The objective is to establish a dynamic model to capture key characteristics of regulatory actions and compliance behaviors, and to allow quantitative measurement of regulatory effectiveness. The model is to only rely on the fundamental concepts and principles of regulation. The present work borrows the terminology and methodology of physics, which provide a helpful abstraction for an otherwise complex regulatory compliance situation.

Dynamics of a Single Firm

A fundamental understanding in physics is that anything observable in daily life represents a stationary state, which in turn indicates the existence of a constraint force that keeps the state stationary. Using such an observable to characterize a firm’s state of compliance may seem overly simplified, as GMP consists of many technical and managerial requirements. For example, a firm may have a state of art facility, but its microbial control system may not be robust. In fact, how to characterize the complex nature of the state of compliance with one or a few metrics is the topic of the quality metric and quality management maturity research [24,25,26,27]. Nonetheless, the regulator takes a yes or no approach in evaluating whether a firm is in compliance with GMP. In this regard, using a single observable to characterize the state of compliance is a reasonable start.

The constraint force is the interplay of two forces. One is to minimize manufacturing cost, and another is to minimize compliance risk for violation of a myriad of GMP requirements. For illustration purposes, the constraint force is assumed to be the derivative of a constraint potential. In reality, not all forces are derivatives, and not all potentials are derivable. While the simulation described in present work requires no such assumption, with the assumption, however, the constraint force can be conveniently visualized with three potentials (curves) in Fig. 1, where the horizontal axis denotes a measure of a firm’s state of compliance. The manufacturing cost potential forces the state to the left. The compliance risk potential forces the state to the right. The net effect is a constraint force that keeps the state near the minimum of the combined constraint potential.

Fig. 1

Illustration of the assumed potentials for the compliance risk force, the manufacturing cost force and the combined constraint force

In addition to the constraint force, which is deterministic in nature, the state of compliance is affected by other factors [2]. For example, while a manufacturer may have a sound compliance management system in place, such a system may not be in use consistently. To represent the impact of such factors, a random term is introduced, where random merely means that the details of the force are beyond scope of the present work.

The dynamics of a firm’s state of compliance is proposed to follow the equation:

$$\frac=f\left(x,t\right)+ \xi \left(t\right)$$

(1)

where \(x\left(t\right)\) denotes the time-dependent state of compliance, \(f(x,t)\) the constraint force, and \(\xi \left(t\right)\) the noise. This is a simplest stochastic dynamic differential equation with a deterministic term and a noise term.

The simplest form of \(f(x,t)\) that constrains \(x\left(t\right)\) to the fully compliant state \(_\), while being a derivative of a potential, is a linear function:

$$f\left(x,t\right)= -k\left(t\right)\left(x-_\right)=-\frac, \text\text\text\text\text \;U\left(x,t\right)= \frack\left(t\right)_)}^,$$

(2)

where \(k\left(t\right)\) is independent of \(x\), and \(U(x,t)\) is a quadratic function with a time-dependent coefficient. An implicit assumption here may seem to be that the driving force to reduce manufacturing cost is the same as the force to reduce regulatory risk, as \(U(x,t)\) is symmetric to \(_\). The reality is a little more complex than this, but the quadratic form in Eq. (2) remains intact even when the two competing forces change their relative strength. The symmetry of \(U(x,t)\) is actually determined by the linearity of the forces, not their relative strength, as discussed in the numerical simulation part of the present work.

With Eq. (2), Eq. (1) can be rewritten as:

$$\frac=-k\left(t\right)(x-_)+ \xi \left(t\right).$$

(3)

The constraint force \(f\left(x,t\right)\) is linearly proportional to the deviation from \(_\). This mathematical representation of proportionality is a key assumption of present work. The dynamics described by Eq. (3) is a generalized Ornstein-Uhlenbeck equation (whose coefficient of the linear term traditionally is a constant) with broad applications in physics, chemistry, biology, engineering, finance and social studies [28, 29].

Equation (3) is a stochastic differential equation, so its solution is best represented by the probability distribution \(p(x,t)\). For a general function \(k\left(t\right)\), the analytical form of \(p(x,t)\) is not available. To gain insight into the nature of this dynamics, let’s make further simplifications.

If \(k\left(t\right)=_\) is a constant, and \(\xi \left(t\right)\) is white noise \(<\xi \left(t\right)\xi \left(^}\right)>=_}^\delta (t-^})\), where \(_\) is a constant, \(p(x,t)\) takes the following analytic form for \(p\left(x, t=0\right)=\delta (x-_)\),

$$p\left(x,t\right)=\frac\sigma \left(t\right)}^\left(t\right)\right)}^/2^\left(t\right)},$$

(4)

where the average \(\overline\left(t\right)=_^_t}+_(1-^_t})\) and \(\overline\left(t\to \infty \right)=_\),

and the variance \(^\left(t\right)=\frac_}^}_}(1-^_t})\) and \(^\left(t\to \infty \right)=\frac_}^}_}\) [28, 29].

Equation (4) show that, over time, \(\overline\left(t\right)\) approaches \(_\), independent of its initial state \(_\). The probability distribution is Gaussian, centered on \(_\). A larger \(_\) represents a stronger constraint force, a steeper potential curve and a narrower Gaussian distribution. \(_\) is a measure of the magnitude of noise. A large \(_\) leads to a broad Gaussian peak. As long as \(k\left(t\right)\) remains positive, the overall shape of \(U\left(x,t\right)\) is qualitatively similar to the dashed curve in Fig. 1, \(p\left(x,t\to \infty \right)\) remains to be bell-shaped, and \(x\left(t\right)\) is very much constrained to \(_\).

The Regress-Wake Cycle

Firms tend to become less vigilant about compliance in the absence of regulatory action [17, 18]. This behavior is modeled with a \(k\left(t\right)\) that regresses over time. A smaller \(k\left(t\right)\) represents a weaker constraint force, meaning the firm is more likely to deviate from \(_\). The effect of a regulatory action is modeled with a sudden jump of \(k\left(t\right)\), resulting in the firm’s heightened vigilance. After a while, the firm regresses again, starting a new regress-wake cycle. This regress-wake cycle of a firm’s compliance vigilance is typical in the pharmaceutical industry, and is a main reason for which the GMP regulation program exists. The dynamic details of the cycle can be conveniently modeled by using various forms of \(k\left(t\right)\), as shown in the numerical simulation part of the present work. For instance, the consistency of regulatory actions can be modeled through the inconsistency of regulatory actions by introducing a random perturbation to \(k\left(t\right)\). Please note that regulatory actions are directly applied to \(k\left(t\right)\), not \(x\left(t\right)\). This makes sense, as a regulatory action does not change a firm’s state of compliance overnight, but does reset its vigilance level. Over time, the heightened vigilance will raise the state of compliance, as \(x\left(t\right)\) is essentially a time-integration of \(k\left(t\right)\).

Dynamics of Multiple Firms

For an industry with \(N\) firms, each firm’s state of compliance \(x\left(t\right)\) follows the dynamic equation in Eq. (3). In principle, the force coefficient for each firm can be dependent on all other firms. A general solution for such a complicated situation is beyond the scope of the present work. Instead, a practical approach is taken as follows.

The FDA doesn’t have the resources to inspect all firms at all times. It is therefore important to maximize the impact of each regulatory action on the entire industry. Suppose a firm receives a regulatory action, feels the pain, and becomes more vigilant about compliance risk. If the FDA keeps this information confidential, all other firms’ vigilances remain down. If the information is made public, other firms’ vigilances can be heightened but to a lesser degree than the firm that received a regulatory action [30]. This is because other firms do not directly suffer the pain caused by the regulatory action, and don’t know all the violation details. A regulatory action, like a Warning Letter, usually lists only a few compliance violations uncovered during the inspection. Firms tend to become more vigilant about those violated compliance requirements. In the real world, words get around. For example, firms whose quality heads are friends of the quality head of the firm receiving the regulatory action, or friend firms, or simply friends, tend to know more about the violations and may become more vigilant than others. In present work, only the direct impact of a regulatory action to the firm and its circle of friends is considered. The secondary and higher order impacts of a friend to its own friends and friends’ friends are left for further studies.

This circle of friends approach is employed in present work to study the effect of transparency as follows. If the regulatory action details are kept strictly confidential to the receiving firm that has no friend, this corresponds to the case of complete lack of transparency. If the regulatory action is published with extensive and detailed information on the compliance violations along with both the firm’s and FDA’s take on the violations and suggested remedies, all firms benefit. This corresponds to the case of total transparency. By gradually increasing the size of circle of friends from 0 to \(N\)-1, one can systematically study the effects of transparency.

Measures of Regulatory Effectiveness

The average state of compliance for the industry \(\overline\left(t\right)\) is defined as the average over all firms’ \(x\left(t\right)s\) at time \(t\). The overall state of compliance \(\bar}\) is defined as the average of \(\overline\left(t\right)\) over the time of a simulation run. The standard deviation \(\sigma\) of \(\bar}\) is calculated over the same time.

The ultimate goal of compliance regulation is to keep the overall state of compliance \(\bar}\) above marginal compliance \(_\), and close to full compliance \(_\). Therefore, the effectiveness of regulatory actions can be quantitatively measured by its impact to \(\bar}\) as follows.

(a)

Efficiency: if regulatory actions lead to a higher increase of \(\bar}\), the regulation is considered as more efficient.

(b)

Cost: if less regulatory actions are needed to achieve the same increase of \(\bar}\), the regulation is considered as more cost effective.

(c)

Quality: if regulatory actions lead to a lower \(\sigma\), the regulation is considered having higher quality.

Summary

The proposed dynamic model has three levels.

Level 1 is the state of compliance of a firm \(x\left(t\right)\).

Level 2 is the constraint force that determines the dynamics of \(x\left(t\right)\), where the force coefficient \(k\left(t\right)\) is a measure of firm’s compliance vigilance.

Level 3 is the constraint potential \(U(x,t)\), which measures the trouble that a firm is potentially in when its vigilance is low.

The three fundamental principles of regulation, i.e. proportionality, transparency and consistency are treated as input by the dynamic model as follows.

Proportionality is built into the linear force \(k\left(t\right)(x-_)\).

Transparency is introduced with the concept of circle of friends.

Consistency is introduced through \(k\left(t\right)\) with random perturbation.

The measures of regulatory effectiveness, i.e. efficiency, cost and quality, are treated as output by the dynamic model, and can be quantitatively calculated based on \(\bar}\) and \(\sigma\). The present work is to show that the dynamic model can link the input to the output in a quantitatively predictable way to generate academically and programmatically meaningful results.