Mathematics In Medical Field
Bacterial growth
Bacterial growth is the asexual reproduction, or cell division,
of a bacterium into two daughter cells, in a process called binary fission.
Providing no mutational event occurs, the resulting daughter cells are
genetically identical to the original cell. Hence, "local doubling"
of the bacterial population occurs. Both daughter cells from the division do
not necessarily survive. However, if the number surviving exceeds unity on
average, the bacterial population undergoes exponential growth. The measurement of an
exponential bacterial growth curve in batch culture was traditionally a part of
the training of all microbiologists; the basic means requires bacterial
enumeration (cell counting) by direct and individual (microscopic, flow
cytometry[1]), direct and bulk
(biomass), indirect and individual (colony counting), or indirect and bulk
(most probable number, turbidity,
nutrient uptake) methods. Models reconcile theory with the measurements.
Growth is shown as L = log(numbers) where
numbers is the number of colony forming units per ml, versus T (time.)
Bacterial growth
curve\Kinetic Curve
In autecological studies, the growth of
bacteria (or other microorganisms, as protozoa, microalgae or yeasts) in batch culture can be modeled with
four different phases: lag phase (A), log phase or exponential phase (B), stationary phase (C), and death phase(D).[3]
1.
During lag phase, bacteria adapt themselves to
growth conditions. It is the period where the individual bacteria are maturing and not yet
able to divide. During the lag phase of the bacterial growth cycle, synthesis
of RNA, enzymes and other molecules occurs.
2.
The log phase (sometimes called the
logarithmic phase or the exponential phase) is a period
characterized by cell doubling.[4]The number of new
bacteria appearing per unit time is proportional to the present population. If
growth is not limited, doubling will continue at a constant rate so both the
number of cells and the rate of population increase doubles with each
consecutive time period. For this type of exponential growth, plotting the
natural logarithm of cell number against time produces a straight line. The
slope of this line is the specific growth rate of the organism, which is a
measure of the number of divisions per cell per unit time.[4] The actual rate of this
growth (i.e. the slope of the line in the figure) depends upon the growth
conditions, which affect the frequency of cell division events and the
probability of both daughter cells surviving. Under controlled conditions, cyanobacteria can double their
population four times a day.[5] Exponential growth cannot
continue indefinitely, however, because the medium is soon depleted of
nutrients and enriched with wastes.
3.
The stationary phase is often due to a
growth-limiting factor such as the depletion of an essential nutrient, and/or
the formation of an inhibitory product such as an organic acid. Stationary
phase results from a situation in which growth rate and death rate are equal.
The number of new cells created is limited by the growth factor and as a result
the rate of cell growth matches the rate of cell death. The result is a
“smooth,” horizontal linear part of the curve during the stationary phase.
4.
At death phase (decline phase), bacteria
die. This could be caused by lack of nutrients, environmental temperature above
or below the tolerance band for the species, or other injurious conditions.
This basic batch culture growth model draws
out and emphasizes aspects of bacterial growth which may differ from the growth
of macrofauna. It emphasizes clonality, asexual binary division, the short
development time relative to replication itself, the seemingly low death rate,
the need to move from a dormant state to a reproductive state or to condition
the media, and finally, the tendency of lab adapted strains to exhaust their
nutrients. In reality, even in batch culture, the four phases are not well
defined. The cells do not reproduce in synchrony without explicit and continual
prompting (as in experiments with stalked bacteria [6]) and their exponential
phase growth is often not ever a constant rate, but instead a slowly decaying
rate, a constant stochastic response to pressures both to reproduce and to go
dormant in the face of declining nutrient concentrations and increasing waste
concentrations.
Batch culture is the most common laboratory
growth method in which bacterial growth is studied, but it is only one of many.
It is ideally spatially unstructured and temporally structured. The bacterial
culture is incubated in a closed vessel with a single batch of medium. In some
experimental regimes, some of the bacterial culture is periodically removed and
added to fresh sterile medium. In the extreme case, this leads to the continual
renewal of the nutrients. This is a chemostat, also
known as continuous culture. It is ideally spatially unstructured and
temporally unstructured, in a steady state defined by the rates of nutrient
supply and bacterial growth. In comparison to batch culture, bacteria are
maintained in exponential growth phase, and the growth rate of the bacteria is
known. Related devices include turbidostats and auxostats.
Bacterial growth can be suppressed with bacteriostats,
without necessarily killing the bacteria. In a synecological,
true-to-nature situation in which more than one bacterial species is present,
the growth of microbes is more dynamic and continual.
Liquid is not the only laboratory
environment for bacterial growth. Spatially structured environments such as
biofilms or agar surfaces present
additional complex growth models.
Blood pressure
Blood pressure (BP) is the pressure of circulating blood on the walls of blood vessels. When used without further specification,
"blood pressure" usually refers to the pressure in large arteries of the systemic circulation.
Blood pressure is usually expressed in terms of the systolic (maximum during one heart
beat) pressure over diastolic (minimum in between two
heart beats) pressure and is measured in millimeters of mercury (mmHg), above
the surrounding atmospheric pressure (considered to be zero for convenience).
Systemic arterial pressure
The risk of cardiovascular
disease increases progressively above 115/75 mmHg.[6] In practice blood pressure is considered
too low only if noticeable symptoms are present.[4]
Observational
studies demonstrate that people who maintain arterial pressures at the low end
of these pressure ranges have much better long term cardiovascular health.
There is an ongoing medical debate over what is the optimal level of blood
pressure to target when using drugs to lower blood pressure with hypertension,
particularly in older people
Mean arterial pressure
The mean arterial
pressure (MAP) is the
average over a cardiac cycle and is determined by the cardiac output (CO), systemic
vascular resistance (SVR),
and central venous
pressure (CVP).
Pulse pressure
Curve of
the arterial pressure during one cardiac cycle. The closing of the aortic valve
causes the notch in the curve.
The pulse pressure is the difference between the measured
systolic and diastolic pressures,[24]
The up and down fluctuation of
the arterial pressure
results from the pulsatile nature of the cardiac output, i.e. the heartbeat. Pulse
pressure is determined by the interaction of the stroke volume of the heart, the compliance (ability
to expand) of the arterial system—largely attributable to the aorta and large elastic arteries—and the resistance to
flow in the arterial tree. By expanding under pressure,
the aorta absorbs some of the force of the blood surge from the heart during a
heartbeat. In this way, the pulse pressure is reduced from what it would be if
the aorta were not compliant.[24] The loss of arterial compliance that occurs
with aging explains the elevated pulse pressures found in elderly patients.
Measurement
Taking another persons blood pressure with a
sphygmomanometer
For each heartbeat, blood pressure varies between systolic and diastolic
pressures. Systolic pressure is peak pressure in the arteries, which occurs
near the end of the cardiac cycle when the ventricles are contracting.
Diastolic pressure is minimum pressure in the arteries, which occurs near the
beginning of the cardiac cycle when the ventricles are filled with blood. An
example of normal measured values for a resting, healthy adult human is
120 mmHg systolic and 80 mmHg diastolic (written as
120/80 mmHg, and spoken as "one-twenty over eighty").Color vision
Color vision is the ability of an organism or machine to distinguish
objects based on the wavelengths (or frequencies) of the light theyreflect, emit, or transmit. Colors can be measured and quantified in
various ways; indeed, a person's perception of colors is a subjective process
whereby the brain responds to the stimuli that are produced
when incoming light reacts with the several types of cone cells in
theeye. In essence, different people see the same
illuminated object or light source in different ways.
Detection
Isaac Newton discovered
that white light splits into its component colours when passed through a dispersive prism. Newton also found that he
could recombine these colours by passing them through a different prism to make
white light.
The characteristic colours are, from long
to short wavelengths (and, correspondingly, from low to high frequency), red,
orange, yellow, green, blue, indigo, and violet. Sufficient differences in
wavelength cause a difference in the perceived hue;
the just-noticeable
difference in
wavelength varies from about 1 nm in
the blue-green and yellow wavelengths,
to 10 nm and more in the longer red and shorter blue wavelengths. Although
the human eye can distinguish up to a few hundred hues, when those pure spectral colours are mixed together or diluted with
white light, the number of distinguishable chromaticities can be quite high.
In very low light levels, vision is scotopic: light is detected by rod cells of
the retina. Rods are maximally sensitive to
wavelengths near 500 nm, and play little, if any, role in colour vision.
In brighter light, such as daylight, vision is photopic: light is detected by cone cells which
are responsible for colour vision. Cones are sensitive to a range of
wavelengths, but are most sensitive to wavelengths near 555 nm. Between
these regions, mesopic vision comes into play and both rods and
cones provide signals to the retinal ganglion
cells. The shift in colour perception from dim light to daylight
gives rise to differences known as the Purkinje effect.
Detection
- The modern model of human color perception as it occurs in the retina, pertaining to both the trichromatic andopponent process theories introduced in the 19th century.
- Normalized response spectra of human cones, to monochromatic spectral stimuli, with wavelength given in nanometers.
Perception of color begins with specialized
retinal cells containing pigments with different spectral sensitivities,
known as cone cells. In humans, there are three types
of cones sensitive to three different spectra, resulting in trichromatic
color vision.
Each individual cone contains pigments
composed of opsin apoprotein, which is covalently linked
to either 11-cis-hydroretinal or more rarely 11-cis-dehydroretinal.[2]
The cones are conventionally labeled according
to the ordering of the wavelengths of the peaks of their spectral sensitivities:
short (S), medium (M), and long (L) cone types. These three types do not correspond
well to particular colors as we know them. Rather, the perception of color is
achieved by a complex process that starts with the differential output of these
cells in the retina and it will be finalized in the visual cortex and associative areas of the brain.
For
example, while the L cones have been referred to simply as red receptors, microspectrophotometry has shown that their peak sensitivity
is in the greenish-yellow region of the spectrum. Similarly, the S- and M-cones
do not directly correspond to blue and green,
although they are often described as such. The RGB color model, therefore, is a convenient
means for representing color, but is not directly based on the types of cones
in the human eye.
The peak response of human cone cells
varies, even among individuals with so-called normal color vision;[3] in some non-human species this polymorphic variation is
even greater, and it may well be adaptive.[4]
Theories
Two complementary theories of color vision
are the trichromatic theory and the opponent process theory. The trichromatic theory, orYoung–Helmholtz
theory, proposed in the 19th century by Thomas Young and Hermann von Helmholtz,
as mentioned above, states that the retina's three types of cones are
preferentially sensitive to blue, green, and red. Ewald Hering proposed
the opponent process theory in 1872.[5] It states that the visual system interprets color in an
antagonistic way: red vs. green, blue vs. yellow, black vs. white. Both
theories are now accepted as valid, describing different stages in visual
physiology, visualized in the diagram on the right.[6]Green ←→ Magenta and Blue ←→ Yellow are
scales with mutually exclusive boundaries. In the same way that there cannot
exist a "slightly negative" positive number, a single eye cannot
perceive a blueish-yellow or a reddish-green. (But such impossible colors can be perceived due to binocular rivalry.)
Cone cells in the human eye
Cone type
|
Name
|
Range
|
|
S
|
β
|
400–500 nm
|
420–440 nm
|
M
|
γ
|
450–630 nm
|
534–555 nm
|
L
|
ρ
|
500–700 nm
|
564–580 nm
|
A range of wavelengths of light stimulates
each of these receptor types to varying degrees. Yellowish-green light, for
example, stimulates both L and M cones equally strongly, but only stimulates
S-cones weakly. Red light, on the other hand, stimulates L cones much more than
M cones, and S cones hardly at all; blue-green light stimulates M cones more
than L cones, and S cones a bit more strongly, and is also the peak stimulant
for rod cells; and blue light stimulates S cones more strongly
than red or green light, but L and M cones more weakly. The brain combines the
information from each type of receptor to give rise to different perceptions of
different wavelengths of light.
The opsins (photopigments) present in the L
and M cones are encoded on the X chromosome; defective encoding of these leads
to the two most common forms of color blindness. The OPN1LW gene,
which codes for the opsin present in the L cones, is highly polymorphic (a recent study by Verrelli and
Tishkoff found 85 variants in a sample of 236 men).[9] A very small percentage of women may have an extra type
of color receptor because they have different alleles for the gene for the L
opsin on each X chromosome. X chromosome
inactivationmeans that only one opsin is expressed in each cone
cell, and some women may therefore show a degree of tetrachromatic color vision.[10]Variations in OPN1MW, which codes the opsin expressed in M
cones, appear to be rare, and the observed variants have no effect onspectral sensitivity.
