The new work, led by Martin Lubich of University Hospital Sofiamed together with Krassimir Atanassov, Peter Vassilev, Vassia Atanassova, Chavdar Slavov, Elenko Popov and Lyudmila Todorova, uses a modelling framework known as generalized nets. These structures, an extension of Petri nets developed by Atanassov and colleagues, are designed to represent parallel, concurrent and interdependent processes. That makes them unusually well suited to physiology, where material flows such as blood and urine run simultaneously with informational flows such as the signals exchanged between the bladder, the nervous system and the muscles. The model is conceptually aligned with network physiology, the perspective that views the body as interacting subsystems rather than isolated organs.

The physiological basis of the model is drawn from established urological literature. The kidneys, two bean-shaped organs, receive roughly twenty percent of cardiac output through the renal arteries, which branch down to the glomerular capillary plexus. Each kidney contains on the order of one to one and a half million nephrons, the functional units where filtration begins. Blood enters the glomerulus through the afferent arteriole, and hydrostatic pressure drives water and small solutes across the filtration barrier, which consists of the fenestrated capillary endothelium, the glomerular basement membrane and the filtration slits between podocyte foot processes. Blood cells and most plasma proteins are held back and leave through the efferent arteriole to rejoin the bloodstream.

The numbers involved are striking. In a healthy young adult male, the glomerular filtration rate is approximately 180 liters per 24 hours, or about 125 milliliters per minute. With a typical blood volume of around five liters and a plasma volume near three liters, renal blood flow runs at about 1.1 liters per minute. Of the roughly 605 milliliters of plasma entering the glomeruli each minute, about 125 milliliters, nearly a fifth, is filtered, while the remaining 480 milliliters continues through the efferent arterioles. The filtration rate depends on the water permeability of the filtration membrane, the available filtration surface area and the net filtration pressure, the algebraic sum of hydrostatic and oncotic pressures. In many diseases, the reduction in filtration rate stems not from pressure changes but from the loss of functioning nephrons, which shrinks the total glomerular filtration surface.

In the generalized net model, these nephron-level processes, including glomerular filtration, tubular reabsorption, tubular secretion and urine concentration, are aggregated within each kidney transition. The model contains eight transitions, nineteen places and seventeen types of tokens. The transitions represent the cardiovascular system, the left and right kidneys, the bladder, the nervous system, the prostate gland, the muscle system and the urethra. Tokens carry the material and informational content: blood tokens flow from the cardiovascular system to each kidney and return as purified blood, while urine tokens travel toward the bladder. Status tokens monitor and report the current state of each organ, including the quantity of urine in the bladder and, in the pathological case, the residual urine left after incomplete emptying.

The essence of the model can be described in a few sentences. Blood from the cardiovascular system enters the left and right kidneys and, after being purified, returns to the cardiovascular system, while the substances excreted through the ureters go to the bladder. When the bladder is full, a signal is sent from it through the nervous system to the bladder muscles, which causes urine to be pushed out of the bladder into the urethra. When the prostate gland is functioning normally, urine passes through the prostatic urethra without significant obstruction. In the pathological branch, prostate-related bladder outlet obstruction reduces urinary outflow and may result in incomplete bladder emptying and residual urine.

The prostate itself is the pivot of the pathological branch. The gland, part of the male reproductive system that produces seminal fluid, typically has a volume of about 20 milliliters in young adult men. It sits beneath the bladder and in front of the rectum, surrounding the proximal portion of the urethra. With advancing age, particularly in association with benign prostatic hyperplasia, the gland may enlarge, and a volume greater than approximately 30 milliliters is commonly used as a clinical threshold for prostatic enlargement. An enlarged prostate compresses the urethra and impedes the normal flow of urine, producing a weak urinary stream, difficulty initiating urination, nocturia, a sensation of incomplete emptying, double voiding and the need to apply intra-abdominal pressure during urination. These symptoms are collectively known as lower urinary tract symptoms.

What makes the model technically interesting is how it encodes the difference between normal and pathological states. In the bladder transition, when there is enough urine, a token splits and a signal token enters the nervous system with one of two characteristics: a signal from stretch receptors if there is no problem with the prostate, or an increased frequency of bladder emptying impulses if there is. That detail mirrors a real clinical phenomenon: obstructed bladders tend to send more frequent, more urgent signals, which is why men with prostate obstruction often experience nocturia and frequency. The nervous system then relays a contraction signal to the muscle system, the detrusor contracts, and urine leaves the bladder. In the pathological case, the residual quantity of urine is reflected in the updated characteristic of the bladder-status token, which remains in place, capturing the clinical reality of incomplete emptying.

The construction was deliberately analytical. No patient-level dataset, automated model-generation tool or proprietary software was used; the anatomical entities, physiological processes and interactions were identified on the basis of the cited medical literature and the authors’ clinical expertise. The formal specification was examined for internal consistency, checking the correspondence between input and output places, the continuity of token pathways, the use of declared token types, the consistency of transition predicates and the symmetry of the left- and right-kidney representations. The authors note that the model extends earlier generalized net research on the human body and complements more focused models concerning nephrological diseases, prostate cancer development and the evaluation of lower urinary tract symptoms.

The limitations are acknowledged candidly. The model is intentionally qualitative and operates at an aggregated organ-system level; it does not represent detailed intrarenal, urodynamic, neurological or muscular mechanisms, nor does it distinguish among different degrees of obstruction. No clinical or data-driven validation has been performed, and such validation would require temporally aligned measurements of renal function, bladder filling and emptying, neural signalling, detrusor activity, obstruction and urine flow in individual patients. Despite this, the framework provides a formal basis for the qualitative comparison of normal and pathological states, and future development may include clinically defined degrees of obstruction, enabling scenario-based simulation and comparison with patient observations. The work was supported by the Bulgarian National Science Fund under a grant on prognostic modelling in prostate cancer patients. For a field where benign prostatic hyperplasia affects a large share of aging men, a formal, extensible map of the urinary relay, with the prostate explicitly in the path of flow, offers a foundation that educators, modellers and, eventually, clinical decision-support developers can build upon.

Subject of Research: Mathematical modelling of the male urinary system with prostate-related outflow obstruction

Article Title: Generalized net modelling of the male urinary system with prostate-related outflow obstruction

Article References: Lubich, M., Atanassov, K., Vassilev, P., Atanassova, V., Slavov, C., Popov, E., & Todorova, L. (2026). Generalized net modelling of the male urinary system with prostate-related outflow obstruction. Frontiers in Network Physiology, 6, Article 1888801. https://doi.org/10.3389/fnetp.2026.1888801

Image Credits: AI Generated

DOI: 10.3389/fnetp.2026.1888801

Keywords: generalized nets, mathematical modelling, prostate gland, benign prostatic hyperplasia, urinary system, network physiology, Generalized, modelling, male, urinary, system, prostate-related