Fraktale w anatomii
Powszechnie znanym fenomenem występującym w anatomii jest niezwykle efektywne wykorzystanie dostępnej przestrzeni- stosunkowo małe struktury mogą mieć bardzo dużą powierznię, na której zachodzą właściwe im procesy. Wiadomym jest, iż dzieję sie dzięki stopniowemu podziałowi na coraz mniejsze części. Jednakże spojrzenie na te struktury poprzez pryzmat zwykłej geometrii nie dostarcza kompletnych informacji na ich temat. Okazuje się, iż w wielu przypadkach dopiero analiza pod kątem fraktalności potrafi wyjaśnić szereg anatomicznych fenomenów. Jedną z cech fraktala jest to, iż jego struktura nie daje się łatwo opisać językiem tradycynej geometrii euklidesowej. To właśnie właściwości fraktalne, wyznaczone m.in. przez wymiar fraktalny, nadają strukurom najlepsze, najbardziej optymalne zdolności do wypełnienia przestrzeni (space-filling). Zgadza się to z tymi cechami fraktali jakimi są samopodobieństwo (każdy fragment podobny do siebie, zawiera wielokrotnie pomniejszone kopie) oraz nietrywialność struktury (zmiana skali, np. powiększenie ukazuje tak samo złożone i skomplikowane struktury jak wyjściowo). Zatem geometria fraktalna stoi jako podstawowa zasada projektowania biologicznych form, zapewniającą im m.in. najlepszą produktywność (1).
Najlepiej to zjawisko zobrazuje system rozgałęzień płuc, będący klasycznym przykładem fraktala (2,3) i matematyczny model drzewka fraktalnego.
źródło: Fractals in Biology and Medicine, vol IV (3). Kształty typu drzewko Kocha powstają za pomocą równań matematycznych. Zastanawiające czy za tworzenie struktur organicznych także odpowiadają równania matematyczne?
Następnym znakomitym przykładem struktury efektywnie wypełniającej przestrzeń jest system naczyń krwionośnych. Obliczono wymiar fraktalny tętnic płycnych: 2,71 oraz żył płucnych: 2,64 (3, 4) co zaskakująco zgadza się z przewidywaniami B. Mandelbrota, który wcześniej oszacował wymiar fraktalny drzewa krwionośnego na 2,7 (3)Istnieje ponadto wiele badań charakteryzujących fraktalną geometrię systemu krwionośnego oraz fraktalną dynamikę przepływu krwi. (5, 6, 7, 8, 9)
źródło: Fractals in Biology and Medicine, vol IV (3)
Fratalną organizację materii odkryto także w tkance kostnej (10, 11) dokładniej w istocie gąbczastej i budujących ją beleczkach kostnych. Tu również najczęściej mierzonym parametrem był wymiar frakatalny oraz, w niektórych pracach, lakunarność (fractal lacunarity) (12, 14) Znaczenie tego odkrycia w diagnostyce chorób kości, zwłaszcza w osteoporozie oraz w pomiarach gęstości mineralnej kości, została opisana pracach: 12, 13, 15
Pomiary wartości fraktalnych można prowadzić także w skali mikro. Rozwiązaniem problemu otrzymywania rozbieżnych danych przy tradycyjnym obliczaniu powierzchni retikulum endoplazmatycznego komórki wątroby okazało się dopiero ustalenie jej wymiaru fraktalnego (wynoszącego 2,7), obliczono także wymiar fraktalny wewnętrznej błony mitochondrium na (2,54) (3)
W neuronauce temat fraktali jest bardzo obszerny ze względu na złożoność budowy jak i skomplikowaną funkcjonalność układu nerwowego, którego rozwijające się modele stale odkrywają nowe, nieznane dotąd cechy. Jednakże wychodząc od podstawowej jednostki organizacyjnej jaką jest neuron, okazuje się, iż już w 1990 roku obliczono wymiar fraktalny komórek nerwowych siatkówki oka badając wzór rozgałęzień drzewa dendrytycznego (16)
Następnie podobne obserwacje poczyniono z dendrytycznymi rozgałęzieniami komórek nerwowych wzgórza (17) oraz róznymi typami komórek rdzenia kręgowego (18). Komórki piramidalne przeanalizowano także pod względem ich multifraktalności (19), która opisuje systemy fraktalne nie tylko pod względem pojedynczego parametry (np. najczęstszego wymiaru fraktalnego) a całego spektrum parametrów i ukazuje połączenia wielu fraktali ze sobą, będąc jeszcze bardziej skomplikowaną strukturą niż pojedynczy fraktal. Jest to jedynie niewielki wycinek z ogromnej ilości prac jakie zostały przeprowadzone w tej tematyce. Obszerną pracą podsumowującą znacenie fraktali w badaniach nad układem nerwowym oraz w neuronaukach jest „Fractals in the Nervous System: Conceptual Implications for Theoretical Neuroscience” (20) Przedstawione są w niej szczegółowo nie tylko badania nad fraktalną morfologią układu nerwowego ale także, co ciekawsze, koncepcje jego fizjologicznych funkcji w ujęciu fraktalnym. Jednym z wielu przykładów jest chociażby fraktalna analiza sygnałów EEG oraz rozpatrywanie fraktalnych właściwości jako grających rolę w funkcjonalnej integracji różnych poziomów neuronalnej organizacji, odnosząc się także nawet do funkcji poznawczych i emocjonalych, co rozpatruje praca „From Molecules to Mindfulness- how vertically convergent fractal time fluctuations unify cognition and emotion” (21)
Powyższe przykłady stanowią jedynie fragment ukazujący związki fraktali z medycyną koncentrując się na prawidłowych warunkach anatomicznych. Sugeruje się, iż wiedza na temat fraktali przyczyni się do stowrzenia gruntu pod bardziej holistyczne spojrzenie na funkcjonowanie organizmów żywych i ich środowiska oraz rozwoju nowoczesnych metod diagnostyki.
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