طراحی زنجیره تامین پزشکی هسته ای : مدل شبکه قابل انعطاف و رویکرد محاسباتی

In this paper, we develop a tractable network model and computational approach for the design of medical nuclear supply chains. Our focus is on the molybdenum supply chain, which is the most commonly used radioisotope for medical imaging utilized in cardiac and cancer diagnostics. This topic is of special relevance to healthcare given the medical nuclear product's widespread use as well as the aging of the nuclear reactors where it is produced. The generalized network model, for which we derive formulae for the arc and path multipliers that capture the underlying physics of radioisotope decay, includes total operational cost minimization, and the minimization of cost associated with nuclear waste discarding, coupled with capacity investment costs. Its solution yields the optimal link capacities as well as the optimal product flows so that demand at the medical facilities is satisfied. We illustrate the framework with a case study. The framework provides the foundation for further empirical research and the basis for the modeling and analysis of supply chain networks for other very time-sensitive medical products.

Medical nuclear supply chains are essential supply chains in healthcare and provide the conduits for products used in nuclear medical imaging, which is routinely utilized by physicians for diagnostic analysis. For example, each day, 41,000 nuclear medical procedures are performed in the United States using technetium-99m, a radioisotope obtained from the decay of molybdenum-99. Such supply chains have unique features and characteristics due to the products' time-sensitivity along with their hazardous nature. In this paper, we take on the challenge of developing a model for supply chain network design of medical nuclear products, which captures some of the salient issues surrounding such supply chains today, from their complexity, to the economic aspects, the underlying physics of radioactive decay, and the inclusion of waste management. We focus on molybdenum-99 due to its importance in medical diagnostics, its time-sensitive nature, and the fact that there are only a handful of production and processing facilities for this radioisotope globally. In order to appropriately ground our framework, we first describe the underlying features of medical nuclear supply chains, and provide the necessary background for their understanding. For example, to create an image for medical diagnostic purposes, a radioactive isotope is bound to a pharmaceutical that is injected into the patient and travels to the site or organ of interest. The gamma rays emitted by the radioactive decay of the isotope are then used to create an image of that site or organ (Berger et al., 2004). Technetium, 99mTc, which is a decay product of molybdenum-99, 99Mo, is the most commonly used medical radioisotope, accounting for over 80% of the radioisotope injections and representing over 30 million procedures worldwide each year. Over 100,000 hospitals in the world use radioisotopes. (World Nuclear Association, 2011). In 2008, over 18.5 million doses of 99mTc were injected in the US with 2/3 of them used for cardiac exams, with the other uses including bone scans, functional brain imaging, sentinel-node identification, immunoscintigraphy, blood pool labeling, pyrophosphates for identifying heart damage, and sulfur colloids for spleen scans (Lantheur Medical Imaging, 2009). Through this most widely used medical radioisotope, health professionals can enable the earlier and more accurate detection of cardiac problems as well as cancer, the two most common causes of death (see Kochanek et al., 2011). It is estimated that the global market for medical isotopes is 3.7 billion US$ per year (Kahn, 2008). The production cycle of 99mTc, typically, begins with the fission of highly enriched uranium, HEU, to produce 99Mo, with a half-life of 66.7 h. The 99Mo, in turn, decays to 99mTc, whose half-life is approximately 6 h. The relatively short half-life of 99mTc, as compared to metabolism and human activity, makes it a suitable isotope for imaging. In addition, the gamma rays that are emitted due to the 99mTc decay have roughly the same wavelength as common X-rays allowing detection by detectors similar to those used for medical X-rays. The production of 99Mo occurs at only nine reactors in the world, with one in Canada, five in Europe, one in Australia, one in South Africa, and one in Argentina (see OECD, 2010a). The reactors irradiate targets, aluminum blocks or foil containing uranium-235, 235U, to produce multiple fissions products, including 99Mo. The irradiated targets containing the 99Mo are then shipped to processing facilities where the 99Mo is extracted and purified. The extracted 99Mo is further transported to generator manufacturing facilities. There, generators, which are containers of 99Mo in a chemical form that allows easy extraction of 99mTc are produced. The generators, which are relatively radioactively safe, are then shipped to the hospitals and medical imaging facilities where the 99mTc is eluted by a saline solution and the pharmaceutical injections prepared and administered. Since the decay of a single atom of 99Mo produces a single atom of 99mTc, the activity of the generator is determined by the quantity of the 99Mo present. Since 99Mo decays with a 66.7 h half-life, approximately 99.9% of the atoms decay in 27.5 days, making its production, transportation, and processing all extremely time-sensitive. In fact, the production of 99Mo is quantified in Six-day curies end of processing denoting the activity of the sample 6 days after it was irradiated to highlight this (see OECD, 2010a). In addition to the time-sensitivity, the irradiated targets are highly radioactive, significantly constraining transportation options between the reactor and the processing facilities to only trucks that can transport the heavily shielded transportation containers. While the extracted 99Mo continues to be constrained by its decay, its shielding requirements are reduced, allowing for transportation by modes other than trucks, including by air (cf. de Lange, 2010). Although the maximum possible production from current reactors in 2010 was well over twice the current demand, it has been predicted that, with a 5% annual growth rate for imaging, the demand will exceed the supply by the end of the decade. However, this assumes that all reactors are capable of irradiating the necessary targets at all times. Due to routine maintenance, unexpected maintenance, and shutdowns due to safety concerns, the actual supply has been much closer to the demand. In 2009, in fact, the demand exceeded the supply and created a worldwide shortage of 99Mo. Furthermore, several of the reactors are reaching the end of their lifetimes, since they are 40 to over 50 years old (cf. OECD, 2010a, Seeverens, 2010). Between 2000 and 2010, there were six unexpected shutdowns of reactors used for medical imaging products due to safety concerns (Ponsard, 2010) with the Canadian one shutdown in May 2009 due to a leak in the reactor with its return to service more than a year later in August 2010. It is also important to note that the number of processors that supply the global market is only four, and that they are located in Canada, Belgium, The Netherlands, and South Africa. Australia and Argentina produce bulk 99Mo for their domestic markets but are expected to export small amounts in the future. Amazingly, there are parts of the world in which there are no processing facilities for 99Mo, including the Unites States, parts of South America, and Japan. Such limitations in processing capabilities limit the ability to produce the medical radioisotopes from regional reactors since long-distance transportation of the product raises safety and security risks, and also results in greater decay of the product. The number of generator manufacturers, in turn, with substantial processing capabilities, is under a dozen (OECD, 2010a). Furthermore, in 2016, the Canadian reactor is scheduled for complete shutdown, raising critical questions for supply chain network design, since its processing facility will also need to be shutdown (OECD, 2010a). This paper is organized as follows. In Section 2, we develop the multitiered supply chain network design model for molybdenum, 99Mo. The framework may be used, with minor modification, for other radioisotopes. We describe the various tiers of the supply chain network, beginning with the nuclear reactors, moving on to the processors, then on to the generator manufacturing facilities, and, finally, to the hospitals and medical facilities, where the medical radioisotopes are injected into the patients. The supply chain network is quite complex since it consists of multiple activities of production, transportation, and processing, coupled with the physics of the radioisotope and its decay, along with regulatory restrictions as to transportation, due to the hazardous nature of the medical nuclear product. We model the supply chain network design problem as an optimization problem on a generalized network. We identify the specific losses on the links/arcs through the use of the time decay of the radioisotope. We consider total cost minimization associated with the operational costs, along with the waste management costs, since we are dealing with nuclear products. Medical nuclear waste management issues have not received much attention in recent reports (cf. OECD, 2010a). The model captures the investment in capacities through the construction of new links. Its solution provides the optimal investments along with the optimal levels of production, transportation, and processing, given the demands at the various hospitals and medical imaging facilities. We use a variational inequality formulation since such a formulation results in an elegant computational procedure. Moreover, the theory of variational inequalities has been applied to a plethora of supply chain modeling, analysis, and design problems (see Zhang, 2006, Nagurney, 2006, Nagurney, 2010, Qiang et al., 2009, Liu and Nagurney, 2011 and Cruz and Liu, 2011). Furthermore, it provides a rigorous mathematical and computational framework to enable the exploration of alternative economic behaviors among the medical nuclear supply chain stakeholders, including competition (see Nagurney, 2006). Such a modeling approach is in concert with recent studies that have focused on the security and reliability of medical nuclear supply chains that also emphasize that governments ultimately have the responsibility for establishing an environment conducive to investment in such supply chains (cf. OECD, 2010a). However, to the best of our knowledge, our model is the first mathematical one to include the operational, engineering, economic, and physics aspects of medical nuclear products. Indeed, the model is sufficiently general to capture the economic aspects of medical nuclear supply chain network design, which is an important issue since it has been recognized that usually governments run the reactors, which are research reactors, and the prices associated with the radioisotope may fail to capture the associated costs and, as a consequence, the pricing may be below marginal costs resulting in market failure; see OECD (2010a) and Seeverens (2010). For references to other generalized nonlinear network models and applications, see Nagurney and Aronson (1989), Nagurney et al. (2012), and the references therein. Nagurney and Masoumi (2012) recently developed a supply chain network design model for a sustainable blood banking system but the demands therein were uncertain. In our model and applications the demands are fixed since the associated medical procedures need to be scheduled. In Section 3, we propose a computational approach for the new model, along with the accompanying theory, which resolves the supply chain network design problem into subproblems that can be solved explicitly and exactly at each iteration. In Section 4, we present a case study. In Section 5, we summarize our findings, present our conclusions, and provide suggestions for future research. 2. The medical nuclear supply chain network design model In this section, we develop the supply chain network design model for a medical nuclear product, that of 99Mo, referred to, henceforth, as Mo. The model is general and can be applied, with appropriate data, to evaluate the design of such supply chains in the cognizant organization's nation/region. In Section 4, we illustrate how this framework can be applied to the Canada–United States and other countries supply chain for this product. For definiteness, refer to Fig. 1. Fig. 1 depicts a possible network topology of the medical nuclear supply chain. In this network, the top level (origin) node 0 represents the organization and the bottom level nodes represent the destination nodes. Every other node in the network denotes a component/facility in the system. A path connecting the origin node to a destination node, corresponding to a demand point, consists of a sequence of directed links which correspond to supply chain network activities that ensure that the nuclear product is produced, processed, and, ultimately, distributed to the hospitals and medical imaging facilities, where they are administered to the patients. We assume that, in the initial supply chain network topology, as in Fig. 1, which serves as a template upon which the optimal supply chain network design is constructed, there exists at least one path joining node 0 with each destination node: View the MathML sourceH12,…,HnH2. This assumption guarantees that the demand at each demand point will be met.

In this paper, we developed a rigorous framework for the design of medical nuclear supply chains. We focused on the most widely used radioisotope, molybdenum, 99Mo, which is used in medical diagnostics for cancer and cardiac problems. Medical nuclear supply chains have numerous challenging features, including the time-sensitivity of the product, which is subject to radioactive decay, the hazardous nature of production and transportation as well as waste disposal. In addition, such radioisotopes are produced globally only in a handful of reactors and the same holds for their processing. Moreover, the nuclear reactors where they are produced are aging and have been subject to failures creating shortages of this critical healthcare product. The specific contributions of the findings in this paper are: (1) A theoretically sound, based on physics principles, methodology to determine the loss, due to time-decay, of the radioisotope on the various links of the supply chain network, through the use of arc multipliers. (2) A generalized network optimization model that includes the relevant criteria associated with link construction, coupled with the operational costs and the associated discarding and waste management costs, subject to demand satisfaction at the patient demand points. (3) A unified framework that can handle the design of the medical nuclear supply chain network from scratch, with specific relevance to the existing economic and engineering situation, coupled with the physics underlying the time-decay of the radioisotope. (4) An algorithm which resolves the new supply chain network design problem into subproblems with elegant features for computation, for which we provide explicit formulae and a generalized exact equilibration algorithm to handle multipliers. We note that the contributions in the paper can serve as the foundation for the investigation of other medical nuclear product supply chains. In addition, the framework can serve as the basis for the exploration of alternative behaviors among the various stakeholders, including competition. Finally, it can be used to assess the vulnerability of medical nuclear supply chains and to explore alternative topologies and the associated costs. Since it has been recognized that some of such supply chains are presently operating without recovering the costs at the reactors, resulting in market failure and a lack of incentive investment, plus that the need for such medical diagnostics is expected to grow with the aging population, we believe that this paper, in emphasizing a new research agenda, has made a valuable contribution.