- Email: [email protected]
Application of optimization methods to the hematological support of patients with disseminated malignancies
Application of Optimization Methods to the Hematological Support of Patients with Disseminated Malignancies
RICHARD SIMON National
Cancer Institute,
Bethesda,
Maryland
20014
Communicated by S. Azen
ABSTRACT The problem of providing long-term platelet transfusion support for patients being treated for malignancies is described. Optimization techniques are identified for one important aspect of this problem.
INTRODUCTION
Most drugs used in the chemotherapy of human cancers are toxic to normal tissue as well as to malignant cells. The degree of success achieved in the treatment of drug-sensitive tumors is related to the amount of drug which may be administered before being limited by toxicity [l]. Because most of the drugs used are selectively lethal to proliferating cells, the bone marrow is frequently the site of this dose-limiting toxicity. In order to permit the aggressive utilization of such drugs, the transfusion of blood products to such patients has become an integral part of the chemotherapy of many disseminated malignancies [2]. This is particularly true for the acute leukemias, because the disease itself usually precludes normal bonemarrow function. In the acute leukemias, infection and hemorrhage account for approximately 75 percent of causes of death. These complications are the result of reductions in the levels of circulating granulocytes and platelets, respectively. In this paper application of mathematical techniques to one aspect of the problem of the hematological support of patients with malignant disease will be discussed. The models and methods to be described were developed to provide a system for the selection of platelet donors, but this approach may at some point also be relevant to the selection of granulocyte donors. The effective selection of platelet donors is important not only for the prevention of hemorrhage, but also because
MATHEMATICAL
BIOSCIENCES
25, 125-138 (1975)
Q American Elsevier Publishing Company, Inc., 1975
125
126
RICHARD SIMON
recent research has indicated that incompatible platelet transfusions may lower the patient’s granulocyte count and thereby render the patient more susceptible to infection. Before discussing the mathematical methods, it is necessary to describe certain aspects of the problem of providing hematological support. The period of aggressive chemotherapy may be rather long, and hence some patients require transfusion of blood products for periods ranging from weeks to months. Providing red-blood-cell support to such patients is a much simpler problem than that of providing compatible platelets or granulocytes [2]. Because the immunology of the red-cell system is well understood, and because there are few major antigen groups, the identification of a compatible donor is easily accomplished. Red-blood-cells can be effectively stored for later use, and hence the logistics of the blood banking and delivery of red-cells is a manageable problem. In contrast, the immunology of the platelet and granulocyte systems is rather poorly understood today, there are many major antigens, and these cells cannot be effectively stored for later use [2,3]. Research within the last few years has demonstrated the importance of the system of HLA antigens for predicting platelet compatibility [4,5]. Because of the complexity of this system of antigens, extremely large populations of donors are generally required in order to find an individual matched to the patient unless the patient should be fortunate enough to have a matched sibling. The fact that the cells cannot be stored means that the donor identified must donate on the day on which the cells are needed, and the donation of 4 units of platelets is typically a 2: hour procedure [2]. Clearly this mode of supporting more than a very few patients is beset with problems. There are alternative approaches, however. A patient not previously transfused will initially receive as compatible the platelets of any donor. After a period ranging from days to several weeks of retransfusions with the same foreign antigens, the patient will develop antibodies and will not respond to transfusions from donors whose platelets contain these antigens. The antibodies developed are specific for the foreign antigens which have been transfused, and the patient will still respond well to donors containing different and new foreign antigens. Thus by properly selecting donors, one can extend the period of effective hematological support [4,5]. Because of the logistical problems produced by the inability to effectively store platelets, and because many patients require support for long periods of time, a common mode for supporting large numbers of patients is the utilization of a panel of HLA-typed donors each of whom donates 2 units of platelets once a week. On any day, the platelets collected will be allocated to the patients who require support. Because a patient will usually
MATHEMATICAL
METHODS FOR HEMATOLOGICAL
SUPPORT
127
require 4 units on a given day, the patient will receive platelets from two different donors at the same time. Each day, the physician in charge of platelet support determines which individuals have donated platelets and which patients require transfusions. He then allocates donated platelets to patients in a non-formalized manner which attempts to provide a patient with platelets from a donor to whom he has recently shown a good transfusion response or to whom compatibility might be predicted on the basis of HLA type and in vitro antibodies. This manual assignment is performed sequentially for each patient without any attempt at overall optimization. In supporting a large number of patients, it becomes a considerable operational problem to record what patients were transfused from which donors and what the responses were, and to utilize this information together with HLA typing information and information concerning in vitro antibody screening in order to allocate the donors available to the patients in a reasonable manner. For this reason it is desirable to store in a computer the transfusion history of patients, HLA typing data, and in vitro antibody screening data, to update this information daily, and to mechanize the allocation procedure. An approach to one aspect of this problem will be presented here; that is, the development of methods for the allocation of donors to patients in reasonable ways, given estimates of the probabilities of a compatible response for each patient-donor pair. The problem of the estimation of these probabilities is itself an interesting area for the application of biomathematical methodology. An approach to this estimation is outlined here, but a detailed investigation is deferred to a subsequent paper. THE ALLOCATION
PROBLEM
In the notation to follow, indices i and j will denote a particular patient and donor, respectively. Let a, denote the number of units required by patient i, 4 the number of units contributed by donor j, pG the probability that donor j is compatible with patient i, and x0 the number of units transfused to patient i from donor j. We shall let I and J denote the number of patients and donors respectively. An allocation specified by a set of x!, will be called feasible if it satisfies the following constraints: for each j, J jT, xij > a,
for each i,
xii > 0 and integer.
(1)
128
RICHARD SIMON
In most cases we wish to require that each donor be assigned to at most one patient. This can be accomplished by requiring that the xV satisfy the following constraints rather than those above: I
for each;,
Xx& i=l
5 bjxu >
a,
(2)
for each i,
j-1
xii > 0 and integer. Since the xti are non-negative integers, it follows that x0 = 1 or 0, to be interpreted as donor j contributing or not contributing all 6, units to patient i. Usually, in practice, all b, will be equal, and each a, is an integer multiple of the common b. Furthermore, the number of units required by a patient is specified as an equality in order that a patient not be exposed to too many donors on any one day. This is done because each donor to which a patient is exposed carries the risk of an incompatible transfusion reaction and increases the rate at which antibodies are developed. This system can be written
i,x,, 1
for each j:
for each i, xti > 0 and integer. In all of these constraints, the right-hand side consists entirely of integers. Three criteria functions to be studied representing the adequacy of allocation are as follows:
f*(x) = mjn
1-
I [
h(x)= i: i-1
II Ulxb>O)
(1 -IQ
, 1
[I- (j,~>o)o]. I,
MATHEMATICAL
METHODS FOR HEMATOLOGICAL
129
SUPPORT
Functionf, represents the expected number of compatible units transfused when x is a feasible solution to the constraint set (1). If the constraint set (2) or (3) is used, thenf, should be modified to: f(x)
= i
i
i=l
j=I
pgbjxg.
The function f2 represents the probability of a compatible transfusion for that patient with the smallest such probability. The function f3 represents the expected number of patients receiving a compatible transfusion. If x is a feasible solution to the constraint set (2) or (3) thenf, andf, can be written more simply as
f2(x) = mm
1[
6 (1 -piixij) j=l
f&4= i: 1 - fi (I i=l
F(x) = i i=l
2
wkPr[patient
-pijxii)
/=I
We are concerned here with methods of maximize a chosen criterion function. All some intuitive appeal. The selection of somewhat upon the degree of difficulty in Criteria fi and f3are special cases of a
, I
. I
finding feasible solutions which of these criterion functions have a particular function depends finding good solutions. more general function
i receives exactly k compatible
units].
k-0
F is equivalent to f,when w,= kw, and F is equivalent to f3 when w,=O, w, = w for k > 0. The maximization of F for general w, is quite difficult, and no intuitively meaningful choices of w, other than those embodied in f, and fa are apparent. Consequently, F will not be dealt with in this paper. Before proceeding, it should be noted that complete enumeration of all feasible allocations is not a viable approach for problems of practical size. For example, if J = 2I, a, = 4 for all i, and bj = 2 for all j, then the number of feasible solutions to (2) or (3) is (21)!/2’; for I= 10, this number exceeds 10’5. MAXIMIZING THE EXPECTED NUMBER OF COMPATIBLE UNITS TRANSFUSED Maximization
off, over the set (1) orfl
over the set (3) is a form of the
130
RICHARD
SIMON
classical transportation problem and can be efficiently solved by algorithms for this problem such as those given in [6]. Maximization of fl over the set (2) is an integer linear program with binary variables. Such problems may be computationally quite difficult to solve, though numerous algorithms are available [7-131. In [14] and [15] specialized algorithms are developed for a variant of the constraint set (2), and these algorithms are easily adapted to our problem. The method of [14] is an implicit enumeration technique, whereas [15] presents a branch-and-bound approach with bounds obtained by solving a related simple transportation problem. MAXIMIZING THE MINIMUM PROBABILITY OF A COMPATIBLE TRANSFUSION In maximizing j2 over one of the constraint sets, we will utilize the results of [16]. Given a finite set E, a “clutter” on E is a family F of subsets of E such that no member of F is contained in another member of F. Let C , . . . ,.I }. Let E denote the class denote the set of pairs {(i,j)li=l,...,Z;j=l of subsets of C of the form {(i,j,), (i,j,), . . . , (i,jp)} where ],,~z,. . . ,jp E { 1,2,. . . ,J}. Thus, thinking of C as a grid with I rows and J columns, E is the class of sets of cells in the same row. Let F denote the family of subsets S of E such that S contains one subset corresponding to each row and such that if ((i,jJ, (ij,), . . . , (i,j,)} is the subset of S corresponding to row i, then
5 b, > a,
(4)
k-l
and there are no two distinct indices i,i, such that (i,J) and (i2,j) are in the union of the elements of S. An element S of F determines a solution to the constraint set (2) by setting xii= 1 if (ij) is a member of the union of elements of S. It can be seen that F is a clutter on E, and thus an x which maximizesj, over the set (2) can be obtained from the threshold algorithm of [ 161. The threshold algorithm proceeds as follows. For an element e of E, (i.e. a set of cells in the same row), let s(e)=l-
II (l-p,,). (i,j)Ef?
Choose elements of E in non-increasing magnitude of g until the set of chosen elements first contains a member S of F. Any such S will produce a solution x which maximizes jz over the set (2). The proof of the threshold algorithm is given in [16]. Maximizing j2 over the set (3) is very similar to maximizing ji over the set (2). Equation (4) becomes the requirement that the subset corresponding to row i be of cardinality ai/b, and this makes the
MATHEMATICAL
METHODS
FOR HEMATOLOGICAL
SUPPORT
131
algorithm considerably more rapid. The computational effort of the threshold algorithm is primarily that of determining whether the selected elements of E contain a member of F. This can be represented as determining when a set partitioning problem has a feasible solution [17,18]. The algorithm of Garfinkel and Nemhauser [ 171 can be used, and they have reported excellent efficiency for their procedure. In maximizing fi over the constraint set (1) the threshold method may again be used, but the difficulty of determining when a feasible solution exists makes this approach impractical. Suppose that instead of maximizing f2, we wish to maximize
A very simple form of the threshold algorithm can be used in which single cells (ij) are selected in non-increasing order of po. For the constraint set (1) or (3) we determine whether a feasible solution involving only selected cells exists by solving a maximal flow problem for the two-terminal graph of Fig. 1. All arcs are directed to the right. The capacity of the arc connecting donor j with patient i is zero if element (ij) of E has not yet been selected by the threshold algorithm, and is infinite otherwise. The capacity of the arc connecting T, to donor j is b,, and the capacity of the arc connecting patient i to T, is a,. If the maximal flow from T, to T2 is at least Ecf_,q, then a feasible (and therefore optimal) solution involving only the selected cells exists and is identified by the maximal flow. Very efficient algorithms are available for maximal flow problems [19], and Garfinkel [20] has reported excellent results in using this procedure for the special case where all a, and b, are unity. In maximizing f4 over (2) the threshold algorithm selecting single cells may be used, but the existence of a feasible solution cannot be determined as a maximal flow problem. With the constraint set (2) there is no computational advantage in selectingf, over f2 as the criterion function. In general, the maximization of f2 over the set (2) or (3) and the maximization of f4 over any of the sets is computationally feasible and intuitively attractive. It is attractive because it is the only criterion considered here which does not assume that benefits to individual patients are substitutable. However, when some patient has a very small probability of response for all donors, then almost all feasible solutions will be optimal by criteria f2 or f4. For such recipients, it is suggested that the maximization should be performed excluding them, and they be assigned donors not already allocated. In general, there may be a number of feasible solutions which maximize fi, and it would be desirable to select the one which maximizes f, or f3.
132
RICHARD
SIMON
Patisntr
FIG. 1. Sample
two-terminal
graph
for maximal
flow problem.
MAXIMIZING THE EXPECTED NUMBER OF PATIENTS RECEIVING AT LEAST ONE COMPATIBLE UNIT We shall not address ourselves to the problem of maximizing& over the set (1). This appears to be a problem of great computational complexity. In the special case a,/b = 1 for all i, maximizing fj over the set (3) is equivalent to maximizing f, over the set (3). In general, however, these two problems are not equivalent. In order to maximize f3 over the set (2) or (3) the dynamic-programming approach of Bellman [21] has been employed. For any patient i and any set of donors D, define r,(D)=
1-
n
(l-p,-)>
jGD
b(D)=
c
bj.
jED
The dynamic-programming of donors: h,(D)=
recursion
is defined as follows for any subset D
(D,c~~~.D.,>~,)[‘i(D’)+hi+~(D-D’)]
and h,+,-0.
The optimal value of f3 is h,({ l,..., J}), and the allocation of donors to patient i is specified by the subset D’ selected at stage i in calculating over the set (3) the maximization at stage i h&(1,..., J}). In maximizing&
MATHEMATICAL
METHODS FOR HEMATOLOGICAL
SUPPORT
133
is performed over the set {D’c Dlb(D’)= q/b}, and hi need only be J-Ci-L,u,/6. This tabulated for subsets of { 1,2,. . . , J} of cardinality considerably reduces the dimensionality of the problem, though the storage requirements can still be considerable for all but small problems unless specialized methods are used such as described in [22]. An approximation to maximizingf, over the set (3) can be obtained as follows. Maximizingf, over the set (3) is equivalent to minimizing -f3(x)
= i i=i
ir
(1 -p&-
over the set (3). One could approximate
A(x) = i i=l
over the set (3). This simplifies
1
j=l
this by minimizing
log
ITi (1 -pijx,>
j=l
to
_m)= ,g, i, [log(l -P,j)l% which yields a standard transportation problem. Methods of geometric programming [23] or complementary geometric programming [24] could possibly be applied to the problem of maximizing fJ over the sets (2) or (3) as was done for a similar problem in [25], but these methods would not insure integer solutions. SPECIAL
CASES
When using the constraint set (3), if all of the pij are equal, then any feasible solution is optimal with regard to each of the criteria f,, fi, and f3. For this constraint set, if for some patient i, p,, =p, for all donorsj, then the problem may be solved by maximizing the chosen criterion with patient j omitted, and assigning to patient i any ai/b donors not already allocated. A similar decomposition is generally not possible when for some donor j, pii =pi for all patients i. When using the constraint sets (1) or (2), these simplifications also do not apply. NUMERICAL
EXAMPLE
Table 1 shows a hypothetical example of the probabilities of compatible transfusions for three patients and six donors. It will be assumed that bj= 1 for all donorsj, that a, = 2 for all patients i, and that the constraint set (3) is used. Thus, the problem is to assign two donors to each patient, with each
134
RICHARD
SIMON
donor assigned to a single patient. Table 2 indicates optimal assignments for each of the three optimization criteria. It can be seen that the three optimal solutions differ. Table 3 shows the values of the criteria functions for the optimal solutions (denoted f,), for purely random assignment, and for a procedure denoted “sequential maximum” assignment. Using the sequential maximum procedure, one randomly orders the patients and sequentially assigns to each patient the best two donors among those not previously assigned. For the purely random and sequential maximum procedures, expected values and standard deviations are presented in Table 3, both figures having been estimated by computer simulation with 100 replications. It can be seen that the methods of random or sequential maximum assignment are quite sub-optimal for all criteria functions. Also, the assignment produced by maximizing f, is very poor with regard to criteria fi andf,. Table pi, for Numerical
1 Example
I i
jl
1
0.9 0.7 0.8
2 3
Optimal
0.9
0
0.5 0.8
0.5 0.8
Assignments
5
0.15 0
0.4 0.4
0.2
0.5
Table 2 for Numerical Donors
Patient
4
3
2
6 0 0.6 0.9
Example
Assigned
f1
f2
Values of Criteria
Table 3 for Numerical
f3
Example
Assignment Criterion
f3
i,
L
f,
3.90 0.40 2.37
3.65 0.88 2.70
3.65 0.85 2.72
Pure random 3.00~0.50 0.49 rt 0.24 2.24-c 0.33
Sequential max 3.57a0.19 0.56a0.14 2.41 kO.13
MATHEMATICAL
ESTIMATION
METHODS
FOR
HEMATOLOGICAL
SUPPORT
13.5
OF pij
The usefulness of the allocation methods which have been presented depends upon the adequacy of procedures used to estimate the probability of a compatible transfusion for each patient-donor pair. The term “probability” is used in the sense of subjective probability, for the biology of the patient and donor at any time determine the transfusion response. In some cases, this response can be predicted prior to transfusion using in vitro antibody detection methods. In cases where an antibody to a donor j is found in the serum of patient i, pij = 0. In many cases, however, no antibody is detected by in vitro means to explain a poor transfusion response. As in vitro methods improve, all of the pii will be determined as zero or unity. For the present, however, mathematical methods are required for the estimation of these parameters. The development of effective mathematical estimation methods is a challenging area for biometric research. In this section, an approach to this problem will be outlined. If a patient i has not been previously transfused, then pu = 1 for all donors j. If the patient has on a recent day received a transfusion from a single donor j and exhibited a compatible response, then pi, = 1. If the patient has recently on some day received a transfusion from a set D of donors and exhibited a poor response, thep, =0 for all donors j in D unless the transfusion response can be attributed to a concurrent infection or drug effect which interferes with the evaluation. If in uitro testing indicates the presence of an antibody in the patient against the platelets of a donor, then pil=o. The elucidation of the HLA system of antigens as determinants of platelet compatibility makes possible a general model for estimation of the pii. In this model it is assumed that a patient will respond well to a donor unless the patient possesses an antibody to some HLA antigen of the donor [4]. Each individual contains determinants for four HLA antigens-two determinants per chromosome, with each chromosome containing a selection from each of two allelic series. Focusing attention on a particular patient, we will drop the subscript i. Let pj denote the probability that the patient is compatible with the platelets of donor j. Assume that the HLA antigens are arbitrarily indexed, and let qk denote the subjective probability that the patient does not have an antibody to the k’th HLA antigen. Let H(j) denote the set of HLA antigens of donor j. Then
Pj =
1fl 1
EH(j)
qk.
kgH(A
EHci, denotes
the expected value with regard to the set H(j).
This must be
RICHARD
136
SIMON
included in Eq. (5), because if less than four distinct HLA antigens are determined for donor j, one does not know for certain whether this is due to homozygosity or because the donor has a rare antigen for which no test is made. Using Eq. (5), the estimation of pj is reduced to the estimation of the qk. The parameters qk depend upon the HLA type of the patient, the HLA types of previous donors for the patient and the frequency of their donations, and the rate at which an antibody to antigen k is developed. If the patient has antigen k, then qk = I. If anti-k is detected in vitro, qk = 0. Two approaches to the estimation of the qk will be indicated. In the first approach, for a patient not having antigen k, qk is estimated as f(n, k), a function only of antigen k and the total number n of units of platelets containing antigen k which have been previously transfused to the patient. f(0, k) = 1, lim,,,f(n, k) = 0, and the functional form of f(n, k) is estimated from experimental data. In describing the second approach to the estimation of the qk, the following notation is used. For each transfusion t to the patient under consideration: at t, 1 otherwise,
l&(t)=0 if the patient has an anti-k antibody D (t) = the set of donors for transfusion t, C(f)= 1 if the patient had a good transfusion The transfusion history relations for each t: Rk(t)
of the patient
I( =
response,
is represented
implies
by the following
C(t) if dt>=O ,
> C(t) if m(t)=
&(t)=O
0 otherwise.
&(t’)=O
1
(6)
for all t’> T.
If Rk(t)=O for any t, then qk =O; otherwise, qk = 1. Though some of the &(t) variables are uniquely determined by (6), others are not. Exact and approximate methods for solving the system (6) are currently under investigation. SUMMARY Because most of the drugs used in the chemotherapy of disseminated malignancies are selectively lethal to proliferating cells, the bone marrow is generally the site of dose-limiting toxicity. This toxicity limits the degree of eradiction of malignant cells that can be achieved. For this reason, considerable amounts of effort and resources are devoted to long-term hematolo-
MATHEMATICAL
METHODS
FOR
HEMATOLOGICAL
137
SUPPORT
gical support of patients in cancer centers. In this paper, we have examined one aspect of the problem of improving the quality of such support. Certain of the results developed here are being implemented in a computer system being developed at the National Cancer Institute. It is hoped that this paper will stimulate the interest of other mathematicians in developing improved solutions to the problems presented. The author wishes to express his appreciation to Dr. Ronald A. Yankee of the Childrens Cancer Research Foundation for many useful discussions, and to a referee for his helpful comments. REFERENCES I 2
G. Mathe, Ed., Scientific Basis of Cancer Chemotherapy, Springer, New York, 1969. R. G. Graw, Jr. and R. A. Yankee, Principles of hematologic supportive care, Med.
3
Clin. N. Am. 57, 441 (1973). P. I. Terasaki and D. P. Singal,
4
Ann. Rev. Med. 20, 175 (1969). R. A. Yankee, Importance of histocompatibility
5
Sang 20,419 (1971). R. A. Yankee and K. Graff,
of non-family
matched
6
Congr. Blood Transfus., Washington, D. C., 1972. G. B. Danzig, Linear Programming and Extensions,
Princeton
7
8 9 10 11 12 13 14 15 16 17
Human
Efficacy
1963. R. E. Gomory, Solving linear programming Analysis, Proc. 10th Symp. in Appl. Math. Hall, Jr., Eds.), 1960, pp. 211-216.
histocompatibility in platelet
antigens transfusion platelets,
of leukocytes, therapy,
VOX
Proc. 13th Znf.
U. P., Princeton,
N. J.,
problems in integers, in Combinatorial of Am. Math. Sot. (R. Bellman and M.
M. L. Balinski, Integer programming: methods, uses, computations, Manage. Sri. 12, 253 (1965). E. Bales, An additive algorithm for solving linear programs with zero-one variables, Oper. Res. 13, 517 (1965). S. Kaplan, Solution of the Lorie-Savage and similar integer programming problems by the generalized Lagrange multiplier method, Oper. Res. 14, 1140 (1966). C. E. Lemke and K. Spielman, Direct search algorithms for zero-one and mixedinteger programming, Oper. Res. 15, 892 (1967). E. Balas, Discrete programming by the filter method, Oper. Res. 15, 915 (1967). P. L. Hammer and S. Rudenaunu, Pseudo-Boolean programming, Oper. Res. 17, 233 (1969). A. De Maio and C. Roveda, An all zero-one algorithm for a certain class of transportation problems, Oper. Res. 19, 1406 (1971). V. Srinivasan and G. L. Thompson, An algorithm for assigning uses to sources in a special class of transportation problems, Oper. Res. 21, 284 (1973). J. Edmonds and D. R. Fulkerson, Bottleneck extremes, RM-5375-PR, The Rand Corporation, Santa Monica, California (1968). R. S. Garfinkel and G. L. Nemhauser, The set-partitioning problem: set covering with equality constraints, Oper. Res. 17, 848 (1969).
138 18 19 20 21 22 23 24 25
RICHARD
SIMON
E. Balas and M. W. Padberg, On the set covering problem, Oper. Res. 20, 1152 (1972). L. R. Ford, Jr. and D. R. Fulkerson, Flows in Networks, Princeton U. P., Princeton, N. J., 1962. R. S. Garfinkel, An improved algorithm for the bottleneck assignment problem, Oper. Rex 19, 1747 (1971). R. Bellman, Dynamic Programming, Princeton U. P., Princeton, N. J., 1957. P. J. Wong and D. G. Luenberger, Reducing the memory requirements of dynamic programming, Oper. Rex 16, 1115 (1968). R. J. Duffin, E. L. Peterson, and C. Zener, Geometric Programming-Theory Application, Wiley, New York, 1967. M. Avriel and A. C. Williams, Appl. Math. 19, 125 (1970). U. Passy, Nonlinear assignment Rex 19, 1675 (1971).
Complementary problems
treated
geometric
programming,
by geometric
programming,
SIAM J. Oper.
RICHARD SIMON National
Cancer Institute,
Bethesda,
Maryland
20014
Communicated by S. Azen
ABSTRACT The problem of providing long-term platelet transfusion support for patients being treated for malignancies is described. Optimization techniques are identified for one important aspect of this problem.
INTRODUCTION
Most drugs used in the chemotherapy of human cancers are toxic to normal tissue as well as to malignant cells. The degree of success achieved in the treatment of drug-sensitive tumors is related to the amount of drug which may be administered before being limited by toxicity [l]. Because most of the drugs used are selectively lethal to proliferating cells, the bone marrow is frequently the site of this dose-limiting toxicity. In order to permit the aggressive utilization of such drugs, the transfusion of blood products to such patients has become an integral part of the chemotherapy of many disseminated malignancies [2]. This is particularly true for the acute leukemias, because the disease itself usually precludes normal bonemarrow function. In the acute leukemias, infection and hemorrhage account for approximately 75 percent of causes of death. These complications are the result of reductions in the levels of circulating granulocytes and platelets, respectively. In this paper application of mathematical techniques to one aspect of the problem of the hematological support of patients with malignant disease will be discussed. The models and methods to be described were developed to provide a system for the selection of platelet donors, but this approach may at some point also be relevant to the selection of granulocyte donors. The effective selection of platelet donors is important not only for the prevention of hemorrhage, but also because
MATHEMATICAL
BIOSCIENCES
25, 125-138 (1975)
Q American Elsevier Publishing Company, Inc., 1975
125
126
RICHARD SIMON
recent research has indicated that incompatible platelet transfusions may lower the patient’s granulocyte count and thereby render the patient more susceptible to infection. Before discussing the mathematical methods, it is necessary to describe certain aspects of the problem of providing hematological support. The period of aggressive chemotherapy may be rather long, and hence some patients require transfusion of blood products for periods ranging from weeks to months. Providing red-blood-cell support to such patients is a much simpler problem than that of providing compatible platelets or granulocytes [2]. Because the immunology of the red-cell system is well understood, and because there are few major antigen groups, the identification of a compatible donor is easily accomplished. Red-blood-cells can be effectively stored for later use, and hence the logistics of the blood banking and delivery of red-cells is a manageable problem. In contrast, the immunology of the platelet and granulocyte systems is rather poorly understood today, there are many major antigens, and these cells cannot be effectively stored for later use [2,3]. Research within the last few years has demonstrated the importance of the system of HLA antigens for predicting platelet compatibility [4,5]. Because of the complexity of this system of antigens, extremely large populations of donors are generally required in order to find an individual matched to the patient unless the patient should be fortunate enough to have a matched sibling. The fact that the cells cannot be stored means that the donor identified must donate on the day on which the cells are needed, and the donation of 4 units of platelets is typically a 2: hour procedure [2]. Clearly this mode of supporting more than a very few patients is beset with problems. There are alternative approaches, however. A patient not previously transfused will initially receive as compatible the platelets of any donor. After a period ranging from days to several weeks of retransfusions with the same foreign antigens, the patient will develop antibodies and will not respond to transfusions from donors whose platelets contain these antigens. The antibodies developed are specific for the foreign antigens which have been transfused, and the patient will still respond well to donors containing different and new foreign antigens. Thus by properly selecting donors, one can extend the period of effective hematological support [4,5]. Because of the logistical problems produced by the inability to effectively store platelets, and because many patients require support for long periods of time, a common mode for supporting large numbers of patients is the utilization of a panel of HLA-typed donors each of whom donates 2 units of platelets once a week. On any day, the platelets collected will be allocated to the patients who require support. Because a patient will usually
MATHEMATICAL
METHODS FOR HEMATOLOGICAL
SUPPORT
127
require 4 units on a given day, the patient will receive platelets from two different donors at the same time. Each day, the physician in charge of platelet support determines which individuals have donated platelets and which patients require transfusions. He then allocates donated platelets to patients in a non-formalized manner which attempts to provide a patient with platelets from a donor to whom he has recently shown a good transfusion response or to whom compatibility might be predicted on the basis of HLA type and in vitro antibodies. This manual assignment is performed sequentially for each patient without any attempt at overall optimization. In supporting a large number of patients, it becomes a considerable operational problem to record what patients were transfused from which donors and what the responses were, and to utilize this information together with HLA typing information and information concerning in vitro antibody screening in order to allocate the donors available to the patients in a reasonable manner. For this reason it is desirable to store in a computer the transfusion history of patients, HLA typing data, and in vitro antibody screening data, to update this information daily, and to mechanize the allocation procedure. An approach to one aspect of this problem will be presented here; that is, the development of methods for the allocation of donors to patients in reasonable ways, given estimates of the probabilities of a compatible response for each patient-donor pair. The problem of the estimation of these probabilities is itself an interesting area for the application of biomathematical methodology. An approach to this estimation is outlined here, but a detailed investigation is deferred to a subsequent paper. THE ALLOCATION
PROBLEM
In the notation to follow, indices i and j will denote a particular patient and donor, respectively. Let a, denote the number of units required by patient i, 4 the number of units contributed by donor j, pG the probability that donor j is compatible with patient i, and x0 the number of units transfused to patient i from donor j. We shall let I and J denote the number of patients and donors respectively. An allocation specified by a set of x!, will be called feasible if it satisfies the following constraints: for each j, J jT, xij > a,
for each i,
xii > 0 and integer.
(1)
128
RICHARD SIMON
In most cases we wish to require that each donor be assigned to at most one patient. This can be accomplished by requiring that the xV satisfy the following constraints rather than those above: I
for each;,
Xx& i=l
5 bjxu >
a,
(2)
for each i,
j-1
xii > 0 and integer. Since the xti are non-negative integers, it follows that x0 = 1 or 0, to be interpreted as donor j contributing or not contributing all 6, units to patient i. Usually, in practice, all b, will be equal, and each a, is an integer multiple of the common b. Furthermore, the number of units required by a patient is specified as an equality in order that a patient not be exposed to too many donors on any one day. This is done because each donor to which a patient is exposed carries the risk of an incompatible transfusion reaction and increases the rate at which antibodies are developed. This system can be written
i,x,, 1
for each j:
for each i, xti > 0 and integer. In all of these constraints, the right-hand side consists entirely of integers. Three criteria functions to be studied representing the adequacy of allocation are as follows:
f*(x) = mjn
1-
I [
h(x)= i: i-1
II Ulxb>O)
(1 -IQ
, 1
[I- (j,~>o)o]. I,
MATHEMATICAL
METHODS FOR HEMATOLOGICAL
129
SUPPORT
Functionf, represents the expected number of compatible units transfused when x is a feasible solution to the constraint set (1). If the constraint set (2) or (3) is used, thenf, should be modified to: f(x)
= i
i
i=l
j=I
pgbjxg.
The function f2 represents the probability of a compatible transfusion for that patient with the smallest such probability. The function f3 represents the expected number of patients receiving a compatible transfusion. If x is a feasible solution to the constraint set (2) or (3) thenf, andf, can be written more simply as
f2(x) = mm
1[
6 (1 -piixij) j=l
f&4= i: 1 - fi (I i=l
F(x) = i i=l
2
wkPr[patient
-pijxii)
/=I
We are concerned here with methods of maximize a chosen criterion function. All some intuitive appeal. The selection of somewhat upon the degree of difficulty in Criteria fi and f3are special cases of a
, I
. I
finding feasible solutions which of these criterion functions have a particular function depends finding good solutions. more general function
i receives exactly k compatible
units].
k-0
F is equivalent to f,when w,= kw, and F is equivalent to f3 when w,=O, w, = w for k > 0. The maximization of F for general w, is quite difficult, and no intuitively meaningful choices of w, other than those embodied in f, and fa are apparent. Consequently, F will not be dealt with in this paper. Before proceeding, it should be noted that complete enumeration of all feasible allocations is not a viable approach for problems of practical size. For example, if J = 2I, a, = 4 for all i, and bj = 2 for all j, then the number of feasible solutions to (2) or (3) is (21)!/2’; for I= 10, this number exceeds 10’5. MAXIMIZING THE EXPECTED NUMBER OF COMPATIBLE UNITS TRANSFUSED Maximization
off, over the set (1) orfl
over the set (3) is a form of the
130
RICHARD
SIMON
classical transportation problem and can be efficiently solved by algorithms for this problem such as those given in [6]. Maximization of fl over the set (2) is an integer linear program with binary variables. Such problems may be computationally quite difficult to solve, though numerous algorithms are available [7-131. In [14] and [15] specialized algorithms are developed for a variant of the constraint set (2), and these algorithms are easily adapted to our problem. The method of [14] is an implicit enumeration technique, whereas [15] presents a branch-and-bound approach with bounds obtained by solving a related simple transportation problem. MAXIMIZING THE MINIMUM PROBABILITY OF A COMPATIBLE TRANSFUSION In maximizing j2 over one of the constraint sets, we will utilize the results of [16]. Given a finite set E, a “clutter” on E is a family F of subsets of E such that no member of F is contained in another member of F. Let C , . . . ,.I }. Let E denote the class denote the set of pairs {(i,j)li=l,...,Z;j=l of subsets of C of the form {(i,j,), (i,j,), . . . , (i,jp)} where ],,~z,. . . ,jp E { 1,2,. . . ,J}. Thus, thinking of C as a grid with I rows and J columns, E is the class of sets of cells in the same row. Let F denote the family of subsets S of E such that S contains one subset corresponding to each row and such that if ((i,jJ, (ij,), . . . , (i,j,)} is the subset of S corresponding to row i, then
5 b, > a,
(4)
k-l
and there are no two distinct indices i,i, such that (i,J) and (i2,j) are in the union of the elements of S. An element S of F determines a solution to the constraint set (2) by setting xii= 1 if (ij) is a member of the union of elements of S. It can be seen that F is a clutter on E, and thus an x which maximizesj, over the set (2) can be obtained from the threshold algorithm of [ 161. The threshold algorithm proceeds as follows. For an element e of E, (i.e. a set of cells in the same row), let s(e)=l-
II (l-p,,). (i,j)Ef?
Choose elements of E in non-increasing magnitude of g until the set of chosen elements first contains a member S of F. Any such S will produce a solution x which maximizes jz over the set (2). The proof of the threshold algorithm is given in [16]. Maximizing j2 over the set (3) is very similar to maximizing ji over the set (2). Equation (4) becomes the requirement that the subset corresponding to row i be of cardinality ai/b, and this makes the
MATHEMATICAL
METHODS
FOR HEMATOLOGICAL
SUPPORT
131
algorithm considerably more rapid. The computational effort of the threshold algorithm is primarily that of determining whether the selected elements of E contain a member of F. This can be represented as determining when a set partitioning problem has a feasible solution [17,18]. The algorithm of Garfinkel and Nemhauser [ 171 can be used, and they have reported excellent efficiency for their procedure. In maximizing fi over the constraint set (1) the threshold method may again be used, but the difficulty of determining when a feasible solution exists makes this approach impractical. Suppose that instead of maximizing f2, we wish to maximize
A very simple form of the threshold algorithm can be used in which single cells (ij) are selected in non-increasing order of po. For the constraint set (1) or (3) we determine whether a feasible solution involving only selected cells exists by solving a maximal flow problem for the two-terminal graph of Fig. 1. All arcs are directed to the right. The capacity of the arc connecting donor j with patient i is zero if element (ij) of E has not yet been selected by the threshold algorithm, and is infinite otherwise. The capacity of the arc connecting T, to donor j is b,, and the capacity of the arc connecting patient i to T, is a,. If the maximal flow from T, to T2 is at least Ecf_,q, then a feasible (and therefore optimal) solution involving only the selected cells exists and is identified by the maximal flow. Very efficient algorithms are available for maximal flow problems [19], and Garfinkel [20] has reported excellent results in using this procedure for the special case where all a, and b, are unity. In maximizing f4 over (2) the threshold algorithm selecting single cells may be used, but the existence of a feasible solution cannot be determined as a maximal flow problem. With the constraint set (2) there is no computational advantage in selectingf, over f2 as the criterion function. In general, the maximization of f2 over the set (2) or (3) and the maximization of f4 over any of the sets is computationally feasible and intuitively attractive. It is attractive because it is the only criterion considered here which does not assume that benefits to individual patients are substitutable. However, when some patient has a very small probability of response for all donors, then almost all feasible solutions will be optimal by criteria f2 or f4. For such recipients, it is suggested that the maximization should be performed excluding them, and they be assigned donors not already allocated. In general, there may be a number of feasible solutions which maximize fi, and it would be desirable to select the one which maximizes f, or f3.
132
RICHARD
SIMON
Patisntr
FIG. 1. Sample
two-terminal
graph
for maximal
flow problem.
MAXIMIZING THE EXPECTED NUMBER OF PATIENTS RECEIVING AT LEAST ONE COMPATIBLE UNIT We shall not address ourselves to the problem of maximizing& over the set (1). This appears to be a problem of great computational complexity. In the special case a,/b = 1 for all i, maximizing fj over the set (3) is equivalent to maximizing f, over the set (3). In general, however, these two problems are not equivalent. In order to maximize f3 over the set (2) or (3) the dynamic-programming approach of Bellman [21] has been employed. For any patient i and any set of donors D, define r,(D)=
1-
n
(l-p,-)>
jGD
b(D)=
c
bj.
jED
The dynamic-programming of donors: h,(D)=
recursion
is defined as follows for any subset D
(D,c~~~.D.,>~,)[‘i(D’)+hi+~(D-D’)]
and h,+,-0.
The optimal value of f3 is h,({ l,..., J}), and the allocation of donors to patient i is specified by the subset D’ selected at stage i in calculating over the set (3) the maximization at stage i h&(1,..., J}). In maximizing&
MATHEMATICAL
METHODS FOR HEMATOLOGICAL
SUPPORT
133
is performed over the set {D’c Dlb(D’)= q/b}, and hi need only be J-Ci-L,u,/6. This tabulated for subsets of { 1,2,. . . , J} of cardinality considerably reduces the dimensionality of the problem, though the storage requirements can still be considerable for all but small problems unless specialized methods are used such as described in [22]. An approximation to maximizingf, over the set (3) can be obtained as follows. Maximizingf, over the set (3) is equivalent to minimizing -f3(x)
= i i=i
ir
(1 -p&-
over the set (3). One could approximate
A(x) = i i=l
over the set (3). This simplifies
1
j=l
this by minimizing
log
ITi (1 -pijx,>
j=l
to
_m)= ,g, i, [log(l -P,j)l% which yields a standard transportation problem. Methods of geometric programming [23] or complementary geometric programming [24] could possibly be applied to the problem of maximizing fJ over the sets (2) or (3) as was done for a similar problem in [25], but these methods would not insure integer solutions. SPECIAL
CASES
When using the constraint set (3), if all of the pij are equal, then any feasible solution is optimal with regard to each of the criteria f,, fi, and f3. For this constraint set, if for some patient i, p,, =p, for all donorsj, then the problem may be solved by maximizing the chosen criterion with patient j omitted, and assigning to patient i any ai/b donors not already allocated. A similar decomposition is generally not possible when for some donor j, pii =pi for all patients i. When using the constraint sets (1) or (2), these simplifications also do not apply. NUMERICAL
EXAMPLE
Table 1 shows a hypothetical example of the probabilities of compatible transfusions for three patients and six donors. It will be assumed that bj= 1 for all donorsj, that a, = 2 for all patients i, and that the constraint set (3) is used. Thus, the problem is to assign two donors to each patient, with each
134
RICHARD
SIMON
donor assigned to a single patient. Table 2 indicates optimal assignments for each of the three optimization criteria. It can be seen that the three optimal solutions differ. Table 3 shows the values of the criteria functions for the optimal solutions (denoted f,), for purely random assignment, and for a procedure denoted “sequential maximum” assignment. Using the sequential maximum procedure, one randomly orders the patients and sequentially assigns to each patient the best two donors among those not previously assigned. For the purely random and sequential maximum procedures, expected values and standard deviations are presented in Table 3, both figures having been estimated by computer simulation with 100 replications. It can be seen that the methods of random or sequential maximum assignment are quite sub-optimal for all criteria functions. Also, the assignment produced by maximizing f, is very poor with regard to criteria fi andf,. Table pi, for Numerical
1 Example
I i
jl
1
0.9 0.7 0.8
2 3
Optimal
0.9
0
0.5 0.8
0.5 0.8
Assignments
5
0.15 0
0.4 0.4
0.2
0.5
Table 2 for Numerical Donors
Patient
4
3
2
6 0 0.6 0.9
Example
Assigned
f1
f2
Values of Criteria
Table 3 for Numerical
f3
Example
Assignment Criterion
f3
i,
L
f,
3.90 0.40 2.37
3.65 0.88 2.70
3.65 0.85 2.72
Pure random 3.00~0.50 0.49 rt 0.24 2.24-c 0.33
Sequential max 3.57a0.19 0.56a0.14 2.41 kO.13
MATHEMATICAL
ESTIMATION
METHODS
FOR
HEMATOLOGICAL
SUPPORT
13.5
OF pij
The usefulness of the allocation methods which have been presented depends upon the adequacy of procedures used to estimate the probability of a compatible transfusion for each patient-donor pair. The term “probability” is used in the sense of subjective probability, for the biology of the patient and donor at any time determine the transfusion response. In some cases, this response can be predicted prior to transfusion using in vitro antibody detection methods. In cases where an antibody to a donor j is found in the serum of patient i, pij = 0. In many cases, however, no antibody is detected by in vitro means to explain a poor transfusion response. As in vitro methods improve, all of the pii will be determined as zero or unity. For the present, however, mathematical methods are required for the estimation of these parameters. The development of effective mathematical estimation methods is a challenging area for biometric research. In this section, an approach to this problem will be outlined. If a patient i has not been previously transfused, then pu = 1 for all donors j. If the patient has on a recent day received a transfusion from a single donor j and exhibited a compatible response, then pi, = 1. If the patient has recently on some day received a transfusion from a set D of donors and exhibited a poor response, thep, =0 for all donors j in D unless the transfusion response can be attributed to a concurrent infection or drug effect which interferes with the evaluation. If in uitro testing indicates the presence of an antibody in the patient against the platelets of a donor, then pil=o. The elucidation of the HLA system of antigens as determinants of platelet compatibility makes possible a general model for estimation of the pii. In this model it is assumed that a patient will respond well to a donor unless the patient possesses an antibody to some HLA antigen of the donor [4]. Each individual contains determinants for four HLA antigens-two determinants per chromosome, with each chromosome containing a selection from each of two allelic series. Focusing attention on a particular patient, we will drop the subscript i. Let pj denote the probability that the patient is compatible with the platelets of donor j. Assume that the HLA antigens are arbitrarily indexed, and let qk denote the subjective probability that the patient does not have an antibody to the k’th HLA antigen. Let H(j) denote the set of HLA antigens of donor j. Then
Pj =
1fl 1
EH(j)
qk.
kgH(A
EHci, denotes
the expected value with regard to the set H(j).
This must be
RICHARD
136
SIMON
included in Eq. (5), because if less than four distinct HLA antigens are determined for donor j, one does not know for certain whether this is due to homozygosity or because the donor has a rare antigen for which no test is made. Using Eq. (5), the estimation of pj is reduced to the estimation of the qk. The parameters qk depend upon the HLA type of the patient, the HLA types of previous donors for the patient and the frequency of their donations, and the rate at which an antibody to antigen k is developed. If the patient has antigen k, then qk = I. If anti-k is detected in vitro, qk = 0. Two approaches to the estimation of the qk will be indicated. In the first approach, for a patient not having antigen k, qk is estimated as f(n, k), a function only of antigen k and the total number n of units of platelets containing antigen k which have been previously transfused to the patient. f(0, k) = 1, lim,,,f(n, k) = 0, and the functional form of f(n, k) is estimated from experimental data. In describing the second approach to the estimation of the qk, the following notation is used. For each transfusion t to the patient under consideration: at t, 1 otherwise,
l&(t)=0 if the patient has an anti-k antibody D (t) = the set of donors for transfusion t, C(f)= 1 if the patient had a good transfusion The transfusion history relations for each t: Rk(t)
of the patient
I( =
response,
is represented
implies
by the following
C(t) if dt>=O ,
> C(t) if m(t)=
&(t)=O
0 otherwise.
&(t’)=O
1
(6)
for all t’> T.
If Rk(t)=O for any t, then qk =O; otherwise, qk = 1. Though some of the &(t) variables are uniquely determined by (6), others are not. Exact and approximate methods for solving the system (6) are currently under investigation. SUMMARY Because most of the drugs used in the chemotherapy of disseminated malignancies are selectively lethal to proliferating cells, the bone marrow is generally the site of dose-limiting toxicity. This toxicity limits the degree of eradiction of malignant cells that can be achieved. For this reason, considerable amounts of effort and resources are devoted to long-term hematolo-
MATHEMATICAL
METHODS
FOR
HEMATOLOGICAL
137
SUPPORT
gical support of patients in cancer centers. In this paper, we have examined one aspect of the problem of improving the quality of such support. Certain of the results developed here are being implemented in a computer system being developed at the National Cancer Institute. It is hoped that this paper will stimulate the interest of other mathematicians in developing improved solutions to the problems presented. The author wishes to express his appreciation to Dr. Ronald A. Yankee of the Childrens Cancer Research Foundation for many useful discussions, and to a referee for his helpful comments. REFERENCES I 2
G. Mathe, Ed., Scientific Basis of Cancer Chemotherapy, Springer, New York, 1969. R. G. Graw, Jr. and R. A. Yankee, Principles of hematologic supportive care, Med.
3
Clin. N. Am. 57, 441 (1973). P. I. Terasaki and D. P. Singal,
4
Ann. Rev. Med. 20, 175 (1969). R. A. Yankee, Importance of histocompatibility
5
Sang 20,419 (1971). R. A. Yankee and K. Graff,
of non-family
matched
6
Congr. Blood Transfus., Washington, D. C., 1972. G. B. Danzig, Linear Programming and Extensions,
Princeton
7
8 9 10 11 12 13 14 15 16 17
Human
Efficacy
1963. R. E. Gomory, Solving linear programming Analysis, Proc. 10th Symp. in Appl. Math. Hall, Jr., Eds.), 1960, pp. 211-216.
histocompatibility in platelet
antigens transfusion platelets,
of leukocytes, therapy,
VOX
Proc. 13th Znf.
U. P., Princeton,
N. J.,
problems in integers, in Combinatorial of Am. Math. Sot. (R. Bellman and M.
M. L. Balinski, Integer programming: methods, uses, computations, Manage. Sri. 12, 253 (1965). E. Bales, An additive algorithm for solving linear programs with zero-one variables, Oper. Res. 13, 517 (1965). S. Kaplan, Solution of the Lorie-Savage and similar integer programming problems by the generalized Lagrange multiplier method, Oper. Res. 14, 1140 (1966). C. E. Lemke and K. Spielman, Direct search algorithms for zero-one and mixedinteger programming, Oper. Res. 15, 892 (1967). E. Balas, Discrete programming by the filter method, Oper. Res. 15, 915 (1967). P. L. Hammer and S. Rudenaunu, Pseudo-Boolean programming, Oper. Res. 17, 233 (1969). A. De Maio and C. Roveda, An all zero-one algorithm for a certain class of transportation problems, Oper. Res. 19, 1406 (1971). V. Srinivasan and G. L. Thompson, An algorithm for assigning uses to sources in a special class of transportation problems, Oper. Res. 21, 284 (1973). J. Edmonds and D. R. Fulkerson, Bottleneck extremes, RM-5375-PR, The Rand Corporation, Santa Monica, California (1968). R. S. Garfinkel and G. L. Nemhauser, The set-partitioning problem: set covering with equality constraints, Oper. Res. 17, 848 (1969).
138 18 19 20 21 22 23 24 25
RICHARD
SIMON
E. Balas and M. W. Padberg, On the set covering problem, Oper. Res. 20, 1152 (1972). L. R. Ford, Jr. and D. R. Fulkerson, Flows in Networks, Princeton U. P., Princeton, N. J., 1962. R. S. Garfinkel, An improved algorithm for the bottleneck assignment problem, Oper. Rex 19, 1747 (1971). R. Bellman, Dynamic Programming, Princeton U. P., Princeton, N. J., 1957. P. J. Wong and D. G. Luenberger, Reducing the memory requirements of dynamic programming, Oper. Rex 16, 1115 (1968). R. J. Duffin, E. L. Peterson, and C. Zener, Geometric Programming-Theory Application, Wiley, New York, 1967. M. Avriel and A. C. Williams, Appl. Math. 19, 125 (1970). U. Passy, Nonlinear assignment Rex 19, 1675 (1971).
Complementary problems
treated
geometric
programming,
by geometric
programming,
SIAM J. Oper.