster form. This is only effective for small sets, however; for larger sets, this would be time-consuming and error-prone. Instead of listing every single element, sometimes an ellipsis ("...") is used to represent many elements between the starting element and the end element in a set, if the writer believes that the reader can easily guess what ... represents; for example, {1, 2, 3, ..., 100} presumably denotes the set of integers from 1 to 100. Even in this case, however, it is still possible to list all the elements, because number of elements in the set is finite.

Some sets are infinite; these sets have more than n elements where n is any integer that can be specified. (No matter how large the specified integer n is, such as n = 9 × 10³², infinite sets have more than n elements.) For example, the set of natural numbers, denotable by {0, 1, 2, 3, 4, 5, ...}, has infinitely many elements, and we cannot use any natural number to give its size. Nonetheless, it turns out that infinite sets do have a well-defined notion of size (or more properly, cardinality, the technical term for the number of elements in a set), and not all infinite sets have the same cardinality.

Bijective mapping from integer to even numbers

To understand what this means, we first examine what it does not mean. For example, there are infinitely many odd integers, infinitely many even integers, and (hence) infinitely many integers overall. However, it turns out that the number of even integers, which is the same as the number of odd integers, is also the same as the number of integers overall. This is because we can arrange things such that, for every integer, there is a distinct even integer:

$${\displaystyle \ldots \,-\!2\!\rightarrow \!-\!4,\,-\!1\!\rightarrow \!-\!2,\,0\!\rightarrow \!0,\,1\!\rightarrow \!2,\,2\!\rightarrow \!4\,\ldots }$$

or, more generally, ${\displaystyle n\rightarrow 2n}$ (see picture). What we have done here is arrange the integers and the even integers into a one-to-one correspondence (or bijection), which is a function that maps between two sets such that each element of each set corresponds to a single element in the other set.

However, not all infinite sets have the same cardinality. For example, Georg Cantor (who introduced this concept) demonstrated that the real numbers cannot be put into one-to-one correspondence with the natural numbers (non-negative integers), and therefore that the set of real numbers has a greater cardinality than the set of natural numbers.

Formal overview

By definition, a set S is countable if there exists an injective function f : S → N from S to the natural numbers N = {0, 1, 2, 3, ...}. It simply means that every element in S has the correspondence to a different element in N.

It might seem natural to divide the sets into different classes: put all the sets containing one element together; all the sets containing two elements together; ...; finally, put together all infinite sets and consider them as having the same size. This view is not tenable, however, under the natural definition of size.

To elaborate this, we need the concept of a bijection. Although a "bijection" may seem a more advanced concept than a number, the usual development of mathematics in terms of set theory defines functions before numbers, as they are based on much simpler sets. This is where the concept of a bijection comes in: define the correspondence

a ↔ 1, b ↔ 2, c ↔ 3

Since every element of {a, b, c} is paired with precisely one element of {1, 2, 3}, and vice versa, this defines a bijection.

We now generalize this situation; we define that two sets are of the same size, if and only if there is a bijection between them. For all finite sets, this gives us the usual definition of "the same size".

As for the case of infinite sets, consider the sets A = {1, 2, 3, ... }, the set of positive integers, and B = {2, 4, 6, ... }, the set of even positive integers. We claim that, under our definition, these sets have the same size, and that therefore B is countably infinite. Recall that to prove this, we need to exhibit a bijection between them. This can be achieved using the assignment n ↔ 2n, so that

1 ↔ 2, 2 ↔ 4, 3 ↔ 6, 4 ↔ 8, ....

As in the earlier example, every element of A has been paired off with precisely one element of B, and vice versa. Hence they have the same size. This is an example of a set of the same size as one of its proper subsets, which is impossible for finite sets.

Likewise, the set of all ordered pairs of natural numbers (the Cartesian product of two sets of natural numbers, N × N) is countably infinite, as can be seen by following a path like the one in the picture:

 

The Cantor pairing function assigns one natural number to each pair of natural numbers

The resulting mapping proceeds as follows:

0 ↔ (0, 0), 1 ↔ (1, 0), 2 ↔ (0, 1), 3 ↔ (2, 0), 4 ↔ (1, 1), 5 ↔ (0, 2), 6 ↔ (3, 0), ....

This mapping covers all such ordered pairs.

This form of triangular mapping recursively generalizes to n-tuples of natural numbers, i.e., (a₁, a₂, a₃, ..., a_n) where a_i and n are natural numbers, by repeatedly mapping the first two elements of a n-tuple to a natural number. For example, (0, 2, 3) can be written as ((0, 2), 3). Then (0, 2) maps to 5 so ((0, 2), 3) maps to (5, 3), then (5, 3) maps to 39. Since a different 2-tuple, that is a pair such as (a, b), maps to a different natural number, a difference between two n-tuples by a single element is enough to ensure the n-tuples being mapped to different natural numbers. So, an injection from the set of n-tuples to the set of natural numbers N is proved. For the set of n-tuples made by the Cartesian product of finitely many different sets, each element in each tuple has the correspondence to a natural number, so every tuple can be written in natural numbers then the same logic is applied to prove the theorem.

Theorem: The Cartesian product of finitely many countable sets is countable.

The set of all integers Z and the set of all rational numbers Q may intuitively seem much bigger than N. But looks can be deceiving. If a pair is treated as the numerator and denominator of a vulgar fraction (a fraction in the form of a/b where a and b ≠ 0 are integers), then for every positive fraction, we can come up with a distinct natural number corresponding to it. This representation also includes the natural numbers, since every natural number is also a fraction N/1. So we can conclude that there are exactly as many positive rational numbers as there are positive integers. This is also true for all rational numbers, as can be seen below.

Theorem: Z (the set of all integers) and Q (the set of all rational numbers) are countable.

In a similar manner, the set of algebraic numbers is countable.

Sometimes more than one mapping is useful: a set A to be shown as countable is one-to-one mapped (injection) to another set B, then A is proved as countable if B is one-to-one mapped to the set of natural numbers. For example, the set of positive rational numbers can easily be one-to-one mapped to the set of natural number pairs (2-tuples) because p/q maps to (p, q). Since the set of natural number pairs is one-to-one mapped (actually one-to-one correspondence or bijection) to the set of natural numbers as shown above, the positive rational number set is proved as countable.

Theorem: Any finite union of countable sets is countable.

With the foresight of knowing that there are uncountable sets, we can wonder whether or not this last result can be pushed any further. The answer is "yes" and "no", we can extend it, but we need to assume a new axiom to do so.

Theorem: (Assuming the axiom of countable choice) The union of countably many countable sets is countable.

For example, given countable sets a, b, c, ...

Enumeration for countable number of countable sets

Using a variant of the triangular enumeration we saw above:

• a₀ maps to 0
• a₁ maps to 1
• b₀ maps to 2
• a₂ maps to 3
• b₁ maps to 4
• c₀ maps to 5
• a₃ maps to 6
• b₂ maps to 7
• c₁ maps to 8
• d₀ maps to 9
• a₄ maps to 10
• ...

This only works if the sets a, b, c, ... are disjoint. If not, then the union is even smaller and is therefore also countable by a previous theorem.

We need the axiom of countable choice to index all the sets a, b, c, ... simultaneously.

Theorem: The set of all finite-length sequences of natural numbers is countable.

This set is the union of the length-1 sequences, the length-2 sequences, the length-3 sequences, each of which is a countable set (finite Cartesian product). So we are talking about a countable union of countable sets, which is countable by the previous theorem.

Theorem: The set of all finite subsets of the natural numbers is countable.

The elements of any finite subset can be ordered into a finite sequence. There are only countably many finite sequences, so also there are only countably many finite subsets.

Theorem: Let S and T be sets.

1. If the function f : S → T is injective and T is countable then S is countable.
2. If the function g : S → T is surjective and S is countable then T is countable.

These follow from the definitions of countable set as injective / surjective functions.

Cantor's theorem asserts that if A is a set and P(A) is its power set, i.e. the set of all subsets of A, then there is no surjective function from A to P(A). A proof is given in the article Cantor's theorem. As an immediate consequence of this and the Basic Theorem above we have:

Proposition: The set P(N) is not countable; i.e. it is uncountable.

For an elaboration of this result see Cantor's diagonal argument.

The set of real numbers is uncountable, and so is the set of all infinite sequences of natural numbers.

Minimal model of set theory is countable

If there is a set that is a standard model (see inner model) of ZFC set theory, then there is a minimal standard model (see Constructible universe). The Löwenheim–Skolem theorem can be used to show that this minimal model is countable. The fact that the notion of "uncountability" makes sense even in this model, and in particular that this model M contains elements that are:

• subsets of M, hence countable,
• but uncountable from the point of view of M,

was seen as paradoxical in the early days of set theory, see Skolem's paradox for more.

The minimal standard model includes all the algebraic numbers and all effectively computable transcendental numbers, as well as many other kinds of numbers.

Total orders

Countable sets can be totally ordered in various ways, for example:

• Well-orders (see also ordinal number):
  • The usual order of natural numbers (0, 1, 2, 3, 4, 5, ...)
  • The integers in the order (0, 1, 2, 3, ...; −1, −2, −3, ...)
• Other (not well orders):
  • The usual order of integers (..., −3, −2, −1, 0, 1, 2, 3, ...)
  • The usual order of rational numbers (Cannot be explicitly written as an ordered list!)

In both examples of well orders here, any subset has a least element; and in both examples of non-well orders, some subsets do not have a least element. This is the key definition that determines whether a total order is also a well order.