Performing Deconvolution Operations in SAS for Input Rate Computation

Transcription

1 ABSTRACT Paper PO11 Performing Deconvolution Operations in SAS for Input Rate Computation Steven Hege, Rho, Inc., Chapel Hill, NC. Deconvolution is a non-parametric technique used in the quantitative analysis of systems in different scientific fields. For example, in the pharmaceutical industry, the technique can be used in the study of input characteristics across time and bio-equivalence of two or more dosage forms. This paper presents two macros that perform these deconvolution analyses within the SAS System. The first macro provides a framework for processing of data from multiple subjects. The second macro executes the deconvolution operation for each individual using an iterative algorithm. SAS Base was the only SAS product used in this paper and there was a minor limitation for the operating system that can be easily modified as needed. INTRODUCTION A system response over time can be described as the convolution of an impulse response, or kernel, with external inputs. The impulse response is the system reaction when stimulated by a single input pulse and is characteristic to each system. External inputs can consist of a single pulse, continuous input signal or a chain of stimuli across time. The following integral is a mathematical representation of the convolution operation R ( t ) = t A ( s ) K ( t s ) ds where R(t) is the response relative to time t, A(s) is the input rate function, K(t) is the impulse response and s is the integration variable [2]. This form of convolution assumes that the response is linear with respect to the input and the impulse response is invariant in time t. The system linearity implies that a superposition principle is present where the observed response to multiple inputs is the summation of the responses to each individual input. Time-invariance assures that the impulse response is stable with the system giving identical responses to the same inputs across all time. Deconvolution is the reverse operation and can be used to solve two complementary problems in pharmacokinetics (PK) and physiology. The first problem is a system identification where the impulse response K(t) is determined from known input function A(s) and measured responses R(t). In PK studies, the impulse response, K(t), to a compound is typically estimated by administering a bolus dose or rapid infusion to an experimental subject then measuring the compound disposition over time. The complementary and more common task is estimating the input function A(s), given the impulse response K(t) and the observed responses R(t). An example would be assessing the input rates (i.e. absorption rate) A(s) during evaluation of oral formulations. Both problems are identical where the estimates of A(s) or K(t) are found by the deconvolution of R(t) using K(t) or A(s), respectively. In this paper, two SAS macros are presented that can help perform deconvolution analyses on data from both single and multiple subjects. Assuming system linearity, the first SAS macro was developed that calculates the input function over time when given the observed response and the impulse response. The second macro presented provides a framework to systematically process a large number of input-response profiles. SIMULATED DATA Although it is possible to conceive of other cases in various fields, this paper is concentrating on the field of pharmacokinetics for illustration and testing. Several impulse response and response functions were simulated using a one-compartment open model with first-order elimination. Either IV-bolus or first-order input was applied to the impulse response function while the simulated responses had a first-order input for all case. A lag time was applied to test the effects of a time-delayed input. To eliminate the effect and need for interpolation, complete concentrationtime profiles were generated from to 24 hours with time-points incremented by one minute. The deconvolutions were performed on response data simulated according to the design given in Table 1. 1

2 ID Response (First-order Input) Impulse Response 1 No lag time IV-Bolus 2 No lag time First-order Input without lag time 3 Lag time IV-Bolus 4 Lag time First-order input with lag time Table 1. Experimental design. The equation created the IV-Bolus simulated data (IV) while d Dose C ( t) = * exp el * V d [ k ( t t )] { exp [ k * ( t t )] exp [ k ( t t )]} Dose C ( t ) = * el lag a * V generated the first-order input data with time (t) in minutes using constants Dose = (4. response, 1. impulse), and volume of distribution (V d ) = 1.. Table 2 lists the remaining parameters: absorption rate constant (k a ), elimination rate constant (k el ), and lag times for the impulse (t lag,imp ) and response (t lag,res ). ID k el (hr -1 ) k a (hr -1 ) t lag,res (min) t lag,imp (min) NA NA Table 2. Pharmacokinetics parameters used in simulated data. lag lag RESULTS Figure 1 shows the calculated input rates for each test case ID = 1 through 4. The curves for Subjects 1 and 3 show the typical input rate patterns related to oral administration of a compound. After the initial rapid input, each curve displays the exponential decline of the compound at the administration site associated with a first-order absorption. The Subject 3 curve revealed the absorption time lag used in the simulated data. The results for Subject 2 and 4 display the characteristics when the impulse and response functions have first-order inputs. The rationale for examining this model is that for many reasons IV-bolus data are not available but a comparison of two dosage forms is necessary. Both Subject 2 and 4 results indicate that the initial rate is equal to the dose and the time lag for Subject 4. The spreading seen in Subject 4 is caused by the differences on absorption time lag. 2

3 ID = 1 ID = 2 Input Rate (per min) Input Rate (per min) Time ( hr) Time ( hr) ID = 3 ID = 4 Input Rate (per min) Time (hr) Input Rate (per min) Time (hr) Figure 1. Estimated input rates (per minute) SAS MACRO I: DECONVOLUTION OF AN INDIVIDUAL PROFILE The macro presented in this section performs the individual deconvolution operations. If given the response data and impulse response function, the input rates over time are returned. The algorithm employs an iterative approach to computing input rates from the given data and was written so it is not limited by the response data array size. The arguments for this macro are: INDEPVAR: DEPVAR1: RESDS: DEPVAR2: IMPULSE1: INPT: EPS1: EPS2: Independent variable in input datasets Dependent variable in response dataset Name of response dataset Dependent variable in impulse dataset Name of impulse dataset Name of dataset for output of input rates Tolerance of input rates Tolerance of concentrations %macro decon(indepvar,depvar1,resds,depvar2,impulse1,inpt,eps1,eps2); /* The only operating system dependent statement */ 3

4 proc printto log='nul:'; proc summary data=&resds.; var &indepvar.; output out=two max=max1 N=num1; data _null_; set two; call symput('maxtime',put(max1,3.)); call symput('conccnt',put(num1,.)); proc sort data=&resds. out=wrk; by &indepvar.; proc sort data=&impulse1. out=implse1a; by &indepvar.; proc summary data=implse1a; var &depvar2.; output out=five n=n; data _null_; set five; call symput('implcnt',put(n,.)); /* Find first non-zero value in impulse function */ data _null_; set implse1a; retain found ; if (&depvar2. gt ) and not found then do; found=1; call symput('impl1amt',put(&depvar2.,best12.)); call symput('x1',put(&indepvar.,best12.)); call symput('i_diff',put(_n_,best12.)); end; data wrk1; set wrk; i=_n_; c2=&depvar1.; data implse1; set implse1a; i=_n_ ; drop &indepvar.; /* Start of deconvolution iterations */ %LET DONE=; %LET k=; %do %while((&k. LE &CONCCNT.) AND (&DONE. NE 1)); 4

5 %let k=%eval(&k.+1); proc sort data=wrk1; by i; proc sort data=implse1; by i; data wrk1; merge wrk1(in=in1) implse1(in=in2); by i; IF IN2 AND NOT IN1 THEN DELETE; proc sort data=wrk1; by &indepvar.; DATA _null_; SET WRK1; retain FOUNDIT ; IF (FOUNDIT NE 1) and ((c2 gt ) or (&depvar2. gt )) THEN DO; if (&impl1amt. not in (.,)) then pulse=round((c2/&impl1amt.),&eps1.); else pulse=; call symput('pulssize',put(pulse,best12.)); FOUNDIT=1; END; RUN; data wrk1 TEST2; set wrk1; retain testsum ; if i ge &k. then do; if i eq &k. then pulse=&pulssize.; c3=&pulssize.*&depvar2.; c2=sum(c2,-c3); if ABS(round(c2,&eps2.)) le &eps2. then c2=; TESTSUM=max(TESTSUM,ABS(C2)); end; OUTPUT WRK1; IF I EQ &CONCCNT. THEN OUTPUT TEST2; RUN; DATA _NULL_; SET TEST2; call symput('sumtest',put(testsum,best12.)); IF TESTSUM LE &EPS2. THEN call symput('done','1'); RUN; data wrk1; set wrk1; drop &depvar2. testsum; data implse1; set implse1; i=i+1; if i gt &conccnt. then delete;

6 %end; data &inpt; set wrk1; keep &indepvar. pulse; proc printto; %mend; SAS MACRO II: PROCESS MULTIPLE SUBJECT PROFILES This macro allows you to process the time-profile data from multiple subjects by separating the data for each subject before calling the deconvolution macro. The arguments of this macro are: IDSTR: RESPSDS: INDEPVAR: DEPVAR1: IMPLDS: DEPVAR2: INPTDS: DELTIME: DSEPS: CNCEPS: Variables identifying each subject Name of response dataset Independent variable in response and impulse datasets Dependent variable in response dataset Name of impulse dataset Dependent variable in impulse dataset Name of input rate dataset for output Step in time for interpolation Tolerance of input rates Tolerance of concentrations %macro deconby(idstr,respsds,indepvar,depvar1,implds, depvar2,inptds,deltime,dseps,cnceps); data inpt114; set &respsds.; delete; data res114; set &respsds.; data imp114; set &implds.; proc sort data=res114 nodupkeys out=dby1; proc summary data=dby1; var &idstr; output out=dby2 n=n_ids; data _null_; set dby2; call symput('idcnt1',put(n_ids,best3.)); 6

7 %do i=1 %to &idcnt1.; data dby3; set dby1; if _n_ eq &i. then output; keep &idstr.; proc sort data=dby3; proc sort data=res114; proc sort data=imp114; data res114a(rename=(time=x)); merge res114 dby3(in=in2); if in2; y=&depvar1.; keep time y; data imp114a(rename=(time=x)); merge imp114 dby3(in=in2); if in2; imp=&depvar2.; keep time imp; /* Call Deconvolution Macro */ %decon(x,y,res114a,imp,imp114a,inpt114b,&dseps.,&cnceps.); %end; data dby4; merge res114a imp114a inpt114b; by x; tmpkey=1; data dby3; set dby3; tmpkey=1; proc sort data=dby4; by tmpkey x; data dby(drop=tmpkey); merge dby4 dby3; by tmpkey; data inpt114; set inpt114 dby; &indepvar.=x; &depvar1.=y; data &inptds.; set inpt114; 7

8 %mend; A typical invocation of this macro is: %deconby(id,conc1,time,c,impulse1,imp,input8,.1,.1,.1); with the resulting input rates versus time in the input8 dataset. Using the following example, %decon(x,y_hat,res114b,imp_hat,imp114b,inpt114b,.1,.1); this macro returns the input rate versus time in the output dataset inpt114b for an individual subject. CONCLUSION This paper presents two SAS macros for using deconvolution analysis to determine input rates and/or system response; and processes profiles from multiple subjects. The deconvolution algorithm in this presentation uses an iterative technique to non-parametrically estimate the input rates under the assumptions of system linearity and timeinvariance. The resulting input rates can be used during the evaluation of dosage bioavailability or give an estimate of the overall input profile across time. It is hoped that this paper will stimulate further application and development of this types of analyses within the SAS system for pharmacological and physiological analyses. REFERENCES [1] Gibaldi, M., Perrier, D. (1982), Pharmacokinetics, New York: Marcel Dekker. [2] Verotta, D. (23), Volterra Series in Pharmacokinetics and Pharmacodynamics ; Journal of Pharmacokinetics and Pharmacodynamics, 3 (), CONTACT INFORMATION Steve Hege Rho, Inc. 633 Quadrangle Drive, Sutie Chapel Hill, North Carolina 2717 Work Phone: (919) 48-8 ext. 3 Steven_Hege@RhoWorld.com SAS and all other SAS Institute Inc. product or service names are registered trademarks or trademarks of SAS Institute Inc. in the USA and other countries. indicates USA registration. Other brand and product names are registered trademarks or trademarks of their respective companies. 8