REŠAVANJE JEDNOG PROBLEMA U ORGANSKOJ HEMIJI METODOM EKVIVALENTNIH JEDNAČINA

U teorijskoj organskoj hemiji, zakoni molekulsko – orbitalne teorije omogućuju da se odrede energetska stanja π – elektrona u konjugovanim (nezasićenim) jedinjenjima. Pri ovome energetski nivoi elektrona, dobijaju se razvijanjem odgovarajuće determinante koja se formira koristeći tri karakteristične matrice po definiciji. Svakom jedinjenju, pri ovome pridružuje se odgovarajući polinom, dok će energetski nivoi zavisiti od njegovih nula. Kod velike većine jedinjenja koja se susreću u teoriji i praksi, dobijeni polinom uvek ima bar jednu racionalnu nulu pa se problem rešava jednostavno metodom faktorizacije. Problem nastaje kada karakteristični polinom nema ni jednu racionalnu nulu. U ovom slučaju moguća je primena gotovih matematičkih formula koje su međutim nepodesne za praktičnu upotrebu. Isto tako moguća je i primena numeričkih metoda, koje daju samo približno realno rešenje koje nije pogodno za primenu s obzirom da se gotovo uvek traži tačno realno rešenje. Uzimajući u obzir predhodno, u radu biće dat predlog za rešavanje gore navedenog problema na bazi formiranja ekvivalentnih jednačina preko odgovarajućih matematičkih transformacija, koje omogućuju da se na relativno jednostavan način dobiju tačno sve realne nule polinoma. U prikazanom radu, rešavanje matematičkog problema biće ilustrovano na nekoliko karakterističnih primera. Tačnost dobijenih rezultata prema predloženoj metodi biće izvedena poređenjem sa približnim numeričkim rešenjem koristeći odgovarajući računarski program, odnosno numeričke metode

Cumulative, additive benefits of memantine-donepezil combination over component monotherapies in moderate to severe Alzheimer’s dementia: a pooled area under the curve analysis

Introduction: Treatment in moderate or severe Alzheimer’s disease (AD) often involves adding memantine to a cholinesterase-inhibitor (ChEI: donepezil, galantamine, rivastigmine). Evidence from six-month randomized trials and long-term observational studies supports superiority of memantine-ChEI combination to ChEI monotherapy. We utilized area-under-the-curve (AUC) analysis to assess six-month cumulative treatment efficacy of memantine-donepezil combination versus component monotherapies on individual clinical domains and on a composite index. Methods: Data were pooled from 1,408 individuals with moderate to severe AD from four six-month randomized trials of memantine monotherapy (n = 570) or add-on therapy (donepezil-only subset: n = 847). AUC changes from baseline on measures of cognition (SIB), function (ADCS-ADL19), behavior (NPI), global status (CIBIC-Plus), and a composite index (4D-CI: equally weighted composite of four domain measures) were calculated using the trapezoidal rule and evaluated via analysis of covariance (ANCOVA) (2-sided-α = 0.05). AUC results were contrasted with visit-by-visit changes from baseline (“snapshot analysis”), performed using a mixed-effects model with repeated measures (MMRM). Results: Over the entire six-month period, placebo-only treatment was associated with significant cumulative worsening on all outcomes. Memantine-donepezil combination showed significantly greater AUC improvements (point x week) on the SIB, NPI, and CIBIC-Plus than placebo-donepezil (SIB: 68.4 versus 32.0, P = 0.019; NPI: −74.3 versus −28.2, P = 0.003; CIBIC-Plus: −2.5 versus 1.4, P = 0.006) and memantine-only monotherapies (SIB: 68.4 versus 12.0, P <0.001; NPI: −74.3 versus −7.4, P <0.001; CIBIC-Plus: −2.5 versus 2.7, P <0.001), whereas these comparisons were not significant for the ADCS-ADL19 (memantine-donepezil (1.4) versus placebo-donepezil (−0.9), P = 0.407; versus memantine-only (−12.2), P = 0.310). Composite index analysis demonstrated significant cumulative advantages of memantine-donepezil combination (630.0) over placebo-donepezil (344.7, P <0.001) and memantine-only (152.1, P <0.001) treatments. Combining memantine and donepezil had an additive effect. Compared with AUC analysis, baseline-to-endpoint change-score analysis underestimated effects of combination therapy, monotherapies, or both. Conclusions: This large pooled area-under-the-curve analysis of randomized-trial data in moderate to severe AD provides ecologically valid support that adding memantine to stable donepezil results in overall clinical benefits that are additive compared with individual monotherapies, continue to accumulate through six-month treatment, and are at least 50% greater than those of monotherapies