[FRIAM] FW: Mathematical Inquiry

[Nick Thompson]

Dear Mathematical Friammers,

 

What follows is a problem in mathematics, which, of course, has nothing to
do with me.  

 

Jones is a diabetic, and he has a glucose monitor that gives him his exact
blood glucose level moment to moment.  Jones notices at that after
breakfast, his blood sugars behave in in very different manners, even though
he eats exactly the same food every day, doesn't exercise at that time of
day ever, and takes exactly the same amount of insulin.  Some mornings, his
blood sugar rises steadily for several hours after a meal, sometimes it
falls steadily.  Only rarely does it remain steady.  One variable seems left
for Jones to control and that is the exact timing of the relation between
when he take his insulin and the time he begins his meal.  

 

So, Jones imagines a model as follows.  Because Jones always takes exactly
the amount of insulin necessary to account for the amount of sugar he eats,
he assumes that the curves of insulin activity and sugar activity are both
normal curves, with the same median time and the same sd and, therefore, the
same area under the curve.  However, one curve is offset from the other
because sometimes Jones takes his insulin before he eats his sugar and
sometimes he eats his sugar before he takes his insulin.  Bearing in mind
that the Insulin curve SUBTRACTS from the sugar curve, Jones wonders about
the shape of the difference curve that results from different offsets
between eating his meal and taking his insulin.  He wonders if, perhaps,
that this whole dramatic failure of control, could be due to the fact that
on some days he takes his insulin a little too early and the sugar in the
meal is slow to catch up and on other days, he takes it too late and the
insulin is slow to  catch up.  Thus, the correct offset is a tipping point,
an unstable equilibrium which is very difficult to achieve.  

 

Jones is not a mathematician, but he hangs around with mathematicians, and
he suspects that there is a software that is readily available on line for
free that would allow him to display the different curves that result from
the different offsets and, perhaps, even better, display the function that
relates the integral of the difference function as a function of the offset.
This function might have some interesting properties that could be used to
guide Jones's injection behavior.  

 

Does anybody have any thoughts on Jones's predicament?  

 

Not that I care, but still, 

 

Nick 

 

Nicholas S. Thompson

Emeritus Professor of Psychology and Biology

Clark University

http://home.earthlink.net/~nickthompson/naturaldesigns/

 

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