An "Incremental" mathematical model for Immune Thrombocytopenic Purpura (ITP)

K e y w o r d s M a t h e m a t i c a l modeling, Biomathematics, Medical applications, Immune thrombocytopenic purpura (ITP). 1. I N T R O D U C T I O N The integr i ty of blood vessels is t aken in charge by platelets, which are a fract ion of the blood itself. Thei r n u m b e r usual ly ranges between 150 400 • 103#1-1. W h e n their level is less t han the lowest one, we speak abou t thrombocytopenia which, in case of a very low level, may be also cause of death, because of bleeding. Immune (or idiopathic) thrombocytopenic purpura ( ITP) is one of the most common causes of t h r o m b o c y t o p e n i a encounte red in the medical practice. Its incidence is of approximate ly one case per 10,000 in the general popula t ion , and it accounts for 0.16% of hospital admissions [1]. This pathology may be very complex, and its diagnosis remains one of exclusion [1 3]. Two ma in forms of I T P are usual ly encountered: the acute form, which often resolves spon taneous ly after six months or less in dura t ion , and the chronic form, which is the most challenging one, from the point of view of the clinical managemen t . Moreover, The authors are very indebted with the Azienda Ospedaliera CNR, CREAS Malattie Cardiovascolari e Discipline Affini, Laboratorio di Ematologia e Coagulazione, Pisa (Italy), for providing the data shown in Figures 1-3 and 5. In particular, we would like to thank G. Lazzerini and A. Giorgetti. 0895-7177/05/$ see front matter (~) 2005 Elsevier Ltd. All rights reserved. doi: 10.1016/j.mcm.2004.12.003 Typeset by AA~-TEX 1300 L. BRUGNANO et al. a rare form (cyclical thrombocytopenia) is also known, in which the level of platelets displays heavy oscillations [4,5]. Our concern will be the first two forms, in particular the chronic one. The clinical management of ITP has various lines of therapy [1,3,4,6]: the first one is essentially based on high-dose steroids, intravenous immune globulin, and spleneetomy (i.e., the surgical removal of the spleen). Such treatments may have many side-effects and/or be very invasive, in particular splenectomy. Concerning the latter, there is, at the moment, no way of predicting its effectiveness [6]. As a mat ter of fact, though different practices have been proposed in the literature [7-11], nevertheless, their conclusions are often controversial and in conflict each other (see, for example, [9,11]). In any case, most of the arguments used are of statistical nature, based on the follow-up of splenectomized patients (see, e.g., [10,12,131). The main problem with chronic ITP is that its pathogenesis has not been fully elucidated, and its management is primarily empirical [3,14]. In this context, mathematical modeling may be of help to understand the possible involved mechanisms (see, e.g., [15-24]). Unfortunately, very few models are suitably simple to be of practical use in the clinical management of ITP: to our knowledge, only in [19] a model is proposed to be used for transfusion management (and, by the way, its derivation is partially incorrect). We observe that, in the clinical management, it would be useful to rank a thrombocytopenia, depending on: (I) the production of platelets, which might be impaired, (2) the destruction of platelets, which might be increased, (3) the site of platelet destruction. Concerning the first issue, often evidenced in the medical literature [25-28], it is well known that an impaired production can cause thrombocytopenia: as an example, this happens in various clinical settings, such as leukemia, myelodisplasia, as well as in some cases of ITP. The production of platclets is in the bone marrow, and we shall refer to BMP as the "bone marrow production" of platelets. The previous aspect is complementary to the second problem, since the level of platelets is nothing but the difference between the produced ones and the destroyed ones. In order to measure platelet destruction, it is customary to look for their mean life: hereafter, MPL will denote the "mean platelet life". Finally, in case of augmented destruction, the site where platelets are destroyed could be useful, for example, to predict the effectiveness of splenectomy. A useful clinical test to measure the above parameters is the study of thrombokinetic by means of platelets labeled with suitable radionuclides (formerly, 51Cr and, more recently, 1111n [20,22,26-33]). In this test, labeled platelets are reinfused in the patient, and their level in the blood is measured at prescribed time intervals. Moreover, on the third day, a scan of the body is done by using a v-camera, allowing to measure the sites where the radioisotope is accumulated. Nevertheless, it is not completely clear how to "read" the measured data: though some standardization guidelines have been given [34,35], the underlying mechanisms have not yet been made completely clear. In this paper, we propose an "incremental" model of the physiopathology of ITP which has the intent to provide a practical tool to be used in the interpretation of the results of the throEbokinetic via the use of radionuclides. Moreover, at least in theory, the model provides a way to predict the effectiveness of splenectomy in the management of ITP, which we hope to translate into a clinical practice, in the near future. The paper is then organized as follows. In Section 2, we describe the basic model and its subsequent developments, which are introduced in order to take into account a number of aspects linked to the thrombokinetic. In Section 3, we then show some possible utilizations of the model, along with some practical examples of application. Finally, Section 4 contains some concluding remarks. An "Incremental" Mathematical Model 1301 2 . T H E M A T H E M A T I C A L M O D E L To begin with, let us start with the simplest mathematical model which describes the kinetic of platelets, to be "increased", as soon as it will become inadequate to describe the observed data. If we denote by, • p( t ) the level of platelets at time t, • ")' the rate of platelet destruction, • g( t ) the production of platelets at time t (i.e., g = BMP), then we may, at first, state that d ~ p : ~ p + g (1) In particular, with the only exclusion of cyclic thrombocytopenia where the level of platelets may heavily oscillate, we may assume that g is constant and that p is at steady state. That is, the derivative of p vanishes and the equilibrium value of platelets is