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Clinical Investigations |
PDT Center [D. A. B., R. K. P., T. J. D.] and Departments of Cancer Prevention [W. R. G.], Medicine [G. M. L.], Surgery [H. N.], and Dermatology [A. R. O.], Roswell Park Cancer Institute, Buffalo, New York 14263 and Department of Surgery, Tokyo Medical University, Tokyo, Japan [T. T.]
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ABSTRACT |
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 Top ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONAppendixREFERENCES |
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and ß half-lives (95% confidence interval) were 7.77 h (3.46–17.6 h) and 596 h (120–2951 h), respectively. High-performance liquid chromatography analysis of serum showed no circulating HPPH metabolites, and in vitro incubation of HPPH with human liver microsomal preparations resulted in no metabolite or glucuronic acid-HPPH conjugate production. A minimal skin response to the solar simulator was observed, mostly in patients treated with the highest dose of HPPH, 6 mg/m2. All of the HPPH pharmacokinetic parameters were consistent with a highly lipophilic agent that is concentrated in plasma and is nearly 100% bound to plasma proteins; this was verified by plasma protein binding studies. Whereas low concentrations of HPPH can be detected in plasma several months after a single infusion, no instances of cutaneous photosensitivity have been noted in these patients. In general, HPPH pharmacokinetic profiles are readily predictable from the global population model. This is the first comprehensive human population pharmacokinetic/pharmacodynamic study of a clinical anticancer photodynamic therapy agent. |
INTRODUCTION |
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TopABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONAppendixREFERENCES |
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max = 630 nm; molar extinction coefficient 
630 
2000 M-1 cm-1), and light of this wavelength penetrates tissue suboptimally (3)
. In addition, use of this photosensitizer is associated with prolonged but diminishing skin photosensitivity lasting a month or more after injection (4)
. As a result, a number of other drugs with more optimal photophysical properties (e.g., longer wavelength absorption and greater molar extinction coefficients) and shorter durations of skin photosensitivity are being examined as photosensitizing agents (5)
.
One such drug is HPPH (Photochlor), a chlorin-based molecule (6)
(Fig. 1
, inset). HPPH is an extremely hydrophobic compound that was found, in a quantitative structure-activity relationship study that addressed the general property of lipophilicity (7)
, to be the most effective photosensitizer against murine tumors amongst a series of homologues with different numbers of methylene groups on the ether function. This compound strongly absorbs light (665 47,000 M-1 cm-1) at 665 nm, so that penetration into tumor tissue is increased vis-à-vis Photofrin (3)
.
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MATERIALS AND METHODS |
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TopABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONAppendixREFERENCES |
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HPPH was manufactured at the University of California Davis under good manufacturing practice conditions by one of us (R. K. P.). Preparation and characterization of HPPH were as described previously (9) ; the injectable drug was formulated in 5% dextrose in sterile water containing 2% ethanol and 1% polyoxyethylene sorbitan monooleate (Tween 80) and kept frozen before use. Doses were administered i.v. over 1 h at an infusion rate of 110 ml/h. Pharmacokinetic data were available for doses of 3 mg/m2 (n = 4), 4 mg/m2 (n = 7), 5 mg/m2 (n = 3), and 6 mg/m2 (n = 11). Blood samples were collected in anticoagulant-free tubes on 4–13 occasions/patient between the end of infusion and study day 143. A total of 193 blood samples were obtained after HPPH infusion; in some cases, preinfusion blood samples were acquired to establish baseline serum fluorescence levels (see below).
Blood Sampling and Photosensitizer Analysis.
HPPH serum levels were monitored using a fluorescence assay because of the high quantum fluorescence yield of the drug (
F 0.45) and the inherent increased sensitivity of spectrofluorimetry over absorption-based methods (e.g., spectrophotometry, HPLC). Serum was isolated by centrifugation, and samples were kept at -70°C until assayed. Serum concentrations of HPPH were determined by admixture with Solvable (Packard Bioscience, Meriden, CT) and heating at 53°C followed by analysis of fluorescence emission spectra as described previously (10)
. Briefly, after treatment with Solvable and heat, each sample was placed in a quartz cuvet, and a fluorescence emission spectrum (ex = 412 nm) was obtained using a commercial fluorimeter (model FP-777; JASCO, Inc., Easton, MD). Typically, 3 replicates/time point/patient were analyzed. The amplitude of the fluorescence emission maximum (em = 670 nm), following baseline correction, was compared with a standard curve obtained by processing known amounts of HPPH in Solvable-containing serum from volunteers. No patient sera obtained before infusion had measurable fluorescence corresponding to the HPPH fluorescence spectrum.
Studies of the Binding of HPPH to Plasma Proteins.
The binding of HPPH to protein and lipoprotein components in human plasma was assessed using KBr density gradient analysis. This system was a modification of the previously published procedure of Woodburn and Kessel (11)
. Briefly, blood was collected from a donor using EDTA as the anticoagulant to minimize lipoprotein oxidation. Erythrocytes were removed by centrifugation, and the plasma was incubated with HPPH at a concentration of 10 µM. After 24 h at 4°C, the plasma was mixed with 0.15 M NaCl and brought to a density of 1.21 g/ml with solid KBr. A portion of the resulting mixture was layered over a KBr solution (density, 1.27 g/ml) in polyallomer centrifuge tubes (Beckman Instruments Inc., Palo Alto, CA). The tubes were filled with isotonic saline and spun in a Beckman TL-100 tabletop ultracentrifuge for 1 h. The tubes were fractionated from the top; a total of 25 fractions were collected. Each fraction was diluted with 1% polyoxyethylene sorbitan monooleate (Tween 80), and the relative concentration of HPPH was assessed by fluorescence.
HPPH Metabolism Studies.
A study was carried out to identify circulating metabolites of HPPH. For this, serum was isolated, by centrifugation, from blood drawn from patients at various times (1, 7, 24, 48, and 96 h) after the end of the HPPH infusion. The serum was vortex-mixed with methanol in microcentrifuge tubes. Two to four serum samples from different patients were analyzed for each time point. This mixture was then centrifuged at 10°C to pellet the precipitated protein. The supernatant was removed and analyzed by HPLC. Sera isolated from patients who had not yet been injected with photosensitizer were used as controls.
To study whether HPPH could be metabolized by liver enzymes in vitro, the drug was incubated with mixed sex, pooled human liver microsomes according to the provided protocol (In Vitro Technologies, Baltimore, MD); the cryopreserved microsomes had been characterized previously by the supplier for a wide variety of isozyme activities. Briefly, solid HPPH was dissolved in acetonitrile and mixed with potassium phosphate buffer (pH 7.4) and the pooled microsomes, followed by warming. After 5 min, warm NADPH regenerating system was added to the reaction mixture. After shaking at 37°C for 1 h, the reaction was terminated with an equal volume of acetonitrile. The tubes were centrifuged to pellet the precipitated protein, after which the supernatant was removed and analyzed by HPLC.
To study type II conjugation of HPPH, the protocol was modified by the addition of uridine-5'-diphosphoglucuronic acid and 3'-phosphoadenosine 5'-phosphosulfate to the NADPH regenerating system. Samples were shaken in the heated water bath and centrifuged to precipitate protein, as described above. The supernatants were analyzed by HPLC.
Reverse-phase HPLC was used for identification of HPPH and its metabolites and conjugates. Separation was carried out on an RP-8 LiChrospher 100 column (Merk KGaA, Darmstadt, Germany) using mobile phases consisting of 60% methanol in 10 mM sodium phosphate buffer and 90% methanol in 2 mM sodium phosphate buffer, both pH 7.5. The sample was monitored by absorption measurements at 405, 450, and 665 nm with a diode array UV/VIS detector (Hewlett Packard 1100 Series; Agilent Technologies, Palo Alto, CA).
Solar Simulator Studies.
The magnitude and duration of cutaneous photosensitivity in patients who were given HPPH were determined. For this, the skin of the medial forearm was exposed to a simulated solar spectrum (350–2500 nm) produced by a 300 W solar simulator (model 81250 with an AM1.5 direct filter; Oriel Corporation, Stratford, CT). For each test, four 1.0-cm2 spots were exposed to light for 10, 15, 20, or 30 min at a fluence rate of 74 mW/cm2. Skin responses were graded approximately 24 h after exposure. Baseline reactions were obtained by exposing the forearm skin to the solar simulator light before HPPH was injected, typically on the same day. After receiving injection, each patient was then tested daily for 3 days and occasionally upon return for clinical follow-up. The scale for grading skin photosensitivity can be found in the legend to Table 2
.
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), and half-time for second exponential term (t1/2ß), other pharmacokinetic parameters were calculated (Eqs. 2
3
4
5
6
7)
, including rate constant for first exponential term ( = ln(2)/t1/2), rate constant for second exponential term (ß = ln(2)/t1/2ß), area under the concentration-time plasma HPPH curve from time 0 to infinity (AUC), mean residence time (MRT), plasma clearance (CL), distributional clearance (CLD), volume of distribution of the central compartment (VC), and volume of distribution at steady state (VSS). The estimated concentration of HPPH in the plasma (CP) at 24 and at 48 h was calculated from Eq. 1
with the fitted parameters (A, B, t1/2, and t1/2ß) for each patient. The possible relationships between specific parameters and potential covariates, including age, gender, body surface area (BSA), body weight, total dose of administered HPPH (mg), and dose of HPPH (mg/m2), were explored by making bivariate scatter plots.
Nonlinear mixed effects modeling was applied to the complete data set of 193 data points from 25 patients simultaneously with proc NLMIXED (15) , with the first order (firo) option for integration and the default options for all other choices, with SAS version 8.12 on an Intel Pentium IV-based 2.0 GHz microcomputer. A typical run, such as the final run, which included 10 estimable parameters, took approximately 10 s. In contrast, when using the option for integration recommended by the SAS manual, adaptive Gaussian quadrature, no run with 10 estimable parameters ever finished; runs were stopped at 120 h or less.
The actual model that was fitted to the complete data set included 14 equations from the Appendix (Eqs. 16 17 18 19 , 8 9 10 11 12 13 14 15 , 1 , and 20 , in that order). Eqs. 16 17 18 19 scale and transform the set of four physiological parameters, CL, CLD, VC, and VSS, so that they are all in the same order of magnitude. This is important for optimizing the efficiency of the fitting algorithms. Eq. 17 transforms VSS to ln(VSS). This gave the distribution of the parameter more of a Gaussian shape; a normal distribution for each parameter among patients is a common and desirable assumption in nonlinear mixed effects modeling. Eq. 18 expresses the relationship between CL and age; an initial look at the individual patient CL against age hinted at a linear relationship. The inclusion of a covariate in the overall model has the potential to explain some of the variation among patients for that parameter. Eqs. 16 17 18 19 include the random parameters b1, b2, b3, and b4. As is usual, it was assumed that the population mean for each of b1, b2, b3, and b4 is zero, and estimation was attempted for only the population variances and covariances for the set of four random variables. It was further assumed that each of these random variables was normally distributed.
Eqs. 8
9
10
11
12
13
14
15
map the set of four physiological pharmacokinetic parameters to the set of four empirical pharmacokinetic parameters: A, B, t1/2, and t1/2ß. Eq. 1
includes the set of four empirical parameters and is the structural model fit directly to the data. Combining Eqs. 8
9
10
11
12
13
14
15
with Eq. 1
allows the four physiological pharmacokinetic parameters to be estimated directly and facilitates the inclusion of covariates that affect the physiological parameters.
The error variance model used for this study is Eq. 20
. It was assumed that errors were normally distributed with a population mean of zero. Contributions to the total error come from both assay error and model misspecification error. This is a relative error model, and the square root of 
2 is the overall coefficient of variation for the data.
To apply NLMIXED to the raw data, the above equations were coded into the SAS procedure language in the following order: Eqs. 16 17 18 19 ; 8 9 10 11 12 13 14 15 ; 1 ; and 20 . This final model was built up gradually by adding/subtracting logical model components and covariates. The log ratio test (P < 0.05) and the examination of various diagnostic plots were used in the model building process to decide upon the inclusion/exclusion of model components and covariates. The SAS code for the final run of the complete model will be emailed to any interested persons upon request.5
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RESULTS AND DISCUSSION |
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, inset, shows the structure of HPPH. HPPH is a highly lipophilic drug with a log P of 5.6 at physiologic pH (7)
, and it is formulated in 2% ethanol and 1% polyoxyethylene sorbitan monooleate (Tween 80). Its large molar extinction coefficient in the red region of the visible spectrum (665nm 47,000 M-1 cm-1) and singlet oxygen quantum yield (
) of 0.48 (16)
make it an attractive candidate for use in the PDT of malignant tumors.
PDT with HPPH appears to be a relatively effective treatment. In the first Phase I-II clinical study with HPPH, patients with partially obstructive esophageal cancer received a starting dose of 6 mg/m2 HPPH and a light dose of 150 J/cm ( = 665 nm) delivered endoscopically 48 h after injection, based on extensive preclinical pharmacokinetic and toxicological information. This is a palliative treatment meant to relieve difficulty in swallowing. Initially, this low dose, equivalent to approximately 0.15 mg/kg body weight and the lowest of three planned escalating doses, was expected to be ineffective but was required by the Food and Drug Administration based on the data presented to them for obtaining permission to initiate a human clinical trial. However, we found that when examined endoscopically 2 days after PDT, the first and all subsequent patients treated with this dose achieved a brisk response with extensive necrosis of the tumor within the esophagus. Six of eight patients with follow-up experienced palliation of symptoms but no increase in survival as expected (median survival, 8 months); however, one patient with fairly limited disease achieved a complete response for 1.5 years. This result is similar to that for Photofrin-based PDT in such patients.
Three patients with multiple basal cell carcinomas received 3 mg/m2 HPPH (one-half the dose used for the esophageal patients, above) followed (a) 24 h later by 50 or 150 J/cm2 or (b) 48 h later by 150–200 J/cm2 of 665 nm light. Follow-up of these patients to date indicated that most lesions had complete responses with a drug-light interval of 24 h and complete and partial responses with an interval of 48 h.
A third Phase I-II trial is the study of PDT of high-grade dysplasia in Barrett’s esophagus. Clinical information for these patients is still being accumulated, although it appears that a 3 mg/m2 HPPH dose is ineffective, whereas higher doses of 4, 5, or 6 mg/m2 may be capable of destroying both the dysplasia and most of the Barrett’s esophagus when treated with 150 J/cm 665 nm light 48 h after injection.
Up to now, three patients have been treated for lung cancer with HPPH (4 mg/m2), with two complete responses and one recurrence 1 month after treatment. Only 1 patient of 25 patients treated to date has reported a cutaneous reaction to sunlight, which may have been unrelated to the HPPH.
Each set of HPPH pharmacokinetic data, one per patient, was initially fit with both a one-compartment model (data not shown) and a two-compartment model (Eq. 1
in "Appendix"), the four empirical parameters (A, B, t1/2, and t1/2ß) were estimated, and the four physiological parameters (CL, CLD, VC, and VSS) were calculated. It was clear from visual inspection that a one-compartment model was inadequate for fitting the data, that a two-compartment model fit very well, and that there were insufficient data to characterize a three-compartment model. Each 25-patient set of the four empirical parameter estimates was plotted against dose of HPPH, patient weight, patient BSA, and patient age. Because VC was linearly related to BSA (data not shown), it was decided to divide VC (and the other three physiological parameters) by BSA for further analyses. The sum of A + B was linearly related to dose of HPPH, and this allowed us to assume linear pharmacokinetics for further analyses. None of the four physiological parameters was related to dose of HPPH in any obvious manner.
The results for the population pharmacokinetic analysis of HPPH are shown in Figs. 1
2
3
and Table 1
. In Fig. 1
, the final global fitted model is represented by the thick black curve, and the raw data are represented by the thin colored curves (see legend). Overall, from the perspective of Fig. 1
, the final global model fit the raw data extremely well. Note that the assumption that pharmacokinetics are linear in dose is supported from the good fit of the global model to the data in Fig. 1
. The predicted mean Cmax (maximum plasma HPPH concentration achieved at the end of the 1-h infusion) values were 1.13, 1.50, 1.87, and 2.25 µg/ml for doses of 3, 4, 5, and 6 mg/m2, respectively. The vertical dashed lines in the figure represent the two times that light is normally administered for PDT with HPPH. A two-compartment pharmacokinetic population model was shown to be vastly superior to the one-compartment population model via the likelihood ratio test (P << 0.01); a three-compartment population model could not be successfully fit to the data.
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, the parameter estimates for the fits to the individual patient curves are represented by the normalized histograms, and the final fitted global model is represented by the thick black Gaussian curves. Again, the concordance of the individual patient parameters and the global fitted model is excellent. Note that for seven of the patients, adequate HPPH plasma measurements were not observed at later time points to allow neither the accurate estimation of the ß phase half-life nor any of the physiological parameters that are highly dependent on this empirical parameter. The population nonlinear mixed effects modeling approach compensates for this experimental design fault, and therefore, its result, the Gaussian curves, is much more credible than the individual patient parameter estimate histograms. This set of four Gaussian curves provides an excellent prediction of the distribution of physiological parameters (i.e., volumes and clearances) for patients given HPPH by i.v. infusion. When normalized by the BSA, all of the physiological parameters, including VSS, were independent of dose of administered HPPH, at least from 3 to 6 mg/m2. The use of the model building tool (the likelihood ratio test) yielded the unambiguous result that age was a significant covariate for CL but not a significant covariate for CLD, either with or without the age dependency of CL in the global model.
Table 1
includes the 10 parameter estimates from the final global model, along with a SE for each estimate and a 95% confidence interval. The only population parameters that were not statistically significant (P > 0.05) were the population variances for CL and CLD. The estimated population parameter, VSS, and the estimated population variance for VSS, s2(VSS), along with SEs and 95% confidence intervals, were approximated from the results for ln(VSS) and s2(ln(VSS)). Note that the population parameter, CL, itself, was not estimated directly; rather the linear coefficient of Eq. 18
, which relates CL to age, was directly estimated, as is explained in footnote c of Table 1
. Note that only one population covariance term, that for the covariance between VC and ln(VSS), was estimated. After many attempts to include more than one population covariance parameter, it became clear that the data from 25 patients had only enough information for one, and the covariance parameter listed in the table was estimated the best. From knowledge of the dose of HPPH and the age of the patient, along with the information in Table 1
, one can estimate the mean pharmacokinetic profile for any new patient and can also predict confidence bounds around this mean profile.
The mean population and ß half-lives, calculated from the physiological population parameters in Table 1
, were 7.73 and 251 h, respectively. When Eq. 1
was fit directly to the whole 25-patient data set with nonlinear mixed effects modeling, the estimated mean population and ß half-lives (95% confidence interval) were 7.77 h (3.46–17.6 h) and 596 h (120–2951 h), respectively. The discrepancy in the ß half-life estimates from the two different approaches for estimation underscores the uncertainty in the population distribution of the ß half-life. Both the and ß half-lives are relatively long, as compared with many other drugs; note that the Food and Drug Administration-approved photodynamic sensitizer Photofrin has initial and terminal population half-lives of 16.1 h and 155.6 days (17)
. Both these drugs leave the plasma very slowly, which may account for their prolonged clinical activity.
The very low clearances and volumes for HPPH listed in Table 1
are consistent with a highly lipophilic agent that is concentrated in the plasma, where it is almost 100% bound to plasma proteins. The binding of HPPH to serum proteins was measured using KBr density gradient ultracentrifugation. The majority of HPPH partitioned to low density lipoprotein (53%), albumin and other heavy proteins (27.5%), and high density lipoprotein (15%), whereas the remaining drug (4.5%) was either associated with very low density lipoprotein or unbound.
The results from the present human HPPH pharmacokinetic study were compared with results from previous pharmacokinetic studies in the mouse (9) and in the dog (8) . Conversion factors from Freireich et al. (18) were used to calculate volumes and clearances for the mouse, dog, and human in common units. Overall, the concordance of the pharmacokinetic results was very good. For example, the estimated mouse (20 g body weight) total VC was 1.25 ml, which is very close to the estimated mouse plasma volume of 1.40 ml. The estimated mouse CL was 0.019 liter/h/m2, which is close to the value of 0.0296 liter/h/m2 found for the human in the current study. The estimated VC for the dog (24.6 kg) was 0.069 liter/kg, which is almost exactly equal to the 0.07 liter/kg empirical factor (19) for estimating plasma volume. The CL for the dog was estimated to be 0.102 liter/h/m2, which is about three times the average CL estimated for the human in the current study. Hence, in three species, HPPH is cleared very slowly from the central plasma compartment, and the VC is close in size to the estimated plasma volume. Therefore, most of the HPPH in plasma is likely bound to plasma protein.
A possible explanation for the increase in CL with age involves the binding of HPPH to albumin. Because plasma albumin concentration tends to decrease with age (20)
, older patients may have more unbound plasma HPPH, which would result in a faster clearance (20)
. Although not labeled in Fig. 1
, the HPPH concentration-time curves for the patients given 6 mg/m2 (the green lines) were roughly rank-ordered, with the younger patients having the higher concentrations, and the older patients having the lower concentrations at each time point. This is shown convincingly in Fig. 3
.
Table 2
shows the relationship among HPPH dose, individual patient AUC, and 24 h predicted HPPH plasma concentration, and the results of a solar simulator test for 19 of the total 25 patients in this study. The results suggest that increasing the dose of HPPH to 6 mg/m2 increases the likelihood of postinfusion positive solar simulator tests. However, also note that the most striking solar simulator effects, those for patient 17, were still quite mild, consisting of pronounced erythema, but without tissue edema. In addition, note that the two listed individual pharmacokinetic parameters (individual patient AUC and 24 h predicted HPPH plasma concentration) were not related to the solar simulator results in any obvious way. Overall, phototoxicity from exposure to the sun was not a serious problem with HPPH (21)
. The observed phototoxicity from this solar simulator test and from the exposure of these patients to normal sunlight after their HPPH treatment was minimal. This can be contrasted with the phototoxicity from treatment with Photofrin, which induces cutaneous phototoxicity for a minimum of 3 weeks to several months (the usual length of required eye and skin protection from bright light is 4–6 weeks, in our experience).
Examination of patient sera isolated up to 4 days after HPPH infusion showed no circulating metabolites. In addition, the incubation of HPPH with pooled human microsomes did not produce either phase I or II metabolites. For both assays, only HPPH and the few minor by-products (<3%) already present in the injectable preparation appeared on the reverse phase HPLC chromatographs (data not shown). Studies of other synthetic tetrapyrrolic photodynamic sensitizers (22, 23, 24) that also do not contain coordinated metal ions suggest that these compounds may not be easily metabolized in vivo. A comprehensive study, by us, of [14C]HPPH in the tissues, plasma, feces, and urine of rats is planned.
Because this pharmacokinetic study includes patients with different diseases, different possible outcomes, different HPPH doses, and different light exposures, it would not be practical in this study to try to examine the relationship of HPPH pharmacokinetics to patient response. This type of pharmacodynamic analysis will become practical in future Phase II and Phase III studies of patients with the same disease.
A comprehensive population pharmacokinetic study and, if possible, a comprehensive population pharmacokinetic/pharmacodynamic study of an agent that is being introduced into clinical practice can often increase the utility of the new agent in routine clinical use (25) . For the new photodynamic agent HPPH, the pharmacokinetic/pharmacodynamic study reported here will aid in the prediction of individual and population pharmacokinetic profiles. The unexplained variation in both profiles and physiological population parameters is acceptable. The increase of systemic clearance with age is a noteworthy finding and suggests that an upward dose adjustment may be useful for some elderly patients.
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Appendix |
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at the end of the Appendix. Eq. 1
is the structural pharmacokinetic model that was fit to the plasma-concentration-time raw data for each individual patient with SAS NLIN (14)
, with the direct estimation of four parameters for each patient: A, B, t1/2, and t1/2ß.
 | (A1) |
|
, and t1/2ß (12)
. Essentially, Eqs. 2
3
4
5
6
7
map the set of four empirical pharmacokinetic parameters to the set of four physiological pharmacokinetic parameters.
 | (A2) |
 | (A3) |
 | (A4) |
 | (A5) |
 | (A6) |
 | (A7) |
Eqs. 8
9
10
11
12
13
14
15
perform the reverse mapping of the set of physiological pharmacokinetic parameters to the set of four empirical pharmacokinetic parameters. The use of Eqs. 8
9
10
11
12
13
14
15
, along with Eq. 1
, allow the direct estimation of the four physiological parameters, with nonlinear regression.
 | (A8) |
 | (A9) |
 | (A10) |
 | (A11) |
 | (A12) |
 | (A13) |
 | (A14) |
 | (A15) |
When using nonlinear mixed effects modeling, it is usually advantageous to scale the parameters that are directly estimated, so that all of the estimated parameters are of roughly the same magnitude (15)
. This was done in Eqs. 16
17
18
19
. In addition, transformations can be applied to parameters to make the statistical assumption of a normal distribution for each parameter among patients more reasonable. This was done for VSS in Eq. 17
; a logarithmic transformation was applied, so that ln(VSS) is the actual parameter that is estimated. One can introduce additional covariates into the model that can potentially explain and thus reduce the patient to patient variance in parameters. This was done for CL in Eq. 18 ; the covariate, age (years) was introduced. The structural parameters that are actually directly estimated in Eqs. 16
17
18
19
20
are ß1, ß2, ß3, and ß4. Many parameters will vary from patient to patient, such as the four physiological pharmacokinetic parameters, CL, CLD, VC, and VSS. One can introduce a random parameter for each structural pharmacokinetic parameter to quantify the spread of values of the pharmacokinetic parameters among patients. These random parameters are b1, b2, b3, and b4 in Eqs. 16
17
18
19
. As is usual, it was assumed that the population mean for each of b1, b2, b3, and b4 is zero, and estimation was attempted for only the population variances and covariances for the set of four random variables. It was further assumed that each of these random variables was normally distributed. Note that in Table 1
, the appropriate reverse parameter transformations were applied, and that the reported parameter estimates are for the physiological parameters, CL, CLD, VC, and VSS; the four ß and four b parameters are not listed.
 | (A16) |
 | (A17) |
 | (A18) |
 | (A19) |
The error variance model used for this study is Eq. 20
. It was assumed that errors were normally distributed with a population mean of zero. Contributions to the total error come from both assay error and model misspecification error. This is a relative error model, and the square root of 2 is the overall coefficient of variation for the data.
 | (A20) |
To apply NLMIXED to the raw data, the above equations were coded into the SAS procedure language in the following order: Eqs. 16 17 18 19 , 8 9 10 11 12 13 14 15 , 1 , and 20 . This set of 14 equations was fit with SAS NLMIXED (15) using nonlinear mixed effects modeling to the entire data set for all 25 patients simultaneously.
The SAS code for the final run of the complete model will be emailed to any interested persons upon request.5
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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1 Supported in part by NIH Grant P01 CA55791. This work utilized core facilities supported in part by Roswell Park Cancer Institute’s National Cancer Institute-funded Cancer Center Support Grant CA16056. 
2 These authors contributed equally to this work.
3 To whom requests for reprints should be addressed, at PDT Center, Roswell Park Cancer Institute, Elm and Carlton Streets, Buffalo, NY 14263.
4 The abbreviations used are: PDT, photodynamic therapy; HPPH, 2-[1-hexyloxyethyl]-2-devinyl pyropheophorbide-a; HPLC, high-performance liquid chromatography; BSA, body surface area.
5 Available upon request from William R. Greco (william.greco{at}roswellpark.org).
Received 9/13/02. Accepted 2/19/03.
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, patients given 6 mg/m2; 
, patients given 5 mg/m2; 
, 4 mg/m2; the 
, patients given 3 mg/m2. The line was fit with unweighted linear regression to the data from the 11 patients given 6 mg/m2. The parameter estimates ± SE for this fit were as follows: y-intercept = 1.84 ± 0.32; slope = -0.0167 ± 0.0047.