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Final Answers
© 2000-2018   Gérard P. Michon, Ph.D.

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When I woke up just after dawn on September 28, 1928, I certainly didn't plan to revolutionize all medicine by discovering the World's first antibiotic.
Sir Alexander Fleming (1881-1955; Nobel 1945)


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Concentrations of blood glucose (bG) in  mg/dL  (cg/L)  or  mmol/L.
Hemoglobin A1c Test
Wikipedia: Glycemia  |  Glycosylated Hemoglobin
 Pierre Jean Georges Cabanis 
 1757-1808  Jean-Nicolas Corvisart 
 1755-1821  William Harvey 
 1578-1657  Andreas Vesalius 
 1514-1564  Paracelsus 

Medicine by the Numbers

 Guillaume Dupuytren 
 1777-1835  Joseph Lister 
 1827-1912  Emil Adolf von Behring 
It is with medicine as with mathematics:  We should 
occupy our minds only with what we continue to know;
what we once knew is of little consequence.

Charles Augustin  Sainte-Beuve   (1804-1869) 

(2010-08-02)   What is the normal human body temperature?

The usual answer is a simple rule of thumb:  37°C  or  98.6°F  (same thing).  This traditional estimate of normal body temperature is originally based on the findings of Dr. Carl Wunderlich (1815-1877)  who recorded about a million armpit temperature measurements on  25 000  patients.

When Gabriel Farhenheit devised his temperature scale in 1714, he  meant  100°F to be the normal temperature of the human body.  However, this turns out to be only a rough estimate which is not appropriate for clinical thermometry  (either the measurement wasn't accurate or Fahrenheit was running a slight fever at the time).

Modern studies  (1992)  have found the average normal temperature for adults to be close to  98.2°F  (36.8°C).  Older people usually have lower mean body temperatures, which are normally well below  98.6°F.

The traditional threshold for fever is  38°C  (100.4°F).  However, body temperature does depend on the time of day.  For women, it also varies with the phase of the menstrual cycle, which may translate into a higher baseline body temperature.

It's thus more accurate to base a diagnosis on a curve of the average body temperature recorded at different times of the day when the person was known to be in good health.

Note that the normal  body temperatures  of various warm-blooded animals depends strongly on their species  (it's about  42°C  for a healthy chicken).

Normal Body Temperature: Rethinking the normal human body temperature   (Harvard Health Letter)

(2010-08-02)   What is the normal human arterial blood pressure?

During a normal heartbeat, the blood pressure varies between a minimum  (diastolic)  and a maximum  (systolic).  Both numbers are usually expressed in  mmHg  (millimeters of mercury)  or  torrs  (those two units are used interchangeably;  the minute difference between them makes no clinical difference whatsoever).

In some countries  (France, etc.)  medical instruments are usually graduated in  centimeters  of mercury instead  (cmHg).  A typical blood pressure might thus be given as  130/80  in the US and  13/8  in France.

Blood pressure should be measured at the level of the heart itself.  This is one reason why it's usually measured on the upper arm  (the hydrostatic pressure difference between the heart and the lower leg of a standing person is about  80 mmHg).  Readings can be influenced by many begnign factors, including posture and recent physical activity.

The normal blood pressure of a healthy person will typically be between  90/60  and  120/80.

(2010-08-02)   What is the normal pulse rate in humans at rest?

At rest, a healthy human heart beats at a rate of about 60 pulsations per minute  (1 Hz).  The pulse rate of trained athletes can be much lower and the heartbeat of sedentary people is often faster...

During and after a substantial effort, the pulse of a person quickens.

 Coat-of-arms of 
 William Harvey (2012-08-24)   Circulation of the blood
It was first described in 1628, by William Harvey (1578-1657).

For more than a thousand years, the teachings of  Galen (AD 129-217)  were not questioned by medical students or their teachers.  In particular, it was thought and taught that veinous blood was produced by the liver and consumed by the rest of the body.  The first success story of modern experimental medecine was to prove that it ain't so.

In 1628, Harvey (1578-1657)  estimated  extremely  conservatively that the heart pumps no less than 540 pounds of blood per day  (modern estimates are nearly 30 times larger; around  7 tons  per day).  Clearly, the liver could not possibly  produce  that much blood...  Blood  has to  circulate!

 Mammalian Blood Circulation   The heart of mammals is a double pump  (a full circuit of the blood goes through the heart twice).  The right half of the heart pulls the blood from the veinous network through the two large  venae cavae  and pushes it into the  pulmonary artery.  The left part of the heart pulls blood back from the  pulmonary vein  and pushes it into the arterial network  (aorta, carotid, etc.)
In either half of the heart, blood enters into the  atrium  (part of which is the  auricle )  and it's expelled from the  ventricle.

BBC Video :   William Harvey's discovery of the circulatory system  by  Michael Mosley

(2012-08-25)   Respiration  is a slow form of combustion   (1780)
The joint work of  Lavoisier  (1743-1794) and  Laplace  (1749-1827).

It is the role of the lungs to continuously convert veinous blood into arterial blood (with oxygen-rich hemoglobin).  All other human tissues take oxygen from blood and reject carbon dioxide in it.  As the reverse process takes place in the lungs,  respiration  appears chemically to be a form of slow  combustion,  as was first established by Laplace and Lavoisier in 1780.

(2010-08-02)   What dietary caloric intake is considered normal ?

The caloric intake should compensate for the expenditure of energy spent on basic metabolic functions and physical efforts.  If the intake is more than that, then it is stored the form of body fat.  That fat is burned as needed when more energy is required than what is provided by the daily intake.

A very rough rule of thumb is that a person burns about 100 W of power  (power is the energy spent per unit of time).  One watt  (W)  is a joule  (J)  per second.  A  calorie  (cal)  per second is  4.184 W.

The unit of energy used by dieticians is the so-called dietary calorie, large calorie or kilocalorie  whose proper symbol is  kcal  (the confusing capitalized symbol  "Cal"  was once popular but it's now deprecated).

1 kcal   =   1000 cal   =   4184 J

If you burn  100 W  continuously for a full day  (24 h  is  86400 s)  you will have burned  8 640 000 J  or about  2065 kcal.

Typically a reasonably active average man will consume energy at the above rate  (2000 kcal/day)  and should compensate for it by an equal food intake.  The need of other individuals may vary.  For example, a young woman may only need  1200 kcal/day  or less.

Dr. Rita Rae Fontenot (2006-10-19)   Emergency use of an IU rating.
How do I give 125 IU from poorly-labeled 10 mL vials of 10000 units?

This amounts to 1000 IU per mL.  8 doses per mL.  To administer such a small dose (2 drops) with some precision, you may want to dilute it first.  For example 1 vial in 90 cc of an inactive solution yields 100 cc, from which you get 80 doses of 125 IU (1.25 cc each).  If you're a young doctor by herself in a remote area, I'll just pray that you'll know what to do with whatever means you have at your disposal and whatever help you can gather about this emergency.

What is an  IU  worth ?

The IU (International Unit) is a unit of biological activity which is standardized for each substance (fairly arbitrarily) by the World Health Organization.  It's also abreviated UI (from the French locution  unité internationale ).

For a simple chemical (e.g., Vitamin C) the WHO simply assigns a value of 1 IU to a particular mass of that substance.  The rating of biological preparations (e.g., vaccines) is more delicate but it need not be of concern to the practitioner...

If you need to give 125 IU of a substance to a patient, you must first know the concentration of the solution you have at hand.  Normally, this is shown directly in IU/ml, IU/mL or IU/cc (same thing) on the package.  It could also be given as the reciprocal of that:  For example 1mL/40 IU is the same as 40 IU/mL.

(2007-03-29)   Concentration is amount (grams or moles) per volume.
Blood glucose concentration (bG) is thus given in mg/dL or in mmol/L.

A mole of glucose  (CAS 50-99-7)  weighs  180.16 grams.  Therefore, a blood glucose concentration  (bG)  of  1 mmol/L  is equivalent to  18.016 mg/dL.

The blood glucose concentration given in  mg/dL  (the form most commonly used by doctors and diabetic patients across Europe and the US)  is thus about 18 times the number in  mmol/L  (often used in medical research).

Blood Glucose, Plasma Levels   (bG)
mg/dLmmol/L   Interpretation and/or Symptoms  
54030.0 Severe imbalance.
36020.0 Very high blood sugar level.
27015.0 High or very high blood sugar
(depending on patient)
18010.0 Non-diabetic postprandial
(i.e., after meal)
1086.0 Non-diabetic preprandial
(i.e., before meal)
100 5.55 
724.0 Slightly low.  Mild lethargy.
543.0 Low blood sugar level.  Lethargy.
362.0 Extremely low.  Risk of fainting.

Whole blood concentration is actually 15% lower than the plasma level quoted above, but modern portable glucose meters are calibrated to match the plasma readings obtained in lab tests.  Venous blood and capillary blood may have slighlty different compositions only when blood chemistry evolves rapidly (after a meal).

(2007-03-29)   Blood Glucose and HbA1c
Glycated hemoglobin buildup indicates average blood glucose (bG).

The table below gives the rough correspondence between HbA1c results (in %) and  long-term average  blood glucose level (bG in mg/dL).  It is based on the following approximative formula:

(mean bG, in mg/dL)     =     35.6  (% of A1c hemoglobin)  -  77.3

Glucose 65697276798386909497
Glucose 101104108111115119122126129133
Glucose 136140143147151154158161165168
Glucose 172175179183186190193197200204
Glucose 208211215218222225229232236240
Glucose 243247250254257261264268272275
Glucose 279282286289293297300304307311
Glucose 314318321325329332336339343346
Glucose 350353357361364368371375378382
Glucose 386389393396400403407410414418
Glucose 421425428432435439442446450453

(2012-02-18)   Medical Abbreviations
Traditionally used on prescriptions and elsewhere.

Quid quid latine dictum sit, altum videtur.
Anything stated in Latin is perceived as profound.

Abbr.Read as (Latin)English translationUsage notes
q7d once a week 
 every other day 
QDquaque dieonce a day 
SID (veterinary medicine)
BIDbis in dietwice a dayMorning and evening.
TIDter in diethree times a dayMorning, noon, evening.
QIDquater in diefour times a dayMorning, noon, evening, nighttime
q4hquaque 4 hevery 4 hours6 times a day (including sleeptime)
quaque 2 hevery 2 hours
every other hour
12 times a day
PRNpro re nataas neededwhenever symptoms call for it

Scholarly abbreviations

(2016-03-20)   Mosquito-borne diseases.
They're eradicated below a certain ratio of local mosquitoes per human.

Such diseases are transmitted by specific species of mosquitoes.

  • Dengue  (in French: la dengue, le petit palu, la fièvre rouge).  At least five different types of viral infections.  Each infection normally lasts only 2 to 7 days and subsequently provides lifelong immunity to one serotype only  (with short-term immunity to other types).  The risk of life-threatening complications is increased by multiple infections.
  • Yellow fever  (in French: fièvre jaune, vomi noir).  First human virus ever isolated  (1927).  Spread by female  Aedes aegypti.
  • Chikungunya.  In 2005 and 2006,  an epidemic of  chikungunya  infected nearly one third of the population in  French Reunion Island.
  • West Nile virus (WNV). 
  • Zika virus  (2015 outbreak).
  • Malaria.  (French: paludisme)  The key example discussed below.

Malaria  was known in Europe since antiquity.  The parasite responsible for the disease was first identified in the blood of infected patient by the French military physician  Alphonse Laveran in 1880.  Malaria is still causing hundreds of thousands of deaths every year, mostly in Africa.

In 1897 the fact that malaria was actually transmitted by mosquito bites was discovered in India by the British military physician  Ronald Ross (1857-1932; Nobel 1902).

In 1911,  it was Ross himself who first stated a surprising mathematical fact, which he called  the mosquito theorem :  Malaria is locally eradicated as soon as the number of mosquito per inhabitant falls below a certain predictable threshold.  (It's thus not necessary to get rid of every single mosquito!)

Considering a constant local population of  N  humans and  n  mosquitoes,  Ross  assumed that every mosquito  (infected or not)  would inflict an average number of  f dt  bites over a small interval of time  dt.  Both the biting mosquito and the biten human become infected if either one of them was infected before the bite.

The number of infected humans and mosquitoes are respectively approximated by two continuous functions of time,  I(t)  and  i(t).

The Ross Model :

In the time-scale of interest, Ross neglected the occurences of human births and deaths.  The healing rate among humans is a constant  g.  The number  I(t)  of infected humans thus obeys the differential relation  (the first term accounts for infected mosquitoes biting a healthy human and the second term pertains to newly-healed humans):

dI(t)/dt   =   i(t) [ 1 - I(t)/N ] f  -  I(t) g

On the other hand, infected mosquitoes never heal.  However, they're always born uninfected and their mortality rate  m  ends up playing the same mathematical rôle as the healing of humans  (under the assumption that the mosquito population is constant, the mortality-rate equals the birth-rate).

di(t)/dt   =   I(t) [ 1 - i(t)/n ] f  -  i(t) m

Stationary Rates of Infection  (Ronald Ross, 1911) :

Ross remarked that the infected populations were stationary when both of the above variations vanished, namely when:

  • i [ 1 - I/N ] f   =   I g
  • I [ 1 - i/n ] f   =   i m

Dividing those two equations into  i I,  we obtain two linear relations between  1/I  and  1/i:

i  ( N + m )   =   I  ( n + g )

Thus, the first equation multiplied into  ( N + m )  becomes:

I  ( n + g )  [ N - I ] f   =   I g  ( N + m )

This equation in  I  has a trivial solution  (I = 0)  and a nontrivial one:

I   =   ( nN - gm )  /  ( nf + gf )

Dynamics of Malaria  (Alfred Lotka, 1923) :

Mosquito-borne diseases   |   Elephantiasis
"Le modèle de Ross" by N. Bacër  (Tangente Hors Série, 58, pp 34-35. Feb. 2016)

(2016-03-11)   Propagation of epidemics.
Predictions based on mathematical models.

The first simplified theoretical model which explained general epidemics was proposed in 1927 by  William Kermack  (1898-1970)  and  Anderson McKendrick  (1876-1943).  That model considers only three variables whose sum is assumed to be constant:

  • S :   The number of individuals  susceptible  of catching the disease.
  • I :   The number of  infected  individuals.
  • R :   The number of individuals who have  recovered  or died from the disease.

"La propagation des épidémies" by H.L. (Tangente Hors Série, 58, pp 28-30. Feb. 2016)

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