Are cancer studies cases always left-censored? I'm used to traditionnal studies with plain old Cox PH models and right censoring.     
But lately I was wondering about the outcome date in cancer studies. Indeed, the date we usually use as outcome is the diagnosis date, not the "real" cancer declaration date.
Cancers like solid tumors start out as few cells growing out of control and exceeding the immune system. As they grow over time, they will eventually cause symptoms that can lead to cancer diagnosis, but the primary event (as a point of no return) is the "few cells" time. Unfortunately this event is impossible to register, even by the most talented physician, so the event can have already happen unnoticed at the time of enrollment.
As a cancer can evolve for years before being diagnosable, the cancer could have appeared anywhere before the diagnosis date in case events, in the same way that it could appear anywhere after the last contact date in non-case events. 
By using the date of diagnosis as a fixed endpoint, it seems to me that we are ignoring some left-censoring.
I could not find anything about this in the litterature (tried "cancer left censor" on google scholar), is my logic flawed or is my research skill weak ?
If this is a thing, which type of model can take into account such a double censoring ? 

EDIT TO ACCOUNT FOR @andbel ANSWER: as several people talked about left censoring being relative to date of entry, I'll try to explain here why I don't agree. If I say anything wrong, please correct me as this may be the seed of my misunderstanding of the problem.  
TL;DR: cancer appeared anytime before cancer diagnosis if event==1 (left-censoring) and anytime after censoring time if event==0 (right-censoring). I think the common (Oxford) definition about left-censoring is wrong as it is more a definiton of "left-censoring at date of entry". This is not about exposition hence no truncation. Cancer appearance is not positively bound by an interval so it is not interval-censoring.
Left censoring is strictly defined as following:

A left censoring scheme is such that the random variable of interest,
  X, is only observed if it is greater than or equal to a left censoring
  variable L, otherwise L is observed

In my textbook, they gave the example of low limit of quantification when measuring something, like viral load for HIV: the result is lower than L=20 copies/ml but you cannot know if it is 0 or 10 or 15, you only know it is not higher than 20. Hence left censoring.
When dealing with time and events, I guess it becomes a little more complicated. 
In usual cases, each person have either a date of event or a date of right-censoring. I like to translate this latter as, "for anyone who did not experienced the event, it may happen at $T≥t_{rightCensoring}$". The right censoring is thus on the time of event in people who did not experienced it only, whereas there is no censoring in "event" people (the time is well known).
Here is the definition of left censoring from Oxford university, which I think everybody usualy relates:

Left censoring is when the event of interest has already occurred
  before enrolment. This is very rarely encountered.

I think this definition is about the more specific "Left censoring at enrolment point", rather than the wide definition of left censoring. Moreover, since baseline cases are usually excluded from analysis, I find this definition, along with some internet examples quite confusing. This SO example is more like what I'm thinking of. 
Interval censoring is defined as:

By interval censoring, we mean that a random variable of interest is known only to lie within an interval instead of being observed exactly

This is not the case either, sine cancer appearance is not bound by an interval. Conversely, the study time is more an interval of exclusion of cancer if event==0.   
Lastly, I really don't think this is left truncation. Correct me if I'm wrong, but AFAIK, left truncation is when you don't have information on exposition before a date. Oxford university define it as "Left truncation occurs when the subjects have been at risk before entering the study". If you try to measure the association between tobacco and cancer, I think it is left censored if you don't know about cancer before a date, and left truncated if you don't know about tobacco. In this question, it is not a matter of "being at risk", and we know that cancer appeared, but we don't know exactly when.
 A: Left censoring occurs when the outcome has already happened when you begin following the participant, so this is not generally the case. If you are thinking of a classical population-based study in which you follow individuals until the diagnosis of cancer (i.e. cancer diagnosis is the event of interest in the survival analysis), you could technically think of this as an interval censored observation: the event occurs at a specific (unknown) time between date of entry and diagnosis date. 
I say technically, however, because in large cohort study this interval will likely be very long, and the real date of cancer initiation very close to the right bound (diagnosis), making methods for interval censored not really useful. 
And that's why you don't find the problem described as a left/interval censoring issue. In epidemiological literature the problem you describe is referred to as latent (or incubation) period  (https://www.ncbi.nlm.nih.gov/pubmed/7304560).
A: The issue here is what is being modeled in cancer survival studies. You cannot correctly define censoring, truncation, or whatever without specifying the nature of the study.
Cancer survival studies have to do with determining what a clinician can tell a patient about prognosis or can best provide for therapy once a tumor has been detected clinically. In that case what you are modeling is the survival after the time of tumor detection, with the date of pathologic confirmation of a tumor typically set as the time=0 origin for each individual. The event of interest is something related to subsequent outcome (e.g., recurrence after therapy, death from cancer, death from any cause). In this type of study there is no left censoring or truncation of event times: if such an event does happen, it by definition cannot have happened before the time=0 origin as defined for that individual in the study.
As the OP rightly points out, that time of tumor detection can be long after--even decades after--the initiation of the tumor. The origins and evolution of a tumor from a single initiating cell into a clinically detectable tumor with about $10^9$ cells is a fascinating object of study. If you were trying to model survival from the time of cancer initiation to death, then some of the concerns in the OP would have to be dealt with.*
But that is not what clinical cancer studies are about. They are about survival after cancer detection. Neither the patient, family, friends, or clinicians at that point care explicitly about how long the cancer previously grew undetected. A cancer biomarker available at time=0 might in principle contain information about pre-detection tumor evolution and potentially inform prognostication. Even then, the prognosis being modeled is the clinically relevant survival after someone is known to have cancer, not from the time decades past when the tumor began.

*You would also have to deal with just what is meant as "cancer initiation": for a carcinoma do you mean the first dysplastic event, the development of carcinoma in situ, production of the first cell capable of invasiveness, or first invasion through a basement membrane?
