ANOVA.

Make sure you know well what is your Experimental Unit. You may have sub samples, samples and blocks that you may need to define in the model.

You have not designed the experiment to estimate a growth curve. If this was your research question then you should consider a different design with more measurements at more different time points; replicate measurements at one time point are not required.

Neda: no, he can't. Or he can, but it won't characterise the difference in the curves. 

As Jochen points out, if he's interested in growth curves, than he should have set up the experiment differently. 

Agree with the other comments. As Jürgen suggested, for a "real" growth curve you would need more timepoints, about 6-10. Depending if you expect a kind of sigmoidal curve you would have to measure in short intervals at the beginning and end to better estimate the non-linear part. For now, I suppose you data structure is as follows

Outcome    Treatment         Time point    Replicate   Experiment

[...]                Control                       1                   1                   1

[...]                Control                       1                   2                   1

[...]                Control                       1                   3                   1

[...]                Drug                           1                    4                   1

[...]                Drug                            1                   5                  1

.........

[...]                Drug                            6                   6                    3

Are your replicates biological replicates? I suppose so as these are different cells, right? Or did you just measured the OD three times per time point? If its just technical replicates (3x OD-measure) then calculate the mean of your replicates per time point in advance. But then you only have a n=3 = experiment ID

In the first case if true, are your Drug and Control treatments connected in a way that you know which Control belong to which Drug treatment (i.e. connected via relicate ID) or is it just 3 wells with Control and 3 wells with Drug treatment in your cell culture plates?

In the latter case you would have 6 wells (3x Control + 3x Drug) per experiment that you measured at three different time points (1, 3, 6). Well-ID would be 1...6 for each experiment (1...3).

Analysis would look like this:

Outcome ~ factor(Treatment)*factor(Time point) + (1|Experiment/Replicate)

with Treatment as between-subject and Timepoint as within-subject factor. The "*" indicates the interaction which gives you the information if there are different effects of Control vs. Drug at the different time points. The random factor "Replicate nested in Experiment" identifies which repeated measurement belongs to which individual (= replicate, in total n=9).

I have both biological and technical replicates. The aim of the experiment was to test if the drug treatment inhibits the proliferation of the cancer cells. The data structures is as described by stefan above; I had 3x control & 3x drug per experiment that was measured at three time points and repeated a two more times.

Is it accurate to use repeated measure ANOVA? or I have to fit a curve and do ANOVA on the slope values?

There are some points which do not sugest to do ANOVA: if I remember well, ANOVA needs data from normally distributed groups and if you do ANOVA on slopes you will have three case for group (which will result almost certainly in a not significant test, but you could try).

If the control and drug groups are biologically identical, you could consider each replicate-timing as a case (3 cases [replicates] for time 1, 3 cases for time 3 etc.), so obtaining 9 cases and trying to perform a Wilcoxon (non-parametric) test for coupled cases (the null-hypothesis is that the difference between treated and untreated cells at each timing is 0).