Math1 Comput. Modellmg, Vol. I I. pp. 578-582, 1988 Printed in Great Britain

CONTROL THEORY

0895-7177/88 $3.00 + 0.00 Pergamon Press plc

MEASURING DRUG PERFORMANCE WITH A NEW OPTIMIZATION ALGORITHM

William Conley and Harriet Wichowski

Departments of Mathematics Business and Nursing University of Wisconsin - Green Bay Green Bay, Wisconsin 54302 - U.S.A.

Abstract: Multi stage Monte Carlo optimization (MSMCO) is a new computer mathematics solution technique that shows promise in solving the general nonlinear model. (It also works on many linear systems). This new multi- purpose multivariate solution technique allows one to fit response curves by methods other than least squares. This paper will present several hypothetical multivariate druo interaction studies. where the curve fits (modeling the response function) are done using least absolute deviation and mini max deviation with the MSMCO algorithm. (Mini max curve fittinq finds the curve, from the family of functions under consideration, such _ that the maximum error is minimized). Least absolute deviation and mini max curve fitting help to overcome some of the problems inherent in the least squares technique, namely the squaring of the error term tends to overemphasize outlier points. Also least squares tends to use families of functions such that the resulting normal equations are linear in the betas making the curve fit easier to solve.

MSMCO can free one from these problems and allow families of functions for fitting without concern about the form of the normal equations, because the curve fitting is done directly. A sample problem would be a medical condi- tion that has shown signs of remission with six different drugs given to patients separately. Therefore a study is conducted to see if a combina- tion of the six different drugs could provide even better results than just separately. So various amounts of the six drugs (the six independent vari- ables) are given to the patients over time and a response variable (that measures the remission) is recorded. These dependent variable values are then regressed on the six independent variables (using the multivariate nonlinear or linear familv of functions selected for the studvl with the MSMCO algorithm. Least absolute deviation, mini max (or even-least squares) could be used for fitting. The resultinq model is then optimized usinq MSMCO (representing maximum remission) or set equal to a constant, hence an equation, (which is a target goal for remission measurement) and the re- sulting equation is solved with MSMCO.

This technique also allows for multiple health goals for the patient to be pursued simultaneously. This could give rise to systems of nonlinear equa- tions which are also solved with MSMCO.

The MSMCO technique will be illustrated and generalized with the sample problems. MSMCO could give the medical researcher one more tool in trying to understand complex multivariate drug interactions.

Keywords. Drug interaction; Multi Stage Monte Carlo; General optimization.

INTRODUCTION physiological dysfunction.

Cardiovascular disease is one of the foremost causes of disability and death today despite ad-

One frequently used protocol is a combination

vances in both medicine and surgery. Over forty of lanoxin, lasix and Slow K. Lanoxin increases

million Americans have some form of the disease. cardiac output in acute or congestive heart failure. It is a toxic drug. Side effects in-

Congestive heart failure is one such disorder. It is a complex clinical syndrome that results from the heart's inability to increase cardiac output sufficiently to meet the body's metabolic demands. There are many potent chemotherapeutic agents which can be utilized to relieve or de- crease the symptomatology associated with this

elude erratic pulse rate and rhvthm. fluid re- tention, nausea and vomiting, confusion and headaches. Lasix is a diuretic. Diuretics re- duce the body's total volume of water and salt by increasing their unrinary excretion. Lasix is used to treat the edema associated with con- gestive heart failure. Side effects to be moni- tored include volume depletion and dehydration,

578

Proc. 6th Int. Conf. on Mathemarical Modelling 579

orthostatic hypotension, hypokalemia, and fluid and electrolyte imbalances. Muscle cramping, weakness, paralysis, spasms, and dizziness upon arising may be indicative of the above physiolo- gical dysfunctions. Slow K is a potassium re- placement and is given in conjunction with lasix to counteract hvookalemia. Slow K is less toxic than Lanoxin and’ Lasix, but one must still be alert for potential side effects. These could include nausea and vomiting, peripheral vascular collapse, mental confusion, and flaccid paraly- sis.

Its a given that all clients receiving the above medications, which are taken on a daily basis, must be under close medical supervision, and be aware of subjective physiological and psycholo- gical changes which could indicate a toxic reac- tion.

Congestive heart failure is a consequence of myo- cardial dysfunction. Although often controll- able with medication such as the above, it re- mains the ma.ior form of chronic cardiac disabi-

headedness, headache, confusion transient hypo- tension, dry mouth, nausea and vomiting.

As already mentioned these and other chemothera- peutic agents cannot cure schizophrenia. How- ever, by controlling symptomalology, the drugs may help the client to respond more efficaciou- sly to psychotherapy, and increase the poten- tial for the individual to function successfullv in daily life.

Perhaps a new optimization algorithm such as multi stage Monte Carlo optimization could help in quantifying and understanding drug inter- raction. A few examples are presented.

THREE QUANTIFIABLE EXAMPLES

Fifteen patients who have a chemical EQ82 defi- ciency are given varying amounts (in milligrams) of six compounds. Their resulting increase in parts per billion of KQ82 is noted in the chart below. The researchers fit

lity. It is-also one of the most costly in terms of medical and nursing services, repeated insti-

Y=BlXl+B2X2+63X3+B4X4+BSXS+BgXg

tutional services, medication cost, and decreased using least squares and obtain. productivity due to disability.

Xl X2 X3 X4 X5 X6 Schizophrenia is a major crippling mental di- 50 50 50 sease with, as yet, an unknown etiology. It is :: 3;: 50 50 300 Z a psychotic illness which affects thought process, 400 100 20 90 700 30 mood; emotional regulation, behavior, and total personality integration. Symptoms of the ill- ness include autism, ambivalence, affective dis- turbance, illusions, delusions, hallucinations, paranoia withdrawal, and distortion and lack of contact with reality. At this time schizophre- nia is believed to be a broad term covering a group of syndromes.

Medications do not cure but can aid in control- ling the symptoms of people so affected. The efficacity of specific chemotherapeutic agents varies with each individual. Therefore, it is not uncommon to see frequent adjustments or changes in the drugs used for symptom abatement in various individuals.

A sample protocol for a person in an acute or chronic schizophrenic state could include Haldol (a major tranquilizer which helps in thought re- organization); cogentin (a cholinergic blocker) to control extrapyramidal reactions, which are common side effect of the major tranquilizers; and Xanax, a minor tranquilizer, which decreases feelings of anxiety and anger.

Haldol and other major tranauilizers block post- synaptic dopamine receptors'in the brain. Side effects include transient leukopenia and leuko- cytosis, blurred vision, dry mouth, and a high incidence of severe extrapyramidal reactions. Some physicians automatically start clients on cogentin when prescribing Haldol. Others wait for the appearance of the extrapyramidal symtoms such as akinesia, dystonia, rigidity, and tremor before prescribing counteracting medicine.

Though congentin is prescribed to minimize the undesirable effects of the major tranquilizers one must be aware of the side effects of this drug itself. Some of these are disorientation, tachycardia, and bradycardia. Gastrointestional symptoms include constipation, dryness of the mouth, nausea and vomiting. Xanax, though a minor tranquilizer, can cause drowsiness, light-

100 290 600 30 80 500 500 100 200 700 150 400 70 190 800 200 100 300 50 600 250 700 300 400 500 450 325 85 900 500 400 300 800 50 600 900 300 200 100 500 100 800 1:: 400 200 600 300

2:; 85 250 900

200 150 600

100 300 4;:

100 400 7:; 300 100 100 500

300 300 1:;

(1)

Y 134,000 484,000 830,600 697,500 513,500 754,100 688,000

1,266,500 1,869,500

761,750 908,000 794,000 798,000 658,750 672,000

Y=-76.5Xl-17.X2+712X3+32OX4+1214X5+638X6 (2)

with a total absolute deviation of 1,657,974 parts per billion. However using multi stage Monte Carlo optimization (MSMCO) to find a least absolute deviation fit by minimizing

f(Bl~B2,B3,B4~BS,B6)=~~lIBlXli+B2X2i+B3X3i

+B4X4i+BgXSi+B6X6i_Yil

they obtain the equation

(3)

+938.038X5+449.425X6 (4)

with a total absolute deviation of 826,318 parts per billion. Note that the total absolute de- viation is twice as large with the least squares curve fit.

Twelve patients who have a deficiency of com- pound RT75 are given varying amounts (in milli- grams) of five drugs. Their increase in parts per billion of RT75 is noted and least squares fits

Y=162Xl+24.4X2+151X3+168X4+30.6X5 (5)

to the data (below) with a total absolute devia-

580 Proc. 6th Int. Conf. on Mathematical Modelling

tion of 288,150. the practical optimal.

Xl

: 200 100 1;; 100 300 1;; 400 1;: 225 1;; 14SY7D0 1131250 3 600 400 200 100 50 127,900 4 300 200 100 400 55 180,630

z 700 200 300 600 500 400 400 500 300 600 240,400 179,900 7 800 100 800 100 800 279;100 a 300 100 500 600 169;OUO 9 200 100 600

1:: 400 169,500

10 500 300 700 200 300 227,300 ;: 700 100 7;: 700 500 250 700 700 150 326,900 130,375

However the MSMCO least absolute deviation fit for the data yields

+65.9794X5 (6)

with a total absolute deviation of 200,033.

Ten patients with an excess of chemical TDV83 are given varying amounts of four compounds in hopes of reducing the parts per billion of TDV83 in their blood. The results were

Y Resoonse

1 :: :25 735 :: Reduction 114,375

$ 300 30 210 100 250 110 400 25 491,975 364,000 4 400 400 400 400 610.000 5 500 100 500 50 451;750 6 100 400 300 200 417,000 7 400 100 300 700 576,500 8 800 200 600 400 738,000 9 600 500 400 300 629,500

10 300 400 500 700 784,500

The least squares fit of

Y=BlXl+B2X2+B3X3+B4X4 was Y=374Xl+5OOX2

+343X3+389X4

with a minimax error of 227,299 parts per

Using MSMCO to minimize the maximum error obtained

with a minimax error of 155,490 parts per

(7)

The distances in meters to and from each custo- mer and the pharmacy are listed in Table 1 (location 25 is the pharmacy). Next the column numbers are ranked from smallest to larqest. Then they use a FORTRAN IV MSMCO program to find a minimum tour (complete loop). Multi stage (MSMCO) is done on the subscripts of the ranked array. Twenty five dimensional rectan- gles of ever decreasing size rocket across this rank ordered subspace of R25 to the optimal so- lution. Only 38,000 samples were needed to identify the solution and five nearly optimal solutions even though there are 24!=620,450,000, OOO,OOO,OOO,OOO,OOO possible routes. (See Table 3).

Table 1. Distances in Meters

1 2 3 4 5 6 10 1000 2003 462 2379 2620 2 1000 3 2003 199:

1991 724 1885 2719 0 1675 686 755

4 462 724 1675 1947 2335 5 2379 1885 686 19470 0 1085 6 2620 2719 755 2335 1085 0 7 1640 1803 240 1355 716 975 8 1989 2076 111 1701 728 643 9 950 800 1266 516 1520 2005

10 1910 2043 58 1607 740 707 11 1575 580 1703 1300 1452 2425 12 2070 1351 1276 1648 876 1962 13 1499 1303 709 1081 883 1441 14 1375 540 2405 1080 2252 3130 15 3041 2516 1343 2609 664 1548 16 1205 1773 663 1016 1140 1398 17 2355 1580 1091 1855 519 1626 18 2481 2500 612 2197 862 216 19 2400 2420 530 2111 777 309 20 2099 2130 236 1818 745 596 21 673 1231 3489 888 2876 3227

;: 1:;; 830 393 2459 1691 321 969 2555 1960 2430 3199 24 581 1311 2421 807 2951 3166 25 2331 2340 468 2050 642 444

Multi stage Monte Carlo programs are flexible and compact. They can be very competitive with known algorithms. Also they show real promise on problems that are considered "unsolvable", such as the next example involving a pharmacy delivering system.

THE GENERALITY OF THE MSMCO ALGORITHM

A pharmacy offers home delivery of prescriptions to its customers in the earlv eveninq for all orders placed by 5:00 pm that day. Today the delivery boy must make 24 deliveries and return to the pharmacy (location 25). The manager wishes him to drive a route that minimizes the total distance traveled in order to save gaso- line and time. He also would like several near- Iv outimal alternatives in case road construc- t:on'or other traffic problems might make the second or third best solution (mathematically)

7 8 9 10 11 12 billion. 1 1640 1989 950 1910 1575 2070

2 1803 2076 800 2043 580 1351 they 3 240 111 1266 58 1703 1276

4 1358 1701 516 1607 1300 1648

: '9:: 728 1520 740 1452 876

(8) 643 2005 707 2425 1962 7

billion. 8 34: 343 1018 280 1449 1074

0 1377 166 1786 1359 1018 280

1449 1074 508

2190 1389 418 939 817 766 465

2250 1412 2234 2185 697 25

1377 166

1786 1359 798

2488 1367 770 1133 503 426 129 2600

1797 2563 2536 360

0 1301 1301 695 175:

1039 1329 605 751

1190 2447 2127 1395 868 690 1291 1147 1853 635 1772 590 1423 295 1348 2525 494 1727

1268 2490 1427 2455 1706 527 2053 1504

695 1039 1755 1329

0 756 756 0 940 748 781 1565

2069 1307 1379 1183 1101 366 2207 1746 2129 1649 1843 1394 1749 2369 885 1500 1210 1985 1825 2446

Mathematical Modelling 581

13 1 1499 2 1303 3 709 4 1081

7 508 8 798

1: 605 751

11 940 12 748 13 0 14 1632 15 1486 16 436

i; 1E 19 1141 20 857 21 1954 22 1102 23 1901 24 1986 25 1066

19 1 2400 2 2420 3 530 4 2111 5 777 6 309

L 766 426

9 1772 10 590 11 2129 12 1649 13 1141 14 2827 15 1233 16 1180 17 1314 18 103 19 0 20 306 21 3007 22 2194 23 2961 24 2945 25 141

Table 2.

1

3' 46: 581

4 673 5 691 6 950 7 1000 8 1205 9 1211

10 1375 11 1499 12 1575 13 1640 14 1910 15 1989 16 2003 17 2070 18 2099 19 2331 20 2355

14 15 12::

17 18 1375 3041 2355 2481 540 2516 1773 1580 2500

2405 1343 663 1091 612 1080 2609 1016 1855 2197 2252 664 1140 519 862 3130 1548 1398 1626 216 2190 1389 418 939 817 2488 1367 770 1133 503 1190 2127 868 1291 1853 2447 1395 690 1147 635 781 2069 1379 1101 2207 1565 1307 1183 366 1746 1632 1486 436 808 1222

0 2888 2034 1949 2911 2888 1789 882 1326 2034 178; 0 1229 1262 1949 882 1229 0 1407 2911 1326 1262 1407 2827 1233 1180 1314 10: 2545 1261 886 1173 376 1017 3439 1831 2654 3089 705 2623 1310 1736 2276 378 3239 2075 2301 3081

1194 3470 1692 2732 3026 2754 1097 1115 1166 237

20 21 22 23 24 25 2099 673 691 1211 581 2331 2130 1231 393 830 1311 2340 236 2489 1691 2459 2421 468

1818 888 321 969 807 2050 745 2876 1960 2555 2951 642 596 3227 2430 3199 3166 444 465 2250 1412 2234 2185 697 129 2600 1797 2563 2536 360

1423 1348 494 1268 1427 1706 295 2525 1727 2490 2455 527

1843 1749 885 1210 1825 2053 1394 2369 1500 1985 2446 1504 857 1954 1102 1901 1986 1066

2545 1017 705 378 1194 2754 1261 3439 2623 3239 3470 1097 886 1831 1310 2075 1692 1115 1173 2654 1736 2301 2732 1166 376 3089 2276 3081 3026 237 306 3007 2194 2961 2945 141

0 2713 1934 2711 2650 231 2713 0 769 639 174 2942 1943 769 0 731 925 2132 2711 639 731 0 789 2987 2650 174 925 789 0 2881 231 2942 2132 2897 2881 0

Column Ranked Distances in Meters

2 3 4 5 6

39; 0 0 0

321 519 2106 540 1:: 462 642 309 580 236 516 664 444 724 240 724 686 596 800 468 807 716 643 a30 530 888 728 707 1000 612 969 740 755 1231 663 1016 745 975 1303 686 1080 777 1085 1311 709 1081 862 1398 1351 755 1300 876 1441 1580 1091 1355 883 1548 1773 1266 1607 1085 1626 1803 1276 1648 1140 1962 1885 1343 1675 1452 2005 1991 1675 1701 1520 2335 2043 1691 1818 1885 2425 2076 1703 1855 1947 2430 2130 1991 1947 1960 2620

1 2 3 4 5 6 21 2379 2340 2003 2050 2252 2719 22 2400 2420 2405 2111 2379 3130 23 2481 2500 2421 2197 2555 3166 24 2620 2516 2459 2335 2876 3199 25 3041 2719 2489 2609 2951 3227

7 8 9 10 11 12

: 0 0 0 0 0 0

240 111 494 58 580 366 3 280 129 516 166 695 748 4 343 166 605 280 756 756 5 418 343 695 295 781 876 6 465 360 800 527 885 1039 7 508 426 868 590 940 1074 a 697 503 950 635 1101 1183

1: ::: 643 1018 690 1210 1276 728 1039 707 1300 1307

11 817 770 1190 740 1379 1329 12 939 798 1266 751 1449 1351 13 975 1133 1268 1147 1452 1359 14 1018 1359 1291 1301 1575 1394 15 1074 1367 1301 1329 1703 1500 16 1355 1377 1348 1395 1749 1504 17 1389 1701 1377 1607 1755 1565 18 1412 1786 1423 1727 1786 1648 19 1449 1797 1427 1755 1825 1649 20 1640 1989 1520 1910 1843 1746 21 1803 22 2185

1706 2043 2053 1962 1772 2447 2069 1985

23 2190 22fi4

2536 1853 2455 2129 2070 24 2563 2005 2490 2207 2369 25 2250 2600 2127 2525 2425 2446

13 14 15 16 17 18 10

37: 0 0 0 0

3' 436 664 418 366 103 508 540 882 436 519 216

4 605 705 1097 663 808 237 5 709 781 1233 690 882 376

7" 748 1017 1261 770 939 503 751 1080 1307 868 1091 612

8 79x a08

1190 1326 886 1101 635 9 1194 1343 1016 1133 817 lo 857 1375 1367 1115 1147 862 11 883 1565 1389 1140 1166 1222 12 940 1632 1395 1180 1173 1262 13 1066 1949 1486 1183 1229 1326 14 1081 2034 1548 1205 1291 1407 15 1102 2190 1789 1229 1314 1746 16 1141 2252 2069 1262 1407 1853 17 1222 2405 2127 1310 1580 2197 18 1303 2447 2516 1379 1626 2207 19 1441 2488 2609 1398 1736 2276 20 1486 2545 2623 1692 1855 2481 21 1499 2754 2888 1773 1949 2500 22 1632 2827 3041 1789 2301 2911 23 1901 2888 3239 1831 2355 3026 24 1954 2911 3439 2034 2654 3081 25 1986 3130 3470 2075 2732 3089

19 20 21 22 23 24 25 1

: 10: 12; 17:

0 0 0 321 37: 174 141

141 231 639 393 639 581 231 4 306 236 673 494 731 789 237

2 309 295 769 691 789 807 360 426 306 888 705 830 925 444

7 530 376 1017 731 696 1194 468 8 590 465 1231 769 1210 1311 527 9 766 596 1348 885 1211 1427 642

10 777 745 1749 925 1268 1692 697 11 1141 857 1831 1102 1901 1925 1066 12 1180 886 1954 1310 1985 1986 1097 13 1233 1173 2250 1412 2075 2185 1115 14 1314 1261 2369 1500 2234 2421 1166 15 1649 1394 2489 1691 2301 2446 1504

582 Proc. 6th Int. Conf. on Mathematical Modding

19 20 22 16 1423 2525 17 2111 2600 1736

2129 1843 1797 19 1934 2713 20 2400 2876 1960

2420 2130 2132 22 2545 3007 23 2945 3089 2276

2961 2711 2430 25 2713 3439

23 24 CONCLUSION 2459 1706 2490 2050 2555 2053 2563 2132 2711 2331 2897 2340 2961 2754 3081 2881 3199 2897 3239 2942

Table Solution Chart

Meters tour

3 10

;:

292

:4

;:

2 11

17 15

25 19

;0 8

tour 17

19 18

20

!

7 16

;2 4

21 23

2 11

5 15

111

ZO

436 605

321 462

174 639

540 580

366 882

642 141

216 596

0

Meters

141 103

596 129

58 280

436 605

321 462

174 639

540 580

876 664

0

10587

2 14

21 24

Z2 9

16

;

8 20

18

::

15 17

11 2

tour 9

4

:4

14 2

;:

75

i:

;0

3 10

16 13

Meters 10665

540 4 22

639 9 174 13 581 16 462 7 321 3 494 10 605 8 436 418 ;: 240 19 58 18 166 6 129 596 75 216 17 103 141 1: 642 2 664 882 :: 366 21 756 24 580 1 0 4

Meters 10923 tour

494 6 321 18 462 19 581 20 174 639 ! 378 10 540 7 580 16 756 13 366 9 519 22 664 4 1097 1 141 24 103 216 ;: 596 14 129 2 111 11 58 280 :: 418 25 436 5 605 15 0 6

Meters

321 494 605 436 418 240 58 166 129 231 141 103 216 1085 664 882 366 756 580 540 378 639 174 581 462 0

Meters

216 103 306 129 111 58 280 418 436 605 494 321 462 581 174 639 378 540 580 756 366 116 642 664 1548 0

The first multi stage Monte Carlo optimization (MSMCO) program was tested by Conley in 1979. Since then it has been applied to a variety of fields and problems. It is a general purpose linear and nonlinear multivariate solution tech- nique. General optimization or solving systems of equations are its most frequent applications.

It improves on the standard Monte Carlo (random search) optimization technique by essentially treating that as merely stage one in the FOR- TRAN or BASIC program. Then stage 2 is a second Monte Carlo random search in a smaller region centered about this first stage best an- swer so far. Then similarly a third and a fourth stage etc. until convergence to the opti- mal solution. Along this trail of ever improv- ing answers, second, third, fourth and fifth best answers can be printed to give the decision maker an array of options. If MSMCO fails to converge then the considerable limit theory surrounding MSMCO is used to identify the opti- mal.

0

0 Therefore the manager and delivery boy have six FIG. 1. N dimensional spheres crossing the "solutions" between 10.5 and 11 kilometers to choose from. The program is flexible enough for

sampling distribution of an optimization prob- lem to the minimal solution.

everyday use.