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Optimal drug administration for the control of certain cholesterol fats in the blood system
MATHEMATICAL
BIOSCIENCES
183
15, 183-l 86 (1972)
Optimal Drug Administration for the Control Cholesterol Fats in the Blood System
of Certain
N. G. F. SANCHO Department of Mathematics, McGill University, Montreal 110, Quebec Communicated
by Richard Bellman
ABSTRACT A formulation is given for the concentration of certain fatty acids such as cholesterol, in the blood system at sequential intervals. The optimal drug to be administered at optimal time intervals for the control of cholesterol is given as a functional equation. It is formulated by Dynamic Programming.
INTRODUCTION
AND
FORMULATION
It is generally known that North Americans suffer from a high rate of leart attacks. This is due to the fact that cholesterol, in the blood system, has a high level of concentration, i.e. about 250 mg in 100 C.C. of blood ‘or the average North American. It would be ideally suited to have a :holesterol concentration of about 200 mg % but the fact that we eat too nuch animal fats and drink too much alcohol account for the high chol:sterol level. A person with a high cholesterol level would have the choice bf eating a restricted diet, trying to burn up the excess cholesterol through Laily exercises, or taking a drug to control the cholesterol level if he is to emain healthy (at present Atromid-S tablets are taken to control cholsterol content). In general it is not bad to have a low cholesterol level iough if a person eats a normal diet and remains in good health his holesterol level is around 200 mg %. In the case of a diabetic patient ze excess sugar in the blood comes out as sugar in the urine. H.ence it is isy to tell a diabetic patient. It will be assumed that at certain time intervals a blood sample can : taken to determine a person’s cholesterol content. If it is the level of lolesterol at the end of time interval t then we have for a healthy invidual ; it = it-1 + w,,
(1)
Copyright 0 1972 by American Elsevier Publishing Company. Inc.
184
N. G. F. SANCHO
where wt (an independent random variable) is the increase or decrease in cholesterol over the previous reading. On the other hand a person with high cholesterol level would be required to keep the level down through drug administration. If the reduction in cholesterol due to the drugs is assumed to be proportional to i,_i times the reduction in the drug in the system in unit time interval, then we have i, = 11 - a(o,_, - v*))i,_l + w,,
(2) in body at end of time interval t and
where z+ is the drug concentration c( is the proportianality constant. The drug concentration in the body can be given in continuous time as the differential equation u’ = -au, v(0) = c, (3) where u is a positive constant. Drugs will be administered at certain time intervals tl, t2, t3 etc. where 0 < tl < t2 < . . . -c T in certain quantities cO, cl, c2 etc. Initially c is zero or very small. The drug concentration in the organ at end of time t,, is therefare given by v(tJ = u,, = (c + cO)e-at’
(4)
where c0 is the quantity of drug administered at time t,, = 0. OPTIMAL DRUG ADMINISTRATION AT FIXED TIME INTERVALS In Bellman [1] some novel type of control processes in optimal drug administration have been formulated. In what follows we formulate again in Dynamic Programming, so that we minimize the expected value of $e (i, - CfY,
(5)
where id is the desired level of cholesterol concentration, subject to Eqs. 2, 3, and 4. Following Dynamic Programming we write,
Mio) = ;;;
(6)
it,
where E [ -1 is expected value w.r.t. all w,‘s. If the drug to be administered Cwr1 must not exceed a certain dosage k then we have, fJiO) = OmCFsrf II(& - iJ2 + .
=
min OccoSk
fn-I(iI)19
[(i, - iJ2 + f,_i{[l
(7:
- cI(c + c&l
s +
- e-“)]i,
wdl Wwds
(9
DRUG
ADMINISTRATION
FOR CHOLESTEROL
CONTROL
185
for N 2 1, where ~G(w,) is the specified distribution function of wl; and withf,(i,) = (iO - #. In certain instances we can approximate u’r to be Gaussian “white” noise and we have s
w1 dG(w,)
(mean value)
= m
and
(10)
w? clG(w,) = 0’ (variance) s Because of the quadratic nature of Eq. 9, one can solve it analytically to obtain, [(l - i;lc(l - e_“))i, + Ill] - i, c 0=> (11) ai,(l -
e-y
with fN(iO) = i; - 2i,i,
+ b,,
(12)
where 0, = b,_,
+ c2 - m2,
N 2 1,
(13)
and b, = ij. Vote the drug to be administered )PTIMAL DRUG rIME INTERVALS
is dependent
ADMINISTRATION
In this case we follow similarly
on i, and m. AT OPTIMAL
[l] and minimize,
- i,)’ dt, (14) s rhere a quantity of drug co, c,, c2 etc. is injected into the system at time itervals II, t2, t3 etc. Following Dynamic Programming we write T fN(io, T) = min E (i - Q2 dt , (W (ctl Cwtl
id obtain
the functional
1
0
equation t1
,(i,, T) =
min O
OBtl
[S SU
E
(i - i,)’ dt + fN_-l(irl, T - tl)
0
1
11
=
min OSco
O
(i - i,)’ dt + f,-,([l
- a(c + c,)(l
0
+
wl,
T - t,
- e-““)]i,
Ww,),(16)
186
N. G. F. SANCHO
n > 1, with
f&i,,
T) =
T o (i - i,)’ dr
min O
1
(17)
dG(w,).
Note that i(t) is only given as a discrete equation can write
in Eq. 2 in which case we
rf1
(i - id)2 dt, Jo as a stochastic integral of Stratonovich [2] where integrals have the same rules as ordinary calculus. Integrating numerically over a trapezium for a close approximation to stochastic integral we obtain t1 (i - id)’ dt = 2[(io - id)’ + (it, - Q21 s 0 =
:[ii
= :[ig
+ 2ii + 2i,(i,
+ i,,) + if,]
+ 2ii + 2i,i,{2
+2&w, + ([I - c((c + c,)(l
- c((c + c,)(l
- e-atl)]
- eda”)]io + w,}‘].
(18‘
Equation 18 can then be substituted back into Eq. 16, which is now itan algorithm for determination of optimal concentration and times. If on the other hand we consider a more realistic constraint where ; minimum dosage must be administered in a minimum time interval. i.e. O
(19
0 < s < t,+, - ti then the relation
and
T 2 IV’S,
(20
[a * .] dG( WI).
(21
(16) takes the form fN(io9 T) =
min r$coCk sbt,
s
CONCLUSION It would probably be more realistic to express w1 as a Gaussian an Markov process instead of “white” noise. Results could be similar1 obtained except that calculations are slightly more difficult. REFERENCES 1 R. Bellman, “Topics in Pharmacokinetics, III: Repeated Dosage and Impulse Co trol,” Mafhematical Biosciences 12,l (1971). 2 R, L. Stratonovich, “A new representation for stochastic integrals and equations SIAM
J. Control 4, 362 (1966).
BIOSCIENCES
183
15, 183-l 86 (1972)
Optimal Drug Administration for the Control Cholesterol Fats in the Blood System
of Certain
N. G. F. SANCHO Department of Mathematics, McGill University, Montreal 110, Quebec Communicated
by Richard Bellman
ABSTRACT A formulation is given for the concentration of certain fatty acids such as cholesterol, in the blood system at sequential intervals. The optimal drug to be administered at optimal time intervals for the control of cholesterol is given as a functional equation. It is formulated by Dynamic Programming.
INTRODUCTION
AND
FORMULATION
It is generally known that North Americans suffer from a high rate of leart attacks. This is due to the fact that cholesterol, in the blood system, has a high level of concentration, i.e. about 250 mg in 100 C.C. of blood ‘or the average North American. It would be ideally suited to have a :holesterol concentration of about 200 mg % but the fact that we eat too nuch animal fats and drink too much alcohol account for the high chol:sterol level. A person with a high cholesterol level would have the choice bf eating a restricted diet, trying to burn up the excess cholesterol through Laily exercises, or taking a drug to control the cholesterol level if he is to emain healthy (at present Atromid-S tablets are taken to control cholsterol content). In general it is not bad to have a low cholesterol level iough if a person eats a normal diet and remains in good health his holesterol level is around 200 mg %. In the case of a diabetic patient ze excess sugar in the blood comes out as sugar in the urine. H.ence it is isy to tell a diabetic patient. It will be assumed that at certain time intervals a blood sample can : taken to determine a person’s cholesterol content. If it is the level of lolesterol at the end of time interval t then we have for a healthy invidual ; it = it-1 + w,,
(1)
Copyright 0 1972 by American Elsevier Publishing Company. Inc.
184
N. G. F. SANCHO
where wt (an independent random variable) is the increase or decrease in cholesterol over the previous reading. On the other hand a person with high cholesterol level would be required to keep the level down through drug administration. If the reduction in cholesterol due to the drugs is assumed to be proportional to i,_i times the reduction in the drug in the system in unit time interval, then we have i, = 11 - a(o,_, - v*))i,_l + w,,
(2) in body at end of time interval t and
where z+ is the drug concentration c( is the proportianality constant. The drug concentration in the body can be given in continuous time as the differential equation u’ = -au, v(0) = c, (3) where u is a positive constant. Drugs will be administered at certain time intervals tl, t2, t3 etc. where 0 < tl < t2 < . . . -c T in certain quantities cO, cl, c2 etc. Initially c is zero or very small. The drug concentration in the organ at end of time t,, is therefare given by v(tJ = u,, = (c + cO)e-at’
(4)
where c0 is the quantity of drug administered at time t,, = 0. OPTIMAL DRUG ADMINISTRATION AT FIXED TIME INTERVALS In Bellman [1] some novel type of control processes in optimal drug administration have been formulated. In what follows we formulate again in Dynamic Programming, so that we minimize the expected value of $e (i, - CfY,
(5)
where id is the desired level of cholesterol concentration, subject to Eqs. 2, 3, and 4. Following Dynamic Programming we write,
Mio) = ;;;
(6)
it,
where E [ -1 is expected value w.r.t. all w,‘s. If the drug to be administered Cwr1 must not exceed a certain dosage k then we have, fJiO) = OmCFsrf II(& - iJ2 + .
=
min OccoSk
fn-I(iI)19
[(i, - iJ2 + f,_i{[l
(7:
- cI(c + c&l
s +
- e-“)]i,
wdl Wwds
(9
DRUG
ADMINISTRATION
FOR CHOLESTEROL
CONTROL
185
for N 2 1, where ~G(w,) is the specified distribution function of wl; and withf,(i,) = (iO - #. In certain instances we can approximate u’r to be Gaussian “white” noise and we have s
w1 dG(w,)
(mean value)
= m
and
(10)
w? clG(w,) = 0’ (variance) s Because of the quadratic nature of Eq. 9, one can solve it analytically to obtain, [(l - i;lc(l - e_“))i, + Ill] - i, c 0=> (11) ai,(l -
e-y
with fN(iO) = i; - 2i,i,
+ b,,
(12)
where 0, = b,_,
+ c2 - m2,
N 2 1,
(13)
and b, = ij. Vote the drug to be administered )PTIMAL DRUG rIME INTERVALS
is dependent
ADMINISTRATION
In this case we follow similarly
on i, and m. AT OPTIMAL
[l] and minimize,
- i,)’ dt, (14) s rhere a quantity of drug co, c,, c2 etc. is injected into the system at time itervals II, t2, t3 etc. Following Dynamic Programming we write T fN(io, T) = min E (i - Q2 dt , (W (ctl Cwtl
id obtain
the functional
1
0
equation t1
,(i,, T) =
min O
OBtl
[S SU
E
(i - i,)’ dt + fN_-l(irl, T - tl)
0
1
11
=
min OSco
O
(i - i,)’ dt + f,-,([l
- a(c + c,)(l
0
+
wl,
T - t,
- e-““)]i,
Ww,),(16)
186
N. G. F. SANCHO
n > 1, with
f&i,,
T) =
T o (i - i,)’ dr
min O
1
(17)
dG(w,).
Note that i(t) is only given as a discrete equation can write
in Eq. 2 in which case we
rf1
(i - id)2 dt, Jo as a stochastic integral of Stratonovich [2] where integrals have the same rules as ordinary calculus. Integrating numerically over a trapezium for a close approximation to stochastic integral we obtain t1 (i - id)’ dt = 2[(io - id)’ + (it, - Q21 s 0 =
:[ii
= :[ig
+ 2ii + 2i,(i,
+ i,,) + if,]
+ 2ii + 2i,i,{2
+2&w, + ([I - c((c + c,)(l
- c((c + c,)(l
- e-atl)]
- eda”)]io + w,}‘].
(18‘
Equation 18 can then be substituted back into Eq. 16, which is now itan algorithm for determination of optimal concentration and times. If on the other hand we consider a more realistic constraint where ; minimum dosage must be administered in a minimum time interval. i.e. O
(19
0 < s < t,+, - ti then the relation
and
T 2 IV’S,
(20
[a * .] dG( WI).
(21
(16) takes the form fN(io9 T) =
min r$coCk sbt,
s
CONCLUSION It would probably be more realistic to express w1 as a Gaussian an Markov process instead of “white” noise. Results could be similar1 obtained except that calculations are slightly more difficult. REFERENCES 1 R. Bellman, “Topics in Pharmacokinetics, III: Repeated Dosage and Impulse Co trol,” Mafhematical Biosciences 12,l (1971). 2 R, L. Stratonovich, “A new representation for stochastic integrals and equations SIAM
J. Control 4, 362 (1966).