DOOB-MARTIN COMPACTIFICATION OF A MARKOV CHAIN FOR GROWING RANDOM WORDS SEQUENTIALLY.

We consider a Markov chain that iteratively generates a sequence of random finite words in such a way that the nth word is uniformly distributed over the set of words of length 2n in which n letters are a and n letters are b: at each step an a and a b are shuffled in uniformly at random among the letters of the current word. We obtain a concrete characterization of the Doob-Martin boundary of this Markov chain and thereby delineate all the ways in which the Markov chain can be conditioned to behave at large times. Writing N(u) for the number of letters a (equivalently, b) in the finite word u, we show that a sequence (un ) n∈ℕ of finite words converges to a point in the boundary if, for an arbitrary word ν, there is convergence as n tends to infinity of the probability that the selection of N(ν) letters a and N(ν) letters b uniformly at random from un and maintaining their relative order results in ν. We exhibit a bijective correspondence between the points in the boundary and ergodic random total orders on the set {a1, b1, a2, b2, …} that have distributions which are separately invariant under finite permutations of the indices of the a's and those of the b's. We establish a further bijective correspondence between the set of such random total orders and the set of pairs (µ, ν) of diffuse probability measures on [0,1] such that ½(µ + ν) is Lebesgue measure: the restriction of the random total order to {a1, b1,…, an, bn } is obtained by taking X1,…, Xn (resp. Y1,… ,Yn ) i.i.d. with common distribution µ (resp. ν), letting (Z1,…, Z2n) be {X1, Y1,…, Xn , Yn } in increasing order, and declaring that the kth smallest element in the restricted total order is ai (resp. bj ) if Zk = Xi (resp. Zk = Yj ).

Image-based ex-vivo drug screening for patients with aggressive haematological malignancies: interim results from a single-arm, open-label, pilot study.

BACKGROUND: Patients with refractory or relapsed haematological malignancies have few treatment options and short survival times. Identification of effective therapies with genomic-based precision medicine is hampered by intratumour heterogeneity and incomplete understanding of the contribution of various mutations within specific cancer phenotypes. Ex-vivo drug-response profiling in patient biopsies might aid effective treatment identification; however, proof of its clinical utility is limited. METHODS: We investigated the feasibility and clinical impact of multiparametric, single-cell, drug-response profiling in patient biopsies by immunofluorescence, automated microscopy, and image analysis, an approach we call pharmacoscopy. First, the ability of pharmacoscopy to separate responders from non-responders was evaluated retrospectively for a cohort of 20 newly diagnosed and previously untreated patients with acute myeloid leukaemia. Next, 48 patients with aggressive haematological malignancies were prospectively evaluated for pharmacoscopy-guided treatment, of whom 17 could receive the treatment. The primary endpoint was progression-free survival in pharmacoscopy-treated patients, as compared with their own progression-free survival for the most recent regimen on which they had progressive disease. This trial is ongoing and registered with ClinicalTrials.gov, number NCT03096821. FINDINGS: Pharmacoscopy retrospectively predicted the clinical response of 20 acute myeloid leukaemia patients to initial therapy with 88·1% accuracy. In this interim analysis, 15 (88%) of 17 patients receiving pharmacoscopy-guided treatment had an overall response compared with four (24%) of 17 patients with their most recent regimen (odds ratio 24·38 [95% CI 3·99-125·4], p=0·0013). 12 (71%) of 17 patients had a progression-free survival ratio of 1·3 or higher, and median progression-free survival increased by four times, from 5·7 (95% CI 4·1-12·1) weeks to 22·6 (7·4-34·0) weeks (hazard ratio 3·14 [95% CI 1·37-7·22], p=0·0075). INTERPRETATION: Routine clinical integration of pharmacoscopy for treatment selection is technically feasible, and led to improved treatment of patients with aggressive refractory haematological malignancies in an initial patient cohort, warranting further investigation. FUNDING: Austrian Academy of Sciences; European Research Council; Austrian Science Fund; Austrian Federal Ministry of Science, Research and Economy; National Foundation for Research, Technology and Development; Anniversary Fund of the Austrian National Bank; MPN Research Foundation; European Molecular Biology Organization; and Swiss National Science Foundation.