Peripheral Blood Stem Cells Mathematical Model of Peripheral Blood Stem Cell Harvest Kinetics

Peripheral Blood Stem Cells Mathematical model of peripheral blood stem cell harvest kinetics

J Mayer1, Z Pospı´ sˇ il2 and Z Korˇ ı´ stek1

1Department of Internal Medicine – Hematooncology, University Hospital Brno, Brno Czech Republic; and 2Department of Mathematical Analysis, Faculty of Science, Masaryk University, Brno, Czech Republic

Summary: this number can be dramatically increased by a wide variety of mobilization stimuli. However, the precise mechanisms A mathematical model of peripheral blood stem cell responsible for mobilization have not been fully eluci- harvests was developed, taking two new parameters R dated.1 Furthermore, there are patients or PBSC donors

(number of recruited cells/minute) and Ef (efficiency of mobilizing only low amounts of progenitor cells, which are collection) into consideration in addition to concentrations insufficient for transplantation.2,3 and collected amounts of cells. This model was tested on Although PBSC harvest and transplantation have 241 harvest procedures in cancer patients become routine procedures, little is known about the (chemotherapy þ G-CSF stimulation), donors of allo- kinetics of release of progenitor cells from the bone marrow geneic PBSC, and platelet donors, using different collec- into the peripheral blood and their collection during tion procedures, with a Cobe Spectra Cell separator. The leukapheresis. Several authors have used simplified for- relationships between preapheresis concentrations, R, Ef mulas to describe these processes, for example, separation and harvested amounts of cells were complex, and efficiency,4 efficiency ratio,5 extraction efficacy,6 and different for different harvest procedures and populations collection efficiencies (two formulas, one standard and of donors. However, invariably, recruitment played an one novel).7 All these formulas calculated static results important role and contributed significantly to the final from values like absolute cell count in the collected cell harvest in all types of cells studied. For example, for the suspension (harvest product), pre- and post apheresis blood patient group, mean recruitment was 1.3 Â 106 CD34 þ cell counts, total blood volume processed, and total blood cells/min and the amount of recruited cells corresponded volume. These results might be of a general value for to 65% of all collected cells. Recruitment was signifi- comparisons of different harvest procedures; however, they cantly influenced by pretreatment with chemo-therapy do not describe the dynamics of the complex collection and/or radiotherapy. The mean recruitment values for the process. subgroups with limited, moderate, and extensive pretreat- It has been shown that progenitor cells can be recruited ment were 1.65 Â 106, 0.87 Â 106, and 0.32 Â 106 CD34 þ into the peripheral blood during PBSC collection.4,6,8–12 cells released per minute, respectively. The finding of a However, a recruitment phenomenon has not always been quick and massive recruitment phenomenon may stimulate reported13 and it is not known if it applies to all kinds of further research into hematopoiesis in order to maximize cells or only progenitor cells.14 We repeatedly observed harvested cells. patients and donors with higher post than pre-apheresis cell Bone Marrow Transplantation (2003) 32, 749–757. counts despite the fact that a significant amount of cells was doi:10.1038/sj.bmt.1704226 collected. This finding supports the existence of the Keywords: mathematical model; cell separator; cell recruitment and means that the number of cells mobilized collection; peripheral blood stem cells; CD34 þ cells; into the peripheral blood must be even higher than the hematopoiesis number of collected cells. However, the pre- and post apheresis cell counts and the numbers of collected cells are strikingly different between Peripheral blood stem cells (PBSC) have been used for patients and donors. Therefore, in order to better under- transplantation since the early 1990s, and they are stand the processes involved in the separation and increasingly replacing bone marrow as a source of stem collection of the blood cells, we decided to design a cells. It is well accepted that there is a small number of mathematical model of cell harvest kinetics; dynamically circulating progenitor cells in the peripheral blood and that describing movements of cells between the blood, the collection bag, and other compartments, mostly bone marrow.15 This model was tested on a large data set from PBSC collection procedures performed in different groups Correspondence: Dr J Mayer, Department of Internal Medicine – of patients and healthy donors, including data from platelet Hematooncology, University Hospital Brno, Jihlavska´ 20, 625 00 Brno, collections. The results may lead to further improvement of Czech Republic. E-mail: [email protected] blood cell collection and shed new light on the physiology Received 30 January 2003; accepted 22 April 2003 of cell mobilization. Mathematical model J Mayer et al 750

Materials and methods cell output from the separator (collection) by wO. These velocities are expressed by number of cells over time unit.

Mathematical model of the separation process The velocity wI is given by

The model is based on several assumptions: wI ¼ wIðtÞ¼vKðtÞð1Þ (1) For each cell type (particle), the separation process and Let us suppose that the cells are collected with an efﬁciency the harvest kinetics during the apheresis can be Ef , that is, different and independent. w ¼ E w ð2Þ (2) There are three compartments: peripheral blood, O f I

collection bag, and the bone marrow. However, cells The efﬁciency of the cell collection Ef may depend on the

may be in fact mobilized from sites other than the bone cell concentration K, that is, Ef ¼ Ef(K) ¼ Ef(K(t)). Ob- marrow (so the term effectively denotes the entire third viously, 0pEfp1. compartment). Additionally, some cells may be lost by Finally, let us denote by M ¼ M(t) the total number of destruction (eg in the separation tubing set and in the cells collected during the time interval from beginning to a centrifuge) rather than by trafﬁcking into the bone time instant t, that is, marrow or collection bag. This is an important consideration for the clinical interpretation of results Zt obtained by this model. MðtÞ¼ wOðtÞ dt (3) Each cell type can enter or leave the peripheral blood or 0 the bone marrow; however, cells can only enter the collection bag. Let us denote a rate of cell movement The value of M is an observable quantity. Substituting (1) from or into the bone marrow by R (recruitment rate), and (2) into the last equation, we obtain

which can be expressed in terms of the number of cells Zt Zt per minute. R describes the interchanges of cells between peripheral blood and bone marrow during MðtÞ¼ Ef ðKðtÞÞwIðtÞ dt ¼ vKðtÞEf ðKðtÞÞ dt ð3Þ leukapheresis. R is positive if the cell count rises 0 0 in the peripheral blood (flow from the bone marrow On the other hand, M satisfies the obvious relation: M(t) ¼ into the peripheral blood), whereas, in the opposite number of cells in the vascular system at the beginning of case, R is negative. Theoretically, R can vary with collection þ total number of recruited/extinct cells during time. the collection procedureÀnumber of cells in the vascular (4) The separator does not collect all of the cells entering system at the end of the collection procedure. the machine. The cells are collected with an efficiency E f That is, which shows the proportion of cells that is actually harvested. Zt

MðtÞ¼Nð0Þþ RðtÞ dt À NðtÞ¼VðK0 À KðtÞÞ Let us denote the following: V the volume of blood in the 0 ð4Þ vascular system (total blood volume), N the total number Zt of cells under consideration in the blood, and K the þ RðtÞ dt concentration of cells in the blood. These status variables are interconnected by the simple 0 formula N ¼ VK. The values of N and K vary with time t, Equalities (3) and (4) yield the following integral equation: that is, N ¼ N(t), K ¼ K(t). Let us suppose that the concentration at the beginning of collection K(0) equals Zt 1 K0. We consider V and K to be observable quantities. KðtÞ¼K0þ ½RðtÞÀvKðtÞEf ðKðtÞÞ dt ð5Þ R V The recruitment rate may vary with time, that is, 0 R ¼ R(t). Hence, the change of cell number in the vascular system during a short time interval of the length Dt equals For the sake of tractability, we assume that K is a to R(t) Dt. Consequently, the total change of cell number differentiable function. Thus, differentiating equation (5) from the beginning to a time instant t, using standard by time, we obtain the following initial value problem for intergral calculus, is the ordinary differential equation:

Zt dKðtÞ v 1 ¼À KðtÞEf ðKðtÞÞþ RðtÞð6Þ RðtÞ dt dt V V 0 Kð0Þ¼K0 ð7Þ

Now, let us denote by v, a velocity of blood ﬂow through The nonlinear equation (6) with the general functions Ef the blood cell separator (expressed by a volume unit over and R cannot be solved in an explicit way. Hence, let us time unit). We suppose that v is constant, that is, it depends simplify the problem by a ‘zero order approximation’; that

on the separator only. Further, let us denote a velocity of is, let us suppose that both the efﬁciency of collection Ef

the cell input into the separator by wI and a velocity of the and the rise/extinction rate R are constant. In other words,

Bone Marrow Transplantation Mathematical model J Mayer et al 751 we replace the varying quantities Ef and R by some mean heavily pretreated patients (more lines of chemotherapy in values of them. The initial value problem in equations (6) combination with radiotherapy, extensive irradiation, etc). and (7) becomes The second group studied consisted of 54 allogeneic PBSC donors. They were mobilized using ﬁlgrastim 16 mg/ dK Ef v R kg/day administered in two divided doses. Leukaphereses ¼À K þ ; Kð0Þ¼K0 dt V V started on day þ 4. According to the protocol, at least two and it possesses the following solution: leukaphereses were to be performed. Leukaphereses in donors were divided into two subgroups according to the R R separation regimen (see below). The third was a group ÀðEf v=VÞt KðtÞ¼ þ K0 À e ð8Þ comprised of 19 volunteer platelet donors. Ef v Ef v

Moreover, equation (4) reduces to the following: Blood cell collection

MðtÞ¼VðK0 À KðtÞÞ þ Rt ð9Þ All aphereses were performed using a COBE Spectra cell separator (COBE BCT, Inc., Lakewood, CO, USA). Mononuclear cell collections were performed with the Now, a mathematical model of the cell collection using a semiautomated MNC program (software Version 5.1 or blood cell separator (a description of evolution of the cell 6.0) or autoPBSC program (software Version 6.0), platelets concentration in the vascular system and of the numbers of were collected using a leukoreduction system chamber collected cells) consists of the two equations, (8) and (9). It (LRS, software Version 5.1). Collections in all patients were involves the observable quantities v, V, and K0, and the performed using the MNC program, collection in allo- unknown parameters R (recruitment), and Ef (efficiency of geneic PBSC donors were performed using either the MNC collection). These parameters can be determined as the program or autoPBSC program. The decision of which solution of the two simultaneous equations (8) and (9) program to use depended mainly on the availability of a satisfying the condition 0pEfp1. given separator, and on pre-apheresis platelet blood count The parameter R (recruitment) is obtained from equation (the autoPBSC program was chosen preferentially in cases (9): of lower platelet counts). Total blood volume (TBV) was MðtÞÀVðK À KðtÞÞ calculated using the COBE Spectra software. R ¼ 0 t Cell counting and progenitor cell assessment Substituting into eq. (8), we obtain the following equation for the unknown parameter Ef (efficiency of collection): Pre-apheresis samples of peripheral blood for blood count, CD34 þ cell analysis, and CFU-GM assay were drawn just M t V K K t ÀðEf v=VÞt ð ÞÀ ð 0 À ð ÞÞ before morning G-CSF administration (about 2 h before KðtÞÀK0e ¼ Ef vt aphereses). Post apheresis peripheral blood samples (blood þ E v=V t count, CD34 , CFU-GM) were drawn from the inlet line Âð1 À eÀð f Þ Þ of an intravenous catheter or from the access needle just which can be solved numerically. We have used the ‘regula after the collection (before starting the rinse-back proce- falsi’ method for the determination of parameter Ef dure). In platelet donors, the samples for platelet count were collected just before and after each harvest procedure. Patients and donors The post-apheresis samples of harvested material were taken from the collection bags after gentle but vigorous We studied three groups of patients and donors. The first mixing. group comprised of myeloma and lymphoma patients. Cell counts were performed by an electronic counter They were treated and mobilized according to previously (Cell Dyne 3500, Abbott Laboratories, IL, USA) including published protocols.16,17 In 20 myeloma patients, the the five-population differential white blood cell count. mobilization chemotherapy consisted of cyclophosphamide The percentage of CD34 þ cells among total leukocytes 5 g/m2 followed by G-CSF (filgrastim) at a dose of 5 or and the number of CFU-GM was determined as described 10 mg/kg/day given subcutaneously and starting 24 h after previously.17 Later, in the second half of the study, CFU- finishing chemotherapy. In 20 lymphoma patients, etopo- GM assays were performed using standard colony assay side and ifosfamide were administered and followed by systems with a complete methylcellulose-based medium filgrastim (5–16 mg/kg/day). Patients were further stratified MethoCult HCC-4434 (StemCell Technologies Inc., Van- according to pretreatment into three subgroups. Subgroup couver, Canada). A (limited pretreatment) consisted of patients treated using first-line chemotherapy only (vincristine þ adriamycin þ Statistical methods dexamethasone in myeloma patients, ABVD or CHOP in lymphoma patients). Subgroup B (moderate pretreat- We evaluated two kinds of statistical dependence: differ- ment) consisted of patients pretreated by first-line chemo- ences of observed quantities among groups and correla- therapy in combination with limited radiotherapy or tions between these quantities. Significance of the pretreated with more than the first-line chemotherapy differences among groups (ie patients, allogeneic PBSC regimen. Subgroup C (extensive pretreatment) consisted of donors and platelet donors or patient with different

Bone Marrow Transplantation Mathematical model J Mayer et al 752 pretreatments) was tested by means of the Kruskal–Wallis The Spearman rank correlation coefﬁcients between para-

test, and closeness of the correlations was quantiﬁed by the meters R and Ef and the numbers of collected cells or pre- Spearman rank correlation coefﬁcients. apheresis cell concentrations, and between the parameter R

and the parameter Ef, are shown in Tables 2a and b (not all calculated data are given). Several typical kinds of relations Results were found on the basis of data analyses:

We performed 241 aphereses in patients and donors (1) A positive correlation between parameter R (recruit- (n ¼ 103 for patients; n ¼ 92 for donors using the MNC ment) and pre-apheresis cell count. A scatter plot of the program; n ¼ 27 for donors using the autoPBSC program; relationship between R for CD34 þ cells and pre- n ¼ 19 for platelet donors). Data on the pre-apheresis blood apheresis CD34 þ cell count in patient blood is shown cell counts, post-apheresis blood cell counts, and harvested in Figure 1. cell numbers were available for almost all aphereses. From (2) A positive correlation between pre-apheresis blood

these data, values of parameters R (recruitment) and Ef cell concentration and parameter Ef (efﬁciency of (efﬁciency of collection) were computed for the apheresis collection; for example, a correlation between pre- procedures. The arithmetic means of the pre- and post- apheresis platelet count in blood and platelet Ef of apheresis blood cell concentrations (counts), numbers of patients).

collected cells, and values of the corresponding R and Ef (3) On the other hand, a negative correlation between parameters are shown in Tables 1a–c as well as the pre-apheresis cell concentrations and parameter Ef differences between the groups of patients and donors. (efﬁciency of collection) for other cells exists (for

Table 1 Pre- and post apheresis blood cell counts, numbers of collected cells, and corresponding parameters R (recruitment) and Ef (efﬁciency of collection)

CD34+ cells CFU-GM

Pre- Post Yield Yield/TBV R Ef Pre- Post Yield Yield/TBV R Ef apheresis apheresis ( Â 106) ( Â 106) ( Â 106/min) (%) apheresis apheresis ( Â 104) ( Â 104) ( Â 104/min) (%) ( Â 106/L) ( Â 106/L) ( Â 104/L) ( Â 104/L)

Patients 121.5 78.9 479 202 1.3 50.0 473 315 2067 901 6.0 49.7 PBSC donors, 71.5 53.9 348 140 1.0 50.4 401 237 2482 979 2.7 45.4 MNC program PBSC donors, 58.6 41.4 199 99 0.5 37.9 376 263 2824 1440 Not available autoPBSC program Kruskal–Wallis NS NS ** NS ** ** NS NS NS NS * NS test

Platelets Erythrocytes

Pre- Post Yield Yield/TBV R Ef Yield R Ef apheresis apheresis ( Â 109) ( Â 109) ( Â 109/min) (%) ( Â 109) ( Â 109/min) (%) ( Â 109L) ( Â 109/L) Patients 75 63 154 62.4 0.36 18.5 56 À4.5 0.14 PBSC donors, 160 91 484 188.8 0.60 33.8 211 À3.9 0.41 MNC program PBSC donors, 115 87 182 89.9 0.16 16.9 73 À6.7 0.17 autoPBSC program PLT donors 274 196 462 598.4 0.90 51.6 Not examined Kruskal–Wallis ** ** ** ** ** ** ** ** ** test

WBC Neutrophils Lymphocytes Monocytes

R Ef R Ef R Ef R Ef ( Â 109/min) (%) ( Â 109/min) (%) ( Â 109/min) (%) ( Â 109/min) (%) Patients 0.13 22.0 0.09 17.3 0.01 44.5 0.02 42.2 PBSC donors, 0.12 12.5 0.03 7.0 0.04 51.3 0.03 50.0 MNC program PBSC donors, 0.07 9.4 0.01 3.6 0.04 46.1 0.04 49.0 autoPBSC program Kruskal–Wallis * ** ** ** ** * ** ** test

The arithmetic means are shown. TBV ¼ total blood volume. Statistical signiﬁcance of differences between groups: ** ¼ Po0.01; * ¼ Po0.05; NS ¼ not signiﬁcant.

Bone Marrow Transplantation Mathematical model J Mayer et al 753

Table 2 Spearman rank correlation coefﬁcients between parameters R (recruitment) and Ef (efﬁciency of collection) CD34+ cells CFU-GM

Pre-apheresis Yield/TBV Parameter Ef Pre-apheresis Yield/TBV Parameter Ef

Patients R 0.72** 0.81** 0.20 0.28* 0.58** 0.25

Ef 0.05 0.24* À0.23 0.05

PBSC donors, MNC program R 0.16 0.64** 0.38** À0.07 0.59** 0.59**

Ef À0.21 0.27* À0.37** 0.27

PBSC donors, autoPBSC program R 0.08 0.45** 0.20 Not available

Ef À0.15 0.19

WBC Platelets

Pre-apheresis Yield/TBV Parameter Ef Pre-apheresis Yield/TBV Parameter Ef

Patients R 0.36** 0.72** 0.27** 0.10 0.40** 0.39**

Ef À0.56** 0.42** 0.39** 0.80**

PBSC donors, MNC program R À0.17 0.31* 0.42** 0.43** 0.66** 0.64**

Ef À0.44** 0.37** 0.23* 0.60**

PBSC donors, autoPBSC program R À0.22 0.06 0.10 À0.12 0.35 0.46*

Ef À0.37 0.74** 0.08 0.79**

Platelet donors R Not analyzed À0.18 À0.04 À0.01

Ef À0.06 0.02

Pre-apheresis: concentrations of cells presented in peripheral blood before apheresis; Yield/TBV: amounts of cells collected per one processed total blood volume. Statistical signiﬁcance: ** ¼ Po0.01; * ¼ Po0.05.

/L] 700 70% 6 10

× 600 60%

500 50% cells [ + 400 40%

300 30%

200 20% 100 Efficiency of collection 10%

Pre-apheresis CD34 0 −2 0246810 0% 0 5 1015202530354045 R (recruitment) for CD34+ cells [×106/min] Pre-apheresis white blood cells [×109/L] Figure 1 Relationship between parameter R (recruitment) and pre- Figure 2 Relationship between parameter E (efﬁciency of collection) and apheresis CD34 þ cell concentrations (MNC program, group of patients, f pre-apheresis white blood concentrations (MNC program, group of Po0.01). patients, Po0.01).

example, a correlation between WBC or neutrophils total blood volume (TBV). These relations for CD34 þ

and Ef for patients or allogeneic PBSC donors when cells and for CFU-GM in the group of patients are using the MNC program). A scatter plot of the shown in Figures 3 and 4, respectively.

relationship between the pre-apheresis WBC counts (5) A positive correlation between parameter Ef (efﬁciency

and Ef of WBC in patients is shown in Figure 2. Post of collection) and the total amount of collected cells or, apheresis blood cell concentrations correlated with similarly, the total amount of collected cells per one

parameter Ef in the same magnitude as pre-apheresis processed TBV. This relation for platelets for the group blood cell concentrations (data not shown). of patients is shown in Figure 5.

(4) A positive correlation between R (recruitment) and the (6) A positive correlation between parameter Ef (efﬁciency total number of collected cells, or, similarly, between R of collection) and parameter R (recruitment) for and the number of cells collected per one processed various cell types, see Table 2.

Bone Marrow Transplantation Mathematical model J Mayer et al 754 Towards understanding how many cells are recruited In order to assess if the pretreatment of patients with from the bone marrow during apheresis, Table 3 contains chemotherapy and/or radiotherapy inﬂuences the ability of the following data (calculated for patient group): (1) the bone marrow to release cells during leukapheresis (para- number of cells recruited from the bone marrow expressed meter R, recruitment), the correlation between pretreat- as a percentage of the total number of collected cells; (2) the ment and the values of the parameter R was analyzed using number of cells recruited from the bone marrow expressed the Kruskal–Wallis test. The differences between parameter as a percentage of the total number of cells in the peripheral R for CD34 þ cells among subgroups of patients with blood before apheresis. However, there were wide ranges of different pretreatments were statistically signiﬁcant these values. For example, the release of CD34 þ cells (Po0.01) and are shown in Figure 6. The mean of during leukapheresis varied from À60% to 201% of the parameter R of the subgroups with limited, moderate, collected amount. The wide range of parameter R values is and extensive pretreatment were 1.65 Â 106, 0.87 Â 106, and shown in Figure 3. 0.32 Â 106 cells released per minute, respectively.

1000 Discussion ]

6 900

10 800 Several authors have indicated that recruitment of certain × 700 hematopoietic progenitor cells occurs during PBSC collec- 10,18 600 tion. Fliedner described experiments in dogs using 10 500 continuous-ﬂow centrifugation carried out for 12.5 h. It 400 was evident that the number of yielded CFU-C could reach up to 60-times the number of CFU-C normally present in 300

cell yield/1 TBV [ the blood stream. This ﬁnding was explained by the + 200 assumption that these cells were drained from extravascular 100 sites. Similar results were also observed in other animals.8 CD34 0 −2 −1 0 12345678910 This implies that apheresis itself might be a stimulus for the recruitment of progenitor cells into the peripheral blood. + × 6 R (recruitment) for CD34 cells [ 10 /min] In stimulated animals, patients, and donors, recruitment Figure 3 Relationship between parameter R (recruitment) and total was also described and can be even higher than in the amounts of collected CD34 þ cells per one processed total blood volume absence of stimulation.8 Regarding the clinical setting, (TBV; MNC program, group of patients, Po0.01). there are indirect data based on some static formulas and

5000 400

] 4500 4 ]

9 350

10 4000 × 10 3500 × 300 3000 250 2500 200 2000 150 1500 1000 100 Platelet yield/1 TBV [

CFU-GM yield/1 TBV [ 500 50 0 0 −20 −15 −10 −5 0 5 10 15 20 25 30 35 40 0% 10% 20% 30% 40% 50% 60% R (recruitment) for CFU-GM [×104/min] Efficiency of collection

Figure 4 Relationship between parameter R (recruitment) and total Figure 5 Relationship between the values of parameter Ef (efﬁciency of amounts of collected CFU-GM per one processed total blood volume collection) and total amounts of collected platelets per one processed total (TBV; MNC program, group of patients, Po0.01). blood volume (TBV; MNC program, group of patients, Po0.01).

Table 3 Relative numbers of cells recruited from the bone marrow during apheresis in the patient group

WBC Neutrophils Lymphocytes Monocytes CD 34+ CFU-GM PLT (%) (%) (%) (%) cells (%) (%) (%)

Recruited cells/collected cellsa 107 169 67 52 65 70 67 Recruited cells/cells in blood before apheresisb 62 56 106 152 77 263 41

aNumber of cells recruited from bone marrow during apheresis expressed as the percentage of total number of collected cells. bNumber of cells recruited from bone marrow during apheresis expressed as the percentage of total number of cells present in peripheral blood before apheresis.

Bone Marrow Transplantation Mathematical model J Mayer et al 755 10 pretreatment (Figure 6). Therefore, we hypothesize that comparison of recruitments could be used for comparison 8 of different mobilization regimens in the future. /min] 6 For erythrocytes, the corresponding parameter R (re- 10 6 × cruitment) was negative. There are several possible 4 explanations for this observation. First, the negative value cells [

+ of R might mean that some red cells were lost during 2 apheresis, for example by hemolysis. There is at least one other possible explanation, however. The post-apheresis 0 blood samples were drawn at the time of completion of the for CD34

R separation, but just before starting the rinse-back proce- −2 ABC dure, when the tubing set was still ﬁlled with blood and certain erythrocytes were trapped in the separator. How- Pretreatment ever, in our model, we considered the blood volume stable Figure 6 Inﬂuence of pretreatment intensity (A, limited; B, moderate; C, during the apheresis. If this explanation is true, then all þ extensive) on the number of CD34 cells recruited during apheresis values of R parameter (recruitment) should be actually (expressed as a parameter R, P 0.01). o slightly higher. In fact, extreme accuracy requires an exact measurement of the total blood volume before and after each harvest procedure, because the calculation of the total calculations showing that CD34 þ cells must be released blood volume is based only on sex, weight, and height and into the peripheral blood during apheresis.4,6,9 In a study of may be imprecise. A further confounding factor might be eight patients, Smolowicz et al11 showed that not only the infusion of anticoagulant solution during apheresis. CD34 þ cells but also white blood cells and mononuclear Nevertheless, we think that these small possible inaccura- cells are mobilized during stem cell harvest. cies do not overshadow the principle of our model.

Nevertheless, a mathematical model describing the The variability in parameter Ef, which shows the behavior of a complex system consisting of bone marrow percentage of collected cells from the cells coming into (or other storage compartments), peripheral blood, cell the separator (efficiency of collection), can be explained in separator, and collection bag, and validated on clinical several ways. First, the set-up of the separator is a major data, has not been published to date. Based on our data, it influence. For the separation of mononuclear cells, the seems that recruitment is a common process occurring parameter Ef for these cells must be higher than for during the apheresis procedure. To some extent, recruit- neutrophils or erythrocytes (Table 1). For platelet pheresis, ment was apparent for all cell types except erythrocytes it is desirable to accomplish a very high value of Ef for in all groups of patients and donors studied (Table 1a–c) platelets (Table 1). In other words, the parameter Ef should and contributed significantly to the collected amount of be preferably very high for the target cell type and very low cells. Moreover, for certain types of cells, the recruited for the other cell types. As can be seen from routine clinical amount was even higher than the collected amount (see practice and also from our data, this is not always the case. Table 3). For example, thrombocytopenia is a serious side effect of Several articles described a very good correlation PBSC collections.23 The mean collected number of platelets between the pre-apheresis concentration of CD34 þ cells in the group of PBSC donors in whom the MNC program in the peripheral blood and the collected number of CD34 þ was used was higher than in the group of platelet donors cells.19,20 We also found this dependence (data not shown) (Table lb). Therefore, to avoid post-apheresis thrombocy- and our data extend these observations. We consider the topenia and to simplify the whole collection, the autoPBSC pre-apheresis concentration to be a secondary parameter program was developed. It is based on a different collection but the ability to release cells from other compartments, the principle than the semiautomatic MNC program that recruitment phenomenon (here characterized by parameter includes operator adjustment. Nevertheless, this change in R), to be the primary one. We found significant correlations the procedure could influence other separation parameters. between the recruitment (parameter R) and the pre- Concerning our data (Table 1a) and also some data in the apheresis concentrations of several types of cells, especially literature,24,25 it seems that harvests using the autoPBSC in the group of patients who were mobilized by chemother- program, albeit causing less thrombocytopenia, might also apy and G-CSF (Table 2, Figure 1). Moreover, significant be less effective for CD34 þ collection. correlations between values of parameter R (recruitment) Together with the set-up of the separator, an additional and collected amounts existed for several types of cells important cause for the variability in parameter Ef (Table 2, Figures 3, 4). Therefore, we suppose that (efficiency of collection) is the concentration of separated sufficient recruitment is very important for successful cells in blood. For white blood cells and in particular for collection. Several authors also showed that pretreatment neutrophils, we found that a higher blood concentration of with cytostatic drugs and/or radiotherapy negatively cells correlated with a lower value of Ef (Table 2, Figure 2). influenced the harvested amounts of progenitor cells.21,22 On the other hand, there was a positive correlation between

Since parameter R (recruitment) influenced the pre-apher- the platelet concentration and Ef value for platelets, which esis concentrations as well as the collected amounts of cells, means that the higher the platelet blood count, the higher we analyzed whether the R was influenced by pretreatment. the value of parameter Ef (see Tables 1 and 2, MNC þ Actually, R for CD34 cells significantly depended on program), and the higher the parameter Ef, the higher the

Bone Marrow Transplantation Mathematical model J Mayer et al 756 collected number of platelets (Figure 5). This could explain Acknowledgements why the same MNC program causes less severe post- apheresis thrombocytopenia in patients than in donors of We thank all the physicians and nurses from the Department of PBSC. Patients mobilized by chemotherapy and G-CSF Internal Medicine –Hematooncology, and Blood Transfusion had lower pre-apheresis platelet concentrations than Service, University Hospital Brno, for taking care of patients donors (mobilized by growth factors alone). and donors, and to the staff of the Laboratory of Flow Cytometry and the Tissue Bank, University Hospital Brno, for There is also a correlation between the parameter E f analyses of CD34 þ cells and CFU-GM, respectively. This work (efficiency of collection) and R (recruitment) for some cell was supported in part by Research Grant No. MZ 00065269705 types (Table 2). Theoretically, it might be possible to of the Ministry of Health of the Czech Republic. increase the parameter Ef for the target cell population by adjusting the separator and thereby increase the value of parameter R in order to increase cell yield. There are only a few reports describing changes in the default setting of the References separator in order to influence the collection of a specific cell population.26,27 However, for very high values of 1 Link DC. Mechanisms of granulocyte-colony stimulating parameter E , the recruitment capacity could be exhausted. factor - induced hematopoietic progenitor –cell mobilization. f Semin Hematol 2000; 37 (Suppl. 2): 25–32. This may be the reason for the thrombocytopenia in 2 de la Rubia J, Martı´ nez C, Solano C et al. Administration of healthy donors. recombinant human granulocyte colony-stimulating factor to In the model described here, the bone marrow was normal donors: results of the Spanish national donor registry. considered as the main compartment from which cells are Bone Marrow Transplant 1999; 24: 723–728. recruited. However, other compartments must be also 3 Rick O, Beyer J, Kingreen D et al. Successful autologous bone taken into the consideration (eg spleen and the marginated marrow rescue in patients who failed peripheral blood stem cell pool). Regardless of the main cell storage pool, the mobilization. Ann Hematol 2000; 79: 681–686. mechanisms responsible for such prompt and massive 4 Zingsem J, Zeiler T, Weisbach V et al. PBSC — collection in release of cells upon the start of apheresis are poorly patients using the FRESENIUS-AS 104 Leucollect — proto- Int J Cell Cloning understood and deserve further elucidation. Control of col. 1992; 10 (Suppl. 1): 85–87. 5 Passos-Coelho JL, Machado MA, Lu´ cio P et al. Large-volume recruitment should lead to a higher collected amount. leukaphereses may be more efficient than standard-volume There are wide physiological differences in the recruitment leukaphereses for collection of peripheral blood progenitor capacities between different individuals; for example, in our cells. J Hematother 1997; 6: 465–474. group of platelet donors, despite the relatively stable 6 Humpe A, Riggert J, Munzel U et al. A prospective,

efficiency of collection (parameter Ef; from 45 to 58%), randomized, sequential, crossover trial of large-volume versus the values of parameter R (recruitment) varied considerably normal-volume leukapheresis procedures: effect on progenitor from –2.5 Â 109 to 6.8 Â 109 platelets/min of recruited cells and engraftment. Transfusion 1999; 39: 1120–1127. platelets. 7 Schlenke P, Frohn C, Hennig H et al. Collection efficiencies of Our model is theoretically able to describe variations in CD34+ progenitor cells and mononuclear cells in leukapher- the cell recruitment over time. In this study, however, we esis products quantified by flow cytometry and calculated on the basis of a new formula. Vox Sang 2000; 78: 242–249. were actually able to calculate only the mean values, 8 Hillyer CD, Swenson RB, Hart KK et al. Peripheral blood because we had no blood samples drawn during the stem cell acquisition by large-volume leukapheresis in leukaphereses. Some data from the literature, based on growth factor-stimulated and unstimulated rhesus monkeys: the estimation of CD34 þ cells concentrations in the development of an animal model. Exp Hematol 1993; 21: peripheral blood or in the collection bag during leuka- 1455–1459. phereses, show that cell recruitment, at least for CD34 þ 9 Murea S, Goldschmidt H, Hahn U et al. Successful collection cells, is not necessarily stable during the procedure.9 and transplantation of peripheral blood stem cells in cancer In conclusion, a mathematical model of blood cell patients using large-volume leukaphereses. J Clin Apheresis collection using a continuous flow cell separator was 1996; 11: 185–194. developed taking into consideration the ability of cells to 10 Fliedner TM. The role of blood stem cells in hematopoietic cell renewal. Stem Cells 1998; 16: 361–374. migrate from other compartments into the peripheral blood 11 Smolowicz AG, Villman K, Berlin G, Tidefelt U. Kinetics of (recruitment, parameter R) and the variable efficiency of peripheral blood stem cell harvest during a single apheresis. the separator to collect different populations of cells Transfusion 1999; 39: 403–409. (parameter Ef). We have shown that the recruitment of 12 Rowley SD, Yu J, Gooley T et al. Trafficking of CD34+ cells cells is a very important phenomenon that considerably into the peripheral circulation during collection of peripheral influences the collected cell numbers. The precise mechan- blood stem cells by apheresis. Bone Marrow Transplant 2001; isms responsible for the recruitment are still not known 28: 649–656. and deserve further study. The relationships between the 13 Cassens U, Momkvist PH, Zuehlsdorf M et al. Kinetics of pre-apheresis cell concentrations, parameter R, parameter standardized large volume leukapheresis (LVL) in patients E , and collected numbers of cells are quite complex do not show a recruitment phenomenon of peripheral f blood progenitor cells. Bone Marrow Transplant 2001; 28: and can differ for different cell populations and 13–20. different separation procedures. Our model gives a 14 Knudsen LM, Nikolaisen K, Gaarsdal E, Johnsen HE. Kinetic rationale basis for studying the optimal set-up of the cell studies during peripheral blood stem cell collection show separator in order to optimize the collection of the target CD34+ cell recruitment intra-apheresis. J Clin Apheresis 2001; cell population. 16: 114–119.

Bone Marrow Transplantation Mathematical model J Mayer et al 757 15 Mayer J, Pospı´ sˇ il Z, Korˇ ı´ stek Z. Mathematical model of 21 Haas R, Mo¨hle R, Fru¨haur S et al. Patient characteristics peripheral blood progenitor cell mobilization and harvest: associated with successful mobilization and autografting of theoretical considerations and clinical applications. Hematol J peripheral blood progenitor cells in malignant lymphoma. 2001; 1 (Suppl. 1): 208. Blood 1994; 83: 3787–3794. 16 Ha´ jek R, Kreje` ı´ M, Sˇ cˇ udla V et al. Therapy of multiple 22 Goldschmidt H, Hegenbart U, Wallmeier M et al. Factor myeloma with high dose melphalan followed by maintenance influencing collection of peripheral blood progenitor cells therapy with interferon alpha or sequential maintenance following high-dose cyclophosphamide and granulocyte col- interferon and dexamethasone — interim analysis of the ony-stimulating factor in patients with multiple myeloma. Br J randomized trial. Bone Marrow Transplant 2002; 29 (Suppl. Haematol 1997; 98: 736–744. 2): S98. 23 Anderlini P, Przepiorka D, Seong D et al. Clinical toxicity and 17 Mayer J, Korˇ ı´ stek Z, Va´ sˇ ova´ I et al. Ifosfamide and etoposide- laboratory effect of granulocyte-colony-stimulating factor based chemotherapy as salvage and mobilizing regimens for (filgrastim) mobilization and blood stem cell apheresis from poor prognosis lymphoma. Bone Marrow Transplant 1999; 23: normal donors, and analysis of charges for the procedures. 413–419. Transfusion 1996; 36: 590–595. 18 Kovacs P, Bruch C, Herbst EW, Fliedner TM. Collection 24 Rowley SD, Prather K, Bui KT et al. Collection of peripheral of in vitro colony-forming units from dogs by repeated blood progenitor cells with an automated leukapheresis continuous flow leukaphereses. Acta Haematol 1978; 60: system. Transfusion 1999; 39: 1200–1206. 172–181. 25 Wilke R, Brettell M, Prince HM et al. Comparison of COBE 19 Mo¨hle R, Murea S, Pfo¨rsich M et al. Estimation of the Spectra software version 4.7 and 6.0 Auto PBSC Program. progenitor cell yield in a leukapheresis product by previous J Clin Apher 1999; 14: 26–30. measurement of CD34+ cells in the peripheral blood. Vox 26 Norol F, Scotto F, Duedari N, Beaujean F. Peripheral blood Sang 1996; 71: 90–96. stem cell collection with a blood cell separator. Transfusion 20 Yu J, Leisenring W, Bensinger WI et al. The predictive value of 1993; 33: 894–897. white cell or CD34+ cell count in the peripheral blood for 27 Ravagnani F, Siena S, De Reys S et al. Improved collection of timing apheresis and maximizing yield. Transfusion 1999; 39: mobilized CD34+ hematopoietic progenitor cells by a novel 442–450. automated leukapheresis system. Transfusion 1999; 39: 48–55.

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