ainical Pharmacokinetics 1: 67-78 (19761 CI ADtS Press 1976
Computer Assisted Prescribing of Drugs G.E. Mawer Department of Pharm8(:ologV, Meteria Medica end Tnerapeutics, UniverlitY of Manchelter, Meocheller
Summory
Computer programs for drug dOlllge adjustment moy be rlXed. adapti~e or empirictll. The aminoglycoside antibimic dosage rtiluirementJ of indMdual patitntl are relati~tly pudictable, and it !ums to be adequate to asrume thD! 1'Olume of distribution II a rlXed proportion of body weight and that renol clearance is a rlXed proportion of creatilline clearance. This approach hal bun len ruccesJ/ul with digoxin because patient compliance, the proportion absorbed and /i~er clearance are not yet predictable. Accordingly, adDptivt progTams NH'e been de~elop«J which use fudfHJck from drug concentration mearunment! to predict the future dosage needs of the patient. When individwl/ need! are known for a large patknt group It becomes po$sibk to predict the dolllge requirement! of a new patient from the Jl2me popuiDtion by empirical mnhods, Computer programs for dosage adjustment will not be widely used until their scopt is incretlSed and objtCtl~e evidence of clinictll benefit is obtained.
Computer programs designed to calculate drug dosage schedules to suit individual patients can be of three types - fixed, adaptive and empirical. The I1rst is based on a fixed kinetic model. The model dimensions are calculated from the patient's body weight and creatinine clearance, for example, but the various proportionality constants are fIXed by the designers after a preliminary !rial period. The second, adaptive type, is also based on a kinetic model but includes a learning facility. Drug con· centration data obtained from the individual patient is used 10 trim the model which will be used to predict his future dosage requirements. The program also modifieS the model for a new patient according to the observed dosage require. ments of earlier palients. The third type is not based on a kinetic model but is completely
empirical. The observed dosage requiremen ts of in· dividual patients are recorded alongside other clinical characteristics which cou ld possibly be reo levant. The idenlification of the relevant characterislics and the derivation of equations which predict dosage requirements from these characteristics are both done by the computer. These three types of program will be illustrated mainly by reference to the aminoglycoside anti· biotics and digoxin,
I. Fixed Model Types
The model is usually a simple one compartment open system (fig. I). Most investigators have not considered it necessary or justifiable to use more
68
Comput'r AAilttd Prescribing of Orugl
complex models. The first examples were programs designed to predict antibiotic dosage requirements (JeUirre et aI., 1970a, 1970b; Knowles et at, 1971 ; Koeppe and Horner. 1971; Mawer et at, 1972, 1974a). These groups chose independently to study the aminoglycoside antibiotics, but this was not fortuitous, for there was a clinical need and a well defmed kinetic model. Toxicity to the ear and kidney presented a special problem in patients with impaired kidney function; the drugs were known to be completely absorbed after parenteral injection, distribution was virtually restricted to the extracellular water. and elimination was due almost entirely to filtration at the renal glomerulus. It was a relatively simple matter therefore to produce a model, the dimensions of which could be adjusted to match each individual patient. The absorption rate constant ka was assumed to be the same in all patients because no way of pre· dicting an individual value was known. The constant for each drug was obtained by fitting equations to serum antibiotic concentration/ time
a,,,,, --~)
curves measured in patients or volunteers. The effective volume of distribution (Vd) was assumed to be a fixed proportion of the body weight, although it was recognised that this was unlikely to be the case when considering the obese or wasted patient (Orme and Cutler, 1969). The renal elimination clearance (Cel(r)i was assumed to be a flXed proportion of the creatinine clearance (see appendix) and aU patienu were considered to have the same rate constant kel(a) for non-renal elimination. This was obtained from the serum concentration half time t*.(a) in a group of severely oliguric patients_ Clearly, the deSigners of these programs were obliged to make some naive working assumptions. They nevertheless constructed patient simulations to take account of individual differences in body weight (W) and creatinine clearance (Ccr) which have a major influence on aminosugar dosage needs. An example of such a patient simulation is given in table I. The patient is described by the assumed ka' the Vd based on body weight, and the tota l elimination rate constant kel based on creatinine clearance.
---)~ Co,l/h
F;g. t. Th' simple one compertment kinetic model which" the basis of th, elrlielt progrtms lor drug dosage tdj ustment. Drug it tb50rbed Irom the site of .oninin... tion It I 1""11, which is proportionll to th, u nabtorbed dose Qt. The p roportlonl litv cOOllent or IIIt.orptioo rete constant iii kill. The drug" m illn distributed th roughOU t I coneept~1 compartment of yolu me Vd from which it il eliminated by o ne or more clelllrlllnce p roceSHI coIlecti....ly deKribed by the total ellmlnliltlo n delrenee Cal. The rite of decay o f Ct is proportionel to lCellVdl "';'lch it termed the eli minatiOl'l rtte connlllni keI.
Compul'"
AssKt~
69
PrftCl"ibing of Drugs
The serum drug concentration CtmgJl (or J,lg/ml) t hours after the patient received the first dose of Q mg is given by the expression:
c, •
O' k. le- k• t Vd Ikel
-
e-kejl)
kill)
J,lg/ ml
(Eq.1)
T¥JI, I. Dlmel1l iol1l of a one corr"C)artment open model intended to limulate Ilia IIandlir"lg of intramuKular gentamicin in a Pltient weigning W kg witll a creatinine cleellince o f Ccr 1111
Ablorption rate coostant
k.
-
Volume of diltribution
which represents the resuilanl of two concurrent exponential processes, absorption and elimination. Renal eliminalion The concentration may be calculated at an early clearance time (I to 2h) to estimate the peak and immediately before the second dose to estimate Renal elimination the trough concentration Co. The concentration I rata constant hours after the second dose of Q mg then be· Half time in comes: anuric pat;anl$
c, •
a .k. Ie- kat _ e- kejt) Vd (k el
70h
0.693 - - -O.Oln ' t)l, lal
Eli mination rate connant in anuric
ka )
(Eq.2) A new value for the trough concentration Co at the time of the third dose is calculated and equa tio n (2) is en\ered again with the appropriate values of Q, t and Co. The calculation is re peated again and again in an iterative fashion for as many times and doses as required. Thus, the concen· tration o f antibiotic t hours after the nth dose can be calculated by a kind of pharmacokinetic dead reckoning. The obvious way to check the validity of the calculation is to compare the estimated con· centration wit h the measured value. The American and European gro ups observed strong and significant positive correlations between calculated and measured serum antibiotic concentrations in groups of patients, which in· c1uded all degrees of kidney impairment. The coroUary is clear. If concentration can be calcu· lated when dose is known, then dose can be calculated when the desired concentratio n i s specined (fig. 2). Thus, the patient simulation can be used to test available dose options and to reject those which give excessive or inadequate concen· trations. Similarly, patient simuJations can be con·
Total el imination rate conllant
0.2S"W I
k"
-
kel
tal
-+- kel
Ir) n-'
" Mawer 11976)
sidered which represent a full range of body weight and creatinine clearance values and a com· prehensive drug dosage nomogram can be con· structed. The use o f dosage schedules selected by these methods red uced between patient variation in peak serum antibiotic concentratio n, but toxici ty was still observed in patient s with severe kidney impairment who had relat ively high trough co ncentrations or gentamicin (Mawer et al., 19743). This raises a fun damen tal question. It is not known whether equitherapeutic doses of amino· glycoside antibiotics are obtained by matching peak serum concent rations or mean vlaues. Despite Ihis difficult y, some degree o f success has been achieved in the calculat ion of aminoglycoside dosage requirements using fIXed model programs. This is probably a ttributable to the relatively
7.
Computer A,$$isted Prnc;ribing of Drugs
Conn.uct PAt~t
Ilm ... luion
(.,
F III-.:I p.ogram
Enl"""a d .... g
conc.ntratlon,
(0' Ad.ptl .... p'09,am
Co"
patient "mu'atlon
Trim IMtlen! ,Im""atlon ealCul.t. p.-.:Ilct-.:l f ... I ... ,. do .. Calcul.t. pr.dict.d fut ... ,. dow n . ."
ch.,
Fig. 2 Flow representing the openItion of pr09"an... for drug dotage adjultment. A model simUlating thl handling of the drug in the patient it conrtructed using body WI~I and alatinine cl.'M:(! for IltMTtpil • described in Ubi" I ,nd II. In thl fixed type of program I,) thit model il used to prldict lututl doIIgI needs from the lpICilied desired drug concentrllion. In thl ldaptiw type Ib) the prOYilionl1 model it trimmed in IICcord¥lOl! with the drUIl concentrations YoiI'1id1 I\IWI been n'WaIiurad in IImples from the patient. The trimmed modIl end the desired drug concentration _e then used 10 predict future dosage needL
Computer A,uisted PrllSaibing of Drug5
simple way in which these drugs are handled in the body. Several other drugs have proved less predictable.
2. Adaptive Type of Model
The limitations of the fIXed approach became obvious when the altempt was made to construct computer programs to pred ict serum digoxin concentrations and individual dosage requiremen ts (JeHiffe et aI., 1970c; Bogdanik, 1972; Bogdanik et aI., 1974a,b). The relevant characteristics of the individual patient were the body weight and a measure of renal function, usually creatinine clearance. The serum digoxin concentrations were measured in a group of treated patients representing a wide range of kidney function by Jelliffe et al. (1972) who were able to account for 74% of the observed variance. The use of the program to predict individual doses appeared to reduce the incidence of digitalis toxicity, although this was not tested by a randomised prospective clinical trial. A digoxin dosage nomogram was constructed on the basis of the model (Jelliffe and Brooker. 1974) and the results seemed encouraging. In a group of patients with creatinine clearance values ranging from 0 to lSOml/ min, Dobbs et al. (1976) were able to account for 67% of the between patient variance in digoxin dose requirements on the basis of differences in kidney function alone. The Jelliffe nomogram also accounted for about 60%. Other workers studied patient groups with less varied kidney function, which are probably more representative of the total digOxin-treated popula tion, and their results were less encouraging. Peck et al. ( 1973) in San Francisco were able to account fo r omy 18% of the variance in serum digoxin concentration and their computer program predicted the concent rat ions on1y slightly better than their clinicians. Wagner et al. (1974) similarly, were able to account for only 34% of the variance in serum digoxin concentration on the
71
basis of dose, weight and kidney function. Clearly, there must be other important sources of individual variation which the fIXed model programs do not encompass. It is not difficult to guess what some of these may be. Suitable dimensions for a one companment digoxin mode l are suggested in table II. The absorption value of 0.6 for 'Lanoxin' tablets (WeUcome Medical Division Ltd.) is a group mean (Huffman et aI., 1974; Johnson and Bye, 1975). If the proportion is 0.8 in one patient and 0.4 in another this may cause a 2-fold diffe rence in serum digoxin concen tration which is unrelated to body weight or kidney function. Similarly the effective volume of distribution of 5 I/kg is known to be excessive fo r obese patients (Ewy et aI., 197 1) and for patients with severe impairment of kidney function ( Reuning et aI., 1973). Another variable which the flXed model does not recognise is that of patient compliance. An outpatient prescribed a given dose may have lower serum digoxin concentrations than an otherwise comparable inpatient (Sheiner et a1., 1973). Non-renal elimination also no doubt varies between patients. Sheiner and his co-workers responded to these problems by the development of an adaptive program (Sheiner et al., 1972, 1975). The statistical basis is complex and apparently more fam iliar to economists than pharmacokineticists but the general principle is represented diagrammatically in figure 2. A provisional patient simulation is constructed from the available clin ical data as outlined in table II . This is then used to calculate the serum drug concen trations expected from the doses already received by the patient. These are compared with the measured concentrat ions and the previsional simulation is trimmed in such a way as to reduce the discrepancy between the two sets of concentrations. The new pat ient simulation is then used to predict the future doses needed to produce the desired concentrations. This feedback system has two advantages. The model used for prescribing is matched more closely to the individual pat ient and the data
72
Computer Assisted Prncribing of Drugs
T,bI~ fl. Dimensions o. a one compertment open model intended to .Imulate the handling o. or,lIy administered dlJOllin in , pati.nt _ighing W kg with' creatinine clearance of Cer f/h
•
p
Abtorption rltl constant
'.
Volume of distribution
0.60
Hultman et," (19141; JohnlOn and Bye (1915)
O.41h· '
Shainer It a l. (1972)
,WI
Steady l1atl volume, Reun ing It al. (1913) : Shainer et al. (1912) assume Vd to be pro· portionalto IUr'aci .rea cal· culaled from weight and height
Cer IJh
Renal elimination clear,nce
Cei
Reml' elimination rite constanl
kal Irl
Ceilr)lVd h"'
Non-renal elimination 1"8te constant
kella )
0.004 h"'
Total Ilimination rltl constant
."
kell,). kel Irl h"'
Ir)
acquired from a group of patients trealed in this way can be used to improve the quality of the provisional simulation for a new patient. Thus, this type of program has a learning facility. The use of the feedback system increased the proportion of the serum digoxin concentration variance which can be explained by 25% (Sheiner et ai., 1972).
3. Empirical Type of Model If the learning facility is adequately developed and Ihe group of patients sludied is large, Ihe pharmacokinelic model ceases to be essential. The information available in the studied patient group can be used empirically to predict the dose reo quired by a new patient. Consider the problem of selecting a suitable mainlenance dose of digoxin for an adul! palient. In the experience of the author (Dobbs et aI., 1976; and unpublished observations) a daily dose of 500l-lg digoxin ('Lanoxio' tablets) gives mean steady state serum concentralions of I to 2ng/ml
1.0
Bloom end Nllp 1'966 )
Sheine. II aI.(1912)
in most healthy volunteers with creatinine clearances in the range 90 10 140ml/ min . AI the other end of Ihe scale very few palienls with creatinine clearance values below 5ml/min develop mean digoxin concenlr:lIions above 2ng/ ml whilsl receiving 62.5JJ8 digoxin daily. Thus. the great majority of patients will require a dose which lies within this range for concentrations of lhis order. Using a geomelric progression. and excluding doses which require half table IS, there are only 6 practical daily doses within this range - namely. 62.5, 125, 187.5, 250, 375 and 500~g. The seleclion of a daily digoxin dose for an individual patient therefore involves Ihe allocation of Ihe patient to one of 6 categories. A clinician makes this allocation every day when he provisionally selects a maintenance dose for a new patient. He chooses a dose which has proved neither excessive nor inadequate when given to similar patients he has treated in Ihe past. The wisdom of this choice depends on his ability to match the new palient to the appropriate in· dividuals in his repertoire of therapeulic
73
CofT1JKlt.r Assisted Prescribing of Drugs
experience. The same type of process can be made the basis for a computer program using either parametric o r non-parametric methods.
3. 1 Partial Correlalion Method Wagner et al. (1974) used the parametric tech· nique of partial correlation to identify clinical characteristics of patients which were relevant 10 the serum digoxin concentrations produced by a given dose. Having identified the characteristics, it was then possible to predict the concentration which would resuil from a given dose using the technique of multiple or stepwise linear regression. The failure to account for more than 34% of the variance was probably not due to an inadequate method but to the present lack of knowledge of all the relevant patient characteristics. The same approach was applied by Dobbs et al. (1976) to the problem of pred icting the daily dose of digoxin for a mean steady state concentration of 1.5ng/mI. The single patient characteristic which showed the st ro ngest correlation with daily dose was the creatinine clearance (see appendix). This is consisten t with the Dettli principle (Dettli et aI., 1971) and wilh a specific recommendation for digoxin dosage (Dettli et at., 1972). The daily dose correlated also with body weight, but the combination of weight with creatinine clearance in a multiple linear regression equation did no t significantly improve the level of prediction . The pair of patien t characteristics with the greatest predictive potential was creatinine clearance and serum albumin concentration.
3.2 Bayesian Probability Method The problem of allocating patients to an appropriate dosage category is analogous to that of allocating to an appropria te d iag nostic category. Computer assisted diagnosis (Horrocks et al., 1972; Dombal et a1., 1974) has been based on
Bayesian probability theory. Patients are grouped into sets on the basis of parametric or non· parametric variables and the diagnostic probabili. ties are calculated fo r each sel. Dobbs et al. (1976) applied the Bayesian method to the problem of predicting digoxin dosage requirements from the clinical characteristics of the 43 paUents they had studied. The characterist ics included the severity of cardiac failure reco rded on a non-linear scale. The prediction lay within the error limits of the daily dose in 26 cases, but a larger patient group is needed for a proper evaluation o f this technique. The validity of an empirical method is reo st ricted to patient populations of which the studied patient group is representative. To some extent this is also true o f the model based methods. The patient characteristic (creatinine clearance) which is most relevant 10 daily digoxin dosage requirements in a teaching hospital with a bias towards kidney disease, may be of less impor tance in a district hospital or general practice situation where low clearance va lues are less commonly e ncount ered. Thus, the rela tive importance of different pat ient character istics is to some extent determined by the investigator when he selects a certain group for study.
3.3 Test Dose Occupancy Method There is a third type of empirical method in which the prediction of futu re drug dosage requirements is based upon the ana lysis of the serum concentration/ time pattern after a te!! dose. The simplest of these was described by Orr et al. (1969). The dail y dose Q i s calculated from the follOwing expression: Q '" 24 ca/a 1lg/24 h
(Eq. 3)
where css represents the desired mean steady state concentration (}.Ig/ml) and a, the serum drug occupancy (h/m!). The latter is calculated from
COl'Y'P'lIer Asslnld Presaibil'lg of Dn.Igs
the total area under the test dose serum concentration/time curve (A ~g , him!) and the size of the test dose (0 ~g):
a • AiD h/ml
(Eq.4)
This method has two important limitations. It only applies to drugs which obey flIst order kinetics and it requires that the serum drug con· centration after the test dose shall be allowed to decay to a ol w value. This may be unacceptable with certain antibiotics and anticonvulsanlS, for example. A test dose method has also been applied to the prediction of serum lithium concenlrations and dosage requirements (Bergner et aI., 1973). These authors used a curve-stripping technique to fit a model-free mathematical function to the concentration/time curve. No comparison was made with the simpler occupancy method, however. A single test dose of sodium phenytoin was used by Mawer el al. ( 1974b) in an attempt to determine the handling characteristics of this drug in individual patients. The serum concen tration/ time curves were fitted by the Michaelis-Menten equation. The estimation of an individual value for the serum concentration at which the elimination process is half saturated (Km) is technically very difficult, however. The method is not clinically practical because it requires replicate assays of several data points to obtain adequate precision in the measurement of the serum drug concentration gradient. R ci hens and Dunlop ( 1975) made the simplifying assumpt ion that all patients have the same Km value. They used an estimate based on the detailed study of the relationship between steady slate serum phenytoin concentration and daily dose in 5 patients. This was combined with the group mean value obtained by the previous workers to give a working estimate. Using this value, they were able to construct a nomogram to predict future phenytOin dosage requirements. To enler the nomogram it is necessary to know the steady state serum phenytoin concentration at a single dose leveL
4. The Future Role of Computer Assisted DoSilge Programs The development of programs to predict the future dosage requirements of individual patients is a fascinating and instructive exercise for those concerned. Their understanding of drug kinetics and patient vanallon is greally increased. Similarly, those doctors involved in clinical trials who are motivated to 'beat the computer' (Domba l et aL, 1974; Sheiner et al., 1975) im· prove their clinical technique in the process. II is not difficult therefore to envisage an educational role for these programs. Doctors in training may benefit from the experience of treating computer simulations of patients. Some experienced prescribers will feel that the programs challenge their own professional com· petence and wiU not use them until there is convincing evidence that the use of the computer im· proves the clinical response. This evidence must be sought in busy general practices and district hospitals w here the majority of patients are treat· ed. Trials must be prospective, with random allocation of patients to computer assisted or conventional treatment groups. Since the prescribing doctor cannot be blind, an objective or independent assessment of clinical response is essential. If evidence of greater efficacy and reduced toxicity is obtained it will be necessary to provide access to these methods, perhaps as an adjunct to regional drug information services. The demand will not be great however, until the programs are applicable 10 a wider variety of clinical situations. Up to the present , computer assisted dosage prl> grams have made a substantial contribution only in the management of the adult patient with renal impairment. The methods must be extended to include drugs which are mainly eliminated by non· renal routes and to include children. The need to show that the attainment of desired plasma con· cent rations is accompanied by an improved clinical response will make progress ~ow.
75
Computer Assiued Prescribing of Drugs
(
S.M'
/::.~
)
/
~t~·m V< 151 F
nOI eppllc.bl.
E.d
)
j'.~'j ;Ighl
0- W' 129.30.203'V )
~ _om r~' / I i'.'"' j0 - 0 .8'0
~r,""lnl"'/
Slebl. :
52, 0
-(51 + 52)/2
a - a'
s - 51
(1-0,03 ' 5)
0- A' (1-0.03'5)
A_4 ' W' (51_52)/0
C - '" (14.4 '5 )
a - a - A
t'""·,··
Computer Assisted Prescribing of Drugs G.E. Mawer Department of Pharm8(:ologV, Meteria Medica end Tnerapeutics, UniverlitY of Manchelter, Meocheller
Summory
Computer programs for drug dOlllge adjustment moy be rlXed. adapti~e or empirictll. The aminoglycoside antibimic dosage rtiluirementJ of indMdual patitntl are relati~tly pudictable, and it !ums to be adequate to asrume thD! 1'Olume of distribution II a rlXed proportion of body weight and that renol clearance is a rlXed proportion of creatilline clearance. This approach hal bun len ruccesJ/ul with digoxin because patient compliance, the proportion absorbed and /i~er clearance are not yet predictable. Accordingly, adDptivt progTams NH'e been de~elop«J which use fudfHJck from drug concentration mearunment! to predict the future dosage needs of the patient. When individwl/ need! are known for a large patknt group It becomes po$sibk to predict the dolllge requirement! of a new patient from the Jl2me popuiDtion by empirical mnhods, Computer programs for dosage adjustment will not be widely used until their scopt is incretlSed and objtCtl~e evidence of clinictll benefit is obtained.
Computer programs designed to calculate drug dosage schedules to suit individual patients can be of three types - fixed, adaptive and empirical. The I1rst is based on a fixed kinetic model. The model dimensions are calculated from the patient's body weight and creatinine clearance, for example, but the various proportionality constants are fIXed by the designers after a preliminary !rial period. The second, adaptive type, is also based on a kinetic model but includes a learning facility. Drug con· centration data obtained from the individual patient is used 10 trim the model which will be used to predict his future dosage requirements. The program also modifieS the model for a new patient according to the observed dosage require. ments of earlier palients. The third type is not based on a kinetic model but is completely
empirical. The observed dosage requiremen ts of in· dividual patients are recorded alongside other clinical characteristics which cou ld possibly be reo levant. The idenlification of the relevant characterislics and the derivation of equations which predict dosage requirements from these characteristics are both done by the computer. These three types of program will be illustrated mainly by reference to the aminoglycoside anti· biotics and digoxin,
I. Fixed Model Types
The model is usually a simple one compartment open system (fig. I). Most investigators have not considered it necessary or justifiable to use more
68
Comput'r AAilttd Prescribing of Orugl
complex models. The first examples were programs designed to predict antibiotic dosage requirements (JeUirre et aI., 1970a, 1970b; Knowles et at, 1971 ; Koeppe and Horner. 1971; Mawer et at, 1972, 1974a). These groups chose independently to study the aminoglycoside antibiotics, but this was not fortuitous, for there was a clinical need and a well defmed kinetic model. Toxicity to the ear and kidney presented a special problem in patients with impaired kidney function; the drugs were known to be completely absorbed after parenteral injection, distribution was virtually restricted to the extracellular water. and elimination was due almost entirely to filtration at the renal glomerulus. It was a relatively simple matter therefore to produce a model, the dimensions of which could be adjusted to match each individual patient. The absorption rate constant ka was assumed to be the same in all patients because no way of pre· dicting an individual value was known. The constant for each drug was obtained by fitting equations to serum antibiotic concentration/ time
a,,,,, --~)
curves measured in patients or volunteers. The effective volume of distribution (Vd) was assumed to be a fixed proportion of the body weight, although it was recognised that this was unlikely to be the case when considering the obese or wasted patient (Orme and Cutler, 1969). The renal elimination clearance (Cel(r)i was assumed to be a flXed proportion of the creatinine clearance (see appendix) and aU patienu were considered to have the same rate constant kel(a) for non-renal elimination. This was obtained from the serum concentration half time t*.(a) in a group of severely oliguric patients_ Clearly, the deSigners of these programs were obliged to make some naive working assumptions. They nevertheless constructed patient simulations to take account of individual differences in body weight (W) and creatinine clearance (Ccr) which have a major influence on aminosugar dosage needs. An example of such a patient simulation is given in table I. The patient is described by the assumed ka' the Vd based on body weight, and the tota l elimination rate constant kel based on creatinine clearance.
---)~ Co,l/h
F;g. t. Th' simple one compertment kinetic model which" the basis of th, elrlielt progrtms lor drug dosage tdj ustment. Drug it tb50rbed Irom the site of .oninin... tion It I 1""11, which is proportionll to th, u nabtorbed dose Qt. The p roportlonl litv cOOllent or IIIt.orptioo rete constant iii kill. The drug" m illn distributed th roughOU t I coneept~1 compartment of yolu me Vd from which it il eliminated by o ne or more clelllrlllnce p roceSHI coIlecti....ly deKribed by the total ellmlnliltlo n delrenee Cal. The rite of decay o f Ct is proportionel to lCellVdl "';'lch it termed the eli minatiOl'l rtte connlllni keI.
Compul'"
AssKt~
69
PrftCl"ibing of Drugs
The serum drug concentration CtmgJl (or J,lg/ml) t hours after the patient received the first dose of Q mg is given by the expression:
c, •
O' k. le- k• t Vd Ikel
-
e-kejl)
kill)
J,lg/ ml
(Eq.1)
T¥JI, I. Dlmel1l iol1l of a one corr"C)artment open model intended to limulate Ilia IIandlir"lg of intramuKular gentamicin in a Pltient weigning W kg witll a creatinine cleellince o f Ccr 1111
Ablorption rate coostant
k.
-
Volume of diltribution
which represents the resuilanl of two concurrent exponential processes, absorption and elimination. Renal eliminalion The concentration may be calculated at an early clearance time (I to 2h) to estimate the peak and immediately before the second dose to estimate Renal elimination the trough concentration Co. The concentration I rata constant hours after the second dose of Q mg then be· Half time in comes: anuric pat;anl$
c, •
a .k. Ie- kat _ e- kejt) Vd (k el
70h
0.693 - - -O.Oln ' t)l, lal
Eli mination rate connant in anuric
ka )
(Eq.2) A new value for the trough concentration Co at the time of the third dose is calculated and equa tio n (2) is en\ered again with the appropriate values of Q, t and Co. The calculation is re peated again and again in an iterative fashion for as many times and doses as required. Thus, the concen· tration o f antibiotic t hours after the nth dose can be calculated by a kind of pharmacokinetic dead reckoning. The obvious way to check the validity of the calculation is to compare the estimated con· centration wit h the measured value. The American and European gro ups observed strong and significant positive correlations between calculated and measured serum antibiotic concentrations in groups of patients, which in· c1uded all degrees of kidney impairment. The coroUary is clear. If concentration can be calcu· lated when dose is known, then dose can be calculated when the desired concentratio n i s specined (fig. 2). Thus, the patient simulation can be used to test available dose options and to reject those which give excessive or inadequate concen· trations. Similarly, patient simuJations can be con·
Total el imination rate conllant
0.2S"W I
k"
-
kel
tal
-+- kel
Ir) n-'
" Mawer 11976)
sidered which represent a full range of body weight and creatinine clearance values and a com· prehensive drug dosage nomogram can be con· structed. The use o f dosage schedules selected by these methods red uced between patient variation in peak serum antibiotic concentratio n, but toxici ty was still observed in patient s with severe kidney impairment who had relat ively high trough co ncentrations or gentamicin (Mawer et al., 19743). This raises a fun damen tal question. It is not known whether equitherapeutic doses of amino· glycoside antibiotics are obtained by matching peak serum concent rations or mean vlaues. Despite Ihis difficult y, some degree o f success has been achieved in the calculat ion of aminoglycoside dosage requirements using fIXed model programs. This is probably a ttributable to the relatively
7.
Computer A,$$isted Prnc;ribing of Drugs
Conn.uct PAt~t
Ilm ... luion
(.,
F III-.:I p.ogram
Enl"""a d .... g
conc.ntratlon,
(0' Ad.ptl .... p'09,am
Co"
patient "mu'atlon
Trim IMtlen! ,Im""atlon ealCul.t. p.-.:Ilct-.:l f ... I ... ,. do .. Calcul.t. pr.dict.d fut ... ,. dow n . ."
ch.,
Fig. 2 Flow representing the openItion of pr09"an... for drug dotage adjultment. A model simUlating thl handling of the drug in the patient it conrtructed using body WI~I and alatinine cl.'M:(! for IltMTtpil • described in Ubi" I ,nd II. In thl fixed type of program I,) thit model il used to prldict lututl doIIgI needs from the lpICilied desired drug concentrllion. In thl ldaptiw type Ib) the prOYilionl1 model it trimmed in IICcord¥lOl! with the drUIl concentrations YoiI'1id1 I\IWI been n'WaIiurad in IImples from the patient. The trimmed modIl end the desired drug concentration _e then used 10 predict future dosage needL
Computer A,uisted PrllSaibing of Drug5
simple way in which these drugs are handled in the body. Several other drugs have proved less predictable.
2. Adaptive Type of Model
The limitations of the fIXed approach became obvious when the altempt was made to construct computer programs to pred ict serum digoxin concentrations and individual dosage requiremen ts (JeHiffe et aI., 1970c; Bogdanik, 1972; Bogdanik et aI., 1974a,b). The relevant characteristics of the individual patient were the body weight and a measure of renal function, usually creatinine clearance. The serum digoxin concentrations were measured in a group of treated patients representing a wide range of kidney function by Jelliffe et al. (1972) who were able to account for 74% of the observed variance. The use of the program to predict individual doses appeared to reduce the incidence of digitalis toxicity, although this was not tested by a randomised prospective clinical trial. A digoxin dosage nomogram was constructed on the basis of the model (Jelliffe and Brooker. 1974) and the results seemed encouraging. In a group of patients with creatinine clearance values ranging from 0 to lSOml/ min, Dobbs et al. (1976) were able to account for 67% of the between patient variance in digoxin dose requirements on the basis of differences in kidney function alone. The Jelliffe nomogram also accounted for about 60%. Other workers studied patient groups with less varied kidney function, which are probably more representative of the total digOxin-treated popula tion, and their results were less encouraging. Peck et al. ( 1973) in San Francisco were able to account fo r omy 18% of the variance in serum digoxin concentration and their computer program predicted the concent rat ions on1y slightly better than their clinicians. Wagner et al. (1974) similarly, were able to account for only 34% of the variance in serum digoxin concentration on the
71
basis of dose, weight and kidney function. Clearly, there must be other important sources of individual variation which the fIXed model programs do not encompass. It is not difficult to guess what some of these may be. Suitable dimensions for a one companment digoxin mode l are suggested in table II. The absorption value of 0.6 for 'Lanoxin' tablets (WeUcome Medical Division Ltd.) is a group mean (Huffman et aI., 1974; Johnson and Bye, 1975). If the proportion is 0.8 in one patient and 0.4 in another this may cause a 2-fold diffe rence in serum digoxin concen tration which is unrelated to body weight or kidney function. Similarly the effective volume of distribution of 5 I/kg is known to be excessive fo r obese patients (Ewy et aI., 197 1) and for patients with severe impairment of kidney function ( Reuning et aI., 1973). Another variable which the flXed model does not recognise is that of patient compliance. An outpatient prescribed a given dose may have lower serum digoxin concentrations than an otherwise comparable inpatient (Sheiner et a1., 1973). Non-renal elimination also no doubt varies between patients. Sheiner and his co-workers responded to these problems by the development of an adaptive program (Sheiner et al., 1972, 1975). The statistical basis is complex and apparently more fam iliar to economists than pharmacokineticists but the general principle is represented diagrammatically in figure 2. A provisional patient simulation is constructed from the available clin ical data as outlined in table II . This is then used to calculate the serum drug concen trations expected from the doses already received by the patient. These are compared with the measured concentrat ions and the previsional simulation is trimmed in such a way as to reduce the discrepancy between the two sets of concentrations. The new pat ient simulation is then used to predict the future doses needed to produce the desired concentrations. This feedback system has two advantages. The model used for prescribing is matched more closely to the individual pat ient and the data
72
Computer Assisted Prncribing of Drugs
T,bI~ fl. Dimensions o. a one compertment open model intended to .Imulate the handling o. or,lIy administered dlJOllin in , pati.nt _ighing W kg with' creatinine clearance of Cer f/h
•
p
Abtorption rltl constant
'.
Volume of distribution
0.60
Hultman et," (19141; JohnlOn and Bye (1915)
O.41h· '
Shainer It a l. (1972)
,WI
Steady l1atl volume, Reun ing It al. (1913) : Shainer et al. (1912) assume Vd to be pro· portionalto IUr'aci .rea cal· culaled from weight and height
Cer IJh
Renal elimination clear,nce
Cei
Reml' elimination rite constanl
kal Irl
Ceilr)lVd h"'
Non-renal elimination 1"8te constant
kella )
0.004 h"'
Total Ilimination rltl constant
."
kell,). kel Irl h"'
Ir)
acquired from a group of patients trealed in this way can be used to improve the quality of the provisional simulation for a new patient. Thus, this type of program has a learning facility. The use of the feedback system increased the proportion of the serum digoxin concentration variance which can be explained by 25% (Sheiner et ai., 1972).
3. Empirical Type of Model If the learning facility is adequately developed and Ihe group of patients sludied is large, Ihe pharmacokinelic model ceases to be essential. The information available in the studied patient group can be used empirically to predict the dose reo quired by a new patient. Consider the problem of selecting a suitable mainlenance dose of digoxin for an adul! palient. In the experience of the author (Dobbs et aI., 1976; and unpublished observations) a daily dose of 500l-lg digoxin ('Lanoxio' tablets) gives mean steady state serum concentralions of I to 2ng/ml
1.0
Bloom end Nllp 1'966 )
Sheine. II aI.(1912)
in most healthy volunteers with creatinine clearances in the range 90 10 140ml/ min . AI the other end of Ihe scale very few palienls with creatinine clearance values below 5ml/min develop mean digoxin concenlr:lIions above 2ng/ ml whilsl receiving 62.5JJ8 digoxin daily. Thus. the great majority of patients will require a dose which lies within this range for concentrations of lhis order. Using a geomelric progression. and excluding doses which require half table IS, there are only 6 practical daily doses within this range - namely. 62.5, 125, 187.5, 250, 375 and 500~g. The seleclion of a daily digoxin dose for an individual patient therefore involves Ihe allocation of Ihe patient to one of 6 categories. A clinician makes this allocation every day when he provisionally selects a maintenance dose for a new patient. He chooses a dose which has proved neither excessive nor inadequate when given to similar patients he has treated in Ihe past. The wisdom of this choice depends on his ability to match the new palient to the appropriate in· dividuals in his repertoire of therapeulic
73
CofT1JKlt.r Assisted Prescribing of Drugs
experience. The same type of process can be made the basis for a computer program using either parametric o r non-parametric methods.
3. 1 Partial Correlalion Method Wagner et al. (1974) used the parametric tech· nique of partial correlation to identify clinical characteristics of patients which were relevant 10 the serum digoxin concentrations produced by a given dose. Having identified the characteristics, it was then possible to predict the concentration which would resuil from a given dose using the technique of multiple or stepwise linear regression. The failure to account for more than 34% of the variance was probably not due to an inadequate method but to the present lack of knowledge of all the relevant patient characteristics. The same approach was applied by Dobbs et al. (1976) to the problem of pred icting the daily dose of digoxin for a mean steady state concentration of 1.5ng/mI. The single patient characteristic which showed the st ro ngest correlation with daily dose was the creatinine clearance (see appendix). This is consisten t with the Dettli principle (Dettli et aI., 1971) and wilh a specific recommendation for digoxin dosage (Dettli et at., 1972). The daily dose correlated also with body weight, but the combination of weight with creatinine clearance in a multiple linear regression equation did no t significantly improve the level of prediction . The pair of patien t characteristics with the greatest predictive potential was creatinine clearance and serum albumin concentration.
3.2 Bayesian Probability Method The problem of allocating patients to an appropriate dosage category is analogous to that of allocating to an appropria te d iag nostic category. Computer assisted diagnosis (Horrocks et al., 1972; Dombal et a1., 1974) has been based on
Bayesian probability theory. Patients are grouped into sets on the basis of parametric or non· parametric variables and the diagnostic probabili. ties are calculated fo r each sel. Dobbs et al. (1976) applied the Bayesian method to the problem of predicting digoxin dosage requirements from the clinical characteristics of the 43 paUents they had studied. The characterist ics included the severity of cardiac failure reco rded on a non-linear scale. The prediction lay within the error limits of the daily dose in 26 cases, but a larger patient group is needed for a proper evaluation o f this technique. The validity of an empirical method is reo st ricted to patient populations of which the studied patient group is representative. To some extent this is also true o f the model based methods. The patient characteristic (creatinine clearance) which is most relevant 10 daily digoxin dosage requirements in a teaching hospital with a bias towards kidney disease, may be of less impor tance in a district hospital or general practice situation where low clearance va lues are less commonly e ncount ered. Thus, the rela tive importance of different pat ient character istics is to some extent determined by the investigator when he selects a certain group for study.
3.3 Test Dose Occupancy Method There is a third type of empirical method in which the prediction of futu re drug dosage requirements is based upon the ana lysis of the serum concentration/ time pattern after a te!! dose. The simplest of these was described by Orr et al. (1969). The dail y dose Q i s calculated from the follOwing expression: Q '" 24 ca/a 1lg/24 h
(Eq. 3)
where css represents the desired mean steady state concentration (}.Ig/ml) and a, the serum drug occupancy (h/m!). The latter is calculated from
COl'Y'P'lIer Asslnld Presaibil'lg of Dn.Igs
the total area under the test dose serum concentration/time curve (A ~g , him!) and the size of the test dose (0 ~g):
a • AiD h/ml
(Eq.4)
This method has two important limitations. It only applies to drugs which obey flIst order kinetics and it requires that the serum drug con· centration after the test dose shall be allowed to decay to a ol w value. This may be unacceptable with certain antibiotics and anticonvulsanlS, for example. A test dose method has also been applied to the prediction of serum lithium concenlrations and dosage requirements (Bergner et aI., 1973). These authors used a curve-stripping technique to fit a model-free mathematical function to the concentration/time curve. No comparison was made with the simpler occupancy method, however. A single test dose of sodium phenytoin was used by Mawer el al. ( 1974b) in an attempt to determine the handling characteristics of this drug in individual patients. The serum concen tration/ time curves were fitted by the Michaelis-Menten equation. The estimation of an individual value for the serum concentration at which the elimination process is half saturated (Km) is technically very difficult, however. The method is not clinically practical because it requires replicate assays of several data points to obtain adequate precision in the measurement of the serum drug concentration gradient. R ci hens and Dunlop ( 1975) made the simplifying assumpt ion that all patients have the same Km value. They used an estimate based on the detailed study of the relationship between steady slate serum phenytoin concentration and daily dose in 5 patients. This was combined with the group mean value obtained by the previous workers to give a working estimate. Using this value, they were able to construct a nomogram to predict future phenytOin dosage requirements. To enler the nomogram it is necessary to know the steady state serum phenytoin concentration at a single dose leveL
4. The Future Role of Computer Assisted DoSilge Programs The development of programs to predict the future dosage requirements of individual patients is a fascinating and instructive exercise for those concerned. Their understanding of drug kinetics and patient vanallon is greally increased. Similarly, those doctors involved in clinical trials who are motivated to 'beat the computer' (Domba l et aL, 1974; Sheiner et al., 1975) im· prove their clinical technique in the process. II is not difficult therefore to envisage an educational role for these programs. Doctors in training may benefit from the experience of treating computer simulations of patients. Some experienced prescribers will feel that the programs challenge their own professional com· petence and wiU not use them until there is convincing evidence that the use of the computer im· proves the clinical response. This evidence must be sought in busy general practices and district hospitals w here the majority of patients are treat· ed. Trials must be prospective, with random allocation of patients to computer assisted or conventional treatment groups. Since the prescribing doctor cannot be blind, an objective or independent assessment of clinical response is essential. If evidence of greater efficacy and reduced toxicity is obtained it will be necessary to provide access to these methods, perhaps as an adjunct to regional drug information services. The demand will not be great however, until the programs are applicable 10 a wider variety of clinical situations. Up to the present , computer assisted dosage prl> grams have made a substantial contribution only in the management of the adult patient with renal impairment. The methods must be extended to include drugs which are mainly eliminated by non· renal routes and to include children. The need to show that the attainment of desired plasma con· cent rations is accompanied by an improved clinical response will make progress ~ow.
75
Computer Assiued Prescribing of Drugs
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