The paper is devoted to the investigation of regional observability with constraints on the state of semilinear parabolic systems. The purpose is to reconstruct the initial state between two prescribed functions only on an internal subregion $\omega$ of the system evolution domain $\Omega$. The proofs use two approaches, the subdifferential and HUM approach. Finally, a numerical example is provided to verify the effectiveness of our theory results.

Embryonic genetic mechanisms are present in the brain and ready to be placed into action upon cellular injury termed the response to injury wound healing (RTIWH) mechanism. When injured, regional brain endothelial cells initially undergo activation and dysfunction with initiation of hemostasis, inflammation (peripheral leukocytes and innate microglia and perivascular macrophage cells), proliferation (astrogliosis), remodeling, repair, and resolution phases if the injurious stimuli are removed. In conditions wherein the injurious stimuli are chronic as occurs in obesity, metabolic syndrome, and type 2 diabetes mellitus this process does not undergo resolution and there is persistent RTIWH with remodeling. Indeed, the brain is unique, in that it utilizes its neuroglia: the microglia cell along with peripheral inflammatory cells and its astroglia instead of peripheral scar-forming fibrocytes/fibroblasts. The brain undergoes astrogliosis to form a gliosis scar instead of a fibrosis scar to protect the surrounding neuropil from regional parenchymal injury. One of the unique and evolving remodeling changes in the brain is the development of enlarged perivascular spaces (EPVS), which is the focus of this brief review. EPVS are important since they serve as a biomarker for cerebral small vessel disease and also represent an impairment of the effluxing glymphatic system that is important for the clearance of metabolic waste from the interstitial fluid to the cerebrospinal fluid and disposal. Therefore, it is important to better understand how the RTIWH mechanism is involved in the development of EPVS that are closely associated with and important to the development of premature and age-related cerebrovascular and neurodegenerative diseases with impaired cognition.

Abstract: Pericyte(s) (Pcs) and resident perivascular macrophages (rPVMΦs) are positioned perfectly in the neurovascular unit (NVU) and perivascular spaces to facilitate metainflammation that results in brain endothelial cell activation and dysfunction and neuroinflammation. Their positions within the NVU allow intimate contact with one another as follows: Brain endothelial cells (BECs) and the Pcs via their shared basement membrane and physical contact peg-socket junctions with N cadherins and gap junctions Cx43; Pcs and rPVMΦs intimate contacts. Additionally, rPVMΦs have intimate contact with the astrocyte endfeet (ACef) that form the outermost membrane of perivascular spaces. Importantly, ACef have intimate contact with BECs that have intimate physical contact with neuronal axons and dendrites to complete NVU coupling. The multiplicity of intimate contacts of NVU cells allow for continuous crosstalk communications to provide brain homeostasis. While each of the cells of the NVU play important roles in the development of enlarged perivascular spaces (EPVS), this review focuses on the Pcs and rPVMΦs and discusses each of the intimate contacts and their functional significance in detail with numerous illustrations and transition electron microscopic images to demonstrate their role in the development of EPVS. EPVS are known to be biomarkers for cerebral small vessel disease and impaired glymphatic system waste clearance. Additionally, EPVS associate with mixed dementias of vascular dementia, vascular contributions to cognitive impairment and dementia, and Alzheimer’s disease that results in high economic and psychosocial cost to the global community.

Abstract: The brain endothelial cell (BEC) glycocalyx (ecGCx) is a BEC surface coating consisting of a complex interwoven polysaccharide (sweet husk) mesh-like network of membrane-bound proteoglycans, glycoproteins, and glycosaminoglycans (GAGs) covering the apical luminal layer of the brain endothelial cells. The ecGCx may considered as the 1st barrier of a tripartite blood-brain barrier (BBB) consisting of 1) ecGCx; 2) BECs; (3) extravascular compartment of pericytes, extracellular matrix, and perivascular astrocytes, Perturbations of this barrier allow for increased permeability in the postcapillary venule that will be permissive to both fluids, solutes, and proinflammatory peripherally-derived leukocytes into the perivascular spaces (PVS) to result in enlargement as well as increased neuroinflammation. The ecGCx is known to have multiple functions, which include its physical and charge barrier, mechanical transduction, regulation of vascular permeability, modulation of inflammatory response, and anticoagulation functions. This review discusses each of the listed functions in detail and utilizes multiple transmission electron micrographs and illustrations to allow for a better understanding of the ecGCx structural and functional roles as it relates to enlarged perivascular spaces (EPVS). This is the 5th review of a series quintet that discuss the importance of EPVS from the perspective of the cells of brain barriers. Attenuation and/or loss of the ecGCx results in brain barrier disruption with increased permeability to proinflammatory leukocytes, fluids and solutes, which accumulate in the postcapillary venule perivascular spaces. This accumulation results in obstruction and results in EPVS with impaired waste removal of the recently recognized glymphatic system. Importantly, EPVS are progressively being regarded as a marker of cerebrovascular and neurodegenerative pathology.

Abstract: Astrocytes (ACs) are the most abundant human cell in the brain and importantly, are the master connecting and communicating cells that provide structural and functional support of brain cells at all levels of organization. Further, they are recognized to be the guardians and housekeepers of the brain. Protoplasmic perivascular astrocyte endfeet and their basal lamina form the delimiting outermost barrier (glia limitans) of the perivascular spaces in postcapillary venules and are important for the clearance of metabolic waste and comprise the glymphatic system, which is critically dependent on the proper waste removal by the pvACef polarized aquaporin 4 water channels. Also, the protoplasmic perisynaptic astrocyte endfeet (psACef) are important in cradling the neuronal synapses that serve to maintain homeostasis and serve a functional and supportive role for synaptic transmission. Enlarged perivascular spaces (EPVS) are emerging as important aberrant findings on magnetic resonance imaging (MRI) and associate with white matter hyperintensities, lacunes, aging, accepted to be biomarkers for cerebral small vessel disease, increased in obesity, metabolic syndrome, and type 2 diabetes. Knowledge is exponentially expanding regarding EPVS along with the glymphatic system, since EPVS are closely associated with impaired glymphatic function and waste removal from the brain to the cerebrospinal fluid and systemic circulation. This review intends to focus on how the pvACef play a crucial role in the development of EPVS.

Abstract: The recently described perivascular unit (PVU) resides immediately adjacent to the true capillary neurovascular unit (NVU) in the postcapillary venule and contains the normal-benign perivascular spaces (PVS) and pathological enlarged perivascular spaces (EPVS). The PVS are important, in that, they also have recently been identified to be the construct and the conduit responsible for the delivery of metabolic waste from the interstitial fluid to the ventricular cerebrospinal fluid for disposal into the systemic circulation and has been termed the glymphatic system. Importantly, the outermost boundary of the PVS is lined by protoplasmic perivascular astrocyte endfeet (pvACef) that communicate with regional neurons. As compared to the well-recognized and described neurovascular unit (NVU) and NVU coupling, the PVU is less well understood and remains an emerging concept. The primary focus of this narrative review is to compare the similarities and differences between these two units and discuss each of their structural and functional relationships and how they relate not only to brain homeostasis but also how they may relate to the development of multiple clinical neurological disease states and specifically how they may relate to obesity, metabolic syndrome, and type 2 diabetes mellitus. Additionally, the concept and importance of a perisynaptic astrocyte coupling to the neuronal synapses with pre- and postsynaptic neurons will also be considered as a perisynaptic unit to provide for the creation of the information transfer in the brain via synaptic transmission and brain homeostasis. Multiple electron microscopic images and illustrations will be utilized in order to help explain these complex units.