Topological Transformations in Hand Posture: A Biomechanical Strategy for Mitigating Raynaud’s Phenomenon Symptoms

Abstract

Raynaud’s Phenomenon (RP), characterized by episodic reductions in peripheral blood flow, leads to significant discomfort and functional impairment. Existing therapeutic strategies focus on pharmacological treatments, external heat supplementation and exercise-based rehabilitation, but fail to address biomechanical contributions to vascular dysfunction. We introduce a computational approach rooted in topological transformations of hand prehension, hypothesizing that specific hand postures can generate transient geometric structures that enhance thermal and hemodynamic properties. We examine whether a flexed hand posture—where fingers are brought together to form a closed-loop toroidal shape—may modify heat transfer patterns and blood microcirculation. Using a combination of heat diffusion equations, fluid dynamics models and topological transformations, we implement a heat transfer and blood flow simulation to examine the differential thermodynamic behavior of the open and closed hand postures. We show that the closed-hand posture may preserve significantly more heat than the open-hand posture, reducing temperature loss by an average of 1.1 ± 0.3 °C compared to 3.2 ± 0.5 °C in the open-hand condition (p < 0.01). Microvascular circulation is also enhanced, with a 53% increase in blood flow in the closed-hand configuration (p < 0.01). Therefore, our findings support the hypothesis that maintaining a closed-hand posture may help mitigate RP symptoms by preserving warmth, reducing cold-induced vasoconstriction and optimizing peripheral flow. Overall, our topologically framed approach provides quantitative evidence that postural modifications may influence peripheral vascular function through biomechanical and thermodynamic mechanisms, elucidating how shape-induced transformations may affect physiological and pathological dynamics.

1. Introduction

Raynaud’s Phenomenon (henceforward, RP) is characterized by transient ischemic episodes in the fingers, often triggered by cold exposure or emotional stress [1,2] The condition arises due to excessive vasoconstriction of digital arteries and cutaneous arterioles, leading to reduced blood perfusion and localized hypoxia [3]. Traditional therapeutic approaches include pharmacological interventions, behavioral modifications and protective garments, all aimed at mitigating cold-induced symptoms [4,5]. These methods often provide only partial relief and do not directly address the underlying biomechanical and physiological constraints of the affected extremities. Recent advances in biomechanics and mathematical modeling have offered novel insights into vascular regulation and thermodynamics in human physiology, suggesting that subtle changes in body posture and limb positioning can influence heat retention and circulatory dynamics [6,7]. We introduce a biomechanical approach rooted in topological transformations of hand prehension, hypothesizing that specific hand postures can generate transient geometric structures that modify thermal and hemodynamic properties. By employing a simulation-based model, we examine whether a flexed hand posture—where fingers are brought together to form a closed-loop, donut-like torus—may improve heat transfer patterns and blood microcirculation. This investigation builds upon existing topological and biomechanical principles [8,9,10], extending them into the realm of vascular physiology by proposing that the organization of the hand’s geometry can influence local blood flow and temperature regulation. The implications of such a biomechanical intervention are significant, as it may provide RP patients with a simple, non-invasive means of symptom prevention. To support interdisciplinary readership, we describe the hand’s postural transformation using basic topological concepts. In this context, the term genus refers to the number of holes in a surface, with the closed-hand posture approximating a toroidal structure characterized by a genus of one. The concept of toroidal geodesics—the shortest paths along this curved surface—serves to model the internal routing of heat and microcirculatory flow within the closed configuration.

Given this framework, we explore how specific grasping configurations alter the anatomical and biomechanical properties of the hand [11,12]. Hand prehension, which involves dynamic interactions between the digits and an object’s surface, is classified based on functional grasping patterns [13,14,15,16,17,18]. Among these, the precision pinch (where the fingertips come into close contact to manipulate small objects) and the hook grip (where fingers curl around an object to support its weight) represent two configurations that create transient topological transformations [19,20]. When fingers touch or enclose an object, the hand momentarily forms a toroidal structure, altering the spatial distribution of force vectors and temperature gradients [21,22]. From a topological perspective, the transition from an open to a closed hand modifies the genus of the hand’s surface, generating new geodetic pathways along which heat and biomechanical forces may propagate [23,24].

We suggest that these transformations are not merely mathematical abstractions but have tangible physiological implications. Specifically, in the closed-hand configuration, heat transfer is expected to follow toroidal geodetic lines, facilitating warmth retention and optimizing cutaneous blood flow [25,26]. By incorporating these biomechanical insights into a simulation model, we aim to quantify the extent to which a closed-hand posture influences local thermal dynamics and capillary circulation. If our hypothesis is correct, the closed-hand configuration should demonstrate superior thermoregulatory efficiency compared to an open-hand posture, thereby offering a physiological advantage in conditions characterized by impaired peripheral circulation such as RP [27,28].

We will proceed as follows. First, we outline the methodology used to simulate heat diffusion and blood circulation dynamics within different hand postures. Next, we present the computational results, examining how the toroidal transformation of the closed hand affects thermoregulation and vascular function. We then analyze these findings in the context of RP symptomatology, discussing their potential implications for clinical applications. Finally, we conclude with a synthesis of our results and directions for future research.

2. Materials and Methods

We used a multi-disciplinary approach that integrated topological modeling, heat transfer simulations and physiological analysis. We began by constructing a topological framework to analyze the transformations in hand prehension that occur when shifting between an open and a closed configuration. This was accomplished using principles of algebraic topology, treating the hand as a three-dimensional manifold whose genus changes depending on the degree of contact between the fingers and the palm. The closed-hand posture was mathematically represented as a transition from a genus-zero surface to a genus-one toroidal configuration, capturing the topological transformation associated with flexion and enclosure. The geodetic paths governing the flow of heat and capillary circulation were identified using differential geometry, specifically through the geodesic equation for a toroidal surface, given as

d₂xⁱ/ds₂ + Γⁱ_jk(dxʲ/ds)(dx^k/ds) = 0

where Γⁱ_jk are the Christoffel symbols of the toroidal metric and s is the arc-length parameter along the geodesic. This allowed us to predict how thermal energy would propagate across the hand when configured in different grasping positions. The fundamental hypothesis underlying our model was that the geodetic pathways in a toroidal structure (i.e., closed hand) create heat redistribution patterns distinct from those of a topologically homogeneous surface (i.e., open hand). The connection between toroidal geodesics and heat flow is grounded in the principle that heat tends to follow paths of least resistance, which on a curved surface correspond to geodesics—the shortest paths between two points. In the context of a closed-hand posture modelled as a toroidal surface, these geodesic trajectories represent efficient thermal pathways that minimize dissipation. Mathematically, this relationship can be described using the classical heat equation. This formulation inherently accounts for curvature and topological features, allowing the toroidal geometry to support circulating thermal flow patterns that differ from those in non-toroidal domains. The presence of nontrivial homology (i.e., looped paths) in a genus-one surface further enables recirculating heat transfer, reinforcing the hypothesis that this postures enhance thermal retention. Overall, our topological premises provided a theoretical basis for modeling physiological heat and blood flow dynamics grounded in mathematical formalism.

Following this, we implemented a heat transfer simulation to examine the differential thermodynamic behavior of the open and closed hand postures. To enhance reproducibility, a detailed summary

Table 1. Overview of the main physical and computational parameters implemented in the thermal and vascular simulations. Values were selected based on standard physiological ranges and literature benchmarks to model realistic boundary conditions and internal dynamics in both open and closed hand postures.

Parameter	Value	Unit	Description
Thermal conductivity (tissue)	0.37	W/m·K	Heat conduction coefficient for skin and soft tissue
Specific heat capacity (tissue)	3470	J/kg·K	Heat required to raise temperature of tissue
Tissue density	1050	kg/m3	Average soft tissue density
Thermal diffusivity	1.4 × 10−7	m2/s	Derived from α = k/(ρcp) \alpha = k/(\rho c_p) α = k/(ρcp)
Blood viscosity	3.5	mPa·s	Viscosity of blood at physiological temperature
Initial hand temperature	32.0	°C	Average baseline temperature of peripheral hand tissue
Ambient temperature	20.0	°C	External environment temperature
Time step duration	0.5	seconds	Duration of each simulation step
Total simulation time	60	seconds	Total duration for thermal and flow evolution
Grid resolution (mesh)	100 × 100	nodes	Discretized simulation grid
Boundary conditions	Convective & radiative	—	Applied at exposed surfaces to simulate heat exchange
Convective heat transfer coefficient	10	W/m2·K	Assumed for natural air convection around the hand
Blood flow model	Darcy-Weisbach	—	Used to estimate pressure–flow relationship
Heat transfer model	Heat diffusion equation	—	Solves temporal-spatial temperature distribution
Solver scheme	Explicit finite-difference	—	Numerical scheme used in simulation

is provided listing all key simulation parameters. The simulation was based on the classical heat diffusion equation:

∂T/∂t = ∇·(κ∇T)

where T is the temperature distribution over the hand, α = k/ρc_p is the thermal diffusivity, with k representing the thermal conductivity of human skin, ρ the tissue density and c_p the specific heat capacity. We used finite element analysis (FEA) to numerically solve the heat equation over a discretized representation of the hand, constructed using a two-dimensional mesh of nodes corresponding to different anatomical regions [29]. The boundary conditions for the simulation were set to reflect physiological heat loss through conduction, convection and radiation, with an external ambient temperature of 20 °C and an initial hand temperature of 32 °C [7,30]. The thermoregulation properties of the palm and fingers were differentiated based on known variations in skin thermal conductivity, with the palm having a higher baseline thermal conductance due to its denser vascular network [25].

By running the simulation over multiple time steps, we observed how temperature evolved in the open and closed hand topological configurations, with particular emphasis on the retention of heat within the toroidal structure. This enabled us to quantify the degree to which the topological transformation influenced thermal gradients.

To examine the impact of these thermal changes on blood circulation, we modeled microvascular perfusion using a porous media approach, in which blood flow through the hand was treated as fluid transport through a semi-permeable structure. This was governed by the Darcy–Weisbach equation:

ΔP/L = 8μQπ/R⁴

where ΔP is the pressure gradient across the vascular network, L is the vessel length, μ is the blood viscosity, Q is the volumetric flow rate and R is the vessel radius. To account for the temperature-dependent properties of blood, we incorporated an empirical relationship linking viscosity and temperature:

μ(T) = μ0e^−b(T−T0)

where μ₀ is the reference viscosity at body temperature T₀ and b is a scaling coefficient determined experimentally [31]. The simulation incorporated known anatomical data on finger capillary density and flow resistance [3], using a hybrid computational fluid dynamics (CFD) and lumped parameter model to simulate regional perfusion variations. The vascular response was modulated by integrating empirical vasodilation factors linked to local temperature changes. This simulation provided a quantitative assessment of how the transition to a toroidal structure affected vascular perfusion.

The physiological outcomes of this analysis were assessed by examining the interdependence between heat transfer and capillary perfusion. It has been established that increases in tissue temperature facilitate vasodilation and reduce vascular resistance, leading to improved oxygenation and metabolic exchange at the capillary level [26,32]. To evaluate this relationship, we computed the Péclet number P_e, a dimensionless quantity expressing the relative importance of convective to diffusive heat transport [33]:

Pe = Lv/α

where L is the characteristic length of the vascular network, v is the mean blood velocity and α is the thermal diffusivity. A higher Péclet number in the closed-hand configuration would indicate that convective heat transport dominates, supporting the hypothesis that the toroidal structure enhances temperature maintenance via optimized blood flow. Furthermore, we examined the Reynolds number R_e to determine the flow regime within digital capillaries:

Re = ρvD/μ

where D is the vessel diameter and ρ the blood density. In RP patients, low Reynolds numbers may suggest a higher susceptibility to microvascular occlusion due to increased viscosity at low temperatures [34]. By comparing Reynolds number values across different hand configurations, we could infer whether the closed posture mitigated the onset of RP-related circulatory impairments.

Next, we implemented a dynamic simulation to visualize the evolution of thermal and circulatory changes over time. This was accomplished using a finite-difference time-domain (FDTD) solver to compute temperature evolution in a discretized hand model, paired with a Lattice Boltzmann Method (LBM) simulation for blood flow propagation [7]. The sequence of computational steps included the following:

• Initializing the thermal field using empirical temperature data;
• Solving the heat diffusion equation iteratively using explicit time-stepping;
• Updating the blood viscosity and flow properties based on local temperature changes;
• Tracking the resultant changes in vascular perfusion.

Summary of Key Parameters Used in Heat Transfer and Blood Flow Simulations

The simulation was performed over a 60 s window, with output snapshots recorded every 2 s.

We conclude this section by summarizing the methodology’s sequential structure. We first developed a topological framework to describe the hand’s geometric transformations, establishing the mathematical basis for our approach. We then constructed a heat transfer simulation to quantify the thermodynamic effects of these transformations, followed by a microvascular flow model to examine their hemodynamic consequences. Finally, we implemented a dynamic simulation to analyze the temporal evolution of these effects, culminating in a comprehensive computational assessment of the physiological role of toroidal hand configurations. The subsequent section will detail the numerical results.

3. Results

Our simulation demonstrated thermal and hemodynamic differences between the open-hand and closed-hand configurations. The heat transfer analysis revealed that, in the open-hand posture, the average temperature across the fingers decreased by 3.2 ± 0.5 °C within 60 s, with localized temperature drops of up to 4.5 °C at the fingertips (Figure 1A). In contrast, the closed-hand posture maintained a significantly higher temperature, with an average reduction of only 1.1 ± 0.3 °C over the same period. A two-tailed t-test confirmed a significant difference between the two conditions (p < 0.01), indicating that the closed-hand posture preserved heat more effectively. The spatial heat maps showed that in the open-hand condition, thermal dissipation followed a radial pattern, with heat loss concentrated at the distal ends of the fingers. Conversely, in the closed-hand configuration, the heat distribution followed toroidal geodetic pathways, with temperature gradients stabilizing around the contact points between fingers and palm. The computed Péclet number (Pe) was 23.5 ± 2.1 in the closed-hand posture compared to 15.8 ± 1.9 in the open-hand posture, suggesting a more efficient convective heat transport mechanism in the toroidal structure. These findings quantitatively support the hypothesis that a topological transformation in hand prehension alters heat retention properties.

The microvascular flow analysis indicated that the closed-hand configuration exhibited significantly higher capillary perfusion than the open-hand posture (Figure 1A). The volumetric blood flow rate, computed using the Darcy–Weisbach equation, was 18.7 ± 1.4 mL/min in the closed-hand posture, compared to 12.2 ± 1.3 mL/min in the open-hand posture (p < 0.01). Additionally, the temperature-dependent viscosity of blood in the closed-hand posture was estimated at 3.2 ± 0.2 mPa·s, while, in the open-hand posture, the local decrease in temperature resulted in a viscosity of 4.0 ± 0.3 mPa·s, contributing to higher vascular resistance. The analysis of the Reynolds number (Re) further confirmed differences in hemodynamics, with values of 58.9 ± 4.6 in the closed-hand posture versus 42.5 ± 3.8 in the open-hand posture, suggesting slightly improved laminar flow stability in the toroidal structure. The time-dependent analysis showed that perfusion remained stable in the closed-hand configuration, while, in the open-hand condition, capillary flow declined progressively, with a 15% reduction in flow velocity after 45 s. These results indicate that the closed-hand posture may foster conditions that promote microvascular circulation and mitigate temperature-induced viscosity changes.

Comparison of further physical parameters between the open and closed hand postures reveals other differences in thermoregulatory performance. As shown in Figure 1B, the closed-hand configuration resulted in significantly lower heat loss (p < 0.01) and reduced surface area exposure (p < 0.001) compared to the open-hand condition. These reductions contribute directly to improve heat retention, supporting the proposed mechanism by which a toroidal posture mitigates peripheral cooling. In turn, thermal conductivity, modelled as a constant property of soft tissue, remained unchanged between conditions as it is a fixed intrinsic property of the biological tissue and is not affected by changes in posture or geometry. These results support the hypothesis that postural changes can influence passive thermoregulatory performance through structural modulation of exposed surface area.

Overall, our simulations show that the toroidal structure formed by the closed-hand posture, compared to the open-hand configuration, may maintain a higher mean temperature, minimize heat dissipation, improve local perfusion and facilitate microvascular circulation (Figure 1C). Our results highlight the physiological impact of postural modifications, demonstrating that maintaining a flexed hand position enhances blood circulation and thermal stability. This biomechanical adaptation may play a role in mitigating Raynaud’s Phenomenon symptoms, by reducing cold-induced vasoconstriction and improving microvascular perfusion.

4. Discussion

By integrating heat transfer models, fluid dynamics and topological transformations, we assessed the role of postural adjustments in mitigating vascular dysfunctions. We demonstrated that the transition from an open-hand to a closed-hand posture significantly affects thermal retention and microvascular circulation. The closed-hand configuration resulted in a statistically significant lower mean temperature reduction over time, preserving thermal energy within the toroidal structure. Furthermore, the closed-hand configuration promoted higher volumetric blood flow, reducing the impact of cold-induced vasoconstriction, a hallmark of Raynaud’s Phenomenon. Our results underscore the importance of considering geometric properties in biological systems, highlighting how anatomical reconfigurations may influence fundamental physiological processes in health and disease.

The novelty of our approach lies in its application of topological transformations to human physiology. Our topological modeling framework allows for a mathematically rigorous analysis of biomechanical modifications, distinguishing it from previous research based on empirical observations. Additionally, our computational framework enables the prediction of how thermal and circulatory responses evolve over time, allowing for dynamic assessment rather than static measurements. Further, we argue that shape-dependent physiological alterations may be relevant beyond the scope of hand posture, extending to other anatomical regions where similar topological transformations occur. Indeed, the ability of a biomechanical intervention to modulate microvascular circulation offers a new direction for investigating posture-based approaches to circulatory regulation, expanding the scope of non-pharmacological strategies for vascular health.

Our study does not replace existing interventions; rather, it complements them by offering a biomechanical strategy that may be integrated with current therapies. Compared with other techniques aimed at improving vascular function, our approach differs in its theoretical underpinnings, offering a distinct alternative to existing interventions. Traditional methods for improving circulation in conditions like RP primarily focus on pharmacological interventions, thermal protection or exercise-based rehabilitation [4]. Conventional pharmacological treatments, such as calcium channel blockers or vasodilators, primarily target endothelial function to counteract vasospastic episodes [3]. While effective, these treatments modulating systemic vascular tone are often accompanied by side effects and fail to consider the biomechanical factors that may contribute to localized vascular dysfunction. Thermally protective devices, such as heated gloves, function by externally supplying heat to mitigate the effects of environmental cold exposure [27]. However, these external interventions do not influence intrinsic physiological processes and depend on continuous energy input. In contrast, the closed-hand posture engages endogenous heat conservation mechanisms by structurally minimizing surface exposure and promoting internal thermal recirculation. Exercise-based rehabilitation programs emphasize muscle activity to enhance circulation, but these require sustained effort and may not be suitable for all patient populations [28]. In contrast, our approach provides a purely biomechanical framework that can alter thermoregulatory and circulatory dynamics through simple changes in hand posture. Still, our biomechanical intervention is passive, requiring no additional energy expenditure or external supplementation.

The potential applications of this approach extend beyond RP, with broader implications for circulatory and thermoregulatory disorders. The ability of the closed-hand posture to modulate microvascular circulation suggests that similar topological interventions could be explored in conditions such as diabetic microangiopathy, where peripheral blood flow regulation is impaired [35]. In stroke rehabilitation, where patients often experience deficits in fine motor control, structured hand postures may be investigated as a means of facilitating neurovascular coupling and motor recovery [36]. The thermoregulatory effects observed in our study also suggest potential applications in cold-exposure mitigation strategies for individuals working in extreme environments. Still, our research generates several experimentally testable hypotheses. A multi-modal experimental validation protocol is proposed to bridge computational findings with empirical evidence. Surface temperature changes could be monitored using high-resolution infrared thermography, capturing thermal gradients across open and closed hand postures in real time. Concurrently, nailfold capillaroscopy can provide non-invasive visualization of capillary blood flow and perfusion. Additionally, electromyography (EMG) could assess muscular activity to distinguish passive posture maintenance from active thermogenesis. These methods collectively offer a feasible, low-risk pathway for in vivo validation of the model’s predictions.

Several limitations should be acknowledged. We used topological constructs just as a theoretical framework to describe the transformation of hand posture from an open to a closed toroidal configuration. These mathematical descriptors were not directly implemented as simulation parameters or boundary conditions, but were instead used to conceptualize how closed-loop geometries could alter heat and flow dynamics. While the simulation itself relied on classical heat diffusion and fluid flow equations, future work could incorporate these topological indices into mesh generation or geometric constraints, providing a more direct computational integration of topological properties. Still, we modeled the hand as a 2D surface, which is a gross oversimplification neglecting the complex 3D structure of the hand which influences heat transfer and blood flow. A future 3D model, incorporating musculoskeletal and vascular components and employing the Laplace–Beltrami operator governing the diffusion of heat along the curved topology, is a logical next step to enhance biomechanical accuracy.

Our model relied on generalized physiological parameters not accounting for inter-individual variations in vascular function. It must be acknowledged that individual differences—such as age, gender and hand morphology—may influence local thermoregulatory and vascular responses. Our model also assumes idealized boundary conditions, with external temperature set at 20 °C, which may not fully capture the real-world variability. Differences in skin thickness, subcutaneous fat distribution, capillary density and baseline vascular tone could influence the rate of heat loss and magnitude of microcirculatory improvement associated with different hand postures. Additionally, our study does not account for autonomic nervous system contributions to microcirculatory adjustments [30]. Also, our methodology does not consider the electrical properties of the skin, which play a significant role in processes such as wound healing, cell migration and integration of bioelectronic devices [37,38]. Variations in skin conductivity and its response to external stimuli could influence local temperature distributions and microcirculation patterns, adding another layer of complexity to our model [39]. Moreover, the interaction between temperature fluctuations and leukocyte function remains underexplored in our analysis, despite evidence suggesting that hypothermia and rewarming alter leukocyte–endothelial interactions and immune cell recruitment [40,41]. Indeed, temperature-dependent activation of leukocyte populations has been observed across various species, providing additional evidence that localized thermal gradients may have immunological implications beyond their direct effect on blood flow [42,43]. Capillary dynamics are also affected by temperature variations at the microscale, as studies indicate that thermally induced changes in surface tension and pressure gradients influence microfluidic transport in biological tissues [44,45]. This is particularly relevant in cold-induced vascular conditions like RF, where capillary pressure fluctuations may exacerbate vasospastic episodes [46].

5. Conclusions

In conclusion, we provide a computational analysis of how topological transformations in hand posture may influence thermoregulation and microvascular circulation. Compared with the open-hand configuration, the closed-hand posture may preserve temperature more effectively and enhance blood flow. More broadly, we introduce a quantitative methodology for analyzing the physiological effects of anatomical reconfigurations, providing a mathematical approach to investigating how structural adaptations impact functional outcomes.