4. Methods
4.1 Model
Let \(Y_{ig}\) be the observed expression levels of the raw measured data from clinically derived malignant tumor samples for gene \(g, g = 1, \cdots, G\) and sample \(i, i = 1, \cdots, My\). \(G\) denotes the total number of probes/genes and \(My\) denotes the number of samples. The observed expression levels for solid tumors can be modeled as a linear combination of raw expression levels from three components: \[ {Y_{ig}} = \pi _{1,i}N_{1,ig} + \pi _{2,i}N_{2,ig} + (1 - \pi_{1,i} - \pi _{2,i}){T_{ig}} \label{eq:1} \]
Here \(N_{1,ig}\), \(N_{2,ig}\) and \({T_{ig}}\) are the unobserved raw expression levels from each of the three components. We call the two components for which we require reference samples the \(N_1\)-component and the \(N_2\)-component. We call the unknown component the T-component. We let \(\pi_{1,i}\) denote the proportion of the \(N_1\)-component, \(\pi_{2,i}\) denote the proportion of the \(N_2\)-component, and \(1 - \pi_{1,i}-\pi_{2,i}\) denote the proportion of the T-component. We assume that the mixing proportions of one specific sample remain the same across all genes.
Our model allows for one component to be unknown, and therefore does not require reference profiles from all components. A set of samples for \(N_{1,ig}\) and \(N_{2,ig}\), respectively, needs to be provided as input data. This three-component deconvolution model is applicable to the linear combination of any three components in any type of material. It can also be simplified to a two-component model, assuming there is just one \(N\)-component. For application in this paper, we consider tumor (\(T\)), stromal (\(N_1\)) and immune components (\(N_2\)) in an admixed sample (\(Y\)).
Following the convention that \(\log_2\)-transformed microarray gene expression data follow a normal distribution, we assume that the raw measures \(N_{1,ig} \sim LN({\mu _{{N_1}g}},\sigma _{{N_1}g}^2)\), \(N_{2,ig} \sim LN({\mu _{{N_2}g}},\sigma _{{N_2}g}^2)\) and \({T_{ig}} \sim LN({\mu _{Tg}}, \sigma _{Tg}^2)\), where LN denotes a \(\log_2\)-normal distribution and \(\sigma _{{N_1}g}^2\),\(\sigma _{{N_2}g}^2\), \(\sigma _{Tg}^2\) reflect the variations under \(\log_2\)-transformed data. Consequently, our model can be expressed as the convolution of the density function for three \(\log_2\)-normal distributions. Because there is no closed form of this convolution, we use numerical integration to evaluate the complete likelihood function (see the full likelihood in the Supplementary Materials).
4.2 The DeMixT algorithm for deconvolution
DeMixT estimates all distribution parameters and cellular proportions and reconstitutes the expression profiles for all three components for each gene and each sample. The estimation procedure (summarized in Figure 1b) has two main steps as follows.
Obtain a set of parameters \(\{\pi_{1,i}, \pi_{2,i}\}_{i=1}^{My}\), \(\{\mu_T, \sigma_T\}_{g=1}^G\) to maximize the complete likelihood function, for which \(\{\mu_{N_{1,g}}, \sigma_{N_{1,g}}, \mu_{N_{2,g}}, \sigma_{N_{2,g}}\}_{g=1}^G\) were already estimated from the available unmatched samples of the \(N_1\) and \(N_2\) component tissues. (See further details in our paper.)
Reconstitute the expression profiles by searching each set of \(\{n_{1,ig}, n_{2,ig}\}\) that maximizes the joint density of \(N_{1,ig}\), \(N_{2,ig}\) and \(T_{ig}\). The value of \(t_{ig}\) is solved as \({y_{ig}} - {{\hat \pi }_{1,i}}{n_{1,ig}} - {{\hat \pi }_{2,i}}{n_{2,ig}}\).
These two steps can be separately implemented using the function DeMixT_DE or DeMixT_GS for the first step and DeMixT_S2 for the second, which are combined in the function DeMixT(Note: DeMixT_GS is the default function for first step).
In version 1.8.2, DeMixT added simulated normal reference samples, i.e., spike-in, based on the observed normal reference samples. It has been shown to improve accuracy in proportion estimation for the scenario where a dataset consists of samples where true tumor proportions are skewed to the high end.
Tumor proportion estimation remains consistence with and without spike-in normal, when its distribution is roughly symmetric around 0.5.