Closed System Control: From Defense to Pharma

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.

Conclusions

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.