STATISTICS IN MEDICINE, VOL. 11,685-702 (1992)
MATHEMATICAL MODELS OF COMPLEX DOSE-RESPONSE RELATIONSHIPS: IMPLICATIONS FOR EXPERIMENTAL DESIGN I N PSYCHOPHARMACOLOGIC RESEARCH STEPHEN V. FARAONE AND JOHN C . SIMPSON Section of Psychiatric EpidPmiology and Genetics. Harvard Medical School Department of Psychiatry, Massachusetts Mental Health Center, and Psychiatry Service. Brockton- West Roxburv Veterans Affairs Medical Center, Brockton. Massachusetts 0240I , U .S .A .
AND WALTER A. BROWN Department of Psychiatry and Human Behavior, Brown Universit-yMedical School, and Providence Veterans Affairs Medical Center, Brockton, Massachusetts 02401, U.S.A.
SUMMARY We develop a mathematical model to account for the complex relationship between drug dose and clinical response in psychopharmacologic research. The model specifies relationships among drug dose, drug bioavailability, pharmacokinetic factors, course moderators, clinical response and the heterogeneity of the disorder, and allows for the derivation of results that have implications for experimental design in psychopharmacologic research. These results form the basis for computer simulations which indicate that random assignment to two fixed doses is more powerful and less sensitive to heterogeneity than assignment to clinically determined doses. Fixed dose designs, however, tend to overestimate the magnitude of drug bioavailability4nical response relationships. Clinically determined dose designs are useful in some experimental situations; their effectiveness is enhanced by systematically reducing the clinically determined dose. Larger dose reductions improve the ability to detect bioavailability-clinical response relationships.
INTRODUCTION The interpretation of pharmacologic treatment outcome studies is often obscured by the presence of a complex dose-response relationship, that is, clinical response is not an increasing monotonic function of drug dose within the accepted therapeutic range. The complexity of dose-response relationships in psychopharmacology results from primarily three categories of influence: pharmacokinetic factors; course moderators, and the heterogeneity of the disorder. This article presents a mathematical model that rigorously specifies relationships among these variables and derives results that have implications for experimental design in psychopharmacologic research. We present the model in the context of schizophrenia research because its development was motivated by issues that arose in our empirical work in that area.'-' Nevertheless, the model and its implications readily generalize to other disorders. The first sections of the paper describe the rationale and formal structure of the model. We then use the model to characterize families of
027747 15/92/050685-18$09.00 0 1992 by John Wiley & Sons, Ltd.
Received May 1990 Revised July 1991
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experimental designs useful in exploration of complex dose-response relationships. Next, we employ the model to predict differential results from comparable studies that employ different experimental designs. These predictions form the basis for computer simulations that compare the ability of these experimental designs to detect and describe complex dose-response relationships, thus allowing us to make judgments about the reasonableness of the model and its potential for guiding empirical research in this area. THE ROLE O F MEDIATING FACTORS Pharmacokinetic factors refer to individual differences in drug absorption, metabolism and excretion that result in variability of peripheral neuroleptic bioavailability among patients on the same dose. Owing to these factors, blood serum levels of neuroleptic drugs will vary 10- to 20-fold among patients on the same dose.’ It is also possible that individual differences in pharmacokinetic factors lead to variability in neuroleptic bioavailability in the central nervous system among patients who have identical serum neuroleptic levels. This is consistent with findings that serum neuroleptic concentrations are not highly or consistently predictive of either the initial antipsychotic response9- l 2 or clinical status during outpatient treatment.’,397913314 Gelder and Kolakowska” suggest two pharmacokinetic factors that may mediate the relationship between peripheral and central measures of bioavailability. Since circulating metabolites of neuroleptic drugs vary in their therapeutic potency, individual differences in metabolism of the neuroleptic parent compound that lead to greater or lesser proportions of active metabolites will decrease the relationship between peripheral measures and central antipsychotic activity. Furthermore, neuroleptic assays measure both protein bound and free fractions of the drug. Since the protein bound fraction does not contribute to antipsychotic activity and may vary among patients with the same neuroleptic serum levels, the correlation between neuroleptic serum levels and antipsychotic activity will be less than perfect even if one could control all experimental error. Course moderators refer to fluctuating environmental variables that mediate the course of clinical status. For example, we know that stress in the form of life change events or family pressure exacerbates symptomatology and precipitates relapse. Life change events modify the environment and require the patient to mobilize coping resources to adapt to the modifications. Reviews of the literat~re’~.’’’ indicate a moderate association between life events and subsequent schizophrenic relapse. This association exists even when one considers only those life events likely to be independent of psychopathology. Research on expressed emotion (EE) has demonstrated that patients who live with highly critical and/or emotionally overinvolved relatives compared with those who do not have a greater risk of The final set of factors pertains to the heterogeneity of schizophrenia. Clinically, schizophrenia manifests itself in a wide variety of symptoms, signs and outcomes. Although these attributes do not allow for categorization of the disorder into neat, discrete categories, research has shown that subdivisions such as paranoid/non-paranoid, good premorbid/poor premorbid, brain damaged/ non-brain damaged and familial/non-familial have some predictive validity. Whether or not this clinical heterogeneity corresponds to etiologic or pathophysiologic heterogeneity is unknown. However, the genetic literature is consistent with the hypothesis that schizophrenia has several genetic and non-genetic etiologies.2 THE MATHEMATICAL MODEL The mathematical model provides a formal description of these complex dose-response relationships with specific reference to the phenomenon of relapse in a population of schizophrenic
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M
K
Figure 1. Path model of the relationships among dose (D), bioavailability (B),stability (S), pharmacokinetic factors (K) and moderating variables (M) when doses are randomly assigned to patients. Signs ( or - ) indicate the direction of causal association
+
outpatients who are maintained on neuroleptic medication. The model thus forms the foundation of a formal description of experimental designs used in dose-response research and enables the derivation of characteristics of these designs that affect their usefulness for testing hypotheses about complex dose-response relationships. The model assumes the existence of a hypothetical random variable called ‘stability’. High stability values are associated with protection from relapse episodes such that patients relapse if and only if their stability value falls below the relapse threshold, We define I;. as the point on the stability continuum below which patients will relapse within i years. We assume Ti is a nondecreasing function of i such that T j P Tk for all j > k. Although we have chosen relapse among outpatients as a substantive framework for the mathematical model, the derivations and results apply equally to situations that examine the relationship between treatment and remission of symptoms among acutely disordered inpatients. In the latter case, Tcorresponds to that point on the stability continuum that a patient must surpass for remission of symptoms. Figure 1 presents the relationships among stability ( S ) , neuroleptic dose (D), neuroleptic bioavailability (B),pharmacokinetic factors ( K )and course moderators ( M ) . Our model assumes that drug dose and pharmacokinetic factors directly influence bioavailability. In turn, the bioavailability measure directly influences stability which course moderators also influence directly. In our model the distribution of drug dose for any given experimental population is normal with a mean ( p D )and variance (0;)equal to those observed in the population of patients maintained on clinical doses. However, we assume that the investigator determines doses randomly, not clinically. Although our previous research21 suggests that drug doses are not normally distributed, we can normalize them with an appropriate transformation. We assume that the pharmacokinetic factors that influence the value of B form, in some unknown combination, a variable K that is normally distributed with mean pK and variance 0;. We assume that the variables K and D are stochastically independent (that is, drug dose is not influenced by pharmacokinetic factors when doses are determined by experimental manipulation). We assume that neuroleptic bioavailability, B, is a linear function of drug dose and pharmacokinetic factors:
r.
B = BDB D
+ BKB K .
(1)
Since B is the sum of two independent normal variables, it is normally distributed with mean p B and variance IS;where:
The correlation between dose and bioavailability, p D B , is the standardized partial regression
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coefficient (or path coefficient, P D B ) because there are no indirect paths from D to B (Figure 1). Thus:
We assume stability is a linear function of neuroleptic bioavailability, B, and course moderators, M, such that:
Since M and B are stochastically independent, normally distributed variables, S is normally distributed with mean and variance:
The correlation between bioavailability and stability, pBs, is the corresponding path coefficient:
The model assumes that dose influences stability only through its influence on bioavailability, hence:
Since the model assumes that P D B and pBs are positive but less than one, pDs is always less than pBs. This is consistent with reports that, compared with drug dose, bioavailability is more predictive of outcome. FAMILIES OF EXPERIMENTAL DESIGNS
A family of experimental designs is a group of designs that share a common mathematical formulation but differ in the values chosen for design parameters. We discuss two design families: fixed dose designs and clinically determined dose designs. We describe these design families in terms of the model presented above, which we then use to derive indices that reflect the utility of the designs for detecting bioavailability4inical response relationships.
Fixed Dose (FD) designs We define the kFD design as one in which we assign patients randomly to one of k drug doses that remain unchanged throughout the trial. The 1FD design assigns all patients to the same fixed dose, d. Under lFD, the distribution of stability in the population is the conditional distribution of S given D = d, represented by fsId(s). We assume that the joint distribution of D, B and S is multivariate normal. It follows that S I d is normally distributed with mean psld and variance (Ti,,,. From normal distribution theory:
+ BDSd.
(8) We can see that BDS equals B B s B D B by substituting the right side of equation (1)for B in equation (4). Also, PSld
= PS
- BDSPD
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689
Then, with the knowledge that D = d , we can derive the expected value of the bioavailability measure for patients with stability greater than the threshold (that is, non-relapsers or patients with S > T). From the theory of linear regression for variables that have a bivariate normal distribution:22
E ( B 1 S > T, D = d ) = B B s . D E ( S I S > T, D
=d)
(10) where f i B s . D = ( p B s . D a B I d )and / ~ splBds . Dis the partial correlation between B and S with D held constant. From Johnson and K o ~ z , ’E~( S I S > T, D = d ) is the expected value of the normal distribution of S I d truncated from below at T:
Therefore, the expected value of bioavailability for non-relapsers is:
where F s I d ( Tis) the integral offsId(t)from t = - 00 to T. We use a parallel argument to derive the expected value of bioavailability for patients with S < T (that is, relapsers). The result is:
The difference between equations (12) and (1 3) is the expected difference in measured neuroleptic bioavailability between non-relapsers and relapsers under the assumptions of the model. After substituting the formula for the partial correlation coefficient for p B S . D and noting that u i l d= (1 - p i D ) & the result is:
Expression (14) indexes the degree to which the 1FD design is sensitive to detecting the relationship between neuroleptic bioavailability and relapse. The first term in the expression indicates that the value of the index will increase with increases in the correlation between bioavailability and stability. It will decrease with increases in the correlation between drug dose and bioavailability. If we assume that stability correlates highly with quantitative clinical outcome indicators such as social functioning and psychopathology, we can use the correlation between bioavailability and stability under the IFD assumptions, p B s . D , as an approximate index of the correlation between neuroleptic bioavailability and a,quantitative scale of clinical response. By recalling that pDs = PDB pBs from equation (7), simple algebraic manipulations indicate that:
This index agrees with expression (14)and shows that the correlation between bioavailability and clinical response increases with increases in the correlation between bioavailability and stability and with decreases in the correlation between dose and bioavailability. The 1FD design generalizes to a family of kFD designs by allowing k to take on integer values greater than one. The resulting stability distribution in the general case is a mixture of k normal
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distributions,.fsId,(s)for each of k distinct dosage levels di, i = 1, . . . , k. Under the kFD design the expected value of bioavailability for non-relapsers is a weighted average of the values given by expression ( 1 2 ) for the k values of d i . The expected value for relapsers obtains similarly. By algebraic manipulation. we can show that the difference in bioavailability between non-relapsers and relapsers is
Equation (16) is equal to expression (14)when di = d j for all i and j , that is, when there is only one fixed dose. By inspection of the numerator of expression (16), we see that the expression in brackets and hence the entire numerator is always positive. The numerator will increase as the absolute value of di - d j increases because FSld( T ) decreases monotonically with increases in d. Therefore, in choosing a kFD design the experimenter should maximize the between group dose differences to maximize the expected difference in neuroleptic bioavailability between relapsed and stable patients. An examination of the expected correlation between bioavailability and stability also supports the experimental utility of increasing the differences between the doses chosen for a kFD design. We derive this correlation, P B s . k , by using the definition of covariance and the definitions of S and ps in expressions (4) and (5). OBS.k
=
EIBS1
=
2 flBS Ol3.k
- PEPS =
ECB(flBSB
+ PMSM)I
- pB(flBSPB
+ flMSPM) (17)
where o ; . is ~ the variance of the mixture of the k bioavailability distributions that arise from the k fixed doses. From the definition of correlation, it follows that
The expressions for o ~and. ~ obtain readily from equations given by Johnson and K ~ t z . ' ~ The resulting expression for P B s . k is:
Inspection of equation (19) reveals that p B s equals pBs when d , = d, for all i and j , that is, when there is only one fixed dose. As the mean of squared dose differences, ( d i - d,)2, increases, pBs approaches one. Thus, an increase in the between group dose difference should increase the observed correlation between bioavailability and measures of stability. An examination of expressions (16) and (19) suggests than an increase in the number of dose groups examined will not necessarily improve a design's ability to detect bioavailability/clinical response relationships. The theoretical limit of the kFD design is the aFD design where we would independently assign patients to doses randomly selected from the infinity of possible doses in the entire dose distribution. The derivations for the mFS design are identical to those for the 1FD design given in expressions ( 1 0 ) to (14) with the exception that we use marginal distributions instead of distributions conditional on a fixed dose. The expected difference in measured neuroleptic bioavailability between non-relapsers and relapsers under cc FD assumptions is:
MATHEMATICAL MODELS OF COMPLEX DOSE-RESPONSE RELATIONSHIPS
K
69 1
-M
Figure 2. Path model of the relationships among dose ( D l ) , bioavailability (El), stability ( S , ) , pharmacokinetic factors ( K ) and moderating variables ( M ) when doses are clinically determined. Signs ( + or - ) indicate the direction of causal association
The expected correlation between bioavailability and measures of clinical response is pBswhich is, by definition, greater than pBs,Das indicated by equation (15).
Clinical dose designs In a clinical design, we place patients on a clinically determined dose that remains unchanged throughout the trial except for some initial manipulation (for example, cutting everybody's clinical dose by 50 per cent). In the least complicated case there is no manipulation. We call this the CD design. To derive expected differences between relapsers and non-relapsers for the CD design, we must operationalize mathematically the concept of clinical dosage. We assume that the clinical dose, which results from a psychiatrist's experience with a patient over a sufficient period of time, is a function of two factors, pharmacokinetic ( K ) ,and moderating ( M ) , such that:
This modification of the original model leads to the path diagram presented in Figure 2. Some examples illustrate the influence of K and M on D,.Patients who have pharmacokinetic factors that lead to lower neuroleptic bioavailability for a given dose will likely receive higher clinical doses, that is, there is a correlation between K and D,. We assume that high values of K correspond to pharmacokinetic situations that result in high bioavailability. Therefore, if clinical dosing is accurate, we should find a negative correlation between D, and K . Patients who have moderating factors, M , supportive of a stable clinical course (for example, less severe illness, family low on expressed emotion) will require less medication than those who have less stabilizing background factors. We assume that high values of M correspond to high levels of stabilizing moderating factors. Therefore, with accurate clinical dosing we should find a negative correlation between D, and M . We define bioavailability and stability as before (equations (1) and (4)). The variances of dose, bioavailability and stability are:
We derive the correlation between bioavailability and stability for the CD design, pel sl, by applying the principles of path analysis for the decomposition of correlation^:^^
Since the second term on the right side of equation (23) is always negative, pBlslwill tend to be
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Figure 3. Path model of the relationships among dose, bioavailability, stability. pharrnacokinetic factors, and moderating variables before (subscript 1 ) and after (subscript 2) reduction of a clinically determined dose. Signs ( + or - ) indicate the direction of causal association
less than pBs in equation ( 6 ) . This may not always be true because the ratio cB/asin (6) is not in (23). necessarily equal to ag,/as, The derivation of the expected bioavailability difference between non-relapsers and relapsers under CD assumptions parallels the development for the 1FD design with the exception that the distributions of bioavailability and stability have the above modified variances and correlation. The result is:
We can generalize the CD design to a family of dose reduction designs by manipulating the clinical dose in a structured manner. We create a kCD design by multiplying all clinically determined doses by the fraction k . For example, the 0.5CD design reduces all doses by 50 per cent and the O.1CD design reduces all doses by 90 per cent. The path diagram in Figure 3 describes the kCD design. A Figure 3 indicates:
D,
=
kD,
B2 = b D B D 2 s2 = b B S B 2
+ bKBK + bMSM.
(25)
It follows that the variances of dose, bioavailability and stability under dose modification kCD are: 0 i Z=
k2ai,
ail = b i B k 2 a i ~ + b i B a : + 2 b D B b K B k p D K a D , o K = bkaiz + b,$Sah + 2 b B S b M S P B 2 M u M a B 2 .
(26)
We obtain the correlation between bioavailability and stability under the kCD design, pB2s2,from use of the principles of path analysis. The result is:
MATHEMATICAL MODELS OF COMPLEX DOSE-RESPONSE RELATIONSHIPS
693
A comparison of equations (23) and (27) indicates that the expected correlations between bioavailability and stability will differ among LCD designs due to corresponding differences in the variances of bioavailability and stability and in the value of the dose reduction fraction, k. The expected bioavailability difference between non-relapsers and relapsers will have the form of expression (24) with p B r S Iand with all variances replaced with the values for the LCD design.
The effects of heterogeneity The bioavailability differences and correlations derived so far assume that all patients in an experimental sample are drug responders. A heterogeneous patient sample will likely require one or two model modifications. The first, always required by definition, is that the ‘true’ correlation between bioavailability and stability, pBs, for non-responders is equal to zero. This corresponds to the assumption that neuroleptic bioavailability and clinical outcome are unrelated among nonresponders. The implications of the pBs = 0 assumption for fixed dose designs are straightforward. If we define 6 as the bioavailability difference expected in samples of 100 per cent responders defined by expressions (14), (16) or (20), then 67 is the expected difference when the proportion of responders is equal to t. Similarly, if p is the expected correlation from one of the fixed dose designs, then p t is the expected correlation for a response proportion o f t . From these results we see that bioavailability/stability relationships become increasingly difficult to detect as the proportion of responders in the sample ( 5 ) decreases. The effect of heterogeneity on clinical dose designs is more complicated. Since the assumption that pBs = 0 means that pBs = 0, it is evident from (23) that the bioavailability/stability correlation for non-responders in a CD design is:
This will never be positive since pMD is always negative or zero and the other variables on the right side of the equation are always non-negative. In the situation where t per cent of the sample are responders, the correlation is:
Through algebraic manipulation this simplifies to: P B i S i . r = t P B i S ~ + (l - t ) P B ~ S ~ , O ’ (30) This shows greater attenuation of p in C D than in fixed dose designs for a given T, that is, CD designs are more sensitive than fixed dose designs to t. Expression (24) then gives the expected bioavailability difference between non-relapsers and relapsers with p B l s l,r replacing pB,s I . In the case of C D designs, a second model modification is appropriate. It is likely that heterogeneity will effect the correlation between moderating variables and dose, pMD. For example, the inclusion of the ‘resister’ and ‘remitter’ subtypes suggested by Gelder and K o l a k o ~ s k a ’and ~ reviewed in the introduction will increase the range of pathology in the sample. This will increase the absolute value of pMMD (that is, it will become more negative) because of the assumption that severity of pathology is a moderating variable that influences the clinician’s choice of dose. Furthermore, pMs and pMs will increase in value because responder/ non-responder status is a moderating variable that increases the variance of M associated with stability ( S ) . An examination of equation (29) reveals that these changes in the correlational structure decrease the value of p B l s l, ~ .
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S. V. FARAONE. J. C. SIMPSON A N D W. A. BROWN
The results for kCD designs are similar and obtain in an analogous fashion. Applying the above considerations to equation (27) gives PBzS2.r
=
TpBzSz
-k ( l
- T)pBzSz,O
(31)
where
Again, there is greater attenuation of p in kCD designs than in fixed dose designs for a given t.As the dose reduction fraction k decreases, the degree of attenuation decreases and approaches that expected for fixed dose designs, that is, the impact of T on p is similar for fixed dose and kCD designs when k is small. In summary, the existence of treatment response heterogeneity in an experimental sample has differential effects on fixed dose and clinical dose designs. For fixed dose designs, heterogeneity makes it more difficult to detect bioavailability-clinical response relationships by reducing the size of expected experimental effects. The size of this reduction is proportional to the per cent of non-responders in the sample. For clinical dose designs, experimental effects will lessen and can also become spuriously negative. This latter problem is most likely when treatment heterogeneity increases the correlation between moderators and stability while decreasing the correlation between moderators and dose. Our results in this regard generalize and formalize statements made by May and Van P ~ t t e n who , ~ ~ noted that the inclusion of chronically psychotic nonresponders in experimental samples will lead to spurious negative correlations between bioavailability and clinical response because such patients usually receive high doses of medication. Our mathematical formulation extends their conclusion by demonstrating that any factor that has a similar effect on the correlational structure of the variables will produce spurious negative correlations. As shown by equation (30), this problem becomes more acute as the proportion of responders T decreases. COMPUTER SIMULATIONS Although the above mathematical derivations provide some insight into the differential utility of fixed dose and clinical dose experimental designs, some of their characteristics are best seen by means of computer simulations. Such simulations allow for comparison of the designs across a wide variety of situations by varying the correlational relationships among the variables of the mathematical model.
Methods We developed a computer program using programming statements and statistical functions from the Statistical Analysis System.26 The simulations examined seven experimental designs; their abbreviations and descriptions appear in Table I. In the kFD designs, we give k fixed doses to k equal sized groups of patients. The simulations look at designs using 1 and 2 fixed doses and we compared two designs within the 2FD category. The first of these designs, 2FD(1) assumes that the two different dose groups differ by one standard deviation in their mean level of ‘clinical stability’. ‘Clinical stability’ is a hypothetical construct defined rigorously in the model. Stability indexes a patient’s likelihood to experience an episode of illness. The 2FD(3) design assumes a three standard deviation expected difference in clinical stability. We create these different 2FD designs by varying the difference between the fixed
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695
TABLE I. Experimental designs examined in computer simulations Design
Description
1 FD 2FD( I)
All patients are placed on one fixed dose of medication Two equal sized groups of patients are placed on different fixed doses. The dose for group one is assumed to result in a mean level of ‘clinical stability’ that is one standard deviation greater than that for group two Same as 2FD( 1) except the expected difference in ‘clinical stability’ is three standard deviations Each Patient’s dose is randomly chosen from the ‘infinite’ number of doses in the accepted therapeutic range All doses are clinically determined The clinically determined dose is multiplied by 0.5 The clinically determined dose is multiplied by 0 1
2FD(3) CCI F D CD 0.5CD 0.1 C D
doses for the two groups. We obtain the coFD design by randomly choosing each patient’s dose from the theoretically infinite number of possible doses in the accepted therapeutic range. The CD design is a clinically determined dose design in which all patients receive doses as prescribed by their physician. The 0.5CD and O.1CD designs modify the CD design by multiplying all doses by 0.5 and 0.1, respectively. We refer to the 0.5 CD and 0.1 CD designs as dose reduction designs because they involve the reduction of clinically determined doses. We use mathematical derivations to compute two indices that reflect the effectiveness of each design for uncovering drug bioavailability-clinical response relationships: the correlation between the bioavailability measure and clinical stability and the expected difference in bioavailability between non-relapsers and relapsers. As shown in the preceding section, the observed value of these indices under a given design depends on the correlational relationships among the variables of the model. Thus, different experimental designs can lead to different results and thereby different conclusions. The indices used in the simulations are given by equations (14) and (15) (for IFD). 16 and 19 (for kFD), 20 (for ooFD), 23 and 24 (for CD) and 27 (for kCD). Inspection of the equations used to derive the indices indicates that, for each design, the observed values of the indices depend on two ~;, , 0% sets of parameters: (1) the correlations: pBs,p D B , and p M D ;and (2) the variances: C J ~C , J o and u i . All other parameters can be computed from these parameters using the equations presented above, the definitions of the parameters (for example, flDB = ~,,cJ,/cJ,) and the constraints of the model (for example, p b = 1 - pis). However, to compute the expected difference in bioavailability between relapsers and non-relapsers, it is also necessary to provide the relapse threshold, T(for example, equation (14)). Since the results do not differ substantively for different values of T, we present our results for the case where T is equal to the mean of the stability distribution. In the simulations we computed the indices for each design under different values of the correlations. The variances, which reflect the scale of measurement, are set at one. We evaluated each fixed dose experimental design under the 9 conditions that correspond to all combinations of the values 0.32, 0.71 and 0.95 for pss and P D B . By definition, p M Dis equal to zero for all fixed dose designs. We evaluated each clinical-dose experimental design under the 27 conditions that correspond to all combinations of these values for pBs, P D B , and p M D . We chose the values 0.32, 0.71 and 0.95 because they correspond to correlated variables that share 10,50, and 90 per cent of their variances, respectively.
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S. V. FARAONE, J. C. SIMPSON AND W. A. BROWN
p = .32
Bs
i
1FD 0
.2
.4 .6 .8 Expected Correlation: rBs
1.0
.32
Figure 4. Expected correlations between bioavailability and stability (rBs) for fixed dose designs. Results from four fixed dose designs are given for three levels of the true correlation between bioavailability and stability (pBs)and three levels of the true correlation between dose and bioavailability (pee). The designs are defined in Table I
Results The results appear in Figures 4 and 5 for fixed dose designs and Figures 6 and 7 for clinical dose designs. For low values of the bioavailability-stability correlation (pes = 0.32) virtually none of the experimental designs can detect a bioavailability-stability relationship. The one exception is the 2FD(3) design where we would expect to find bioavailability-stability correlations of approximately 0 8 for most values of the dose-bioavailability correlation. The difference in bioavailability between non-relapsers and rdapsers, however, is only reasonably large for the case in which p D B= 0.32. In this case the expected difference is 0 8 standard deviations. Consistent with the mathematical derivations, the 2FD designs can result in bioavailability-stability correlations that are greater than the ‘true’ value. This occurs because these designs result in bioavailability and stability values that are either high or low. This has the effect of increasing the sum of cross products and results in a larger covariance and a larger correlation. We can see this effect graphically by drawing a scatter plot with a moderate correlation, truncating the two variables to high and low values and drawing an ellipse around the two clouds of points that remain. Several patterns seen in Figures 4 to 7 appear for all values of the ‘true’ bioavailability-stability correlation. We always expect the coFD design to result in accurate estimates of the ‘true’
MATHEMATICAL MODELS OF COMPLEX DOSE-RESPONSE RELATIONSHIPS
697
i
i
0
1 2 Expected Bioavailability Difference
3
Figure 5. Expected bioavailability differences between non-relapsers and relapsers for fixed dose designs. Results from four fixed dose designs are given for three levels of the true correlation between bioavailability and stability (pes)and three levels of the true correlation between dose and bioavailability (pee). The designs are defined in Table I
bioavailability-stability correlation. The clinical dose design with no subsequent reduction (CD) always results in estimates of the bioavailability-stability correlation lower than those estimated by fixed dose designs. In fact, the correlations and bioavailability differences for CD designs are often spuriously negative and large (Figure 6). For example, when the ‘true’ bioavailability-stability correlation equals 071 and the ‘true’ dose-bioavailability correlation equals 071, the expected bioavailability-stability correlation for the CD design ranges from 0.03 to - 095. The dose reduction designs (O.1CD and 05CD) can also result in negative estimates of the bioavailability-stability relationship. The results of these two designs, however, are usually more accurate than those of the CD design. In fact, an increase in the amount of reduction usually increases the design’s ability to detect bioavailability-stability relationships. Although dose reduction designs compared with fixed dose designs have lesser ability to detect such relationships, they appear suitable when the ‘true’ bioavailability-stability correlation is large or moderate and the ‘true’ dose-bioavailability correlation is low. Another consistent trend in the results is that 1FD and 2FD designs become less effective with increases in the ‘true’ correlation between dose and bioavailability (Figures 4 and 5). These designs appear virtually useless for detection of differences between non-relapsers and relapsers when the dose-bioavailability correlation is 0.95. Under such conditions, we can probably detect
698
c -p
S. V. FARAONE. J. C. SIMPSON AND W. A. BROWN
1
1=
.1CD
BS
p = .32 m
p = .95
p = .71
I
I
p = .95
p = .95
.95
!
!
Ee3!
.5CD
t
CD
z
.lCD
t
p = .71
p = .71 BS
g .5cD 8
MATHEMATICAL MODELS OF COMPLEX DOSE-RESPONSE RELATIONSHIPS: IMPLICATIONS FOR EXPERIMENTAL DESIGN I N PSYCHOPHARMACOLOGIC RESEARCH STEPHEN V. FARAONE AND JOHN C . SIMPSON Section of Psychiatric EpidPmiology and Genetics. Harvard Medical School Department of Psychiatry, Massachusetts Mental Health Center, and Psychiatry Service. Brockton- West Roxburv Veterans Affairs Medical Center, Brockton. Massachusetts 0240I , U .S .A .
AND WALTER A. BROWN Department of Psychiatry and Human Behavior, Brown Universit-yMedical School, and Providence Veterans Affairs Medical Center, Brockton, Massachusetts 02401, U.S.A.
SUMMARY We develop a mathematical model to account for the complex relationship between drug dose and clinical response in psychopharmacologic research. The model specifies relationships among drug dose, drug bioavailability, pharmacokinetic factors, course moderators, clinical response and the heterogeneity of the disorder, and allows for the derivation of results that have implications for experimental design in psychopharmacologic research. These results form the basis for computer simulations which indicate that random assignment to two fixed doses is more powerful and less sensitive to heterogeneity than assignment to clinically determined doses. Fixed dose designs, however, tend to overestimate the magnitude of drug bioavailability4nical response relationships. Clinically determined dose designs are useful in some experimental situations; their effectiveness is enhanced by systematically reducing the clinically determined dose. Larger dose reductions improve the ability to detect bioavailability-clinical response relationships.
INTRODUCTION The interpretation of pharmacologic treatment outcome studies is often obscured by the presence of a complex dose-response relationship, that is, clinical response is not an increasing monotonic function of drug dose within the accepted therapeutic range. The complexity of dose-response relationships in psychopharmacology results from primarily three categories of influence: pharmacokinetic factors; course moderators, and the heterogeneity of the disorder. This article presents a mathematical model that rigorously specifies relationships among these variables and derives results that have implications for experimental design in psychopharmacologic research. We present the model in the context of schizophrenia research because its development was motivated by issues that arose in our empirical work in that area.'-' Nevertheless, the model and its implications readily generalize to other disorders. The first sections of the paper describe the rationale and formal structure of the model. We then use the model to characterize families of
027747 15/92/050685-18$09.00 0 1992 by John Wiley & Sons, Ltd.
Received May 1990 Revised July 1991
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experimental designs useful in exploration of complex dose-response relationships. Next, we employ the model to predict differential results from comparable studies that employ different experimental designs. These predictions form the basis for computer simulations that compare the ability of these experimental designs to detect and describe complex dose-response relationships, thus allowing us to make judgments about the reasonableness of the model and its potential for guiding empirical research in this area. THE ROLE O F MEDIATING FACTORS Pharmacokinetic factors refer to individual differences in drug absorption, metabolism and excretion that result in variability of peripheral neuroleptic bioavailability among patients on the same dose. Owing to these factors, blood serum levels of neuroleptic drugs will vary 10- to 20-fold among patients on the same dose.’ It is also possible that individual differences in pharmacokinetic factors lead to variability in neuroleptic bioavailability in the central nervous system among patients who have identical serum neuroleptic levels. This is consistent with findings that serum neuroleptic concentrations are not highly or consistently predictive of either the initial antipsychotic response9- l 2 or clinical status during outpatient treatment.’,397913314 Gelder and Kolakowska” suggest two pharmacokinetic factors that may mediate the relationship between peripheral and central measures of bioavailability. Since circulating metabolites of neuroleptic drugs vary in their therapeutic potency, individual differences in metabolism of the neuroleptic parent compound that lead to greater or lesser proportions of active metabolites will decrease the relationship between peripheral measures and central antipsychotic activity. Furthermore, neuroleptic assays measure both protein bound and free fractions of the drug. Since the protein bound fraction does not contribute to antipsychotic activity and may vary among patients with the same neuroleptic serum levels, the correlation between neuroleptic serum levels and antipsychotic activity will be less than perfect even if one could control all experimental error. Course moderators refer to fluctuating environmental variables that mediate the course of clinical status. For example, we know that stress in the form of life change events or family pressure exacerbates symptomatology and precipitates relapse. Life change events modify the environment and require the patient to mobilize coping resources to adapt to the modifications. Reviews of the literat~re’~.’’’ indicate a moderate association between life events and subsequent schizophrenic relapse. This association exists even when one considers only those life events likely to be independent of psychopathology. Research on expressed emotion (EE) has demonstrated that patients who live with highly critical and/or emotionally overinvolved relatives compared with those who do not have a greater risk of The final set of factors pertains to the heterogeneity of schizophrenia. Clinically, schizophrenia manifests itself in a wide variety of symptoms, signs and outcomes. Although these attributes do not allow for categorization of the disorder into neat, discrete categories, research has shown that subdivisions such as paranoid/non-paranoid, good premorbid/poor premorbid, brain damaged/ non-brain damaged and familial/non-familial have some predictive validity. Whether or not this clinical heterogeneity corresponds to etiologic or pathophysiologic heterogeneity is unknown. However, the genetic literature is consistent with the hypothesis that schizophrenia has several genetic and non-genetic etiologies.2 THE MATHEMATICAL MODEL The mathematical model provides a formal description of these complex dose-response relationships with specific reference to the phenomenon of relapse in a population of schizophrenic
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M
K
Figure 1. Path model of the relationships among dose (D), bioavailability (B),stability (S), pharmacokinetic factors (K) and moderating variables (M) when doses are randomly assigned to patients. Signs ( or - ) indicate the direction of causal association
+
outpatients who are maintained on neuroleptic medication. The model thus forms the foundation of a formal description of experimental designs used in dose-response research and enables the derivation of characteristics of these designs that affect their usefulness for testing hypotheses about complex dose-response relationships. The model assumes the existence of a hypothetical random variable called ‘stability’. High stability values are associated with protection from relapse episodes such that patients relapse if and only if their stability value falls below the relapse threshold, We define I;. as the point on the stability continuum below which patients will relapse within i years. We assume Ti is a nondecreasing function of i such that T j P Tk for all j > k. Although we have chosen relapse among outpatients as a substantive framework for the mathematical model, the derivations and results apply equally to situations that examine the relationship between treatment and remission of symptoms among acutely disordered inpatients. In the latter case, Tcorresponds to that point on the stability continuum that a patient must surpass for remission of symptoms. Figure 1 presents the relationships among stability ( S ) , neuroleptic dose (D), neuroleptic bioavailability (B),pharmacokinetic factors ( K )and course moderators ( M ) . Our model assumes that drug dose and pharmacokinetic factors directly influence bioavailability. In turn, the bioavailability measure directly influences stability which course moderators also influence directly. In our model the distribution of drug dose for any given experimental population is normal with a mean ( p D )and variance (0;)equal to those observed in the population of patients maintained on clinical doses. However, we assume that the investigator determines doses randomly, not clinically. Although our previous research21 suggests that drug doses are not normally distributed, we can normalize them with an appropriate transformation. We assume that the pharmacokinetic factors that influence the value of B form, in some unknown combination, a variable K that is normally distributed with mean pK and variance 0;. We assume that the variables K and D are stochastically independent (that is, drug dose is not influenced by pharmacokinetic factors when doses are determined by experimental manipulation). We assume that neuroleptic bioavailability, B, is a linear function of drug dose and pharmacokinetic factors:
r.
B = BDB D
+ BKB K .
(1)
Since B is the sum of two independent normal variables, it is normally distributed with mean p B and variance IS;where:
The correlation between dose and bioavailability, p D B , is the standardized partial regression
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coefficient (or path coefficient, P D B ) because there are no indirect paths from D to B (Figure 1). Thus:
We assume stability is a linear function of neuroleptic bioavailability, B, and course moderators, M, such that:
Since M and B are stochastically independent, normally distributed variables, S is normally distributed with mean and variance:
The correlation between bioavailability and stability, pBs, is the corresponding path coefficient:
The model assumes that dose influences stability only through its influence on bioavailability, hence:
Since the model assumes that P D B and pBs are positive but less than one, pDs is always less than pBs. This is consistent with reports that, compared with drug dose, bioavailability is more predictive of outcome. FAMILIES OF EXPERIMENTAL DESIGNS
A family of experimental designs is a group of designs that share a common mathematical formulation but differ in the values chosen for design parameters. We discuss two design families: fixed dose designs and clinically determined dose designs. We describe these design families in terms of the model presented above, which we then use to derive indices that reflect the utility of the designs for detecting bioavailability4inical response relationships.
Fixed Dose (FD) designs We define the kFD design as one in which we assign patients randomly to one of k drug doses that remain unchanged throughout the trial. The 1FD design assigns all patients to the same fixed dose, d. Under lFD, the distribution of stability in the population is the conditional distribution of S given D = d, represented by fsId(s). We assume that the joint distribution of D, B and S is multivariate normal. It follows that S I d is normally distributed with mean psld and variance (Ti,,,. From normal distribution theory:
+ BDSd.
(8) We can see that BDS equals B B s B D B by substituting the right side of equation (1)for B in equation (4). Also, PSld
= PS
- BDSPD
MATHEMATICAL MODELS OF COMPLEX DOSE-RESPONSE RELATIONSHIPS
689
Then, with the knowledge that D = d , we can derive the expected value of the bioavailability measure for patients with stability greater than the threshold (that is, non-relapsers or patients with S > T). From the theory of linear regression for variables that have a bivariate normal distribution:22
E ( B 1 S > T, D = d ) = B B s . D E ( S I S > T, D
=d)
(10) where f i B s . D = ( p B s . D a B I d )and / ~ splBds . Dis the partial correlation between B and S with D held constant. From Johnson and K o ~ z , ’E~( S I S > T, D = d ) is the expected value of the normal distribution of S I d truncated from below at T:
Therefore, the expected value of bioavailability for non-relapsers is:
where F s I d ( Tis) the integral offsId(t)from t = - 00 to T. We use a parallel argument to derive the expected value of bioavailability for patients with S < T (that is, relapsers). The result is:
The difference between equations (12) and (1 3) is the expected difference in measured neuroleptic bioavailability between non-relapsers and relapsers under the assumptions of the model. After substituting the formula for the partial correlation coefficient for p B S . D and noting that u i l d= (1 - p i D ) & the result is:
Expression (14) indexes the degree to which the 1FD design is sensitive to detecting the relationship between neuroleptic bioavailability and relapse. The first term in the expression indicates that the value of the index will increase with increases in the correlation between bioavailability and stability. It will decrease with increases in the correlation between drug dose and bioavailability. If we assume that stability correlates highly with quantitative clinical outcome indicators such as social functioning and psychopathology, we can use the correlation between bioavailability and stability under the IFD assumptions, p B s . D , as an approximate index of the correlation between neuroleptic bioavailability and a,quantitative scale of clinical response. By recalling that pDs = PDB pBs from equation (7), simple algebraic manipulations indicate that:
This index agrees with expression (14)and shows that the correlation between bioavailability and clinical response increases with increases in the correlation between bioavailability and stability and with decreases in the correlation between dose and bioavailability. The 1FD design generalizes to a family of kFD designs by allowing k to take on integer values greater than one. The resulting stability distribution in the general case is a mixture of k normal
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distributions,.fsId,(s)for each of k distinct dosage levels di, i = 1, . . . , k. Under the kFD design the expected value of bioavailability for non-relapsers is a weighted average of the values given by expression ( 1 2 ) for the k values of d i . The expected value for relapsers obtains similarly. By algebraic manipulation. we can show that the difference in bioavailability between non-relapsers and relapsers is
Equation (16) is equal to expression (14)when di = d j for all i and j , that is, when there is only one fixed dose. By inspection of the numerator of expression (16), we see that the expression in brackets and hence the entire numerator is always positive. The numerator will increase as the absolute value of di - d j increases because FSld( T ) decreases monotonically with increases in d. Therefore, in choosing a kFD design the experimenter should maximize the between group dose differences to maximize the expected difference in neuroleptic bioavailability between relapsed and stable patients. An examination of the expected correlation between bioavailability and stability also supports the experimental utility of increasing the differences between the doses chosen for a kFD design. We derive this correlation, P B s . k , by using the definition of covariance and the definitions of S and ps in expressions (4) and (5). OBS.k
=
EIBS1
=
2 flBS Ol3.k
- PEPS =
ECB(flBSB
+ PMSM)I
- pB(flBSPB
+ flMSPM) (17)
where o ; . is ~ the variance of the mixture of the k bioavailability distributions that arise from the k fixed doses. From the definition of correlation, it follows that
The expressions for o ~and. ~ obtain readily from equations given by Johnson and K ~ t z . ' ~ The resulting expression for P B s . k is:
Inspection of equation (19) reveals that p B s equals pBs when d , = d, for all i and j , that is, when there is only one fixed dose. As the mean of squared dose differences, ( d i - d,)2, increases, pBs approaches one. Thus, an increase in the between group dose difference should increase the observed correlation between bioavailability and measures of stability. An examination of expressions (16) and (19) suggests than an increase in the number of dose groups examined will not necessarily improve a design's ability to detect bioavailability/clinical response relationships. The theoretical limit of the kFD design is the aFD design where we would independently assign patients to doses randomly selected from the infinity of possible doses in the entire dose distribution. The derivations for the mFS design are identical to those for the 1FD design given in expressions ( 1 0 ) to (14) with the exception that we use marginal distributions instead of distributions conditional on a fixed dose. The expected difference in measured neuroleptic bioavailability between non-relapsers and relapsers under cc FD assumptions is:
MATHEMATICAL MODELS OF COMPLEX DOSE-RESPONSE RELATIONSHIPS
K
69 1
-M
Figure 2. Path model of the relationships among dose ( D l ) , bioavailability (El), stability ( S , ) , pharmacokinetic factors ( K ) and moderating variables ( M ) when doses are clinically determined. Signs ( + or - ) indicate the direction of causal association
The expected correlation between bioavailability and measures of clinical response is pBswhich is, by definition, greater than pBs,Das indicated by equation (15).
Clinical dose designs In a clinical design, we place patients on a clinically determined dose that remains unchanged throughout the trial except for some initial manipulation (for example, cutting everybody's clinical dose by 50 per cent). In the least complicated case there is no manipulation. We call this the CD design. To derive expected differences between relapsers and non-relapsers for the CD design, we must operationalize mathematically the concept of clinical dosage. We assume that the clinical dose, which results from a psychiatrist's experience with a patient over a sufficient period of time, is a function of two factors, pharmacokinetic ( K ) ,and moderating ( M ) , such that:
This modification of the original model leads to the path diagram presented in Figure 2. Some examples illustrate the influence of K and M on D,.Patients who have pharmacokinetic factors that lead to lower neuroleptic bioavailability for a given dose will likely receive higher clinical doses, that is, there is a correlation between K and D,. We assume that high values of K correspond to pharmacokinetic situations that result in high bioavailability. Therefore, if clinical dosing is accurate, we should find a negative correlation between D, and K . Patients who have moderating factors, M , supportive of a stable clinical course (for example, less severe illness, family low on expressed emotion) will require less medication than those who have less stabilizing background factors. We assume that high values of M correspond to high levels of stabilizing moderating factors. Therefore, with accurate clinical dosing we should find a negative correlation between D, and M . We define bioavailability and stability as before (equations (1) and (4)). The variances of dose, bioavailability and stability are:
We derive the correlation between bioavailability and stability for the CD design, pel sl, by applying the principles of path analysis for the decomposition of correlation^:^^
Since the second term on the right side of equation (23) is always negative, pBlslwill tend to be
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Figure 3. Path model of the relationships among dose, bioavailability, stability. pharrnacokinetic factors, and moderating variables before (subscript 1 ) and after (subscript 2) reduction of a clinically determined dose. Signs ( + or - ) indicate the direction of causal association
less than pBs in equation ( 6 ) . This may not always be true because the ratio cB/asin (6) is not in (23). necessarily equal to ag,/as, The derivation of the expected bioavailability difference between non-relapsers and relapsers under CD assumptions parallels the development for the 1FD design with the exception that the distributions of bioavailability and stability have the above modified variances and correlation. The result is:
We can generalize the CD design to a family of dose reduction designs by manipulating the clinical dose in a structured manner. We create a kCD design by multiplying all clinically determined doses by the fraction k . For example, the 0.5CD design reduces all doses by 50 per cent and the O.1CD design reduces all doses by 90 per cent. The path diagram in Figure 3 describes the kCD design. A Figure 3 indicates:
D,
=
kD,
B2 = b D B D 2 s2 = b B S B 2
+ bKBK + bMSM.
(25)
It follows that the variances of dose, bioavailability and stability under dose modification kCD are: 0 i Z=
k2ai,
ail = b i B k 2 a i ~ + b i B a : + 2 b D B b K B k p D K a D , o K = bkaiz + b,$Sah + 2 b B S b M S P B 2 M u M a B 2 .
(26)
We obtain the correlation between bioavailability and stability under the kCD design, pB2s2,from use of the principles of path analysis. The result is:
MATHEMATICAL MODELS OF COMPLEX DOSE-RESPONSE RELATIONSHIPS
693
A comparison of equations (23) and (27) indicates that the expected correlations between bioavailability and stability will differ among LCD designs due to corresponding differences in the variances of bioavailability and stability and in the value of the dose reduction fraction, k. The expected bioavailability difference between non-relapsers and relapsers will have the form of expression (24) with p B r S Iand with all variances replaced with the values for the LCD design.
The effects of heterogeneity The bioavailability differences and correlations derived so far assume that all patients in an experimental sample are drug responders. A heterogeneous patient sample will likely require one or two model modifications. The first, always required by definition, is that the ‘true’ correlation between bioavailability and stability, pBs, for non-responders is equal to zero. This corresponds to the assumption that neuroleptic bioavailability and clinical outcome are unrelated among nonresponders. The implications of the pBs = 0 assumption for fixed dose designs are straightforward. If we define 6 as the bioavailability difference expected in samples of 100 per cent responders defined by expressions (14), (16) or (20), then 67 is the expected difference when the proportion of responders is equal to t. Similarly, if p is the expected correlation from one of the fixed dose designs, then p t is the expected correlation for a response proportion o f t . From these results we see that bioavailability/stability relationships become increasingly difficult to detect as the proportion of responders in the sample ( 5 ) decreases. The effect of heterogeneity on clinical dose designs is more complicated. Since the assumption that pBs = 0 means that pBs = 0, it is evident from (23) that the bioavailability/stability correlation for non-responders in a CD design is:
This will never be positive since pMD is always negative or zero and the other variables on the right side of the equation are always non-negative. In the situation where t per cent of the sample are responders, the correlation is:
Through algebraic manipulation this simplifies to: P B i S i . r = t P B i S ~ + (l - t ) P B ~ S ~ , O ’ (30) This shows greater attenuation of p in C D than in fixed dose designs for a given T, that is, CD designs are more sensitive than fixed dose designs to t. Expression (24) then gives the expected bioavailability difference between non-relapsers and relapsers with p B l s l,r replacing pB,s I . In the case of C D designs, a second model modification is appropriate. It is likely that heterogeneity will effect the correlation between moderating variables and dose, pMD. For example, the inclusion of the ‘resister’ and ‘remitter’ subtypes suggested by Gelder and K o l a k o ~ s k a ’and ~ reviewed in the introduction will increase the range of pathology in the sample. This will increase the absolute value of pMMD (that is, it will become more negative) because of the assumption that severity of pathology is a moderating variable that influences the clinician’s choice of dose. Furthermore, pMs and pMs will increase in value because responder/ non-responder status is a moderating variable that increases the variance of M associated with stability ( S ) . An examination of equation (29) reveals that these changes in the correlational structure decrease the value of p B l s l, ~ .
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The results for kCD designs are similar and obtain in an analogous fashion. Applying the above considerations to equation (27) gives PBzS2.r
=
TpBzSz
-k ( l
- T)pBzSz,O
(31)
where
Again, there is greater attenuation of p in kCD designs than in fixed dose designs for a given t.As the dose reduction fraction k decreases, the degree of attenuation decreases and approaches that expected for fixed dose designs, that is, the impact of T on p is similar for fixed dose and kCD designs when k is small. In summary, the existence of treatment response heterogeneity in an experimental sample has differential effects on fixed dose and clinical dose designs. For fixed dose designs, heterogeneity makes it more difficult to detect bioavailability-clinical response relationships by reducing the size of expected experimental effects. The size of this reduction is proportional to the per cent of non-responders in the sample. For clinical dose designs, experimental effects will lessen and can also become spuriously negative. This latter problem is most likely when treatment heterogeneity increases the correlation between moderators and stability while decreasing the correlation between moderators and dose. Our results in this regard generalize and formalize statements made by May and Van P ~ t t e n who , ~ ~ noted that the inclusion of chronically psychotic nonresponders in experimental samples will lead to spurious negative correlations between bioavailability and clinical response because such patients usually receive high doses of medication. Our mathematical formulation extends their conclusion by demonstrating that any factor that has a similar effect on the correlational structure of the variables will produce spurious negative correlations. As shown by equation (30), this problem becomes more acute as the proportion of responders T decreases. COMPUTER SIMULATIONS Although the above mathematical derivations provide some insight into the differential utility of fixed dose and clinical dose experimental designs, some of their characteristics are best seen by means of computer simulations. Such simulations allow for comparison of the designs across a wide variety of situations by varying the correlational relationships among the variables of the mathematical model.
Methods We developed a computer program using programming statements and statistical functions from the Statistical Analysis System.26 The simulations examined seven experimental designs; their abbreviations and descriptions appear in Table I. In the kFD designs, we give k fixed doses to k equal sized groups of patients. The simulations look at designs using 1 and 2 fixed doses and we compared two designs within the 2FD category. The first of these designs, 2FD(1) assumes that the two different dose groups differ by one standard deviation in their mean level of ‘clinical stability’. ‘Clinical stability’ is a hypothetical construct defined rigorously in the model. Stability indexes a patient’s likelihood to experience an episode of illness. The 2FD(3) design assumes a three standard deviation expected difference in clinical stability. We create these different 2FD designs by varying the difference between the fixed
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TABLE I. Experimental designs examined in computer simulations Design
Description
1 FD 2FD( I)
All patients are placed on one fixed dose of medication Two equal sized groups of patients are placed on different fixed doses. The dose for group one is assumed to result in a mean level of ‘clinical stability’ that is one standard deviation greater than that for group two Same as 2FD( 1) except the expected difference in ‘clinical stability’ is three standard deviations Each Patient’s dose is randomly chosen from the ‘infinite’ number of doses in the accepted therapeutic range All doses are clinically determined The clinically determined dose is multiplied by 0.5 The clinically determined dose is multiplied by 0 1
2FD(3) CCI F D CD 0.5CD 0.1 C D
doses for the two groups. We obtain the coFD design by randomly choosing each patient’s dose from the theoretically infinite number of possible doses in the accepted therapeutic range. The CD design is a clinically determined dose design in which all patients receive doses as prescribed by their physician. The 0.5CD and O.1CD designs modify the CD design by multiplying all doses by 0.5 and 0.1, respectively. We refer to the 0.5 CD and 0.1 CD designs as dose reduction designs because they involve the reduction of clinically determined doses. We use mathematical derivations to compute two indices that reflect the effectiveness of each design for uncovering drug bioavailability-clinical response relationships: the correlation between the bioavailability measure and clinical stability and the expected difference in bioavailability between non-relapsers and relapsers. As shown in the preceding section, the observed value of these indices under a given design depends on the correlational relationships among the variables of the model. Thus, different experimental designs can lead to different results and thereby different conclusions. The indices used in the simulations are given by equations (14) and (15) (for IFD). 16 and 19 (for kFD), 20 (for ooFD), 23 and 24 (for CD) and 27 (for kCD). Inspection of the equations used to derive the indices indicates that, for each design, the observed values of the indices depend on two ~;, , 0% sets of parameters: (1) the correlations: pBs,p D B , and p M D ;and (2) the variances: C J ~C , J o and u i . All other parameters can be computed from these parameters using the equations presented above, the definitions of the parameters (for example, flDB = ~,,cJ,/cJ,) and the constraints of the model (for example, p b = 1 - pis). However, to compute the expected difference in bioavailability between relapsers and non-relapsers, it is also necessary to provide the relapse threshold, T(for example, equation (14)). Since the results do not differ substantively for different values of T, we present our results for the case where T is equal to the mean of the stability distribution. In the simulations we computed the indices for each design under different values of the correlations. The variances, which reflect the scale of measurement, are set at one. We evaluated each fixed dose experimental design under the 9 conditions that correspond to all combinations of the values 0.32, 0.71 and 0.95 for pss and P D B . By definition, p M Dis equal to zero for all fixed dose designs. We evaluated each clinical-dose experimental design under the 27 conditions that correspond to all combinations of these values for pBs, P D B , and p M D . We chose the values 0.32, 0.71 and 0.95 because they correspond to correlated variables that share 10,50, and 90 per cent of their variances, respectively.
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p = .32
Bs
i
1FD 0
.2
.4 .6 .8 Expected Correlation: rBs
1.0
.32
Figure 4. Expected correlations between bioavailability and stability (rBs) for fixed dose designs. Results from four fixed dose designs are given for three levels of the true correlation between bioavailability and stability (pBs)and three levels of the true correlation between dose and bioavailability (pee). The designs are defined in Table I
Results The results appear in Figures 4 and 5 for fixed dose designs and Figures 6 and 7 for clinical dose designs. For low values of the bioavailability-stability correlation (pes = 0.32) virtually none of the experimental designs can detect a bioavailability-stability relationship. The one exception is the 2FD(3) design where we would expect to find bioavailability-stability correlations of approximately 0 8 for most values of the dose-bioavailability correlation. The difference in bioavailability between non-relapsers and rdapsers, however, is only reasonably large for the case in which p D B= 0.32. In this case the expected difference is 0 8 standard deviations. Consistent with the mathematical derivations, the 2FD designs can result in bioavailability-stability correlations that are greater than the ‘true’ value. This occurs because these designs result in bioavailability and stability values that are either high or low. This has the effect of increasing the sum of cross products and results in a larger covariance and a larger correlation. We can see this effect graphically by drawing a scatter plot with a moderate correlation, truncating the two variables to high and low values and drawing an ellipse around the two clouds of points that remain. Several patterns seen in Figures 4 to 7 appear for all values of the ‘true’ bioavailability-stability correlation. We always expect the coFD design to result in accurate estimates of the ‘true’
MATHEMATICAL MODELS OF COMPLEX DOSE-RESPONSE RELATIONSHIPS
697
i
i
0
1 2 Expected Bioavailability Difference
3
Figure 5. Expected bioavailability differences between non-relapsers and relapsers for fixed dose designs. Results from four fixed dose designs are given for three levels of the true correlation between bioavailability and stability (pes)and three levels of the true correlation between dose and bioavailability (pee). The designs are defined in Table I
bioavailability-stability correlation. The clinical dose design with no subsequent reduction (CD) always results in estimates of the bioavailability-stability correlation lower than those estimated by fixed dose designs. In fact, the correlations and bioavailability differences for CD designs are often spuriously negative and large (Figure 6). For example, when the ‘true’ bioavailability-stability correlation equals 071 and the ‘true’ dose-bioavailability correlation equals 071, the expected bioavailability-stability correlation for the CD design ranges from 0.03 to - 095. The dose reduction designs (O.1CD and 05CD) can also result in negative estimates of the bioavailability-stability relationship. The results of these two designs, however, are usually more accurate than those of the CD design. In fact, an increase in the amount of reduction usually increases the design’s ability to detect bioavailability-stability relationships. Although dose reduction designs compared with fixed dose designs have lesser ability to detect such relationships, they appear suitable when the ‘true’ bioavailability-stability correlation is large or moderate and the ‘true’ dose-bioavailability correlation is low. Another consistent trend in the results is that 1FD and 2FD designs become less effective with increases in the ‘true’ correlation between dose and bioavailability (Figures 4 and 5). These designs appear virtually useless for detection of differences between non-relapsers and relapsers when the dose-bioavailability correlation is 0.95. Under such conditions, we can probably detect
698
c -p
S. V. FARAONE. J. C. SIMPSON AND W. A. BROWN
1
1=
.1CD
BS
p = .32 m
p = .95
p = .71
I
I
p = .95
p = .95
.95
!
!
Ee3!
.5CD
t
CD
z
.lCD
t
p = .71
p = .71 BS
g .5cD 8
