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<div id="header">
<h1 class="title toc-ignore">Modeling patient-level outcomes</h1>
</div>
<div id="slide-deck" class="section level2">
<h2>Slide Deck</h2>
<iframe class="speakerdeck-iframe" frameborder="0" src="https://speakerdeck.com/player/3710e7390c4842afb3f85c22e729ecde" title="MI_fda" allowfullscreen="true" style="border: 0px; background: padding-box padding-box rgba(0, 0, 0, 0.1); margin: 0px; padding: 0px; border-radius: 6px; box-shadow: rgba(0, 0, 0, 0.2) 0px 5px 40px; width: 100%; height: auto; aspect-ratio: 560 / 314;" data-ratio="1.78343949044586">
</iframe>
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<div id="overview" class="section level2">
<h2>Overview</h2>
<p>This module focuses on modeling patient-level outcomes, specifically
patient survival time, using spatial summary statistics derived from
multiplex images as model covariates. First we use traditional
regression models and single-number patient summaries. Then, we model
entire spatial summary functions using methods from functional data
analysis.</p>
<p>Load data and libraries.</p>
<pre class="r"><code>library(tidyverse)
library(patchwork)
library(spatstat)
library(survival)
library(refund)
library(refund.shiny)
# load processed ovarian cancer data from GitHub
load(url("https://github.com/julia-wrobel/MI_tutorial/raw/main/Data/ovarian.RDA"))</code></pre>
</div>
<div id="regression-with-patient-level-outcomes" class="section level2">
<h2>Regression with patient level outcomes</h2>
<p>The general workflow is:</p>
<ol style="list-style-type: decimal">
<li>Select a spatial summary measure (e.g. bivariate Ripley’s K).</li>
<li>Choose a particular radius <em>r</em> at which to evaluate Ripley’s
K. This can be selected based on clinical knowledge.</li>
<li>Choose a cell spatial relationship you want to analyze
(e.g. co-occurence of B-cells and macrophages).</li>
<li>Evaluate bivariate Ripley’s K for B-cells and Macrophages at radius
<em>r</em> for all images in the dataset. This will result in a single
number spatial summary for each image.</li>
<li>Select a patient-level outcome of interest (e.g. survival)</li>
<li>Model the outcome, using and the spatial summary statistic as a
covariate. Include other potential confounders.</li>
</ol>
<div id="ovarian-immune-cell-clustering-and-patient-survival"
class="section level3">
<h3>Ovarian immune cell clustering and patient survival</h3>
<p>Let’s say I am interested in the relationship between patient
survival and the clustering of immune cells within the tumor areas in
the ovarian cancer dataset.</p>
<p>For this analysis I will collapse immune cell subtypes (B-cell,
macrophage, CD8 T cell) into one “immune cell” category. I will then
calculate univariate L at radius <em>r = 100</em> for immune cells. It
would also be reasonable to use the K or G function.</p>
<div id="calculating-the-spatial-summary-statistics"
class="section level4">
<h4>Calculating the spatial summary statistics</h4>
<p>First I define a function that can be used to extract the L-function
at a particular radius or set of radii.</p>
<pre class="r"><code>
# function to extract univariate spatial summary statistics
# replace argument func = Lest with func = Kest to get K function estimates
# returns a data frame with estimate L values and L under CSR and different radii
extract_univariate = function(data, x = "x", y = "y",
subject_id = "sample_id",
marksvar,
mark1 = "immune",
r_vec = seq(0, 300, by = 10),
func = Lest){
# create a convex hull as the observation window
w = convexhull.xy(data[[x]], data[[y]])
# create ppp object
pp_obj = ppp(data[[x]], data[[y]], window = w, marks = data[[marksvar]])
# estimate L using spatstat
L_spatstat = func(subset(pp_obj, marks == mark1),
r = r_vec,
correction = c("iso")
)
df = as_tibble(L_spatstat) %>%
select(r, L = iso, csr = theo) %>%
mutate(Ldiff = L - csr,
subject_id = data[[subject_id]][1]) %>%
select(subject_id, r, L, csr, Ldiff)
df
}</code></pre>
<p>Then I organize the data and apply this function to all images in the
dataset.</p>
<pre class="r"><code>df_nest = ovarian_df %>%
# subset to tumor areas
filter(tissue_category == "Tumor") %>%
mutate(id = sample_id) %>%
dplyr::select(sample_id, x, y, immune, id) %>%
nest(data = c(sample_id, x, y, immune))
L_df = map_dfr(df_nest$data, extract_univariate,
marksvar = "immune", mark1 = "immune",
r_vec = c(0, 50, 100))
## Warning: data contain duplicated points
## Warning: 1 point was rejected as lying outside the specified window
## Warning: data contain duplicated points
## Warning: 1 point was rejected as lying outside the specified window
L_df
## # A tibble: 384 × 5
## subject_id r L csr Ldiff
## <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 68 0 0 0 0
## 2 68 50 83.5 50 33.5
## 3 68 100 138. 100 37.6
## 4 69 0 0 0 0
## 5 69 50 114. 50 64.1
## 6 69 100 176. 100 76.1
## 7 70 0 0 0 0
## 8 70 50 62.9 50 12.9
## 9 70 100 115. 100 15.4
## 10 71 0 0 0 0
## # ℹ 374 more rows</code></pre>
<p>We want to use the <code>Ldiff</code> variable, which is the
difference between the observed L and the theoretical L under CSR, in
our model. This value will be positive if cells are clustered in an
image, negative if cells show spatial regularity, and close to zer0 if
cells exhibit CSR. We will select the <code>Ldiff</code> value at <em>r
= 100</em>. I visualize this variable below.</p>
<pre class="r"><code>L_df = L_df %>%
filter(r == 100) %>%
select(subject_id, Ldiff)
L_df %>%
ggplot(aes(Ldiff)) +
geom_histogram() +
geom_vline(xintercept = 0) </code></pre>
<p><img src="MI_fda_files/figure-html/unnamed-chunk-5-1.png" width="90%" /></p>
<p>Let’s examine the subject with the outlying Ldiff value. It appears
that they have very few immune cells, so the Ldiff cannot be well
estimated. We will remove this subject.</p>
<pre class="r"><code>id = L_df %>% filter(Ldiff > 200) %>% pull(subject_id)
ovarian_df %>%
filter(sample_id %in% id,
tissue_category == "Tumor") %>%
ggplot(aes(x, y, color = immune)) +
geom_point(size = .1)</code></pre>
<p><img src="MI_fda_files/figure-html/unnamed-chunk-6-1.png" width="90%" /></p>
<pre class="r"><code>
# remove this subject
L_df = L_df %>% filter(Ldiff <=200)</code></pre>
</div>
<div id="modeling-patient-survival" class="section level4">
<h4>Modeling patient survival</h4>
<p>First I merge the patient level outcome data with the L summary
statistics.</p>
<pre class="r"><code>survival_df = ovarian_df %>%
# rename id variable to be compatible with L_df and select covariates of interest
dplyr::select(subject_id = sample_id,
survival_time, death, age = age_at_diagnosis) %>%
# delete duplicates- should only have one row per subject
distinct() %>%
# remove subject with outlying survival time
filter(survival_time < 200) %>%
# merge with L function data
left_join(., L_df, by = "subject_id")</code></pre>
<p>Below I plot the survival time against Ldiff values, colored by
survival status. It does appear that lower values of Ldiff (indicating
less extreme immune clustering) have longer survival times.</p>
<pre class="r"><code>survival_df %>%
ggplot(aes(Ldiff, survival_time)) +
geom_point(aes(color = factor(death)))
## Warning: Removed 1 rows containing missing values (`geom_point()`).</code></pre>
<p><img src="MI_fda_files/figure-html/unnamed-chunk-8-1.png" width="90%" /></p>
<p>Last we run a proportional hazards model to model this different. We
will run this model using functions from the <code>survival</code>
package.</p>
<pre class="r"><code>phmod = coxph(Surv(survival_time, death) ~ Ldiff + age,
data = survival_df)
summary(phmod)
## Call:
## coxph(formula = Surv(survival_time, death) ~ Ldiff + age, data = survival_df)
##
## n= 126, number of events= 79
## (1 observation deleted due to missingness)
##
## coef exp(coef) se(coef) z Pr(>|z|)
## Ldiff 0.009221 1.009263 0.002911 3.168 0.00153 **
## age 0.014171 1.014272 0.011298 1.254 0.20974
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## exp(coef) exp(-coef) lower .95 upper .95
## Ldiff 1.009 0.9908 1.0035 1.015
## age 1.014 0.9859 0.9921 1.037
##
## Concordance= 0.621 (se = 0.034 )
## Likelihood ratio test= 13.5 on 2 df, p=0.001
## Wald test = 14.9 on 2 df, p=6e-04
## Score (logrank) test = 15.6 on 2 df, p=4e-04</code></pre>
<p>Here the coefficients are log-hazard ratios. The exponentiated
coefficient for <code>Ldiff</code> > 1, indicating that the hazard of
death is greater for a subject with a greater degree of immune
clustering.</p>
<p>Some caveats about this analysis:</p>
<ul>
<li>We cannot distinguish clustering from inhomogeneity, so we may want
to consider calculating L using a metric that takes into account
potential inhomogeneity in the distribution of cells. This could be the
<code>spatstat::Linhom()</code> function, or using the permutation
approach in <code>spatialTIME</code>.</li>
<li>It is important to check validity of model assumptions, which we
have not done here</li>
<li>In a more thorough analysis I would control for other clinical
variables that may be potential confounders.</li>
</ul>
</div>
</div>
<div id="macrophage-and-b-cell-co-occurence-in-ovarian-cancer"
class="section level3">
<h3>Macrophage and B-cell co-occurence in ovarian cancer</h3>
<p>Here I am interested in the relationship between patient survival and
the co-occurence of B cells and macrophages in the ovarian cancer
dataset. I will use bivariate L at radius <em>r = 100</em> to assess the
spatial relationship between B-cells and macrophages.</p>
<div id="calculating-the-spatial-summary-statistics-1"
class="section level4">
<h4>Calculating the spatial summary statistics</h4>
<p>First I define a function that can be used to extract the bivariate
L-function at a particular radius or set of radii.</p>
<pre class="r"><code># function to extract univariate spatial summary statistics
extract_bivariate = function(data, x = "x", y = "y",
subject_id = "sample_id",
marksvar,
mark1 = "B-cell",
mark2 = "macrophage",
r_vec = seq(0, 300, by = 10)){
# create a convex hull as the observation window
w = convexhull.xy(data[[x]], data[[y]])
# create ppp object
pp_obj = ppp(data[[x]], data[[y]], window = w, marks = data[[marksvar]])
# estimate L using spatstat
L_spatstat = Lcross(pp_obj, mark1, mark2, r = r_vec, correction = "iso")
df = as_tibble(L_spatstat) %>%
select(r, L = iso, csr = theo) %>%
mutate(Ldiff = L - csr,
subject_id = data[[subject_id]][1]) %>%
select(subject_id, r, L, csr, Ldiff)
df
}</code></pre>
<p>Then I organize the data to apply this function. However, some
subjects do not have enough B-cells or macrophages to calculate the L
function. We will exclude subjects with < 3 macrophages or < 3
B-cells in the tumor area. In doing so, we lose several subjects.</p>
<pre class="r"><code>df_bivariate = ovarian_df %>%
filter(tissue_category == "Tumor") %>%
# subset to tumor areas
filter(tissue_category == "Tumor") %>%
# define B cell or macrophage variable
mutate(id = sample_id,
phenotype = case_when(phenotype_cd19 == "CD19+" ~ "B-cell",
phenotype_cd68 == "CD68+" ~ "macrophage",
TRUE ~ "other"),
phenotype = factor(phenotype)) %>%
filter(phenotype != "other") %>%
dplyr::select(sample_id, x, y, phenotype, id) %>%
# calculate number of macrophages and B-cells for each subject
group_by(sample_id) %>%
mutate(n_Bcell = sum(phenotype == "B-cell"),
n_mac = sum(phenotype == "macrophage")) %>%
ungroup() %>%
filter(n_Bcell > 2, n_mac > 2) %>%
# nest data
nest(data = c(sample_id, x, y, phenotype))</code></pre>
<pre class="r"><code>Lbivariate_df = map_dfr(df_bivariate$data, extract_bivariate,
marksvar = "phenotype",
r_vec = c(0, 50, 100))
Lbivariate_df
## # A tibble: 198 × 5
## subject_id r L csr Ldiff
## <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 68 0 0 0 0
## 2 68 50 64.3 50 14.3
## 3 68 100 99.3 100 -0.730
## 4 69 0 0 0 0
## 5 69 50 57.6 50 7.61
## 6 69 100 141. 100 41.1
## 7 70 0 0 0 0
## 8 70 50 49.4 50 -0.565
## 9 70 100 140. 100 39.8
## 10 72 0 0 0 0
## # ℹ 188 more rows</code></pre>
<p>Histogram of the bivariate Ldiff variable:</p>
<pre class="r"><code>Lbivariate_df = Lbivariate_df %>%
filter(r == 100) %>%
select(subject_id, Ldiff)
Lbivariate_df %>%
ggplot(aes(Ldiff)) +
geom_histogram() +
geom_vline(xintercept = 0) </code></pre>
<p><img src="MI_fda_files/figure-html/unnamed-chunk-13-1.png" width="90%" /></p>
</div>
<div id="modeling-patient-survival-1" class="section level4">
<h4>Modeling patient survival</h4>
<p>First I merge the patient level outcome data with the L summary
statistics.</p>
<pre class="r"><code>survival_df = ovarian_df %>%
# rename id variable to be compatible with L_df and select covariates of interest
dplyr::select(subject_id = sample_id,
survival_time, death, age = age_at_diagnosis) %>%
# delete duplicates- should only have one row per subject
distinct() %>%
# remove subject with outlying survival time
filter(survival_time < 200) %>%
# merge with L function data
left_join(., Lbivariate_df, by = "subject_id")</code></pre>
<p>Now it appears that larger L values (more clustering between B cells
and macrophages) is associated with better survival prognosis.</p>
<pre class="r"><code>survival_df %>%
ggplot(aes(Ldiff, survival_time)) +
geom_point(aes(color = factor(death)))
## Warning: Removed 61 rows containing missing values (`geom_point()`).</code></pre>
<p><img src="MI_fda_files/figure-html/unnamed-chunk-15-1.png" width="90%" /></p>
<p>Run the proportional hazards regression model with bivariate L as a
covariate.</p>
<pre class="r"><code>phmod2 = coxph(Surv(survival_time, death) ~ Ldiff + age,
data = survival_df)
summary(phmod2)
## Call:
## coxph(formula = Surv(survival_time, death) ~ Ldiff + age, data = survival_df)
##
## n= 66, number of events= 35
## (61 observations deleted due to missingness)
##
## coef exp(coef) se(coef) z Pr(>|z|)
## Ldiff -0.014583 0.985522 0.006935 -2.103 0.0355 *
## age 0.012650 1.012730 0.015464 0.818 0.4133
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## exp(coef) exp(-coef) lower .95 upper .95
## Ldiff 0.9855 1.0147 0.9722 0.999
## age 1.0127 0.9874 0.9825 1.044
##
## Concordance= 0.567 (se = 0.063 )
## Likelihood ratio test= 5.28 on 2 df, p=0.07
## Wald test = 5.35 on 2 df, p=0.07
## Score (logrank) test = 5.39 on 2 df, p=0.07</code></pre>
<p>Now the exponentiated coefficient for <code>Ldiff</code> < 1,
indicating that the hazard of death is smaller for a subject with a
greater degree co-occurence between B-cells and macrophages.</p>
<p>Some caveats about this analysis:</p>
<ul>
<li>We removed a lot of subjects (61) because we didn’t have enough
subjects to calculate the bivariate L function. I would recommend
either:
<ul>
<li>Creating a categorical L variable that includes a “not estimable”
group so as to not lose these subjects</li>
<li>Or, using something like the G function which is more robust to rare
cells and fewer subjects would be lost.</li>
</ul></li>
</ul>
</div>
</div>
</div>
<div id="functional-data-analysis" class="section level2">
<h2>Functional data analysis</h2>
<p>Functional data analysis allows you to leverage information from the
full spatial summary function (e.g. K-function) rather than picking a
specific radius <em>r</em>, which can be an arbitrary choice.</p>
<p>The general workflow is:</p>
<ol style="list-style-type: decimal">
<li>Select a spatial summary function (e.g. bivariate Ripley’s K).</li>
<li>Evaluate spatial summary function for the cell(s) of choice over a
range of biologically meaningful radii. This will result in one spatial
summary function for each image.</li>
<li>Visualize the functions</li>
<li>Select a patient-level outcome of interest (e.g. survival)</li>
<li>Choose a functional data model (functional principal components
analysis or functional regression)</li>
</ol>
<p>I will use the ovarian immune cell clustering analysis above as a
starting point, and use a functional principal components analysis
approach. Resources for functional regression are provided in the
“references” section below. Here I am interested in the relationship
between patient survival and the clustering of immune cells within the
tumor areas in the ovarian cancer dataset. I will calculate a univariate
L function for immune cells.</p>
<div id="estimating-the-spatial-summary-functions"
class="section level4">
<h4>Estimating the spatial summary functions</h4>
<p>I can use the same <code>extract_univariate</code> function I defined
above to compute the L function at several radii across subjects. This
is performed below:</p>
<pre class="r"><code>df_nest = ovarian_df %>%
# subset to tumor areas
filter(tissue_category == "Tumor") %>%
mutate(id = sample_id) %>%
dplyr::select(sample_id, x, y, immune, id) %>%
nest(data = c(sample_id, x, y, immune))
# we are evaluating L at 100 different radii, so this will take some time to compute
L_fda_df = map_dfr(df_nest$data, extract_univariate,
marksvar = "immune", mark1 = "immune",
r_vec = seq(0, 200, length.out = 100))
## Warning: data contain duplicated points
## Warning: 1 point was rejected as lying outside the specified window
## Warning: data contain duplicated points
## Warning: 1 point was rejected as lying outside the specified window
head(L_fda_df)
## # A tibble: 6 × 5
## subject_id r L csr Ldiff
## <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 68 0 0 0 0
## 2 68 2.02 0 2.02 -2.02
## 3 68 4.04 19.3 4.04 15.2
## 4 68 6.06 24.7 6.06 18.6
## 5 68 8.08 31.3 8.08 23.2
## 6 68 10.1 35.4 10.1 25.3</code></pre>
<p>Let’s visualize the L functions across subjects using a spaghetti
plot.</p>
<pre class="r"><code>Lp = L_fda_df %>%
ggplot(aes(r, L, group = subject_id)) +
geom_line(alpha = 0.4) +
geom_line(data = filter(L_fda_df, subject_id == 68), aes(y = csr), color = "red3",
linetype = 2) +
ggtitle("L(r)")
Lp2 = L_fda_df %>%
ggplot(aes(r, Ldiff, group = subject_id)) +
geom_line(alpha = 0.4) +
geom_hline(yintercept = 0, color = "red3", linetype = 2) +
ggtitle("L(r)-r")
Lp + Lp2</code></pre>
<p><img src="MI_fda_files/figure-html/spaghetti-1.png" width="90%" /></p>
</div>
<div id="functional-principal-components-analysis"
class="section level4">
<h4>Functional principal components analysis</h4>
<p>FPCA can be performed for dimension reduction, then functional
principal component scores can be used as covariates in a traditional
regression model. I run fpca using the <code>fpca.face()</code> function
from the <code>refund</code> package.</p>
<pre class="r"><code># convert Ldiff functions from long to wide format for use in refund
Lmat = L_fda_df %>%
dplyr::select(subject_id, r, Ldiff) %>%
pivot_wider(names_from = r,
names_glue = "r_{round(r, 2)}",
values_from = Ldiff) %>%
arrange(subject_id) %>%
dplyr::select(-subject_id) %>%
as.matrix()
# run fpca
set.seed(133)
L_fpc = fpca.sc(Y = Lmat, pve = .99)
# if you have a large dataset, instead use fpca.face:
# L_fpc = fpca.face(Y = Lmat, pve = .99, knots = 50)</code></pre>
<p>We can interactively visualize the results of the fpca using the
<code>plot_shiny</code> function from the <code>refund.shiny</code>
package. Note that you can’t knit this in an .Rmd file, which is why it
is commented out in the code below.</p>
<pre class="r"><code>library(refund.shiny)
plot_shiny(L_fpc)</code></pre>
<p>We’ll use the first 2 principal components since they explain 99% of
the variance in the data. The scores from these 2 PCs will be used as
covariates in a proportional hazards model.</p>
<ul>
<li>PC 1 interpretation: negative scores indicate high L values across
all radii</li>
<li>PC 2 interpretation: negative scores indicate L function is higher
at small radii then gets lower, where high scores indicate L is steadily
increasing over r</li>
</ul>
<p>To use these as covariates we need to merge them with the clinical
data. Let’s also multiple PC1 by -1 so it has the interpretation that
higher PC1 score indicates higher values for the L function.</p>
<pre class="r"><code>score_df = tibble(sample_id = 1:128,
fpc1 = L_fpc$scores[,1] * -1,
fpc2 = L_fpc$scores[,2])
surv_fpca_df = ovarian_df %>%
dplyr::select(sample_id, survival_time, death, age = age_at_diagnosis) %>%
# delete duplicates- should only have one row per subject
distinct() %>%
# remove subject with outlying survival time
filter(survival_time < 200) %>%
# merge with L function data
left_join(., score_df, by = "sample_id")</code></pre>
<p>Let’s do some exploratory analysis first.</p>
<pre class="r"><code>p1 = surv_fpca_df %>%
ggplot(aes(fpc1, survival_time, color = factor(death))) +
geom_point() + ggtitle("fpc1")
p2 = surv_fpca_df %>%
ggplot(aes(fpc2, survival_time, color = factor(death))) +
geom_point() + ggtitle("fpc2")
p1 + p2</code></pre>
<p><img src="MI_fda_files/figure-html/unnamed-chunk-21-1.png" width="90%" /></p>
<p>Finally, we’ll model survival as the outcome with fpc scores as
covariates.</p>
<pre class="r"><code>phmod_fpca = coxph(Surv(survival_time, death) ~ fpc1 + fpc2 + age,
data = surv_fpca_df)
summary(phmod_fpca)
## Call:
## coxph(formula = Surv(survival_time, death) ~ fpc1 + fpc2 + age,
## data = surv_fpca_df)
##
## n= 127, number of events= 80
##
## coef exp(coef) se(coef) z Pr(>|z|)
## fpc1 0.007829 1.007859 0.002526 3.099 0.00194 **
## fpc2 -0.003667 0.996340 0.012310 -0.298 0.76579
## age 0.014329 1.014432 0.011390 1.258 0.20836
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## exp(coef) exp(-coef) lower .95 upper .95
## fpc1 1.0079 0.9922 1.0029 1.013
## fpc2 0.9963 1.0037 0.9726 1.021
## age 1.0144 0.9858 0.9920 1.037
##
## Concordance= 0.626 (se = 0.034 )
## Likelihood ratio test= 13.75 on 3 df, p=0.003
## Wald test = 17.69 on 3 df, p=5e-04
## Score (logrank) test = 18.08 on 3 df, p=4e-04</code></pre>
</div>
</div>
<div id="references" class="section level2">
<h2>References</h2>
<ul>
<li><a
href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4864857/">interactive
visualization of functional data with refund.shiny</a></li>
<li><a
href="https://www.biorxiv.org/content/10.1101/2022.06.17.496475v1.full.pdf">a
paper using a Cox model with FPC scores for multiplex imaging
data</a></li>
<li><a
href="https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1009486">functional
Cox regression with multiplex imaging data</a></li>
<li><a href="https://pubmed.ncbi.nlm.nih.gov/34898969/">additive
functional Cox regression</a></li>
</ul>
</div>
</div>
</div>
</div>
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