RPEM: Randomized Monte Carlo parametric expectation maximization algorithm.

Inspired from quantum Monte Carlo, by sampling discrete and continuous variables at the same time using the Metropolis-Hastings algorithm, we present a novel, fast, and accurate high performance Monte Carlo Parametric Expectation Maximization (MCPEM) algorithm. We named it Randomized Parametric Expectation Maximization (RPEM). We compared RPEM with NONMEM's Importance Sampling Method (IMP), Monolix's Stochastic Approximation Expectation Maximization (SAEM), and Certara's Quasi-Random Parametric Expectation Maximization (QRPEM) for a realistic two-compartment voriconazole model with ordinary differential equations using simulated data. We show that RPEM is as fast and as accurate as the algorithms IMP, QRPEM, and SAEM for the voriconazole model in reconstructing the population parameters, for the normal and log-normal cases.

Interchangeability of generic drugs for subpopulations: Bioequivalence simulation from a nonparametric PK model of gabapentin generic drugs.

Patients are often switched between generic formulations of the same drug, but in some cases generic interchangeability is questioned. For generic drugs to be approved, bioequivalence with the innovator drug should be demonstrated, but evidence of bioequivalence is not required in the intended patient population or relative to other approved generics. AIM: We aim to identify pathophysiological pharmacokinetic subpopulations for whom there is a difference in comparative bioavailability compared to a healthy population. METHODS: We used simulated exposures from a nonparametric model of multiple generics and the originator gabapentin. Exposure was simulated for virtual populations with pharmacokinetic characteristics beyond those of healthy subjects with regard to rate of absorption, volume of distribution and reduced renal function. Virtual parallel design bioequivalence studies were performed using a random sample of 24 simulated subjects, with standard acceptance criteria. RESULTS: Results indicated increased pharmacokinetic variability for patient populations with a lower rate of absorption or a reduced renal function, but no change in the average comparable bioavailability ratio. This increased variability results in a reduced likelihood of demonstrating bioequivalence. Observations were similar for comparisons between all different formulations, as well as between subjects who received the identical formulation in a repeated fashion. No relevant effect was observed for simulations with increased volume of distribution. CONCLUSION: Our simulations indicate that the reduced likelihood of demonstrating bioequivalence for subjects with altered pharmacokinetics is not influenced by a formulation switch, nor does the average comparable bioavailability ratio change, therefore these results support generic interchangeability and current approval requirements for generics.

Combination Therapy to Kill Mycobacterium tuberculosis in Its Nonreplicating Persister Phenotype.

Mycobacterium tuberculosis (Mtb) exists in various metabolic states, including a nonreplicating persister (NRP) phenotype which may affect response to therapy. We have adopted a model-informed strategy to accelerate discovery of effective Mtb treatment regimens and previously found pretomanid (PMD), moxifloxacin (MXF), and bedaquiline (BDQ) to readily kill logarithmic- and acid-phase Mtb. Here, we studied multiple concentrations of each drug in flask-based, time-kill studies against NRP Mtb in single-, two- and three-drug combinations, including the active M2 metabolite of BDQ. We used nonparametric population algorithms in the Pmetrics package for R to model the data and to simulate the 95% confidence interval of bacterial population decline due to the two-drug combination regimen of PMD + MXF and compared this to observed declines with three-drug regimens. PMD + MXF at concentrations equivalent to average or peak human concentrations effectively eradicated Mtb. Unlike other states for Mtb, we observed no sustained emergence of less susceptible isolates for any regimen. The addition of BDQ as a third drug significantly (P < 0.05) shortened time to total bacterial suppression by 3 days compared to the two-drug regimen, similar to our findings for Mtb in logarithmic or acid growth phases.

Evaluating the effect of clofazimine against Mycobacterium tuberculosis given alone or in combination with pretomanid, bedaquiline or linezolid.

In recent years, clofazimine (CFZ) has been regaining prominence for the treatment of tuberculosis. However, it shows limited efficacy as a single drug and optimal combination partners have not been identified. Therefore, the objective of our analysis was to evaluate the efficacy of CFZ-containing two-drug regimens with pretomanid (PMD), bedaquiline (BDQ) or linezolid (LZD) by: (i) determining their pharmacodynamic (PD) mode of interaction against Mycobacterium tuberculosis (Mtb) strain H37Rv in log- phase and acid-phase metabolic states, and against Mtb strain 18b in a non-replicating persister (NRP) metabolic state; (ii) predicting bacterial cell kill of the drugs alone and in combination; and (iii) evaluating the relationship between the interaction mode and the extent of bacterial cell kill. The results of our Greco universal response surface analysis showed that CFZ was at least additive with a clear trend towards synergy when combined with PMD, BDQ and LZD against Mtb in all explored metabolic states under in vitro checkerboard assay conditions. The results further showed that all two-drug combination regimens exerted greater bacterial kill than any of the drugs alone. CFZ alone showed the least antimicrobial efficacy amongst the evaluated drugs, and there was a lack of correlation between the mode of interaction and the extent of bacterial kill. However, we may underestimate the effect of CFZ in this screening approach owing to limited in vitro study duration and neglect of target site accumulation. Clofazimine; Pretomanid; Bedaquiline; Linezolid; Combination chemotherapy; Mycobacterium tuberculosis.

Nonparametric Methods in Population Pharmacokinetics.

Population pharmacokinetic (PK) modeling is a widely used approach to analyze PK data obtained from groups of individuals, in both industry and academic research. The approach can also be used to analyze pharmacodynamic (PD) data and pooled PK/PD data. There are 2 main families of population PK methods: parametric and nonparametric. The objectives of this article are to present an overview of nonparametric methods used in population pharmacokinetic modeling and to explain their specific characteristics to inform scientists and clinicians about their potential value for data analysis, simulation, dosage design, and therapeutic drug monitoring (TDM). Nonparametric methods have several interesting characteristics for population PK analysis, including computation of exact likelihoods, the ability to accommodate parameter probability distributions of any shape (eg, non-Gaussian), and to detect subpopulations and outliers. Nonparametric population methods are also highly relevant for model-based TDM and design of individualized drug dosage regimens. Several algorithms have been developed to estimate model parameter values within an individual and compute that individual's dosage to achieve target drug exposure with maximum precision and accuracy. Nonparametric modeling methods for both population and individual PK analysis are available under user-friendly packages.

Parametric and Nonparametric Methods in Population Pharmacokinetics: Experts' Discussion on Use, Strengths, and Limitations.

Population pharmacokinetics consists of analyzing pharmacokinetic (PK) data collected in groups of individuals. Population PK is widely used to guide drug development and to inform dose adjustment via therapeutic drug monitoring and model-informed precision dosing. There are 2 main types of population PK methods: parametric (P) and nonparametric (NP). The characteristics of P and NP population methods have been previously reviewed. The aim of this article is to answer some frequently asked questions that are often raised by scholars, clinicians, and researchers about P and NP population PK methods. The strengths and limitations of both approaches are explained, and the characteristics of the main software programs are presented. We also review the results of studies that compared the results of both approaches in the analysis of real data. This opinion article may be informative for potential users of population methods in PK and guide them in the selection and use of those tools. It also provides insights on future research in this area.

Parametric and Nonparametric Population Pharmacokinetic Models to Assess Probability of Target Attainment of Imipenem Concentrations in Critically Ill Patients.

Population pharmacokinetic modeling and simulation (M&S) are used to improve antibiotic dosing. Little is known about the differences in parametric and nonparametric M&S. Our objectives were to compare (1) the external validation of parametric and nonparametric models of imipenem in critically ill patients and (2) the probability of target attainment (PTA) calculations using simulations of both models. The M&S software used was NONMEM 7.2 (parametric) and Pmetrics 1.5.2 (nonparametric). The external predictive performance of both models was adequate for eGFRs ≥ 78 mL/min but insufficient for lower eGFRs, indicating that the models (developed using a population with eGFR ≥ 60 mL/min) could not be extrapolated to lower eGFRs. Simulations were performed for three dosing regimens and three eGFRs (90, 120, 150 mL/min). Fifty percent of the PTA results were similar for both models, while for the other 50% the nonparametric model resulted in lower MICs. This was explained by a higher estimated between-subject variability of the nonparametric model. Simulations indicated that 1000 mg q6h is suitable to reach MICs of 2 mg/L for eGFRs of 90-120 mL/min. For MICs of 4 mg/L and for higher eGFRs, dosing recommendations are missing due to largely different PTA values per model. The consequences of the different modeling approaches in clinical practice should be further investigated.

Building Optimal Three-Drug Combination Chemotherapy Regimens To Eradicate Mycobacterium tuberculosis in Its Slow-Growth Acid Phase.

Mycobacterium tuberculosis metabolic state affects the response to therapy. Quantifying the effect of antimicrobials in the acid and nonreplicating metabolic phases of M. tuberculosis growth will help to optimize therapy for tuberculosis. As a brute-force approach to all possible drug combinations against M. tuberculosis in all different metabolic states is impossible, we have adopted a model-informed strategy to accelerate the discovery. Using multiple concentrations of each drug in time-kill studies, we examined single drugs and two- and three-drug combinations of pretomanid, moxifloxacin, and bedaquiline plus its active metabolite against M. tuberculosis in its acid-phase metabolic state. We used a nonparametric modeling approach to generate full distributions of interaction terms between pretomanid and moxifloxacin for susceptible and less susceptible populations. From the model, we could predict the 95% confidence interval of the simulated total bacterial population decline due to the 2-drug combination regimen of pretomanid and moxifloxacin and compare this to observed declines with 3-drug regimens. We found that the combination of pretomanid and moxifloxacin at concentrations equivalent to average or peak human concentrations effectively eradicated M. tuberculosis in its acid growth phase and prevented emergence of less susceptible isolates. The addition of bedaquiline as a third drug shortened time to total and less susceptible bacterial suppression by 8 days compared to the 2-drug regimen, which was significantly faster than the 2-drug kill.

Dose Fractionation of Moxifloxacin for Treatment of Tuberculosis: Impact of Dosing Interval and Elimination Half-Life on Microbial Kill and Resistance Suppression.

The repurposed agent moxifloxacin has become an important addition to the physician's armamentarium for the therapy of Mycobacterium tuberculosis When a drug is administered, we need to have metrics for success. As for most antimicrobial chemotherapy, we contend that for Mycobacterium tuberculosis therapy, these metrics should be a decline in the susceptible bacterial burden and the suppression of amplification of less-susceptible populations. To achieve optimal outcomes relative to these metrics, a dose and schedule of administration need to be chosen. For large populations of patients, there are true between-patient differences in important pharmacokinetic parameters. These distributions of parameter values may have an impact on these metrics, depending on what measure of drug exposure drives the metrics. To optimize dose and schedule choice of moxifloxacin, we performed a dose fractionation experiment in the hollow fiber infection model. We examined 12-, 24-, and 48-h dosing intervals with doses of 200, 400, and 800 mg for each interval, respectively. Within each interval, we had an arm where half-lives of 12, 8, and 4 h were simulated. We attempted to keep the average concentration (Cavg) or area under the concentration-time curve (AUC) constant across arms. We found that susceptible bacterial load decline was linked to Cavg, as we had indicated previously. Resistance suppression, a nonmonotonic function, had minimum concentration (Cmin) as the linked index. The 48-h interval with the 4-h half-life had the largest less-susceptible population. Balancing bacterial kill, resistance suppression, toxicity (linked to peak concentration [Cpeak]), and adherence, we conclude that the dose of 400 mg daily is optimal for moxifloxacin.

The Funnel: a Screening Technique for Identifying Optimal Two-Drug Combination Chemotherapy Regimens.

The Mycobacterium tuberculosis drug discovery effort has generated a substantial number of new/repurposed drugs for therapy for this pathogen. The arrival of these drugs is welcome, but another layer of difficulty has emerged. Single agent therapy is insufficient for patients with late-stage tuberculosis because of resistance emergence. To achieve our therapeutic ends, it is requisite to identify optimal combination regimens. These regimens go through a lengthy and expensive evaluative process. If we have a modest group of 6 to 8 new or repurposed agents, this translates into 15 to 28 possible 2-drug combinations. There is neither time nor resources to give an extensive evaluation for all combinations. We sought a screening procedure that would identify combinations that had a high likelihood of achieving good bacterial burden decline. We examined pretomanid, moxifloxacin, linezolid, and bedaquiline in log-phase growth, acid-phase growth, and nonreplicative persister (NRP) phase in the Greco interaction model. We employed the interaction term α and the calculated bacterial burden decline as metrics to rank different regimens in different metabolic states. No relationship was found between α and bacterial kill. We chose bacterial kill as the prime metric. The combination of pretomanid plus moxifloxacin emerged as the clear frontrunner, as the largest bacterial declines were seen in log phase and acid phase with this regimen and it was second best in NRP phase. Bedaquiline also produced good kill. This screening process may identify optimal combinations that can be further evaluated in both the hollow-fiber infection model and in animal models of Mycobacterium tuberculosis infection.

Building Optimal Three-Drug Combination Chemotherapy Regimens.

Multidrug therapy is often required. Examples include antiviral therapy, nosocomial infections, and, most commonly, anti-Mycobacterium tuberculosis therapy. Our laboratory previously identified a mathematical approach to identify 2-drug regimens with a synergistic or additive interaction using a full factorial study design. Our objective here was to generate a method to identify an optimal 3-drug therapy. We studied M. tuberculosis isolate H37Rv in log-phase growth in flasks. Pretomanid and moxifloxacin were chosen as the base 2-drug regimen. Bedaquiline (plus M2 metabolite) was chosen as the third drug for evaluation. Total bacterial burden and bacterial burden less-susceptible to study drugs were enumerated. A large mathematical model was fit to all the data. This allowed extension to evaluation of the 3-drug regimen by employing a Monte Carlo simulation. Pretomanid plus moxifloxacin demonstrated excellent bacterial kill and suppressed amplification of less-susceptible pathogens. Total bacterial burden was driven to extinction in 3 weeks in 6 of 9 combination therapy evaluations. Only the lowest pretomanid/moxifloxacin exposures in combination did not extinguish the bacterial burden. No combination regimen allowed resistance amplification. Generation of 95% credible intervals about estimates of the interaction parameters α (αs, αr-p, and αr-m) by bootstrapping showed the interaction was near synergistic. The addition of bedaquiline/M2 metabolite was evaluated by forming a 95% confidence interval regarding the decline in bacterial burden. The addition of bedaquiline/M2 metabolite shortened the time to eradication by 1 week and was significantly different. A model-based system approach to evaluating combinations of 3 agents shows promise to rapidly identify the most promising combinations that can then be trialed.

Population Pharmacokinetics of Imipenem in Critically Ill Patients: A Parametric and Nonparametric Model Converge on CKD-EPI Estimated Glomerular Filtration Rate as an Impactful Covariate.

BACKGROUND: Population pharmacokinetic (popPK) models for antibiotics are used to improve dosing strategies and individualize dosing by therapeutic drug monitoring. Little is known about the differences in results of parametric versus nonparametric popPK models and their potential consequences in clinical practice. We developed both parametric and nonparametric models of imipenem using data from critically ill patients and compared their results. METHODS: Twenty-six critically ill patients treated with intravenous imipenem/cilastatin were included in this study. Median estimated glomerular filtration rate (eGFR) measured by the Chronic Kidney Disease Epidemiology Collaboration (CKD-EPI) equation was 116 mL/min/1.73 m2 (interquartile range 104-124) at inclusion. The usual dosing regimen was 500 mg/500 mg four times daily. On average, five imipenem levels per patient (138 levels in total) were drawn as peak, intermediate, and trough levels. Imipenem concentration-time profiles were analyzed using parametric (NONMEM 7.2) and nonparametric (Pmetrics 1.5.2) popPK software. RESULTS: For both methods, data were best described by a model with two distribution compartments and the CKD-EPI eGFR equation unadjusted for body surface area as a covariate on the elimination rate constant (Ke). The parametric population parameter estimates were Ke 0.637 h-1 (between-subject variability [BSV]: 19.0% coefficient of variation [CV]) and central distribution volume (Vc) 29.6 L (without BSV). The nonparametric values were Ke 0.681 h-1 (34.0% CV) and Vc 31.1 L (42.6% CV). CONCLUSIONS: Both models described imipenem popPK well; the parameter estimates were comparable and the included covariate was identical. However, estimated BSV was higher in the nonparametric model. This may have consequences for estimated exposure during dosing simulations and should be further investigated in simulation studies.

An Algorithm for Nonparametric Estimation of a Multivariate Mixing Distribution with Applications to Population Pharmacokinetics.

Population pharmacokinetic (PK) modeling has become a cornerstone of drug development and optimal patient dosing. This approach offers great benefits for datasets with sparse sampling, such as in pediatric patients, and can describe between-patient variability. While most current algorithms assume normal or log-normal distributions for PK parameters, we present a mathematically consistent nonparametric maximum likelihood (NPML) method for estimating multivariate mixing distributions without any assumption about the shape of the distribution. This approach can handle distributions with any shape for all PK parameters. It is shown in convexity theory that the NPML estimator is discrete, meaning that it has finite number of points with nonzero probability. In fact, there are at most N points where N is the number of observed subjects. The original infinite NPML problem then becomes the finite dimensional problem of finding the location and probability of the support points. In the simplest case, each point essentially represents the set of PK parameters for one patient. The probability of the points is found by a primal-dual interior-point method; the location of the support points is found by an adaptive grid method. Our method is able to handle high-dimensional and complex multivariate mixture models. An important application is discussed for the problem of population pharmacokinetics and a nontrivial example is treated. Our algorithm has been successfully applied in hundreds of published pharmacometric studies. In addition to population pharmacokinetics, this research also applies to empirical Bayes estimation and many other areas of applied mathematics. Thereby, this approach presents an important addition to the pharmacometric toolbox for drug development and optimal patient dosing.

Interchangeability of Generic Drugs: A Nonparametric Pharmacokinetic Model of Gabapentin Generic Drugs.

Substitution by generic drugs is allowed when bioequivalence to the originator drug has been established. However, it is known that similarity in exposure may not be achieved at every occasion for all individual patients when switching between formulations. The ultimate aim of our research is to investigate if pharmacokinetic subpopulations exist when subjects are exposed to bioequivalent formulations. For that purpose, we developed a pharmacokinetic model for gabapentin, based on data from a previously conducted bioavailability study comparing gabapentin exposure following administration of the gabapentin originator and three generic gabapentin formulations in healthy subjects. Both internal and external validation confirmed that the optimal model for description of the gabapentin pharmacokinetics in this comparative bioavailability study was a two-compartment model with absorption constant, an absorption lag time, and clearance adjusted for renal function, in which each model parameter was separately estimated per administered formulation.

Prospective Trial on the Use of Trough Concentration versus Area under the Curve To Determine Therapeutic Vancomycin Dosing.

We hypothesized that dosing vancomycin to achieve trough concentrations of >15 mg/liter overdoses many adults compared to area under the concentration-time curve (AUC)-guided dosing. We conducted a 3-year, prospective study of vancomycin dosing, plasma concentrations, and outcomes. In year 1, nonstudy clinicians targeted trough concentrations of 10 to 20 mg/liter (infection dependent) and controlled dosing. In years 2 and 3, the study team controlled vancomycin dosing with BestDose Bayesian software to achieve a daily, steady-state AUC/MIC ratio of ≥400, with a maximum AUC value of 800 mg · h/liter, regardless of trough concentration. For Bayesian estimation of AUCs, we used trough samples in years 1 and 2 and optimally timed samples in year 3. We enrolled 252 adults who were ≥18 years old with ≥1 available vancomycin concentration. Only 19% of all trough concentrations were therapeutic versus 70% of AUCs (P < 0.0001). After enrollment, median trough concentrations by year were 14.4, 9.7, and 10.9 mg/liter (P = 0.005), with 36%, 7%, and 6% over 15 mg/liter (P < 0.0001). Bayesian AUC-guided dosing in years 2 and 3 was associated with fewer additional blood samples per subject (3.6, 2.0, and 2.4; P = 0.003), shorter therapy durations (8.2, 5.4, and 4.7 days; P = 0.03), and reduced nephrotoxicity (8%, 0%, and 2%; P = 0.01). The median inpatient stay was 20 days among nephrotoxic patients versus 6 days (P = 0.002). There was no difference in efficacy by year, with 42% of patients having microbiologically proven infections. Compared to trough concentration targets, AUC-guided, Bayesian estimation-assisted vancomycin dosing was associated with decreased nephrotoxicity, reduced per-patient blood sampling, and shorter length of therapy, without compromising efficacy. These benefits have the potential for substantial cost savings. (This study has been registered at ClinicalTrials.gov under registration no. NCT01932034.).

Sofosbuvir (SOF) Suppresses Ledipasvir (LDV)-resistant Mutants during SOF/LDV Combination Therapy against Genotype 1b Hepatitis C Virus (HCV).

Our objective was to identify drug interactions between ledipasvir (LDV) and sofosbuvir (SOF) against a genotype 1b replicon to determine optimal exposures for each agent that will maximize antiviral activity against susceptible and drug-resistant subpopulations. LDV and SOF were evaluated using a fully factorial experimental design in the BelloCell system. Replicon levels and drug-resistant variants were quantified at various times post-therapy for 14 days and a high-dimensional mathematical model was fit to the data. Mutations associated with SOF resistance were not detected; but LDV-resistant mutants were selected and mutant subpopulations increased as exposure intensity increased. Combination therapy was additive for the total replicon population and the LDV-resistant population, but a threshold concentration of 100 ng/ml of SOF must be attained to suppress LDV-resistant subpopulations. These novel findings hold important implications for not only improving therapeutic outcomes, but also maximizing the clinical utility of LDV and SOF combination regimens.

Two general methods for population pharmacokinetic modeling: non-parametric adaptive grid and non-parametric Bayesian.

Population pharmacokinetic (PK) modeling methods can be statistically classified as either parametric or nonparametric (NP). Each classification can be divided into maximum likelihood (ML) or Bayesian (B) approaches. In this paper we discuss the nonparametric case using both maximum likelihood and Bayesian approaches. We present two nonparametric methods for estimating the unknown joint population distribution of model parameter values in a pharmacokinetic/pharmacodynamic (PK/PD) dataset. The first method is the NP Adaptive Grid (NPAG). The second is the NP Bayesian (NPB) algorithm with a stick-breaking process to construct a Dirichlet prior. Our objective is to compare the performance of these two methods using a simulated PK/PD dataset. Our results showed excellent performance of NPAG and NPB in a realistically simulated PK study. This simulation allowed us to have benchmarks in the form of the true population parameters to compare with the estimates produced by the two methods, while incorporating challenges like unbalanced sample times and sample numbers as well as the ability to include the covariate of patient weight. We conclude that both NPML and NPB can be used in realistic PK/PD population analysis problems. The advantages of one versus the other are discussed in the paper. NPAG and NPB are implemented in R and freely available for download within the Pmetrics package from www.lapk.org.

Accurate detection of outliers and subpopulations with Pmetrics, a nonparametric and parametric pharmacometric modeling and simulation package for R.

INTRODUCTION: Nonparametric population modeling algorithms have a theoretical superiority over parametric methods to detect pharmacokinetic and pharmacodynamic subgroups and outliers within a study population. METHODS: The authors created "Pmetrics," a new Windows and Unix R software package that updates the older MM-USCPACK software for nonparametric and parametric population modeling and simulation of pharmacokinetic and pharmacodynamic systems. The parametric iterative 2-stage Bayesian and the nonparametric adaptive grid (NPAG) approaches in Pmetrics were used to fit a simulated population with bimodal elimination (Kel) and unimodal volume of distribution (Vd), plus an extreme outlier, for a 1-compartment model of an intravenous drug. RESULTS: The true means (SD) for Kel and Vd in the population sample were 0.19 (0.17) and 102 (22.3), respectively. Those found by NPAG were 0.19 (0.16) and 104 (22.6). The iterative 2-stage Bayesian estimated them to be 0.18 (0.16) and 104 (24.4). However, given the bimodality of Kel, no subject had a value near the mean for the population. Only NPAG was able to accurately detect the bimodal distribution for Kel and to find the outlier in both the population model and in the Bayesian posterior parameter estimates. CONCLUSIONS: Built on over 3 decades of work, Pmetrics adopts a robust, reliable, and mature nonparametric approach to population modeling, which was better than the parametric method at discovering true pharmacokinetic subgroups and an outlier.

Nonlinear dynamic neural network for text-independent speaker identification using information theoretic learning technology.

In this paper we present a novel design for a nonlinear dynamic neural network to implement text-independent speaker recognition without the benefit of exact voice signatures. The dynamic properties between the input neuron and the output neuron make use of a nonlinear high-order synaptic neural model with memory of previous input signals. The dynamic neural network is realized in the short-term-frequency long-term-temporal domain. Informatics metric is used to overcome the challenge of performing blind learning for the nonlinear network. The goal of this study is not only to improve the recognition performance but also to amplify the distinctiveness among different speakers.