British Journal of Haematology. 1988, 70, 31-36

Upper and lower time limits in the decision to recommend marrow transplantation for patients with chronic myelogenous leukaemia

WILLIAM SIMON, GEORGE B. SEGEL AND MARSHALL A . LICHTMAN Departments of Biophysics, Medicine, Pediatrilcs and Division of Genetics, University of Rochester School of Medicine, Rochester, N. Y.

Received 7 December 1987; accepted for publication 10 March 1988

Summaiy. Long-term survival of patients with chronic myelogenous leukaemia (CML) requires marrow transplan- tation from a histocompatible donor. The optimal timing of the transplant is difficult to determine because of the high peritransplant mortality of 20-35% and the existence of a group ofpatients who can have the disease controlled by drug treatment for prolonged periods. We have developed a mathematical model implemented with a computer program which calculates lower and upper time limits for the timing of

marrow transplantation. The lower time limit for transplan- tation is derived from the loss of life expectancy with delay, and the upper time limit is calculated by comparing the transplant survival probability with the probability of surviv- ing an additional year without a transplant. Thus, an objective basis is provided for bracketing the most appropriate time for transplantation. This analysis suggests that the decision to transplant can be postponed in some patients for periods longer than may generally be recommended.

Marrow itransplantation from a histocompatible donor is the only curative treatment for patients with chronic myeloge- nous leukaemia (CML). The timing of transplantation is influenced by the high peritransplant mortality. In a recent paper we presented a mathematical model to assist in determining the optimal time for marrow transplantation for patients with CML (Segel el al. 1986). This model considered the patient’s age, prognostic group, and the success rate of transplantation to produce a parameter that we called the ‘calculated life expectancy’. The model assesses the overall loss of life expectancy as transplantation is delayed and provides a lower time limit for delaying transplantation: transplantation can be postponed until there is a drop from the maxilmum expectation value for remaining life time. However, it became clear as physicians applied the model to their patients that it does not define an upper time limit beyond which delay in transplantation carries a greater risk than the transplantation procedure itself. In this paper we have extended the model and the computation to define an upper as well as a lower time limit for performing marrow transplantation in patients with CML.

Correspondence: Dr George Segel, Box 777, University of Rochester Medical Center, 601 Elmwood Avenue, Rochester, NY 14642, U.S.A. The authors will provide a copy of this program that will run on IBM- compatible computers to interested readers who send us a blank 5; inch magnetic disk.

METHODS Calculation of mean life expectancy The calculated life expectancy, CLE. is the integral of the product of the fractional survival and the number of surviving years (Segel et al. 1986). This parameter must be adjusted for those patients who develop advanced disease before elective transplantation and are successfully trans- planted during this accelerated phase.

Calculated Life Expectancy =

The following equation encompasses these concepts:

T I

LT + ( K - L ) S ( T , ) (T-Tl) + (1-L) J S(t)dt (1) 0

where Tl is the interval in years between diagnosis and transplantation: T is 70 years (normal life span) minus the patient’s age; K is the transplant success rate: L is the salvage rate for transplantation during accelerated phase: S(TI) is the survival fraction at time T1, and S ( t ) is the survival curve.

This equation appears complicated, but the calculation is easily done on a computer (Segel et al, 1986).

The calculated life expectancy (CLE) parameter is divided by the normal life expectancy (NLE) (patients’ presumptive life span of 70 years minus their current age) to generate a CLEINLE ratio which can be compared among patients. We interpret the time at which the CLE/NLE curve turns downward as the minimum time for delaying a marrow

31

32 transplantation for definitive treatment of CML. This down- turning is easily discerned as illustrated in the following examples (results: Figs 1-4). It represents an increasing loss of potential living time and hence defines a lower limit for delaying transplantation.

W . Simon, G . B. Segel and M . A. Lichtman

Probability of surviving one additional year An upper limit for delaying transplantation can be defined as that time beyond which the patients’ probability of surviving one more year is less than the probability of surviving the transplant procedure. To a first approximation, this time is derived from the CML survival curves by finding a time, Ta, such that the ratio of the survival fraction at T,+1 year, which is S(T,+ 1 year), to the survival fraction at Ta, which is S(Ta), is less than K , the survival fraction ofelective transplan- tation.

The choice of 1 year additional survival is to some degree arbitrary. It was chosen as being about the maximum length of time that most people can realistically visualize in their own lives. Times much shorter than 1 year do not seem to be reasonable gambles against the hope of full life time survival. We have considered using 1; and 2 year additional survival comparisons but in general they do not yield significantly different results, and the lack of sufficient long-term untrans- planted survival data make these comparisons less convinc- ing.

As in the life expectation calculation, the probability of surviving one additional year must be slightly modified to take into account the possibility that a successful transplant can be performed if an accelerated phase develops during the year. Using the factor C as the transplant survival fraction for transplantation in the accelerated phase, the CML population survival curves are modified so that the fractional survival between time T, and T, + 1 year is increased, and the point at which this fraction becomes lower than the chronic phase transplant survival fraction occurs at a later time.

Probability of one additional year survival (ADDYR) =

S(T,+ 1 year) S(TJ (2)

C + ( l - L )

Cumulative survival to any given year following diagnosis This parameter (CUMSUR) represents the fraction of patients surviving to any time after diagnosis without elective marrow transplantation. It is derived for the three prognostic groups defined by Sokal and coworkers based on spleen size, platelet count, haematocrit, sex and percentage of blood myeloblasts in a cohort of 625 chronic phase Phl positive patients with CML (Sokal et al, 1985). The resultant CUMSUR curves (which are the same as the probability of not dying) are less steep than Sokal’s survival curves because a survival improvement from transplantation in the accelerated phase of the disease has been included. The calculation of the cumulative survival (CUMSUR) to time t is as follows:

S’ ( t )=L+( l -L) S( t ) (3)

where S ( t ) is Sokal’s survival curve and S’(t ) is the corres- ponding parameter assuming that a fraction, L. of patients entering accelerated phase can be cured.

We have written a computer program which will construct the CLE/NLE. ADDYR and CUMSUR curves for patients with CML at the time of diagnosis. The curves represent average values for a group of patients with the specified character- istics. The program is written in BASIC for the IBM PC and compatible computers. The patient’s age, mean life span and the projected survival after transplantation in the chronic and accelerated phases of CML are requested by the program. The patient’s prognostic group then may be entered directly or calculated by providing the spleen size, platelet count, percentage of blasts, the haematocrit and the sex of the patient. The appropriate curves defining the lower (CLEINLE) and upper (ADDYR and CUMSUR) limits for the delay of bone marrow transplantation are produced using any colour graphics adaptor (CGA) and a colour monitor. We will provide a copy of the program to interested readers who send us a blank 5; inch magnetic disk.

RESULTS

Application of the mathematical model We have extended the utility of our mathematical model to assist in deciding the optimal time of marrow transplantation in patients with CML by computing both a lower and upper limit for delaying transplantation. The model is based on the survival statistics for CML without transplantation and parameters that we have established to define the lower and upper limits for delay of transplantation. The survival statistics may be limited to specific prognostic groups as defined initially by Tura (Tura et al, 198 1) and Sokal (Sokal et al, 1984). The data used in this analysis were compiled by Sokal from a group of patients with CML age 5-45 years at the time of diagnosis as this provides the age group most amenable to transplantation (Sokal et al. 1985). However, if future therapy, for example interferon, alters the survival curves, the revised data can be incorporated easily into the analysis. Moreover, the use of this mathematical model is not dependent on any specific transplant survival statistics: thus, these can correspond to the data from the centre performing the transplant.

We have considered the influence of post-transplant relapse on this analysis. Relapses occur primarily within a 3- year period after marrow transplantation, and they can be considered by introducing a further decrement in the overall transplant success rate. The transplant success rate also can be adjusted for complications such as severe graft-versus-host disease which result in a fatal outcome. Thus, the analysis is applicable under conditions that heighten this risk such as partially mismatched donors or female donors for male patients. Further, recent data suggest that patients trans- planted in accelerated phase have a high probability of early relapse (Thomas et al, 1986: McGlave et al, 1987). Hence the published value for transplant success in accelerated phase/ blastic crisis of approximately 15-20% has been lowered to 5% in the following analysis.

The calculated parameters CLE/NLE. ADDYR and CUM- SUR require further adjustment if delay of transplantation, itself, results in a decreased transplantation success rate

AGE=33 MEAN LIFE SPAN =70 PROG GROUP= 3 K=72% L=5%

loo f+,

80 N -D CLElNLE

40 1 CUMSUR \ -#- ADDYR

2o t 0 0 1 2 3 4 5

YEARS AFTER DX

Fig 1. Calculated parameters for the upper and lower limits for delay of transplantation. This example shows the analysis for CML patients aged 33 in prognostic group 3 . The parameters are plotted as a function of the time of transplant planned at the time of diagnosis. CLE/NLE is the ratio of life expectancy to normal life expectancy. ADDYR ir; the probability of surviving one more year without the transplant, but includes those patients who are successfully trans- planted in the accelerated phase. CUMSUR, the cumulative survival, is the problability of surviving to any given time. K is the percentage of patients who survive the transplant. L is the percentage who survive a transplant during the accelerated phase. The lower and upper time limits defined in the text are indicated by vertical arrows.

(Thomas et nl, 1986). Therefore, we have provided for the entry of a number less than 1 to represent the proportional decrement per year due to this effect of delay.

Calculated parameters An example of the parameters calculated by the computer program for our model are shown in Fig 1 for a 33-year-old patient in prognostic group 3 for whom the success rate for initial transplantation, K. is 72% and the success rate for transplantation in the accelerated phase, L, is 5%. Three curves are shown which represent the CLE/NLE ratio, the probability of surviving one additional year (ADDYR) and the cumulative survival to any given time (CUMSUR) (Fig 1).

In order to assess the effect of postponing transplantation, we have compared the life expectancy (CLE) from the time of diagnosis as a function of the time of planned transplantation to the normal life expectancy (NLE). We have designated this quantity the CLE/NLE ratio. Maximum survival probability requires transplantation at the time of diagnosis, but there is little loss in life expectancy if the CLEINLE curve is flat. We interpret a down turn of this curve as indicating an increased loss of life expectancy which defines the lower limit for the delay of transplantation. In the example shown in Fig 1 the curve turns downward at 6 months after diagnosis. It appears

100

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40

20

Time Limits for BMT in CML

AGE = 20 MEAN LIFE SPAN =70

PROG GROUP =1 K=60% L=5%

+- CUMSUR

-#- ADDYR

3 3

!

0

YEARS AFTER DX

Fig 2. Parameters for CML patients aged 20 years, in prognostic group 1. In this analysis the survival is 60% with planned transplantation and 5% with transplantation in the accelerated phase. CLE/NLE is the ratio ofcalculated life expectancy to normal life expectancy. ADDYR is the probability of surviving one more year without transplantation, and CUMSUR is the cumulative survival to the indicated time. The lower and upper time limits for delay of transplantation are indicated by vertical arrows. In this example, the lower limit for delay as defined by the downward turning of the CLE/ NLE curve is 1; years. The upper limit as defined by the intersection of the CUMSUR curve with the survival from transplantation, K, is between 4; and 5 years.

reasonable to postpone the transplant until this time since there is little loss of life expectancy.

There was concern among users of our original model that the interpretation was somewhat difficult. There appeared to be a need to know an upper limit for delay beyond which transplantation actually carried less risk than further delay, as well as the lower limit for delay which was provided as the CLE/NLE. For this purpose we have computed the probability of surviving one additional year. In the example in Fig 1 the ADDYR curve indicates that at 2; years after diagnosis the probability of surviving the next year is 72%. the same probability for surviving a transplantation procedure (K) which offers the expectation of a full life span. In most cases this can be regarded as an upper limit for delay, as the risk of delay for another year equals or exceeds the risk of the transplant.

We also have computed CUMSUR. the cumulative survival to any specified time after diagnosis. This is a useful alternative upper limit for delay in those circumstances when the ADDYR does not fall below the transplantation success rate during the initial 5 years for which we have good survival statistics. This computation establishes the total delay that results in the same risk as initial transplantation. In delaying to this time the patients have accrued years of life,

34 W. Simon, G. B. Segel and M . A. Lichtman

AGE =20 MEAN LIFE SPAN=70

PROG GROUP =3 K=60% L=5%

looh

80 k ILOWER P

4- CLElNLE

+ CUMSUR .-%A t '" t+ ADDYR

t 0 1 I I ' ' I ' I I I I

0 1 2 3 4 5 YEARS AFTER DX

Fig 3. Effect of prognostic group on the timing of transplantation. The parameters for CML patients aged 20 in prognostic group 3 are shown for comparison to those for patients in prognostic group 1 in Fig 2. The lower limit for delay in this example is approximately 6 months. The upper limit for delay as defined by the intersection of the CUMSUR curve with the survival for transplantation, K. is approxi- mately 2; years.

but they still have a fatal disease (the few successfully transplanted patients during accelerated phase should have long-term survival), whereas transplantation at the lower time limit involves the same risk with the potential for long- term survival. Under some circumstances (as in Fig 1) the CUMSUR curve falls below K before the ADDYR curve within the initial 5 years after diagnosis. We then would interpret this earlier time as the upper limit for delay.

Factors influencing the analysis Hypothetical example. In the illustrations that follow we

have assessed the impact of the prognostic group and of the success rate of transplantation on the lower and upper limits for the delay of transplantation. The three prognostic para- meters are shown in Fig 2 for a cohort of patients age 20 years in prognostic group 1 with a transplant survival, K, of 60% and a success rate for transplant during the accelerated phase, L, of 5%. These statistics for the success rates of transplantation are in the range of those reported for patients transplanted in chronic phase CML (Thomas et al, 1986; McGlave et al, 1987: Speck et al. 1985: Goldman et al, 1986).

For patients in prognostic group 1 (Fig 2) the CLEINLE ratio turns downward at approximately 14 years during which there is little loss of life expectancy with delay of transplantation. The ADDYR curve (probability of surviving one additional year) does not fall below the transplant survival, 60%, during the first 4 years of analysis. However, the CUMSUR curve (the cumulative survival to any specified

100

80

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W o 60 a n

40

2c

C

AGE=20 MEAN LIFE SPAN=7O

PROG GROUP=3 K=80% L=5%

$ UPPER

s 43- CLEiNLE

CUMSUR n +- ADDYR

I 1 l 1 1 1 1 1 1

0 1 2 3 4 5

YEARS AFTER DX

Fig 4. Effect of transplantation success rate on the timing of transplantation. The parameters for CML patients aged 20 in prognostic group 3 with an 80% survival from a transplant are shown for comparison to Fig 3. The lower limit for delay of transplantation is approximately 6 months. The upper limit is indicated by the intersection of the ADDYR curve and the success rate for transplantation, K, at 1 year. The CUMSUR curve intersects K at approximately 1: years.

time) falls below the transplant success rate between 44 and 5 years. This analysis indicates transplantation could be postponed for 1; years without loss of life expectancy but should probably not be postponed beyond 5 years.

Effect ofprognostic group on the timing of transplantation. In contrast, the CLE/NLE ratio turns downward at approxima- tely 6 months for patients in prognostic group 3 (Fig 3 ) and the CUMSUR curve falls below the transplantation success rate, K , at approximately 24 years. Again, the ADDYR curve does not fall below the transplantation success rate during the first 4 years of analysis. Thus, earlier transplantation is indicated in prognostic group 3 patients to maximize the calculated life expectancy, and the upper limit for transplan- tation delay is earlier, 24 years compared to that for patients in prognostic group 1, that is, 5 years (Fig 2).

Effect oftransplantation success rate on the timing of transplan- tation. An improved transplantation success rate also impacts significantly on the analysis (Fig 4). For 20-year-old patients in prognostic group 3 an improved transplantation success rate of 80% would result in little change in the lower limit as the CLE/NLE curve turns downward at 6 months or earlier, but the ADDYR curve falls below the transplantation success rate of 80% at 1 year, and the CUMSUR curve falls below 80% at 14 years. Although the lower limit for delay has not changed, the improvement in transplantation success reduced the upper limit beyond which transplantation should be delayed to 1 year.

Time Limits for BMT in CML 35 the involved transplant centre. In most cases lower and upper limits can be established for delay, and the patient should weigh the importance of survival during a specific time period in order to decide where within the interval to undergo the transplant. The lower limit defines a delay time with minimal loss of calculated life expectancy as indicated by the down- ward turning of the CLE/NLE curve. It is more difficult to define the upper limit. We have provided a parameter which calculates the chance of surviving one additional year, because we feel that if this chance is less than the chance of cure by transplantation, then transplantation is indicated. However, the data are such that this upper limit can be defined for only certain groups of patients during the first 5 years after the diagnosis of CML. At lower transplantation success rates, the probability of surviving one additional year does not fall below the transplantation success rate until the second 5 years. In these cases we feel that the cumulative survival curve (CUMSUR) best reflects the upper limit for transplantation delay.

These computations allow quantitation of lower and upper limits for transplant delay so that the patients can assign a utility function or value to the specific times of survival. For example, if an individual feels it is most important to see their child graduate from college, they might elect to postpone transplantation toward the upper limit for delay, recognizing that they are incurring added risk and potential loss of life expectancy. In contrast, young persons might feel that the risk of transplant is worth taking earlier, since they can plan on longer term goals after a successful transplant. In this case they may choose the minimal delay to permit the optimal chance of long-term survival. The algorithm should provide the data to assist the physician and the patient in deciding the best time for transplantation.

20 40:

0

AGE=PO MEAN LIFE SPAN=7O INITIAL K=60% L=5%

DECAY F A G .7 - 3\, PROG GROUPz1

CLE/NLE CUMSUR ADDYR \

, 1 , 1 , 1 , 1 , 1

c z w

W 2 n

Effect of delay (iftransplant success is reduced) on the timing of’ transplantation. Some data have suggested that delay itself may reduce the success rate of transplantation, although this is a matter of controversy and may be related to specific treatment. The computer program permits entry of a factor less than 1 to indicate the decreased success rate per year. The impact of this delay factor is seen primarily on the CLE/ NLE ratio which is markedly influenced by the transplant success rate (compare Fig 5 to Fig 2). The downward slope of the CLE/NLE curve indicates the loss of life expectancy or the price for delaying the transplantation procedure. If delay itself decreases the probability of transplant success, the loss of life expectancy (CLEINLE) with time may be too great to warrant any delaiy. This fall in CLEINLJ? must be considered if any delay beyond the lower limit is contemplated. An upper limit for delay cannot be defined if the delay factor is substantially less than 1.

DISCUSSION

Analyses of these curves suggest the following practical approach to the consideration of transplantation timing. The most curTent prognostic curves should be incorporated in the algorithm, and the transplantation success rates in chronic and accelerated phase should be adjusted for early disease relapse and other fatal complications and be obtained from

ACKNOWLEDGMENT

Supported by US Public Health Service Grants CA12790, CA34691 and the University of Rochester Blood Research ‘Jimmy Fund.’

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Goldman, J.M., Apperley, J.F., Jones, L. et a1 (1986) Bone marrow transplantation for patients with chronic myeloid leukemia. New England Journal of Medicine 314, 202-207.

McGlave, P.. Arthur. D., Haake, R.. Hurd. D.. Miller, W.. Vercellotti, G., Weisdorf, D.. Kim, T.. Ramsay. N. & Kersey. J. (1987) Therapy of chronic myelogenous leukemia with allogeneic bone marrow transplantation. Journal of Clinical Oncology. 5, 1033-1040.

Segel, G.B.. Simon. W. & Lichtman, M.A. (1986) Variables influenc- ing the timing of marrow transplantation in patients with chronic myelogenous leukemia. Blood. 68, 1055-1064.

Sokal, J.E., Baccarani, M.. Tura. S.. Fiacchini. M.. Cervantes, F., Rozman, C., Gomez, G.A.. Galton. D.A.G. Canellos, G.P., Braun. T.J.. Clarkson, B.D.. Carbonell. F., Heimpel, H., Extra. J.M.. Fiere. D., Nissen, N.I., Robertson, J.E. & Cox, E.B. (1985) Prognostic discrimination among younger patients with chronic granulocytic

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Speck. B.. Bortin. M.M., Champlin, R.. Goldman, J.M.. Herzig, R.H., McGlave. P.B., Messner, H.A.. Weiner, R.S. & Rimm, A.A. (1985) Allogeneic bone marrow transplantation for chronic myelogenous leukaemia. Lancet, i, 665-668.

1352-1 357. Thomas, E.D.. Clift. R.A.. Fefer, A., Applebaum, F.R., Beatty, P.,

Bensinger. W.I.. Buckner. C.D., Cheever, M.A., k e g . H.J., Doney, K., Flournoy, N.. Greenberg, P., Hansen, J.A.. Martin, P., McGuffin, R., Ramburg, R., Sanders, J.E., Singer, J., Stewart, P.. Storb, R., Sullivan. K.. Weiden, P.L. & Witherspoon, R. (1986) Marrow transplantation for the treatment of chronic myelogenous leuke- mia. Annals oflnternuf Medicine, 104, 155-163.

Tura, S.. Baccarani, M. & Corbelli, G. (1981) Staging of chronic myeloid leukaemia. British Journal of Hematology. 47, 105-1 19.