Efficacy Model for Antibody-mediated Pre-erythrocytic Malaria Vaccines

Additional Supplement

1. Interaction between CSP and TRAP antibodies

In our analysis of the effects of CSP and TRAP antibodies on the opsonisation and phagocytosis of P. falciparum sporozoites, our null hypothesis (independent model) is that CSP and TRAP antibodies act independently of each other. We test this against the alternative hypothesis (interaction model) that CSP and TRAP antibodies interact with each other, either synergistically or antagonistically.

Interaction Model

In a pharmacological review on the effects of combinations of interacting drugs, Greco et al(Greco et al. 1995)outline methods for modelling combinations of interacting drugs from a response surface perspective. The outlined framework allows both synergy and antagonism between pairs of drugs to be modelled. We adapt their equations so as to be applicable to exponential dose-response relationships between the probability of sporozoite survival and titres to CSP and TRAP antibodies. The relationship between CSP and TRAP antibody titres and the probability of sporozoite survival can be defined by the following equation:

(1)

where r is the probability of sporozoite survival, is the titre to antibody i, is the titre to antibody i which gives a 50% reduction in the probability of sporozoite survival, and is an interaction term.

Rearranging (1) gives

(2)

Now solving for gives

(3)

And isolating r and choosing the positive solution gives

(4)

We can test for consistency by setting

(5)

corresponding to our null hypothesis that CSP and TRAP titres act independently of each other.

Dependence of the probability of sporozoite survival on the interaction term γ

By looking at the special case whereand we can investigate the dependence of the response on the interaction term

(6)

Setting gives which once again corresponds to the assumption of independence of antibody titres. Setting gives which corresponds to the antagonistic case where a combination of titres is only as effective as each of the titres on their own. Note that mutual inhibition where each antibody actively inhibits the other resulting in a combined effect less than either of the individual effects cannot be modelled in this framework. Figure 1 shows the dose-response curves for a combination of two antibodies with interaction ranging from strongly antagonistic to strongly synergistic.

Figure 1: Dose-response curves for a combination of two antibodies of equal titre. Β is the titre required for a single antibody response to kill half the sporozoites.

Likelihood ratio test

As the null hypothesis is nested within the alternative hypothesis, they can be compared using a likelihood ratio test. Fitting both models gives the results shown in Table 1. The likelihood ratio statistic is (p=0.35) and thus we cannot identify an interaction effect from these data.

Table 1: Comparison of independent and interaction models.

Model / MLE Parameter Estimates / Model Fit
/ / / / / AIC
Independent / 0.011 / 13.07 / 11.64 / - / -223.7 / 453.5
Interaction / 0.011 / 151.84 / 172.23 / 962.7 / -223.2 / 454.6

Interestingly for the interaction model we get a maximum likelihood estimate of for the interaction term suggesting very strong synergy.

To further investigate the dependence of our model fit on we plot the profile log-likelihood in Figure 2. We see that the profile log-likelihood surface is very flat apart from a small fall in the region. Thus it is not possible to identify significant interactions between the antibody responses and outcome in these data.

Figure 2: Profile log-likelihood for γ.

2. Correlation between multiple antibody titres

Table 2: Analysis of Spearman rank correlation between antibody titres to the studied antigens. NS = not significant. a Not significant after Bonferroni correction.

Relationship between antibody response and antigens
Antigen / CSP / TRAP / LSA-1 / AMA-1 / EBA-175 / MSP-1
CSP / _ / _ / _ / _ / _ / _
TRAP / 0.0629 / _ / _ / _ / _ / _
NS
LSA-1 / 0.0641 / 0.5033 / _ / _ / _ / _
NS / 1.21e-05
AMA-1 / -0.1075 / 0.3974 / 0.3753 / _ / _ / _
NS / 7.89e-04 / 0.0016
EBA-175 / 0.1820 / 0.0545 / 0.1998 / 0.1447 / _ / _
NS / NS / NS / NS
MSP-1 / 0.1338 / 0.3629 / 0.3107 / 0.1499 / 0.2599 / _
NS / 0.0023 / 0.0099a / NS / 0.0322a

References

Greco, W. R., Bravo, G. & Parsons, J. C. 1995 THE SEARCH FOR SYNERGY - A CRITICAL-REVIEW FROM A RESPONSE-SURFACE PERSPECTIVE. Pharmacological Reviews47, 331-385.