We study Poisson Galton--Watson trees on a standard Borel type space when the offspring kernel is multiplied by a scalar parameter. On finite trees, we identify the Radon--Nikodym derivative between two parameter values and show that it remains measurable after projection to the total progeny measure. Under a uniform bound on the offspring intensities, differentiation yields exact differential and integral equations for the projected laws without irreducibility, reversibility, or a positive eigenfunction. With an additional positive eigenfunction bounded above and away from zero, we relate these equations to an infinite spinal tree, uniform pruning, the Doob transform, and the Aldous--Pitman ascension process. For a uniformly bounded offspring kernel, we also prove uniform exponential integrability of the total progeny throughout the spectrally subcritical regime. As an application, under the graphical-kernel assumptions of Bollobas, Janson and Riordan, the number $K_n$ of connected components satisfies $K_n/n \to {\mathbb E}_{\pi}[1/T_u]$ in probability and in $L^1$, where $T_u$ is the total progeny of the associated branching process and $1/\infty=0$. If $q_u(x)$ is its extinction probability from type $x$, re-rooting and extinction duality give the explicit limit $$ \int_S q_u(x)\,\pi(dx) - {u\over 2}\int_{S\times S}\kappa(x,y)q_u(x)q_u(y)\,\pi(dx)\pi(dy). $$ This extends the finite-type and compact-continuous formulas to the full BJR graphical-kernel setting, allowing separable noncompact type spaces and kernels that may be unbounded or reducible.
Osteoporosis is a systemic metabolic bone disease characterized by decreased bone mass, destruction of bone microstructure, and increased risk of fractures. Traditional views mainly attribute it to an imbalance between osteoblast-mediated bone formation and osteoclast-mediated bone resorption, but increasing evidence suggests that the pathogenesis and progression of osteoporosis are also regulated by complex interactions among the nervous system, immune system, and skeletal system. The neuro-immune-bone axis provides an important framework for integrating the bidirectional regulation between innervation, immune microenvironment, and bone remodeling process. This article systematically reviews the basis of interactions among sensory nerves, sympathetic nerves, immune cells, and bone cells in bone homeostasis, with a focus on the roles of sympathetic nerve activation, sensory neuropeptide imbalance, and immune inflammatory remodeling in the progression of osteoporosis. Furthermore, this article compares the differential imbalance patterns of this axis in postmenopausal osteoporosis, age-related osteoporosis, and secondary osteoporosis, and accordingly discusses its potential significance for disease classification and risk identification. At the therapeutic level, this article further distinguishes between clinically well-established management strategies and mechanistic interventions still in the exploratory stage. The former includes guideline-recommended anti-osteoporosis drugs, exercise, nutritional support, fall prevention, and control of primary diseases; the latter includes strategies such as sympathetic nerve regulation, CGRP-related interventions, Treg/Th17 balance regulation, and macrophage polarization, which are currently mainly based on animal experiments, mechanistic studies, observational evidence, or early translational research and cannot yet be considered routine clinical treatment. Therefore, the neuro-immune-bone axis is currently more suitable as a theoretical framework to explain the heterogeneity of osteoporosis and guide mechanistic research, and its practical value in precise classification, treatment selection, and clinical decision-making still requires further validation through prospective cohort and intervention trials. By integrating existing evidence, this article aims to offer a theoretical foundation for understanding the multi-system interaction mechanisms of osteoporosis, establishing an evidence stratification framework, and exploring future individualized interventions.
Yupeng Zhang, Junchen Lu, Guojun Deng et al.· Frontiers in Immunology· 0 citations
