For pharmaceutical manufacturers, process analytical technology, or PAT, has been the unachievable Holy Grail for almost two decades. The definition sounds simple enough: The Food and Drug Administration (FDA) describes PAT as a framework for designing, analyzing and controlling manufacturing through timely measurements of raw and in-process materials to help ensure desired product quality.
But despite the promise of PAT – primarily, to create an adaptable control process that would allow greater flexibility, innovation and real-time release of drugs – the industry has not widely accepted and deployed the technology. Companies view PAT as untested by precedent, and they struggle with the imprecise guidance language provided by the FDA. As a result, the industry has shied away from implementing PAT, believing the potential for approval delay to be more costly than the expected benefits.
This paper will discuss a proposed process-control design methodology could create a robust and compliant PAT system using established tools developed for the nuclear industry, computer modeling/gaming software, and existing PAT technologies, such as data- and computer-driven chemical analyses.
This methodology mathematically creates a closed system that describes, simulates and provides a completely defensible pharmaceutical-control process. By explicitly incorporating the objectives of PAT into a closed mathematical system, we have addressed PAT semantic uncertainty, thereby paving the way for timely and cost effective PAT implementation.
The FDA acknowledges that the drug-approval process is expensive, complex and constrains manufacturing innovation. PAT was developed as an attempt to address this problem. Unfortunately, the goal of the FDA and the prerequisites for PAT appear diametrically opposed. The primary charter of the FDA is to ensure patient safety by reducing risk, while PAT’s flexibility and adaptability objectives require an increase in process degrees of freedom. Since each degree of freedom represents a stochastic variable, implementing PAT must theoretically increase potential process variance.
The concept behind PAT is that sufficient process understanding – coupled with extensive analytical instrumentation and quality-control-measurement procedures – will produce the desired compound repeatedly, reliably and with risk quantification. Certainly on the surface, there is little to argue with this concept.
At the same time, however, the risk-aversion mantra, “you don’t know what you don’t know,” is continually at the forefront of any discussion regarding PAT implementation. Therefore, the potential increase in process variability brought about by the degree of freedom expansion involved with PAT results in a circular argument, and requires negotiation between the FDA and the manufacturer regarding risk management and process knowledge.
This dichotomy between the FDA’s goals and the preconditions of PAT contains an inherent logical fallacy: that the process developer must have perfect knowledge of the process, because the source and implication of any variation must be known a priori. However, if this were true, there’d be no need for PAT.
To cost-effectively implement PAT, manufacturers require a process control scheme that is adaptive to inherent process variability, but is a mathematically closed system. Such a system ensures that the process always resides within a known, mathematically robust and regulatory-defensible state.
While this may sound impossible, the U.S. Defense Department faced a similar challenge at the dawn of the nuclear age. Control systems for nuclear weapons cannot be subject to the unknown-unknowns. Such a control system must be a mathematically provable system in which every possible state and state transition can be explicitly calculated. The lack of any nuclear-weapons control-system accidents over the past 60 years attests to the success of this design scheme and mathematical proof.
Compared to a nuclear weapon, a chemical or biological synthesis is considerably more difficult, for two reasons. First, the number of variables is substantially higher. Second, most of the variables are stochastic and non-linear, while a nuclear control system is dominated by binary logic.
This complexity has resulted in a lack of exact solutions to the process equations. As a result, control systems have typically been developed through the traditional abstract transfer function approach, which is consistent with pre-PAT concepts. Implementing a PAT control process, in the absence of exact solutions, requires a different approach. We have elected to take a geometric approach and view of chemical and biological processes.
Any chemical or biological reaction can be fully described by a series of partial differential equations. These equations include expressions for items such as rate, entropy, concentration, metabolism, glucose feeding, PH and viscosity – to mention a few of the hundreds of variables that influence the reaction. Adding to the complexity is the fact that many of these equations are nonlinear, recursive and the variables are stochastic.
Simultaneously solving a large number of such equations is not only extremely difficult – it does not converge to a unique solution, making it difficult for control system design.
Imagine each variable that influences a reaction as a separate dimension, just as x, y and z represent standard Euclidian three spatial dimensions. A reaction can therefore be projected in N dimensional space, where N is the number of influence variables. We refer to this as the N dimensional Process Space PSn.
We will illustrate the concepts and process using simplified, virtual 3-D graphics because we are limited in print.
A differential equation representing some aspect of the reaction can then be considered a hyperplane within the process space. By “successively plotting” each equation’s hyperplane, an N dimensional object is created by the intersection of the hyperplanes.
This N dimensional polygon created by the hyperplane intersections is the reaction domain, RDn. The reaction domain is the geometric set of all possible states for the reaction. The reaction domain is a subspace of the process space, which in turn is a subspace of a set of real numbers. Due to conservation of mass and energy, negative quantities are not permitted, and because we are not working at the quantum level, imaginary numbers can be ignored.
The reaction domain is a closed set, because all possible state vectors are contained within the domain and the result of any operation performed on any process variable is also contained within the domain. Because the reaction domain contains all possible states, some states or geographic regions are undesirable and need to be excluded from consideration. We accomplish this by constraining the reaction domain with additional explicit constraint equations or by applying the Lagrangian method to relevant equations.
The result is an N dimensional closed polygon of acceptable state vectors for the reaction.
The acceptable domain, “ ” provides the adaptability and flexibility that is desired for PAT. The variability of the process is captured by the breadth of the domain.
From a different perspective, we can consider the acceptable domain as representing the known reaction space, while state vectors outside the acceptable domain represent the unknown-unknowns.
This state vector space is closed under all operations as defined by the equations that describe the constrained N dimensional polygon. The role of a control system is to navigate the acceptable domain, given the stochastic nature of the process, and direct the reaction to the most desired state.
In performing this action, the control system may not permit the reaction to escape the acceptable domain, even during state transitions. As mentioned previously, the acceptable domain is closed for all state operations. However, a control system can cause a momentary violation if the trajectory of the state transition takes it outside the acceptable domain.
A control system move that is transitioning from two acceptable states creates a transient unacceptable state, or unknown-unknown. This situation violates our goal of creating an a priori PAT-compliant control system with no unknown-unknowns.
Currently the three traditional methods to address this problem are:
1. Closest end-point, with single valid trajectory.
2. Boundary skimming.
3. Shortest in-boundary multistep route.
Each of the above will result in reaching either the terminal state or a point in close proximity without boundary violation. Unfortunately, these methods are computationally complex, time-consuming and must be determined in real time due to the stochastic nature of the variables.
This conflicts with our goal of a prior definition, design and automated creation of a reusable control system strategy.
We propose an alternative method by which we transform the acceptable domain into a convex hull:
In fact, according to Caratheodory’s theorem, if ADN is a subset of N dimensional vector space, or in this context, a subset of the process and reaction space, then combinations of most N+1 points are sufficient in the definition above.
By restricting the operable vector space to a convex hull, we know that any state transition will remain within the acceptable domain at all times, inclusive of the path. This is the definition of a convex hull. Any spatial coordinate within the convex hull can be reached from any other coordinate by a straight line. This also allows the closure of the reaction domain, as all states and transitions are both contained and uniquely defined.
Clearly, creating a convex hull comes at the cost of reducing the size of the acceptable domain and could pose performance penalties as N increases or the order of governing equations become more nonlinear. Luckily, this typically is not an issue in chemical and biological applications.
Because time is a nonreversible domain dimension, we can view the reaction space as evolving in time. Therefore, rather than creating a single convex hull subscribing the entire acceptable domain, we can transform the acceptable domain into a series of sequential convex hulls over time. For example, consider the nonconvex vector space below:
The complete space is not convex, but can be decomposed into a series of convex space segments along the time dimension. Consider the space as a Rieman integral:
The N dimensional spatial difference between acceptable domain and the sum of the sequential convex hulls tends to zero as . Process kinetics will place a lower limit on which for pharmaceutical and biological reactions, tends to be rather modest, making computation simple and fast. For most reaction dimensions, <<1 the small number of convex segments does not result in a significant loss of process efficiency.
The sequential convex hull space defines the temporal-path domain of acceptable reaction performance. Once the sequential convex hull space is determined, the control strategy and program can be directly extracted, based upon solving the governing Hamiltonian equation. The control strategy and program is then just a vector in convex hull space.
This construction greatly simplifies the control process, because control is now simply a linear vector in the convex hull space and operations on the control domain are affine transformations. By recording a sparse number of process variables during the batch; the reaction vector (and the control system vector) can be uniquely determined and recreated. It is important to recall that the process domain has been made closed and convex, ensuring that the process was in control and in a known, acceptable state at all times.
This methodology addresses the specific challenges regarding real time release. Assume that release is determined by a specific titer assay, but the batch time to the current release point is statistically different than the mean of the previous runs. According to the FDA’s guidance for PAT, there is no resolution for this dilemma, because no explanatory model exists for the time variance nor does the empirical data resolve the temporal uncertainty.. Applying closed convex-domain control ensures the process has been in acceptable reaction states throughout, and the reaction vector can be uniquely determined and compared to the history of other batch reaction vectors. The batch time difference can be calculated and verified, because the time to reach the real time release point is simply the length of the reaction vector, which is easily calculated in standard matrix algebra.
More importantly, the variance window for acceptable real time release for all parameters can be calculated a priori and an acceptable release-domain predefined. Because the reaction domain is closed and convex, and the reaction kinetics slow, relative to the number of convex domains per unit time <<1, there are only a finite number of compliant reaction vectors. Simplifying this even more, since all sequential segments are convex, we only need to calculate those vectors compliant with the last convex hull segment. The projection of this set of vectors onto N-space defines the domain of acceptability for release for all parameters. The projection also represents the probability density function for real time release.
The application of closed convex reaction domains provides pharmaceutical companies and the FDA what they have long been seeking in PAT – a system that is:
1. Adaptable and flexible.
2. Defensible, closed, robust and defined.
3. Recognizes that an end point is stochastic, yet deterministic and constrained.
4. Supports real time release with known variance limits.
5. Allows validation of adaptive and model predictive process optimization because optimization resides inside the closed convex domain and all control moves are linear.
Most readers intuitively understand the concept of modeling and describing a process and control system as a set of vectors in N space, but shy away because of the apparent complexity.
Still, spatial (N dimensional polygon) decomposition into convex hulls has been a mainstay in computer graphics for decades. Computer rendering of complex multidimensional shapes, virtually in real time, relies on the ability to rapidly and precisely decompose spatial objects into their convex hull constituents.
It is now as simple as pointing one of many convex decomposition algorithms to a data matrix that defines the N-dimensional shape (reaction domain) and getting the sequential convex hull set returned. The determination of the reaction and control vector sets is straightforward Lagrangian linear algebra. That leaves the initial definition of the process and reaction N space to be populated with experimental data. But this is exactly what process chemists have been doing all along, though they may not have viewed it in this context.
One of the great benefits of this approach is that by solving the problem geometrically, we avoid having to solve complex nonlinear differential equations and then convert them to transfer functions. Rather, simply run sufficient experiments to generate a representative data set for each domain. The data set then defines the N space polygon, which is decomposed to the convex hull set. All solutions are constructed in the much simpler convex hull.
With this robust yet economical mathematical framework as a foundation, PAT is adaptable, flexible and supports real time release with a definitive path to resolve the problem of stochastic indeterminism. Once designed, simulated and verified with empirical data, the PAT control system is completely deterministic, closed and mathematically verifiable.
We believe we have succeeded in developing a reaction and control-domain definition that is powerful and simple. It allows for the automated creation of a control system, provides greater insight into the process, and allows pharmaceutical companies to reap the benefits of PAT.