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Pain can be present in up to 50% of people with post-COVID-19 condition. Understanding the complexity of post-COVID pain can help with better phenotyping of this post-COVID symptom. The aim of this study is to describe the complex associations between sensory-related, psychological, and cognitive variables in previously hospitalized COVID-19 survivors with post-COVID pain, recruited from three hospitals in Madrid (Spain) by using data-driven path analytic modeling. Demographic (i.e., age, height, and weight), sensory-related (intensity or duration of pain, central sensitization-associated symptoms, and neuropathic pain features), psychological (anxiety and depressive levels, and sleep quality), and cognitive (catastrophizing and kinesiophobia) variables were collected in a sample of 149 subjects with post-COVID pain. A Bayesian network was used for structural learning, and the structural model was fitted using structural equation modeling (SEM). The SEM model fit was excellent: RMSEA < 0.001, CFI = 1.000, SRMR = 0.063, and NNFI = 1.008. The only significant predictor of post-COVID pain was the level of depressive symptoms (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>=</mo><mn>0.241</mn></mrow></semantics></math></inline-formula>, <i>p</i> = 0.001). Higher levels of anxiety were associated with greater central sensitization-associated symptoms by a magnitude of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>=</mo><mn>0.406</mn></mrow></semantics></math></inline-formula> (<i>p</i> = 0.008). Males reported less severe neuropathic pain symptoms (−1.50 SD S-LANSS score, <i>p</i> < 0.001) than females. A higher level of depressive symptoms was associated with worse sleep quality (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>=</mo><mn>0.406</mn></mrow></semantics></math></inline-formula>, <i>p</i> < 0.001), and greater levels of catastrophizing (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>=</mo><mn>0.345</mn></mrow></semantics></math></inline-formula>, <i>p</i> < 0.001). This study presents a model for post-COVID pain where psychological factors were related to central sensitization-associated symptoms and sleep quality. Further, maladaptive cognitions, such as catastrophizing, were also associated with depression. Finally, females reported more neuropathic pain features than males. Our data-driven model could be leveraged in clinical trials investigating treatment approaches in COVID-19 survivors with post-COVID pain and can represent a first step for the development of a theoretical/conceptual framework for post-COVID pain.