Numerical analysis of in vivo platelet consumption data from ITP patients

Modeling in vivo platelet turnover

The numerical model used here [10] posits that an in vivo platelet population can be visualized in a spreadsheet as a series of small platelet cohort concentrations. The cohorts are assumed to be produced at a constant rate (PR, K/ul/h) in short sequential time periods, and individually consumed, by both random and lifespan-dependent processes, at the end of each such time period. The consumption curve for individual cohorts is determined by a random destruction rate constant (RD, %/h), by the lognormally distributed cohort lifespan (LS, hr), and by the standard deviation of ln LS (SD). Population platelet consumption curves are generated by summing the cohort values at sequential time points. Optimal theoretical consumption curves generated by a large range of possible parameter values are identified via quantitative comparison to each data set (summed squared residual values, or SS).

For the 41 patient studies analyzed here, each patient’s platelet consumption data was normalized to the first (baseline) measurement of circulating ¹¹¹In. Visual inspection of the data strongly suggests that many of the labeled platelet preparations contained long lived species, as others have described in similar studies [15, 16]. This is evident in the plateau phase seen at late times in the consumption data (for example, patients 35 and 27, Fig. 4). To take this into account, we evaluated a fourth parameter: the fraction of the labeled cells/platelets consisting of long lived (species (LL, %). This parameter simply “shrinks” the scope of the analysis to the consumption of platelets from 100 % of the time zero value to an optimizable minimum percentage (LL). For modeling purposes, lifespans of these long lived species are assumed to be infinite.

Optimal parameter value search process

Optimal parameter value searches were performed as shown schematically in Fig. 1. For a given data set, SS values are determined for each possible set of parameter values in a four-dimensional parameter space defined by RD, LS, SD, and LL. The core component of the search is an evaluation of SS for each point in a 20 × 20 plane of possible LS and LL values at fixed values of RD and SS. The resultant minimum “planar” SS values are visually identifiable (see examples in Fig. 2). This process is repeated over a range of 20 RD values, yielding in most cases single “volume” SS minima as shown in Fig. 3a. Finally, the entire process is repeated at a series of SD values, and the resultant volume minima are compared in order to identify a “global” minimum SS value and its associated parameter values. Distinguishable alternative volume minima showing SS values greater than those of the global minima were also seen in some data sets (see below). Searches were performed for only three SD values, as this generated a plausible range of distribution widths for the resultant lifespan-dependent consumption rates (see examples in Fig. 3b) while significantly reducing computation time. Examples of the consumption curves generated by the optimal parameter values are shown in Fig. 4.

Data quality evaluation and optimal parameter value search results

This process is outlined in Fig. 5. Of the 41 originally reported ITP patient data sets, one was excluded due to lack of initial time point data. One patient demonstrated an initial platelet clearance rate of 46 % in the first 1.5 h of the study (>4 standard deviations faster than the mean). Because this value suggests the type of platelet activation during labeling/processing that we have on occasion seen in murine platelet clearance studies (TS, unpublished), this data was also excluded. For the remainder, quality of parameter value optimization was evaluated in terms of the ratio of SS to n (the number of data points per patient data set), where n ranged from 5 to 9 (Table 1). One case (patient 31) with an SS/n value of 143 (over four standard deviations from the mean value of 18.5) was then excluded. No other SS/n values fell beyond two sd from the mean.

A single “global” minimum SS value, with its associated (optimal) parameter values, was identified in 24 of the 39 evaluable data sets. The optimal consumption curves show a large amount of inter-patient variation, as the examples in Fig. 4a demonstrate. Of those showing more than one minimum, convincing global minima were identified in four data sets on the basis of goodness-of-fit. Specifically, the global minima in these cases showed SS values which were less than 50 % of those defining the alternative (local) minima. Three data sets showed global minima for which comparison of absolute vs. squared residuals provided additional support for their significance (see Additional file 1). Seven data sets, however, showed local minima that could not be distinguished from the global minima on these bases. In sum, we were able to identify convincing global minima in 31 of the 39 evaluable data sets (79 %) shown in Table 1.

Data quality was further evaluated by performing “jackknife” resampling studies on each of the patient data sets in Table 1 [17]. Optimal parameter values were obtained for each of the n subsets for each data set via the same process used to analyze each complete data set (at the SD value of the complete data set’s global minimum). We found CV values for calculated RD and LL parameters to be under 0.5 for over 75 % of these cases (Table 1). Quantification of platelet lifespan was more difficult, with only 52 % of our cases showing CV values for the LS parameter of under 0.5. That is expected, however, because LS value estimates showed a larger variance in cases where random destruction predominated (Fig. 4b).

Predicting the effects of reduced platelet production and increased random destruction

The effect of reduced platelet production (PR) on RD and platelet count was modeled as shown in Fig. 6. The process begins (step A) with the optimal (baseline) parameter values for pooled data from the three patients who transiently normalized their platelet counts (Table 1), using an initial “f” value of 1.0. From this set, a “target” reduced platelet production rate (PR₁) is generated, corresponding to 90 % of PR₀. Using that value, the model generates the expected (reduced) aHRD value (aHRD₁). We then (step B) incrementally increase HRD until the model generated value of aHRD (aHRD_i) = aHRD₀. The associated RD_i value (=HRD_i + NHRD₀) and platelet count values are those predicted to occur at PR₁. Finally (step C), we repeat steps A and B with a series of reduced platelet production rates (PR_i). This generates predicted HRD and platelet count values for each PR_i value. We then repeated this analysis at f values of 0.5 and 0.2.

We note that for our baseline parameter values, aRD₀ (RD × platelet count) is equal to 45 % of PR₀. We make no quantitative predictions for the effect of reducing PR below aRD₀ because the assumptions underlying the current numerical analysis model may not hold in that case. This is because the number of hemostatic targets is expected to increase below platelet counts at which hemostasis begins to be impaired. That in turn would invalidate the assumption of a constant absolute HRD rate, which is one of the bases for our iterative predictive method (Fig. 5). A model incorporating a dynamic hemostatic target population will be needed to predict platelet consumption rates in these circumstances.

To model the effect of increased random platelet consumption (RD), we generated a series of incrementally reduced target platelet counts (P_i)(range: 90 % to 10 % of baseline), and to achieve each we incrementally increased RD from its baseline value until the model generated value of P (P_m) was equal to P_i. To model the concurrent effects of increased RD and homeostatically increased PR, we used the same series of target platelet counts (P_i), and for each we increased PR in a manner proportional to the reduction in platelet count (to a maximum of twice the baseline PR value, a conservative theoretical starting point) before, again, empirically identifying RD_i.

The results of these three modeling approaches are plotted with the values obtained for the patients in Fig. 7.

Optimal patient parameter values in comparison to modeled values

Surprisingly, only four patients in the study showed a platelet production rate that is even modestly increased (>50 %) in comparison to the presumed normals (Fig. 7c). The latter showed a mean platelet production rate (2.12 K/ul/h) comparable to the 1.7 K/ul/h rate estimated for normals in a previous study [8]. The finding of predominantly low to normal production rates in the thrombocytopenic cases (Fig. 7a) is corroborated by the distribution of random destruction rates (Fig. 7b), where rates consistent with no increase in platelet production are seen for, again, all but a handful of the patients. A surprisingly large number of cases (at least 12 of 31) fall near the RD rates predicted to result solely from impaired platelet production. The predicted rates vary significantly, however, as a function of aHRD₀/RD₀ (‘f’).

Because we don’t know the normal value of ‘f’, our ability to predict the increase in RD at low platelet counts is limited. Future studies in patients with thrombocytopenias due to impaired platelet production could resolve that problem.

Modeling of immature platelet fraction values

An ability to take up fluorescent marker dyes such as thiazole orange (a marker of “reticulated platelets”, RP) or the proprietary dyes used in Sysmex hematology analyzers (marking the “immature platelet fraction”, IPF) is thought to be characteristic of those platelets which have recently been released into the bloodstream. The age threshold (T) at which “young’ platelets stop taking up these marker dyes is not known. Because the numerical analysis model generates a platelet age distribution for any given set of parameter values, it can be used both to estimate T and to predict the effect of altered production and consumption rates on the fraction of platelets of age less than T (i.e. the IPF).

Specifically, the normal range for the IPF is approximately 4.5 % (each clinical laboratory typically establishes its own range; this is the value in use at the Memphis VA Medical Center). Per the age distribution predicted by the model for our normalized controls (Fig. 8a), the youngest 4.5 % of platelets corresponds to those aged less than 4 h (i.e. T = 4 h). Application of that cutoff to the age distributions generated during modeling of the effects of altered production and/or consumption (Fig. 7), generates the predicted IPF and absolute IPF (aIPF) for thrombocytopenias induced by those mechanisms, as shown in Fig. 8b.

We note that this analysis depends on the assumption that all nascent platelets below a given age (T) take up the fluorescent markers used in the RP and IPF assays. Our measurements of mass turnover for mature and reticulated murine platelets suggest that this may not be the case [19].