Going beyond “same-for-all” testing of infectious agents in donated blood

Abstract

Blood products, derived from donated blood, are essential for many medical treatments, and their safety, in terms of being free of Transfusion-Transmitted Infections (TTIs)—i.e., infectious agents that can be spread through their use—is crucial. However, blood screening tests are not perfectly reliable and may produce false negative or false-positive results. Currently, blood donations are tested using a same-for-all testing scheme, where a single test set is used on all blood donations. This article studies differential testing schemes, which may involve multiple test sets, each applied to a randomly selected fraction of the donated blood. Thus, although each blood donation is still tested by a single test set, multiple test sets may be used by the Blood Center. This problem is modeled within an optimization framework and a novel solution methodology is provided that allows important structural properties of such testing schemes to be characterized. It is shown that an optimal differential testing scheme consists of at most two test sets, and such a dual-test scheme can significantly reduce the TTI risk over the current same-for-all testing. The presented analysis leads to an efficient greedy algorithm that generates the optimal differential test sets for a range of budgets to inform the decision-maker (e.g., Blood Center). The differential model is extended to the case where different test sets can be used on sub-sets of donations defined by donation characteristics (e.g., donor demographics, seasonality, or region) that differentiate the sub-set’s TTI prevalence rates. The risk reduction potential of differential testing is quantified through two case studies that use published data from Sub-Saharan Africa and the United States. The study generates key insight into public policy decision making on the design of blood screening schemes.

Keywords:

• Blood screening
• blood safety
• differential testing

Appendix

Charnes-Cooper Transformation of Problem (D-RMP)

We apply the Charnes-Cooper Transformation (Charnes and Cooper, 1962) to transform the LFP of D-RMP in Equations (1(1) ) to (4(4) ) to an LP problem. Let

Then, an equivalent formulation of D-RMP follows: (A1) (A2) (A3) (A4) (A5) (A6) Multiplying both sides of Constraints (A2(A2) ) and (A3(A3) ) by t and letting x_k ≡ p_kt, k = 0, …, f, we obtain the following LP problem: (A7) (A8) (A9) (A10) (A11) (A12) Finally, by replacing t with ∑^f_{k = 0}x_k using Equation (A9(A9) ), the LP in D-RMP2 Primal (Equations (5(5) ) to (8(8) )) follows.

Proof of Proposition 1 The proof of part (1) follows by contradiction. Consider an optimal solution to D-RMP given by p* and suppose, to the contrary, that Constraint (2(2) ) is not binding; that is, , equivalently, . (The equivalence follows by multiplying each side of Equation (2(2) ) by t ( > 0) and noting that by definition x*_k = p*_kt, , and ). Then, by the complementary slackness conditions for D-RMP2, w*₁ = 0. Since (by Assumption 2), the feasible region of D-RMP2(D) reduces to , which implies that . Then, by the complementary slackness conditions, x*_f ⩾ 0 and x*_k = 0, ∀k ≠ f, with the corresponding optimal solution to D-RMP given by , leading to p*_f = 1 and . Since Constraint (2(2) ) is not binding, this implies that ∑_{j ∈ Ω}c_j < B, contradicting with the assumption that B < ∑_{j ∈ Ω}c_j.

If the feasible set of an LFP is bounded, then its objective function attains its optimal value at an extreme point of the feasible region (Bajalinov, 2003, Theorem 4.3). Then, part (2) follows because any extreme point of D-RMP is characterized by two basic variables (Bazaraa et al., 2010, p. 93). Therefore, at most two p*_k can be strictly positive in an optimal solution. Part (3) follows directly from part (1). To prove part (4), by definition of set K⁺ and from Equation (3(3) ), we have that and . From part (1), . Suppose, to the contrary, that . Then, by Assumption 1, we must have Without loss of generality, suppose Then, from Equation (1(1) ), the objective function value to D-RMP can be further increased by increasing to one and decreasing to zero, which contradicts with K⁺ = {k₁, k₂}. Therefore, we must have and or vice versa. Finally, to prove the last part of (4), assume, without loss of generality, that . Since and , by definition and , which implies by the complementary slackness conditions, that in the dual solution, Constraints (10(10) )(k₁) and (10(10) )(k₂) are binding; that is, Since , this implies that , completing the proof.

Proof of Property 1 Since the optimal solution to D-RMP occurs at an extreme point of the feasible region (Bajalinov, 2003, Theorem 4.3), from Constraints (2(2) ) and (3(3) ) it is given by

Then, as the budget (right-handside of Constraint (2(2) )) is perturbed by some δ (positive or negative), the current basic solution remains optimal (with values perturbed to and ) as long as it is feasible (see, for example, Bazaraa et al., 2010, p. 293); that is, and the result follows.

Proofs of Property 2 and Proposition 2 Corresponding to any break point, B_P(s), s ∈ Z⁺, with , we define the following mutually exclusive sets: At break point B_P(s), we also define, for each , that is, w₁(i, j) denotes the value of w₁ at the intersection of the hyperplanes corresponding to Constraints (10(10) )(i) and (10(10) )(j).

Proof of Property 2 Consider some break point B_P(s), s ∈ Z⁺, with (hence, by Proposition 1), which implies that . By the complementary slackness conditions, Constraint (10(10) )(k_P(s)) in D-RMP2(D) must be binding at optimality; that is, We prove the result by contradiction.

We first prove that any Constraint (10(10) )(k) for is redundant. Consider . First, we show that we must have , for . Suppose not, that is, for some , . (Assumption 2 rules out the case where . Then, Constraint (10(10) )(k) in the dual problem, given by , renders the current optimal solution, , infeasible, contradicting with its optimality. Therefore, we must have , for , and it readily follows that Constraint (10(10) )(k), , is redundant at budget level B_P(s). Similarly, it follows, by definition of sets and , that Constraint (10(10) )(k), for , is redundant in the dual problem at budget level B_P(s). This completes the proof.

Property 4.

Consider any break point, B_P(s), s ∈ Z⁺, with . Then, for any and , we have w₁(k_P(s), j) < w₁(k_P(s), i).

Proof of Property 1 The proof follows by contradiction. Suppose the property does not hold. Then, the current optimal solution, , cannot be in the feasible region, contradicting with its optimality.

We are ready to prove Proposition 2.

Proof of Proposition 2 Consider break point B_P(s), with . To simplify the subsequent notation, denote B_P(s) by B. From Proposition 1, if the next break point, B_P(s + 1), exists within the range (B_P(s), ∑_{j ∈ Ω}c_j), then there must exist . Furthermore, from Property 1, because the optimal solution is an extreme point solution, for sufficiently small , the optimal basis must be , which then implies that . In addition, by complementary slackness conditions, the corresponding dual solution at B + δ is given by In what follows, we show that this solution is dual feasible, which implies the optimality of the corresponding primal solution. For dual feasibility, from Constraint (10(10) ), we must have: (A13) By Property 2, the new feasible region for the dual problem at budget B + δ is given by Constraints (10(10) )(k), for k ∈

, where and . Consider any . By Property 1, w₁(k_P(s), k) < w₁(k_P(s), k₂) = w*₁. Then, it trivially follows that Then, if we let we have dual feasibility for any . Equivalently, This completes the proof.

Proof of Proposition 3 The result trivially follows because any break point solution to D-RMP is also optimal for RMP at the break point budget levels.

Notes

¹False-positive and false-negative test results are possible due to technical errors, biological factors (e.g., the presence of antibodies, other infections, or immunizations cross-reacting with test agents), immunological window periods, or imperfect level of knowledge (e.g., poorly understood cross-reactions in healthy individuals); see Dow (2000); Bihl et al. (2007); and Moore et al. (2007).

²The NAT for HBV was approved by the FDA in 2010.

³The significantly longer window periods for serological screening protocols are major contributors to TTI risk.

⁴Another example of such an investigational protocol that occurred in the past is for the parasite that causes babesiosis in New England and the upper Midwest.

⁵Typically the variable testing cost includes the cost of reagents and test kits, supplies (e.g., insulated shipping container, packaging, gel bags, specimen bags), and consumables (e.g., tubes, labels, specimen stoppers, bleach, gloves), as well as cost of shipment supplies and shipping (often paid on a per donation basis) (personal communication with the American Red Cross).

⁶Note the use of the expression, , for Overall Risk, rather than the expression, . The former expression is the one that corresponds to the TTI risk under the differential testing scheme, as it represents the proportion of infected blood within the blood pool classified as infection-free.

⁷Each unit of blood is tested with only one test set. In other words, , where N denotes the total number of donations per year, need to be integers. Since N is very large (e.g., in the order of 16 000 000 per year for the United States), we can obtain integer units of blood tested by each test set while keeping a six-point decimal accuracy. Hence, we omit the integrality constraints for ease of obtaining the solution and analysis.

⁸It trivially follows that for B = 0, p*₀ = 1 (corresponding to set S₀ = ∅), while for B ≥ ∑_{j ∈ Ω}c_j, p*_f = 1 (corresponding to set Ω; by Assumption 2).

⁹We have found counter-examples in which both of the aforementioned properties fail (as well as many others); see Section 5.

¹⁰This assumption is reasonable, especially in the developed world, as pre-donation screening (physical exam and donor questionnaire), uniformly used in all Blood Centers, is effective in deferring co-infected donors.

¹¹The highest budgets allow for multiple tests for each TTI; the $52 and $56 budget for Sub-Saharan Africa and the United States, respectively, corresponds to a break point.

Additional information

Notes on contributors

Douglas R. Bish

Douglas Bish is an Associate Professor at Virginia Tech in the Department of Industrial and Systems Engineering, with a secondary appointment at the Virginia Tech–Carilion School of Medicine. His research focuses on healthcare and emergency management. He is the recipient of the National Science Foundation CAREER award and the INFORMS 2011 Pierskalla Award for the best operations research-based paper in healthcare.

Ebru K. Bish

Ebru Bish is an Associate Professor at Virginia Tech in the Department of Industrial and Systems Engineering. Her research focuses broadly on the design and management of healthcare systems. She is the recipient of the INFORMS 2011 Pierskalla Award for the best operations research–based paper in healthcare. This paper came out a body of work, funded by the National Science Foundation, that studies screening strategies for donated blood.

Ryan S. Xie

Ryan S. Xie earned his Ph.D. from Virginia Tech, Department of Industrial and Systems Engineering. He was advised by Doug Bish and Ebru Bish for his Ph.D., and this work came out of his dissertation. He is the recipient of the INFORMS 2011 Pierskalla Award for the best operations research–based paper in healthcare. Currently, he is working at amazon.com.

Susan L. Stramer

Susan Stramer is an Executive Scientific Officer at the American Red Cross, where she provides scientific leadership in the area of infectious disease for the national testing laboratories and blood collection regions. She is also a liaison to the Transfusion Medicine Committee of the College of American Pathologists and is a member of the Scientific Advisory Committee of HemaQuebec. Along with collaborators, she was a recipient of the Charles C Shepard Science award from the CDC in 1999 and was nominated for the same award in 2004 and 2009.