**Design of survival studies for red blood cells**

Julia Korell, Carolyn V. Coulter, Stephen B. Duffull

School of Pharmacy, University of Otago, Dunedin, New Zealand

**Background: **

The lifespan of red blood cells (RBCs) is unknown. The primary methods for determining RBC lifespan involve labelling with a radioactive marker. Two labelling techniques have been developed: cohort labelling, where cells of a certain age are labelled, and random labelling, where all cells present at a moment in time are labelled. Of these the random labelling technique has been more commonly used. All current labelling methods contain significant flaws including loss of the label from viable RBCs or reincorporation of the label into new RBCs after the death of the originally labelled cells. Loss of label may occur from decay of the radioactive compound, dissociation of the radioactive compound from the target and loss by vesiculation. From a modelling perspective, previously proposed models for the lifespan of RBCs either assume a fixed lifespan for all cells [1], or a continuous distribution of lifespans where the cells are thought to die solely due to senescence [2,3]. Recently, Kalicki *et al.* have shown that combining a finite lifespan with random destruction improves the performance of these models [4].

**Objectives:**

1. To develop a model for RBC survival based on statistical theory that incorporates known physiological mechanisms of RBC destruction.

2. To assess the local identifiability of the parameters of the lifespan model under ideal cohort and random labelling techniques.

3. To evaluate the precision to which the parameter values can be estimated from an *in vivo* RBC survival study using a random labelling technique with loss of the label.

**Methods:**

1) A statistical model for the survival time of RBCs with respect to the physiology* *of RBC destruction was developed. The model was derived from established models that were developed to describe the lifespan of humans [5].

2) The local identifiability of the parameters was determined informally using the theory of design of experiments. In this method the information matrix was constructed for an experiment based on ideal cohort and ideal random labelling and it was assessed whether the matrix was positive definite for a given fixed design, indicating local identifiability. Measurement noise was included as a combined error model, with an additive variance of 1.73 (counts per minute/mL)^{2} and a coefficient of variation of 2.32% for the proportional error, based on i*n vitro* experiments in our laboratory.

3) A D-optimal design was applied to determine optimal blood sampling times for *in vivo* RBC survival studies using a random labelling method with loss of label. A hypothetical *in vivo* study with 100 patients was assumed that uses radioactive chromium as a label for RBCs. A dose of radioactive label was determined that provided an initial concentration of 400 counts per minute (cpm) per mL of blood sample. The lower limit of detection was 0.8 cpm per sample analysed. The percentage standard errors (%SE) of the parameter estimates were determined from the inverse diagonal entries of the corresponding Fisher Information matrix. Measurement noise was the same as in (2).

**Results & Discussion**

1) The model was described by a combination of flexible and reduced additive Weibull distributions. The underlying combined distribution of RBC lifespans accounts for the known processes of RBC destruction, including death due to senescence, random loss during circulation, as well as death due to early or delayed failures. These processes are controlled by five parameters in the model, while a sixth parameter combines the two underlying Weibull distributions. The resulting survival model was used to simulate *in vivo* RBC survival studies using different RBC labelling techniques. Predictions from the model agreed well with models from the literature for cohort labelling techniques as well as for random labelling techniques. Furthermore, the decay of radioactive chromium with a half-life of 27.7 days was included into the model, together with the dissociation of the chromium-haemoglobin complex with an approximate half-life of 70 days and a vesiculation-related loss of 20% of the total haemoglobin together with bound label from the cells during their median lifetime. These values are in accordance with the literature.

2) The Fisher information matrix was positive definite for both the ideal cohort and random labelling studies, indicating that the model was locally identifiable for a given finite design. For the ideal cohort labelling study with 100 patients the percentage standard error (%SE) values for all but one parameter were <5%. Only the parameter controlling the initial destruction of unviable RBCs was associated with a slightly higher %SE of 18%. The precision of the parameter estimates was lower for the ideal random labelling study. However, all parameter values can be estimated from such a study. The %SE for five of the parameters were approximately 20% or less, and the highest %SE, again for the initial destruction of unviable RBCs, was 120%.

3) The D-optimal design was located for the random labelling method including the various loss mechanisms of label from RBCs. Optimal sampling times were on day 1, 28, 55, 56, 78 and 112 after labelling. One blood sample per day was taken at each of these days from each of the 100 patients in the hypothetical study. The %SE for the parameter estimates were as follows: 54% and 49% for the two main parameters controlling the senescence component of RBC survival, 36% for the parameter controlling random destruction, 43% for the parameter controlling death due to delayed failures, and 4% for the mixing parameter that combines the two underlying Weibull distributions. The %SE of the parameter controlling the initial destruction was not estimable (%SE >200%). This initial destruction is the only parameter in the model that cannot be estimated from a study using a random labelling technique with radioactive chromium.

**Conclusions: **

The developed model incorporates plausible processes of RBC destruction in the body. Simulations of RBC survival studies using cohort labelling techniques as well as random labelling techniques are plausible. The model accounts for the known shortcomings of radioactive chromium as the most commonly used random label for RBCs. The model shows local identifiability of all parameter values under ideal labelling techniques. Using a random labelling technique with loss of the label, all but one parameter can be estimated with reasonable precision. The model and design are intended to be used for setting up and interpretation of current *in vivo *studies of RBC survival. However, there is a clear need for better labelling techniques for RBCs in the future.

**References**[1] Krzyzanski W, Ramakrishnan R, Jusko W (1999) Basic pharmacodynamic models for agents that alter production of natural cells. J. Pharmacokinet. Biopharm. 27(5):467-489

[2] Krzyzanski W, Woo S, Jusko W (2006) Pharmacodynamic models for agents that alter production of natural cells with various distributions of lifespans. J. Pharmacokinet. Pharmacodyn. 33(2):125-166

[3] Freise K, Widness J, Schmidt R, Veng-Pedersen P (2008) Modeling time variant distributions of cellular lifespans: Increases in circulating reticulocyte lifespans following double phlebotomies in sheep. J. Pharmacokinet. Pharmacodyn. 35(3):285-323

[4] Kalicki R, Lledo R, Karlsson M (2009) Modeling of red blood cell life-span in a hematologically normal population.

*PAGE 18 (2009) Abstr 1677 [www.page-meeting.org/?abstract=1677]*, St. Petersburg, Russia.

[5] Bebbington M, Lai C, Zitikis R (2007) Modeling human mortality using mixtures of bathtub shaped failure distributions. J. Theor. Biol. 245(3):528-538