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          ARRoGANT                CoURiERS      WiTH     ESSaYS

Grade Level:       Type of Work           Subject/Topic is on:
 [ ]6-8                 [ ]Class Notes    [Essay - Math & Medicine.]
 [x]9-10                [ ]Cliff Notes    [                        ]
 [x]11-12               [x]Essay/Report   [                        ]
 [ ]College             [ ]Misc           [                        ]

 Date: 06/94  # of Words:2,401 School:Public - COED   State:NY
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                        Math and Medicine
     Every physician has experienced the group of numerical data,
for measurement is as much a part of medicine as it is of any
other science. Measurement is the way a size is determined from
a standard unit. Measurements are explicit by such things as: a
table of results, a graph, or converted into a certain form such
as percentage. The mathematics that I am about to perform is that
which is into pattern, and we link these patterns to medicine
which then in turn give us results. Some of the examples that I
will be giving will also be similar to biological data.
     A list of results that have data leads to block diagrams of
a single system. Block diagrams have single scales of
units. The chemical composition of milk, for example is
calculated in it's scale of percentages and can be put in a chart
form:
______________________________________________
Composition                        g per 100 g
----------------------------------------------
Water                              87.58
Protein                            2.01
Fat                                3.80
Carbohydrate                       6.50
Other                              0.11
----------------------------------------------
The block diagram shows the actual percentages by the amount of a
particular substance, meaning that if I were to a draw a picture
of a block diagram, it would contain more water which can be seen
in the next figure:
                   Calculations of two factors
     While it is common to see the measurement of a variable, it
is the measurement of two variables simultaneously is 
shown more often. There are two variables that are used as
factors. The first is x or the independent variable, the second
is called y or the dependant variable. This means that the y
variable on a graph  maybe fluctuating or may be stable, but the
fact is the x or dependent variable is increasing/decreasing
proportionally. 
     
     Data that is collected in the study of two variable scan be
presented in a table of two columns or figures. Tables of
results, do not show the relationship between the variables. This
can be seen as so on a cartesian graph.
     Two lines, instead of being arranged as  parallel (arrow
 diagram), can be set at right angles and so arranged that the
lines increase in numerical value in the direction of the arrow.
A cartesian graph is as follows:
The a and b represent corresponding data.
The c and d represent corresponding data.
                       Frequency Diagrams
There is a step by step difference from the bar graph to the
frequency diagram. In the bar chart, only one variable is
measured, and so the order on one of the axes is biased.
     If the variable being measured is a frequency, then the
figure which results is a frequency chart. Like the bar char,
each category is like a line or a narrow column, with spaces to
show no connection.
     The frequency charts show how "frequent" a event happens.
An example would be:
Frequency chart of the number of flowers on the dicotyledons
growing in a square meter of a roadside verge in april.
                   Positive Direct Proportion
The most basic pattern in the use of science is that of direct
proportion. In, medicine, direct proportion is involved, for
instance, in converting measurements take in one set of units
into their equivalents in another; in preparing solutions of
required strength; calculations of correct dosages for variable
human beings.
     Direct proportion had a number of features, there are 2
kinds of direct proportion, one where the variables increase and
decrease together; in the other, one variable increases while
the other decreases. The first is a positive direct proportion,
the second is a negative direct proportion.
     Positive direct proportion is a relationship that  exists
between two variables and so data relating to it can be presented
on a cartesian graph. From this graph, we can find if the results
are in direct proportion.
This graph show the effect of drug dosage on the number of
bacteria killed. As you can see from the graph, this is a
positive direct proportion because both variables( x,y) are
increasing.
The ratio:
The data formed on the cartesian graph is formed on the evidence 
of a direct proportion. We can also make ratio's from raw data:
-------------------------------------------
Drug dosage                   Estimated 
100mg                         number killed
                              
x                             y
-------------------------------------------
1                             1500
2                             3000
3                             4500
4                             6000           
5                             7500
-------------------------------------------
Two ratios can be gotten from these tables.
First, estimated number killed/drug dosage (y/x), or the
reciprocal of that which is drug dosage/estimated number killed 
(x/y). If we were to make these ratio for y/x they would be as
follows:
1500      3000      4500      6000      7500
----      ----      ----      ----      ---- 
 1         2         3         4         5
While the ratio's are not identical, they are equivalent.
                        Equivalent ratios
     One feature of positive direct proportion to appear is that
the ratios formed by the data are equivalent.
Equivalent ratios themselves have a number of characteristics
which are also of great importance in science, because they are
the foundation of the transformation of equations, of
calculations and constants. 
This first example will be of how much of a substance is needed
to make up a specific volume of a solutions of specific strength.
     150 cm3 of 0.5 M(Moles). The molecular weight of sucrose is
342.
     Calculation
Sucrose requires 171 g (a) per dm3 (b), giving a ratio of:
a    171
- = -----
b   1000  
0.5 M solution of sucrose requires x amount of g (c) in 150 dm3
(d),giving a ratio of:
c    x
- = ---
d   150
Then we make the ratios equivalent, provided the initial and
final conditions are in direct proportion:
171      x
---- = -----
1000    50
x = 150 X 171
    ---------
      1000
This in turn gives us a answer of 25.65 grams.       
Constants:
Any pair of equivalent ratios is equivalent to another ratio
whose denominator is 1. For example, the units for length are
such that 8 inches are equal to 20.32 cm, and 20 inches are equal
to 50.8 cm. Since the relationship between these ratios are in a
positive direct proportion, the equivalent ratios are:
20.32   50.8    
----- = ----
  8      20
These two ratios are from a common ratio of 2.54 cm to 1 inch.
This ratio is known as a constant, because it's denominator is
equal to 1.
In dentistry, a number of factors have to be considered, before a
decision is reached upon the amount of anesthesia to a particular
patient. One popular formula for determining the relative dosage
for a child is to take the child's age and divide it by 20 to
have a proportion to a adult dosage. This can be made into a
equation which is x = (p * q)/h where x is the correct dosage for
the child; p is the age of the child; q is the correct dosage for
a adult; and k is a constant who is 20. This is done by using a
direct proportion and making it equivalent to a constant.
The respiratory quotient:
One ratio that is important in biology is the  respiratory
quotient. Where the ratio is the volume of carbon dioxide
produced by a organism to the volume of Oxygen consumed. Or:
volume of CO2 eliminated
------------------------
volume of O2 consumed
The relationship between these factors deals with the organic
molecule metabolized as the source of energy and it gives a
important to find where the source is.
     A ratio of 1/1 is when the volume of carbon dioxide being
breathed out is equal to the volume of oxygen breathed in, this
shows the patient is metabolizing carbohydrates as it's source
of energy. 
     
                  The Enzyme Substrate Complex
Living organisms depend on chemical reactions to survive. 
The chemical reactions are done by something called a enzyme
substrate complex. There are 2 factors that contribute to the
variation of the reaction. 
Because this relationship is a positive direct proportion, one of
the equations that is used is a = k*b where a is the initial
velocity of the reaction and b is the concentration of the enzyme
and the k is the constant velocity. This is a positive direct
proportion because, as the initial velocity of the reaction
increases, the enzyme concentration increases proportionally.
                          Extrapolation
     One of the most important reasons for getting a
relationship, is because it can predict results in specific and
related cases. For example, the number of bacteria killed by a
range of concentrations of a drug, for instance, because the
relationship is a positive direct proportion, then one knows that
by whatever proportion is changed, the number of bacteria is
changed by the same proportion.  Increase the concentration of
the drug by 10% and, then because of the proportion, the number
of bacteria killed increases by 10%.
     This relationship does not stay constant. The relationship
can change, suppose we took the example of the substrate-enzyme
complex, if we can anticipate that one of the pieces of data
falls out of range, we have reason to extrapolate, and throught
further experimentation, we find:
 
As you can see from extrapolation, the initial velocity remains
the same while the substrate concentration increases, this is
because, there is a limited number of substrates, that can form a
complex with a infinite amount of enzymes.
     If a rat weighing 300 g needs a dose a to effect a cure,
then the dose y needed to effect the same cure in a man of 75 kg
is not given by the equivalence of the ratio between dossage and
mass:
      a         y 
     ---  =  ------
     300     75,000
then:
     y = 75,000a    
         ------- = 250a
           300 
Givivn the man 250 times the dosage of the cured rat would be
giving a overdose to the man.
     Exeptions to extrapolation in the mid 1950s, some
psychologists in attempting to study a male elephant iinjected
one with the drig LSD. The dosage that produced a similar effect
in a cat. The scientists figured out the correct dossage by
taking 0.1 mg kg -1. The cat weighed 2.6 kg, and the elephant 
7725kg. So the researchers calculated the necessary dose (x) by
direct proportion.
0.1    x
--- = ----
2.6   7725
x= 7725 X 0.1
   ---------- = 297 mg.
       2.6 
Upon the injection of LSD into the elephant, is ran around,
stopped, swayed, collapsed and died within 5 min of the
injection. This showes that the elephant is extremely sensative
to certain drugs that man can survive with. Thus the mass of a
animal isn't proportional to it's effects, you see you expect the
elephant to be able to take the effects of the drug, but actually
the human can survive better. Extrapolation has many exceptions.
On certain occasions, the mathematics cannot be always correct,
therefore conclusions that are drawn from the computation of
data, will not always give out accurate data.
                       Inverse Proportion
     Th relationship betweeen two variables is not aloways a
direct proportion. A graph of corresponding data may not always
be a straight line. One relationship that commonly occurs in
biology is known as a inverse proportion. When a given volume is
dilited with water, it's concentration decreases while the volume
increases. As the hydrogen ion concentration increaes, so the
hydroxyl concentration decreases. These involve a negative
proportion, but they are actually inverse proportions.
From the cartesian graph:
This graph is a slope downwards from left to right. This
indicates a inverse direct proportion.
              The algebra of a inverese proportion
In a relationship, ratios are those of one variable (y) and the
reciprocal of the other variable (1/x).
 y
---
1/x
Being equal. these ratios have a constant named k. 
We would form a table of inverse variables as:
----------------------------------------------
Variable A                    Variable B
----------------------------------------------
     a                             b
     c                             d
A equation that helps us interpret this diagram is a * b = c * d.
An example of the dilution of sucrose:
Sucrose is a important component of the saline solution for IV's.
-----------------------------------------------------------------
                              Strength  Volume                   
                              of        of
                              Solution  Solution
-----------------------------------------------------------------
Initial Condition             95%       Unknown
                              (a)       (b)
Final Condition               70%       500cm3
                              (c)       (d)
-----------------------------------------------------------------
Now, the calculations are as follows for the equation of a*b=c*d
Since the (b) is the unknown we shall set up the equation as:
     
     c * d
b=   -----
       a
 
 = 70 X 500
   --------
      95
 = 368.4 cm3
Diluting 386.4 cm3 of Sucrose with water to make a volume of 500
cm3 will give the result. Inverse proportions are very helpful
for measuring the amount of distillation of a certain liquid.    
                    Surface area/volume ratio
     This ratio is familiar to doctors and is  important when
interpreting the growth of animals. It effects the heat balance
or homeotherms and in bones. The needs of a living organism
relate to the mass if the body but the materials it must her, or
must get rid of,if it is to stay alive. This relationship is not
constant but it changes with increases in :size, only if shape is
not changed.  Consider a shape of a cube.  Here are various
measurements/calculations:
---------------------------------------------------------------
Length    Surface Area   Volume    Surface area/
(cm)      (cm2)          (cm3)     volume ratio
a         c              d         b
---------------------------------------------------------------
1         6              6         6
2         24             8         3
3         54             27        2
4         96             64        1.5
---------------------------------------------------------------
---------------------------------------------------------------
          Surface area/
Length    volume ratio   Product
a         b              a * b
--------------------------------------------------------------
1         6              6
2         3              6                        
3         2              6
4         1.5            6
----------------------------------------------------------------
As the sides of the cube increase, the surface area/volume ratio
decreases. The product of a and b is constant of 6.
This can derived from the following equations. A cube that has a
length of a side (a) will have a surface area of 6a2, it having 6
sides. The same cube will have a volume of a X a X a = a3.
It's surface area/volume ratio,b, therefore will be:
  
    6a2   6
b = --- = -
     a3   a
Since k = 6, when the object is a cube, the inverse proportion of
this would result in a equation is a * b = 6.
When a is less that 6, then b is greater that 1. It follows that
b becomes very large as a becomes very small.
Thereby proving that there is a inverse proportional
relationship. Inverse proportions are important in medicine in
that they provide accurate conclusions to data that varies in one
variable increasing while the other decreases. 
                           Conclusions
     There are many applications to medicine, I have only covered
a infinitesimal amount of them. Throughout, there are
relationships between mathematics and medicine that exhibit
phenomenon. Today medicine is growing and thriving throughout the
world, with out the help of mathematics, this would not be
possible, because mathematics gives concrete conclusions in the
world of medicine. Medicine and math have special interrelations
that can be shown  graphically and then can result in conclusions
that help medicine prosper.
Work Cited
