For the values used to predict the synthesis and clearance two types of uncertainty exist:
• Measurement error in the data provided (e.g. plasma concentration).
• Inter- and intra-patient rate constant variation (e.g. FCR, intercompartmental
transfers, natural clearance).
The measurement error for antibody measurements is provided by the clinical staff and an estimate of ±5% has been given by haematological staff through unpublished dilution studies. Patient variability is more difficult to quantify. Each patient will have different parameter estimates than those of the parameter values used; there may also be variation between patients and in the same patient on different days. However, from previous work carried out with regards to FLC clearance (Chapter 4) a coefficient of variation (CoV) of ±10% would appear to give a reasonable estimate.
To estimate the deviation in the values a simple iterative Monte Carlo scheme has been implemented. For each value calculated one thousand iterations were performed to re-calculate the estimated values. At each iteration, values for the model parameters and data measurement were selected from a normally distributed parameter set, with mean value (µi) equal to the initial estimate and a standard deviation given by µi×
Section: 6.1.5 128
6.1.5
Results
The above algorithm was run for each set of patient data. This results in an estimate for the synthesis of the antibody production over the time course of the treatment. Ex- ample sets of results are shown in Figures 6.3 and Figures 6.4. Referring to Figure 6.3, for each antibody type two figures are shown, the lower figure is the estimated produc- tion using the above technique; the upper figure shows the simulated output given the estimated input. The circles in the top figure represent pre-apheresis measurements, whilst the triangles denote the post-apheresis observations. The estimated input is shown as a piecewise constant graph to encourage the recognition that a constant pro- duction is assumed for the discretisation and the estimate should be considered a mean estimate for the antibody synthesis. For the simulated output the solid line in the top figure represents the concentration of the antibody in the plasma compartment. The vertical line shown in each figure shows the time of organ transplant.
Patient 1, as seen in Figure 6.3, shows a classic, and successful, response to the combination of immune suppression and plasma exchange treatment. Pre-transplant, all immune complexes (IgG, IgA and IgM) decrease within two or three days of treat- ment. During this time the patient is given a series of plasma exchange treatments to reduce the basal level of antibodies to as low as possible. In some patients an immune response can be generated against the apheresis treatment which would be recognised by an increase in the antibody synthesis; this can often be masked by reductions in concentration in plasma due to the ongoing apheresis treatments. Post-transplant the patient is still on the same amount of immuno-suppressing drug, but little evidence is seen for this from the plots.
An alternative patient response can be seen in Figure 6.4. The patient received extended apheresis treatment sessions prior to surgery, ten sessions in total, indicat- ing that the clinicians did not believe the change in plasma concentration to be great enough for a successful transplant. However, from viewing the underlying synthesis it can be seen that this is probably due to reaching basal levels of concentration due to production. Overall very little change can be seen in production of each of the anti- body types, approximately 30% at best, as can be seen in Figure 6.3; in a patient with a successful response this can be considerably higher, often 80% or greater. For IgM a
0 2 4 6 8 10 12 14 0.5 1 1.5 2 g/L 0 2 4 6 8 10 12 14 0.4 0.6 0.8 1 1.2 g/day Time (days) (a) IgM 0 2 4 6 8 10 12 14 5 10 15 20 g/L 0 2 4 6 8 10 12 14 0 2 4 6 g/day Time (days) (b) IgG 0 2 4 6 8 10 12 14 0.5 1 1.5 2 g/L 0 2 4 6 8 10 12 14 0 0.5 1 g/day Time (days) (c) IgA
Figure 6.3: Estimates for synthesis of antibodies from patient 1. Top - simulated plasma concentration and measurements (circles pre-apheresis, triangles post-apheresis. Bottom -
solid-line predicted synthesis, dashed-line ± one standard deviation from estimated value.)
typical response to the immuno-suppressing drugs is evident, whilst IgG rises indicat- ing an increased immune response, and IgA remains relatively constant throughout. Unfortunately, patient data were not available post-operatively.
The discrete deconvolution as described presents difficulties that should be noted. Firstly, assuming the production is constant over the discretisation period, e.g. two days for patient two, may be unrealistic as the antibody synthesis is suppressed by drug combinations which are administered throughout the treatment. Whilst the piecewise constant values suggest a trend in the synthesis, it would be beneficial to consider the function as continuous over the treatment range. In addition, there is information regarding the functional form of the synthesis that is known a priori which could be used to determine the generation rate more accurately. The input signal cannot, by definition, be negative; whilst this did not occur for the patient data seen it is
Section: 6.1.5 130 0 2 4 6 8 10 12 14 16 0.2 0.4 0.6 0.8 1 g/L 0 2 4 6 8 10 12 14 16 0.2 0.3 0.4 0.5 g/day Time (days) (a) IgM 0 2 4 6 8 10 12 14 16 2 4 6 8 10 12 g/L 0 2 4 6 8 10 12 14 16 0 2 4 6 g/day Time (days) (b) IgG 0 2 4 6 8 10 12 14 16 0.5 1 1.5 2 2.5 g/L 0 2 4 6 8 10 12 14 16 0.8 1 1.2 1.4 1.6 g/day Time (days) (c) IgA
Figure 6.4: Estimates for synthesis of antibodies from patient 2 (pre-transplant only). Top - simulated plasma concentration and measurements (circles pre-apheresis, triangles post-
apheresis. Bottom - solid-line predicted synthesis, dashed-line ±one standard deviation from
estimated value.
not prevented by the discrete method. An extension of this problem can be seen in Figure 6.4(b), with the confidence interval generated by the Monte-Carlo method suggesting a negative production as the minimum bound between the second to fourth day. Further, as it is a natural signal, the synthesis will not contain discontinuities and will exhibit a degree of smoothness. This issue will be addressed in Chapter 7, where a more complex deconvolution method is developed.
In addition to offering prediction on a patient’s response to immunosuppression the algorithm presented provides information on the different apheresis treatments (plasma exchange, plasma absorption and plasmapheresis). Table 6.1 shows the clearance values obtained for each of the antibody types (IgM, IgG and IgA) for the 13 patients observed during the studies. Each patient was treated with only one of the apheresis methods
Table 6.1: Estimates for the clearance of IgM, IgG and IgA of different apheresis techniques (DFPP - Double Filtration Plasmapheresis, PE - Plasma Exchange, PA - Plasma Absorp- tion). The estimate for each patient is averaged over all the treatments, as specified in the number column.
Id Type Number Clearance mins−1 (std. dev.)
IgM IgG IgA
1 PE 11 6.94E-04 (3.71E-05) 6.44E-03 (3.98E-03) 2.94E-03 (1.19E-03) 2 PA 4 6.90E-04 (2.82E-05) 6.31E-04 (3.44E-04) 5.06E-04 (4.84E-04) 3 PA 3 6.75E-04 (7.37E-06) 1.03E-03 (2.20E-04) 9.21E-04 (3.54E-04) 4 DFPP 4 7.03E-04 (4.53E-05) 3.06E-03 (8.74E-04) 3.88E-03 (1.56E-03) 5 DFPP 8 6.87E-04 (3.12E-05) 3.37E-03 (2.44E-04) 4.61E-03 (1.15E-03) 6 DFPP 3 6.84E-04 (2.32E-05) 4.84E-03 (1.14E-03) 6.46E-03 (1.48E-03) 7 DFPP 3 7.16E-04 (5.09E-05) 3.62E-03 (5.43E-04) 3.68E-03 (3.62E-04) 8 DFPP 10 6.85E-04 (2.95E-05) 5.10E-03 (1.09E-03) 6.19E-03 (1.30E-03) 9 DFPP 5 6.81E-04 (4.80E-05) 5.33E-03 (1.90E-03) 6.35E-03 (2.01E-03) 10 DFPP 5 6.81E-04 (2.30E-05) 5.51E-03 (1.46E-03) 6.91E-03 (1.41E-03) 11 DFPP 7 6.90E-04 (3.20E-05) 5.32E-03 (1.07E-03) 5.85E-03 (1.19E-03) 12 DFPP 7 6.77E-04 (4.25E-05) 4.50E-03 (6.43E-04) 5.02E-03 (1.09E-03) 13 DFPP 5 7.05E-04 (6.95E-04) 5.01E-03 (4.93E-03) 6.12E-03 (5.87E-03) during their entire period, pre- and post-transplant, the values shown in the table are averaged over all sessions. Unfortunately, the treatment used was based on clinical requirements and resources and did not provide equal observations, therefore plasma- pheresis measurements predominate. However, it can still be seen from Table 6.1 that all three treatments appear to clear IgM with similar effectiveness, whilst IgG and IgA have much less predictability. As seen with FLC (Chapter 4) clearance of material that flows freely between plasma and EVF is limited by the exchange rate between these two compartments as well as the rate of clearance. It is therefore surprising that IgG and IgA are cleared more rapidly than IgM, even though IgM is limited to plasma pool distribution. This may be down to the size difference between the three isotypes, with IgM being approximately five times larger, which may influence the filter technology
Section: 6.2.0 132 used in the apheresis method.
Apheresis Clearance mins−1 (std. dev.)
IgM IgG IgA
PA 6.87E-04 (2.51E-05) 7.79E-04 (2.99E-04) 6.62E-04 (4.04E-04) PE 6.94E-04 (3.71E-05) 6.44E-03 (3.98E-03) 2.94E-03 (1.19E-03) DFPP 6.90E-04 (3.18E-05) 4.57E-03 (1.46E-03) 5.57E-03 (1.55E-03)
Table 6.2: Clearance estimates for apheresis treatments, averaged across all patients.
To allow direct comparison of the apheresis methods the results of Table 6.1 have been further condensed. Table 6.2 shows the average clearance values of each treat- ment independent of the patient. It should be noted that for plasma exchange only a single patient over eleven sessions is monitored, and for plasma absorption data for two patients from seven sessions were available. This limited sample size should be kept in mind when considering the predicted outcomes, however, from Table 6.2 some in- teresting conclusions can be drawn. It is believed by clinicians that plasma absorption (PA) is less effective at clearing due to the limited ability of the device to continue ab- sorbing antibodies over the entire treatment period. Whilst this is not proven here, it seems that PA performs relatively consistently across the three isotypes with clearance values in the range 6.62×10−4 to 7.79×10−4 min−1 for IgM, IgG and IgA. Although
PE and DFPP clear IgM with similar rates to PA (≈ 6.9× 10−4 min−1), IgG and
IgA have clearance values an order of magnitude greater, with DFPP offering a slight advantage when clearing IgA. This may allow clinical staff to customise the aphere- sis used depending on the patient’s condition, the cost of treatment and the facilities available. For example, it is known (see Chapter 2) that the immune response consists of two phases: primary, which consist of IgM antibodies and secondary which consist predominately IgG; therefore, during the primary phase any of the treatments would be appropriate but for the secondary response PA should be avoided. Alternatively, if the donor-specific response to the implanted organ is measured and contains a majority of a single antibody isotype, the appropriate apheresis method could be used, e.g. PP for IgA, PE for IgG.