Cutaneous circulation is a major effector of human thermoregulation [14]. Cutaneous vessels dilate or constrict in response to either thermal stress, i.e. temperature changes, arose exogenously from variations in environmental temperatures or endogenously from the body itself, as occurs during isometric exercise [14, 15]. The initiation of isometric hand grip exercise has a little effect on the cutaneous circulation in nonglabrous (hairly) skin, whereas the activation of vasodilator system at skin overlying the PSA inflamed joint, causes vasoconstriction due to withdrawal of the active vasodilator activity [16, 17]. In PsA, heat production of active joint, muscle and elevation of inflammatory blood flow in tendons (as shown usually by Eco power Doppler in this disease [1]), may increase the finger’s skin temperatures [15] (as visually evident in figure two). In fact, according to Johnson and colleagues [15] many factors can modulate control mechanisms of the cutaneous vasculature, such as gender, aging, and clinical conditions. Cutaneous vasoconstriction and vasodilation are vasomotor responses mediated by a sympathetic control action from the simulated temperature regulating center in the anterior hypothalamus [14]. Homeostasis is basically maintained by a negative feedback loop, similar to a thermostat [18], which regulates the energy exchange with the environment at the cutaneous level through metabolic and hemodynamic processes that determine finger temperature at any given time [19]. Employing Control System Theory, the homeostatic process can be seen as a feedback controlled system. This kind of system considers a reference signal to produce the desired output. The reference signal indicates the value that the output has to assume. The reference value is represented by superficial basal temperature that can be considered steady during the experiment, while the output is the superficial finger temperature. The controlled isometric exercise induces a finger temperature (plant controlled output) change from the basal value (reference value). The difference between the plant controlled output and the reference value (i.e., the output error) prompts the thermoregulatory reaction in order to restore the basal value by steering the output error to zero. The time-evolution of the finger temperature can be recorded by means of thermal IR imaging [6, 7, 11–13]. Examples of temperature versus time curves, captured at finger joints (shown in Figure 1) are reported in Figure 2. According to Control System Theory, differences in the temperature recovery curves depend on the efficacy of the cutaneous thermoregulatory effectors, which in turn can be represented by the actual values of a given set of functional modeling parameters.

### Problem statement

Experimental evidence (Figure 2) showed that finger cutaneous thermoregulatory response after the isometric exercise for PsA plaque skin regions has different dynamic characteristics with respect to the healthy skin regions [11]. The wide number of complex processes potentially involved in temperature control and in its alteration suggests considering the overall control system as a ‘black box’, whose overall structure can be investigated by analyzing the input-output time responses either in the healthy or in the pathological conditions [6, 7, 12, 13, 20]. Mariotti et al. [6] proposed a feedback thermoregulatory model through two hierarchical control units: a higher level unit (supervisor) and a feedback lower level executor, driven by the supervisor as shown in Figure 3. These two hierarchical control units were proposed to model both local/peripheral, and systematic/central thermoregulatory effectors known to respond to the isometric exercise attempting to restore the basal temperature [16]. In fact, the supervisor sets the reference signal on the basis of the basal pre-stress temperature and the onset time. The overall performance of the thermoregulatory effecting processes depends on the activity of both the supervisor and the executor. Besides the contribution of the thermoregulatory effector mechanisms, the finger temperature (i.e., system output) is also influenced by the thermal exchange between the finger and the surrounding environment. This thermal exchange depends on the temperature difference which constitutes the external input to the thermoregulatory system [18, 19].

### Model structure

Figure 3 shows the overall architecture of the model proposed by Mariotti et al. [6]. The only observable output is the finger cutaneous temperature [6, 7] y(t), obtained through thermal IR imaging. No information about internal variables is available. Some assumptions can be made about the general structure and the order of the thermoregulatory control system identified with a grey box approach, with the aim of introducing functional parameters to both quantitatively and qualitatively describe the thermoregulatory effector mechanisms [21]. The system is characterized by an external input (room temperature) and a steady state regime reference signal (r) (basal finger cutaneous temperature T). The reference signal can be measured by IR imaging before the initiation of the isometric exercise, and averaged over time to provide a constant reference value T. Visual inspection of the thermal recovery after the isometric exercise confirmed that skin thermoregulatory cutaneous effector system could be assumed as a second-order time-invariant feedback system [22]. In particular, the executor (feedback lower level unit) is composed of a controller and a plant block in sequence (Figure 3), both assumed to be time invariant systems described by first-order transfer functions. Therefore, the plant output y(t) (i.e., the finger temperature) is governed in the time domain by the following differential equation:

\stackrel{\u0307}{y}\left(t\right)=-a\stackrel{\u0307}{y}\left(t\right)+b\stackrel{\u0307}{u}\left(t\right)

(1)

Where u is the plant input, a and b are constant coefficients. The post-exercise temperature y(0) (i.e., the temperature measured immediately after the end of the isometric exercise) constitutes the initial condition for the response of the control system. The plant input u (t) is then the sum of the feedback controller output m (t) plus the additional external input d as shown in Figure 3:

u\left(t\right)=m\left(t\right)+d

(2)

Input d represents passive heat exchange with the environment. Therefore, it depends on room temperature and y(t). In other words, input d can be seen as the uncontrolled effect of environmental conditions on the finger temperature [6]. The feedback controller block generates the signal m(t) stimulated by the difference between the system output and the reference signal r, namely output error e(t):

e\left(t\right)=r-y\left(t\right)

(3)

The feedback controller acts on the plant by the signal m(t) to steer the output error to zero. Common approaches for modeling homeostatic processes are based on an integral-type feedback controller system, which nullifies step-wise variation of the error signal [23]. The differential equation that describes the controller behavior in the time domain is:

\stackrel{\u0307}{m}\left(t\right)=\mathrm{K\u0117}\left(t\right)

(4)

Where K is a proportionality constant. The supervisor unit activates this controller by means of logic signals (on/off transition). When the supervisor unit logical output is “on”, the feedback is closed on the integral type controller and then the active temperature recovery can start. Otherwise, when the supervisor unit logical output is “off” (during the lag time LT), the controller is disabled to restore the initial condition, while the external input d is independent of this switching logic. The evolution of the system can be described more easily in the Laplace domain. The Laplace transform (L-transform) was performed with the assumptions : i) zero initial conditions y(0), and ii) the plant is unitary gain process with b =a in eq.1 [24], since allowing for gain both the plant and the inputs to the controller would result in degenerate parameters [6].Therefore, the overall model works in open loop for the time instance t <LT [6]:

Y\left(S\right)=\frac{a}{S+a}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{0.3em}{0ex}}\stackrel{\u0307}{\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}d}

(5)

and in closed loop for the time instance t >LT [6]:

Y\left(S\right)=\frac{a\stackrel{\u0307}{K}}{S(S+a)+a\stackrel{\u0307}{K}}\stackrel{\u0307}{T}+\frac{a\stackrel{\u0307}{S}}{S(S+a)+a\stackrel{\u0307}{K}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{0.3em}{0ex}}\stackrel{\u0307}{\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}d}

(6)

Where s is the Laplace variable, Y(s), r and d are the output, reference input, and the disturbance inputs, respectively. Moreover, the set of parameters (i.e. a, k, d, and LT) could provide an insight on the dynamics and activity level of thermoregulatory effector mechanisms during both healthy state and the presence of a disease. In fact, the reciprocal of the plant time constant (a) represents the speed of the response of the thermal process to external and internal stimuli. The integral gain (k) could be considered as a descriptor of an active and systemic vasodilation process in restoring and maintaining the reference basal temperature conditions [6], since it refers to the control action and determines the efficiency of the feedback control system in achieving the steady state. The disturbance input (d) represents a passive heat exchange with the environment and, therefore, depends on room temperature and y(t). LT is a time required for the thermoregulatory processes to access the internal re-warming process after the end of the isometric exercise. During this time, the thermal variations are mostly attributable to the passive heat exchange with the environment. Once LT is finished, there is the onset of the re-warming process and the controller starts to restore the reference basal conditions T.

Since the purpose of applying control theory is to offer a model that can fit the sample data well, which means making the calculated system output y* approach the actual/experimental system output y^{e} as closely as possible. The closer those two values are, the better the fitting effect will be. Therefore, the least squares criterion function f [25] can be taken as the fitness function:

f\left(x\right)=\sum _{i}^{\mathit{\text{NE}}}{\left({y}_{i}^{\ast}-{y}_{i}^{e}\right)}^{2}

(7)

Where {y}_{i}^{\ast} is the vector of experimental finger re-warming curves’ data points and {y}_{i}^{e} is the vector of the estimated model’s data points. The data points are defined from i =1 to number of data points NE, and is the vector of the model parameters, i.e. a, k, d, and LT. From equations 5 and 6, the finger thermoregulatory model (Figure 3) is uniquely described by a, k, d, and LT, which can be estimated based on measurements of T and y(t) [6, 7] by solving the optimization problem defined by the cost function stated in equation 7.